Monte Carlo Transport of Electrons and Photons ETTORE MAJORANA INTERNATIONAL SCIENCE SERIES Series Editor: Antonlno Zlchlchi European Physical Society Geneva, Switzerland

(PHYSICAL SCIENCES) Recent volumes in the series: Volume 27 DATA ANALYSIS IN ASTRONOMY II Edited by V. Oi Gesu, l. Scarsi, P. Crane, J. H. Friedman, and S. Levialdi Volume 28 THE RESPONSE OF NUCLEI UNDER EXTREME CONDITIONS Edited by R. A. Broglia and G. F. Bertsch Volume 29 NEW TECHNIQUES FOR FUTURE ACCELERATORS Edited by M. Puglisi, S. Stipcich, and G. Torelli Volume 30 SPECTROSCOPY OF SOLID-STATE LASER-TYPE MATERIALS Edited by Baldassare Oi Bartolo Volume 31 FUNDAMENTAL SYMMETRIES Edited by P. Bloch, P. Pavlopoulos, and R. Klapisch Volume 32 BIOELECTROCHEMISTRY II: Membrane Phenomena Edited by G. Milazzo and M. Blank Volume 33 MUON-CATALYZED FUSION AND FUSION WITH POLARIZED NUCLEI Edited by B. Brunelli and G. G. Leotta Volume 34 VERTEX DETECTORS Edited by Francesco Villa Volume 35 LASER SCIENCE AND TECHNOLOGY Edited by A. N. Chester, V. S. Letokhov, and S. Martellucci Volume 36 NEW TECHNIQUES FOR FUTURE ACCELERATORS II: RF and Microwave Systems Edited by M. Puglisi, S. Stipcich, and G. Torelli Volume 37 SPECTROSCOPY OF LIGHT AND HEAVY QUARKS Edited by Ugo Gastaldi, Robert Klapisch, and Frank Close Volume 38 MONTE CARLO TRANSPORT OF ELECTRONS AND PHOTONS Edited by Theodore M. Jenkins, Walter R. Nelson, and Alessandro Rlndi

A Continuation Order Plan is available for this series. A continuation order will bring delivery of each new volume immediately upon publication. Volumes are billed only upon actual shipment. For further information please contact the publisher. Monte Carlo Transport of Electrons and Photons

Edited by Theodore M. Jenkins and Walter R. Nelson SLAC Stanford, California and Alessandro Rindi Sincrotrone Trieste Trieste, Italy

Technical Editors: A. E. Nahum and David W. O. Rogers Royal Marsden Hospital National Research Council of Canada Sutton, Surrey, United Kingdom Ottawa, Ontario, Canada

Plenum Press • New York and London Ubrary of Congress Cataloging in Publication Data

International School of Radiation Damage and Protection (8th: 1987: Erice, Italy) Monte Carlo transport of electrons and photons / edited by Theodore M. Jenkins, Walter R. Nelson, and Alessandro Rindi. p. cm.-(Ettore Majorana international science series. Physical sciences; v. 38) "Proceedings of the International School of Radiation Damage and Protection, eighth course ... held September 24-Qctober 3, 1987, in Erice, Sicily, ltaly"-T.p. verso. Includes bibliographies and index. ISBN-13: 978-1-4612-8314-0 e-ISBN-13: 978-1-4613-1059-4 001: 10.1007/978-1-4613-1059-4 1. Electron transport-Congresses. 2. Photon transport theory-Congresses. 3. Monte Carlo method-Congresses. I. Jenkins, Theodore M. II. Nelson, Walter R. (Walter Ralph), 1937- III. Rindi, Alessandro. IV. Title. V. Series. aC176.8.E41537 1987 88-31147 530.4'l-dc19 CIP

Proceedings of the International School of Radiation Damage and Protection Eighth Course: Monte Carlo Transport of Electrons and Photons Below 50 MeV, held September 24-Qctober 3, 1987, in Erice, Sicily, Italy © 1988 Plenum Press, New York Softcover reprint of the hardcover 1st edition 1988 A Division of Plenum Publishing Corporation 233 Spring Street, New York, N.Y. 10013

All rights reserved

No part of this book may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording, or otherwise, without written permission from the Publisher Foreword

For ten days at the end of September, 1987, a group of about 75 scientists from 21 different countries gathered in a restored monastery on a 750 meter high piece of rock jutting out of the Mediterranean Sea to discuss the simulation of the transport of electrons and photons using Monte Carlo techniques. When we first had the idea for this meeting, Ralph Nelson, who had organized a previous course at the "Ettore Majorana" Centre for Scientific Culture, suggested that Erice would be the ideal place for such a meeting. Nahum, Nelson and Rogers became Co-Directors of the Course, with the help of Alessandro Rindi, the Director of the School of Radiation Damage and Protection, and Professor Antonino Zichichi, Director of the "Ettore Majorana" Centre.

The course was an outstanding success, both scientifically and socially, and those at the meeting will carry the marks of having attended, both intellectually and on a personal level where many friendships were made. The scientific content of the course was at a very high caliber, both because of the hard work done by all the lecturers in preparing their lectures (e.g., complete copies of each lecture were available at the beginning of the course) and because of the high quality of the "students", many of whom were accomplished experts in the field. The outstanding facilities of the Centre contributed greatly to the success.

This volume contains the formal record of the course lectures. The order has been rearranged to flow in terms of the content, and they have been revised by the authors, both in view of what each learned at the course, and in view of comments by the Technical Editors (Nahum and Rogers). The mammoth task of actually putting the book together, including getting all the lectures typeset in a uniform format, was undertaken by the General Editors (Jenkins, Nelson and Rindi).

One of the major purposes of organizing the course was to produce a book which could be a reference book for those working in this field since the literature is very spread out. Thanks to the diligent efforts of the individual authors and editors, we believe we have succeeded in that goal.

We also believe that this book demonstrates that Monte Carlo techniques for sim• ulating electron and photon transport have become a reliable and valuable tool in many aspects of radiation dosimetry and medical physics in general. There is still much work to be done, but this book demonstrates that a solid foundation has been established.

July 1988 A. E. Nahum David W. O. Rogers

v Preface

The world of electron and photon physics is rapidly becoming more complex, more pragmatic, and more interesting every day. High-energy electron-positron physics has shifted toward colliding beams with center-of-mass energies in the tens to hundreds of GeV. Medical accelerators have proliferated such that they are almost as common as the neighborhood pharmacy. Accelerators and radioactive sources are being used for material and food processing, as well as for structure analysis. With all this infusion of electrons and photons into our everyday lives, there comes a concommitant need to know more about these radiations, how they react, and how they are used. More to the point, it is important to know how they can be used correctly and wisely. To satisfy this need, the Monte Carlo transport code has become one of the tools of choice for the physicist, as well as the medical physicist, and justifiably so, as the chapters of this book so eloquently attest.

Any tool, no matter how popular and useful, has its limitations and its strengths, and the Monte Carlo electron transport code is no exception. In the biological field, for example, the Monte Carlo calculation is able to model the physical processes (e.g., dose) that lead to measurable biological responses, but with strong limitations. On the cellular level, interaction distances are small. On the level of a human body, the system is very complex. Both require trade-offs to achieve answers in a reasonable amount of computer time. Sometimes, short-cuts, such as the pencil-beam method, may be used. Or, one may elect to use electron track simulation in lieu of full analog Monte Carlo.

In high-energy physics, the shower process can be so vast that the computer is swamped with particles to follow. One must know how to speed up the process• i.e., how to determine which segment of the calculation is the most important, and to suppress the less important segments. So-called "variance-reduction" techniques become increasingly important.

These are all manageable parts of this tool, the Monte Carlo transport code, and they lead not only to answers, but also to a deeper understanding of the processes that are occurring in nature. Quite often, it is the Monte Carlo code that tells when there was something wrong with a measurement, and even gives clues as to wh€re to look. And sometimes it is the other way around; the Monte Carlo code has introduced an artefact, such as step sizes near boundaries, and is itself giving false numbers.

These topics were important enough to warrant convening a course in Erice, Sicily, in 1987, for the purpose of teaching and studying electron-photon Monte Carlo trans• port, and to produce a book which could be a reference for those working in this field. A quick reading of the subject index will give a clue as to the complexity, utility, and the importance of Monte Carlo electron-photon transport codes.

VII viii Preface For scientists working with electrons and photons, the Monte Carlo transport code is one of the most important tools they can call upon, and we believe it will become even more so in the future.

Acknowledgements

The Eighth Course of the International School of Radiation Damage and Protection was held from 24 September through 3 October 1987 at the "Ettore Majorana" Centre for Scientific Culture in Erice, Sicily, Italy. The primary sponsors were the Italian Ministry of Education, the Italian Ministry of Scientific and Technological Research, and the Sicilian Regional Government. Support was also provided by the Atomic Energy of Canada Limited.

This book was computer generated using the 'lEX-based macro package called P~, developed by Art Ogawa at the Stanford Linear Accelerator Center. Dr. Ogawa's support is greatly appreciated, as is the immense amount of help given to us by our friend and colleague, Dr. Ray Cowan of the M.LT. Laboratory for Nuclear Science. Ray always found time to help "fine tune" P~ for our particular, often demanding, needs.

A considerable amount of touch-up work had to be done to the figures submitted by the 14 authors from the eight countries represented. We would like to thank the SLAC Publications Department, and particularly Sylvia MacBride in the Technical Illustrations section for her cheerful help.

Finally, on behalf of all the participants of the course, we would like to thank the staff of the "Ettore Majorana" Centre for the exceptionally fine job they did in running the course. All of us will also remember the hospitality shown by the citizens of Erice.

July 1988 Theodore M. Jenkins Walter R. Nelson Alessandro Rindi Contents

INTRODUCTION AND FUNDAMENTALS

1. Overview of Photon and Electron Monte Carlo 3 A. E. Nahum 1.1 Introduction, 3 1.2 Some History, 3 1.3 Photons, Electrons and Medical Physics, 4 1.4 Interesting Electrons, 6 1.5 The Ultimate (Radiotherapy) Problem, 10 1.6 Computer Technology, 14 1. 7 The Appeal of Monte Carlo, 16

2. Multiple-Scattering Angular Deflections and Energy-Loss Straggling 21 Martin J. Berger and Ruqing Wang 2.1 Introduction, 21 2.2 Elastic-Scattering Cross Section, 22 2.2.1 Factorization, 22 2.2.2 Spin and Relativity Effects, 23 2.2.3 Screening Effects, 23 2.2.4 Characteristic Screening Angle, 24 2.2.5 Calculations by the Partial-Wave Method, 25 2.2.6 Comparisons of Elastic-Scattering Cross Sections, 26 2.2.7 Molecular and Solid-State Effects, 31 2.3 Calculation of Multiple-Scattering Deflections, 32 2.3.1 Moliere Multiple-Scattering Distribution, 33 2.3.2 Goudsmit-Saunderson Multiple-Scattering Distribution, 34 2.3.3 Contribution of Inelastic Collisions to Multiple Scattering, 35 2.3.4 Number of Elastic Collisions and Mean Deflection Angle, 37 2.3.5 Comparison of Multiple-Scattering Distributions, 39 2.4 Energy-Loss Straggling, 44 2.4.1 Landau's Distribution: Applicability, Refinements, 45 2.4.2 More Elaborate Treatment of Straggling, 47 2.4.3 Energy-Loss Straggling in Water, 47

IX x Contents

3. Electron Stopping Powers for Transport Calculations 57 Martin J. Berger 3.1 Introduction, 57 3.2 Definition of Stopping Power, 57 3.3 Continuous-Slowing-Down Approximation, 58 3.4 Stopping-Power Formulas and Tables, 61 3.5 Mean Excitation Energies, 63 3.5.1 I-Values from Stopping-Power Data, 63 3.5.2 I-Values from Photon Cross Sections, 64 3.5.3 Survey of Mean Excitation Energies for Elements, 65 3.5.4 Mean Excitation Energies for Compounds, 66 3.6 Density-Effect Correction, 69 3.7 Comparisons with Experiments, 71 3.8 Stopping-Power Ratios, 73 3.9 Stopping Powers at Low Energies, 75 3.10 Concluding Remarks, 76

4. Cross Sections for Bremsstrahlung Production and Electron-Impact Ionization 81 Stephen M. Seltzer 4.1 Introduction, 81 4.2 Bremsstrahlung Production, 81 4.2.1 Electron-Nucleus Bremsstrahlung, 83 4.2.2 Electron-Electron Bremsstrahlung, 89 4.2.3 Comparisons of Calculated and Measured Cross Sections, 90 4.2.4 Radiative Stopping Power, 96 4.2.5 Positron Bremsstrahlung, 99 4.3 Electron-Impact Ionization, 103 4.3.1 Cross-Section Formulas, 103 4.3.2 Input Data, 105 4.3.3 Illustrative Examples, 106

5. Electron Step-Size Artefacts and PRESTA 115 Alex F. Bielajew and David W. O. Rogers 5.1 Introduction, 115 5.2 Electron Step-Size Artefacts, 115 5.2.1 What Is An Electron Step-Size Artefact?, 115 5.2.2 Path-Length Correction, 119 5.2.3 Lateral Displacement, 122 5.2.4 Boundary Crossing, 123 5.3 PRESTA,124 5.3.1 The Elements of PRESTA, 124 5.3.2 Constraints of the Moliere Theory, 126 5.3.3 PRESTA's Path-Length Correction, 128 5.3.4 PRESTA's Lateral-Displacement Algorithm, 130 Contents XI

5.3.5 Accounting for Energy Loss, 131 5.3.6 PRESTA's Boundary-Crossing Algorithm, 133 5.3.7 Caveat Emptor, 135

6. 20 MeV Electrons on a Slab of Water 139 David W. O. Rogers and Alex F. Bielajew 6.1 Introduction, 139 6.2 A Thin Slab, 139 6.2.1 The CSDA Calculation, 139 6.2.2 More Realistic Calculations, 139 6.3 A Thick Slab, 143 6.3.1 Typical Histories, 143 6.3.2 Depth-Dose Curves, 144 6.3.3 Fluence vs Depth, 145 6.4 Conclusions, 147

THE ETRAN SYSTEM

7. An Overview of ETRAN Monte Carlo Methods 153 Stephen M. Seltzer 7.1 Introduction, 153 7.2 Monte Carlo Methods, 154 7.2.1 Photon Transport, 154 7.2.2 Electron Transport, 156 7.3 Organization and Description of the Code System, 173 7.3.1 Data Preparation, 173 7.3.2 Monte Carlo Calculations, 174 7.4 Future Improvements, 177

8. ETRAN - Experimental Benchmarks 183 Martin J. Berger 8.1 Introduction, 183 8.2 Comparisons, 186 8.3 Discussion, 187

9. Applications of ETRAN Monte Carlo Codes 221 Stephen M. Seltzer 9.1 Introduction, 221 9.2 Response of Photon Detectors for Spectrometry, 221 9.2.1 NaI Detectors, 222 9.2.2 High-Purity Ge Detectors, 229 9.3 Space Shielding Calculations, 235 9.4 Bremsstrahlung Beams for Radiation Processing, 241 9.5 Liquid-Scintillation Counting of Beta Emitters, 242 XII Contents

THE INTEGRATED TIGER SERIES

10. Structure and Operation of the ITS Code System 249 J. Halbleib 10.1 Introduction, 249 10.2 History of the TIGER Series, 249 10.3 Structure of the ITS Code System, 251 10.3.1 The Source Files, 252 10.3.2 The UPEML Processor, 252 10.4 Operation of the ITS Code System, 254 10.4.1 Input, 254 10.4.2 Output, 259 10.5 Concluding Remarks, 260

11. Applications of the ITS Codes 263 J. Halbleib 11.1 Introduction, 263 11.2 Verification, 264 11.2.1 Van de Graaff Deposition Profiles, 264 11.2.2 Van de Graaff Electron Backscatter, 264 11.2.3 Van de Graaff Electron Deposition in Film, 264 11.2.4 Van de Graaff X-Ray Production and Dosimetry, 264 11.2.5 Low-Energy Electron Backscatter, 265 11.2.6 2-D Electron Energy Deposition in Water at Intermediate Energies, 265 11.2.7 High-Energy 2-D Profiles, 265 11.2.8 BGO Pulse-Height Distribution, 265 11.2.9 Charge Profiles in Plastic, 266 11.3 Verification/Corroboration, 266 11.3.1 Radial Electron Beam Diode for Gas Laser Excitation, 266 11.3.2 Helia, 267 11.3.3 Proto II 6-Beam Overlap, 267 11.3.4 REB/Multiple-Foil Interaction, 267 11.3.5 18 Blades, 267 11.3.6 REB Pumping of Noble-Gas Halide Laser, 268 11.3.7 Gradient-B Drift Transport, 268 11.3.8 Printed Circuit Boards, 269 11.3.9 Voyager Electron Telescope, 269 11.3.10 SPEED/Triaxial-Diode Flash X-Ray Source, 269 11.3.11 Inverse Ion Diode, 270 11.4 Predictions, 270 11.4.1 Bremsstrahlung Radiation Environment of PBFA-II, 270 11.4.2 PBFA-I, MITL, and Gamma-Ray Telescope Plots, 270 11.4.3 RAYO: REB-Pumped Gas Laser in Rectangular Geometry, 270 11.4.4 Sector X-Ray Converter with Gradient-B Transport, 271 11.4.5 TIGER vs TIGERP Line Radiation, 271 11.4.6 Falcon, 271 11.5 Research, 272 11.5.1 Time-Dependent Response of the Atmosphere to X-Ray Energy Deposition, 272 Contents XIII

11.5.2 Electric Fields in Materials, 272 11.5.3 Hidden Lines, 273 11.5.4 Self-Consistent Alfven Problem, 273 11.6 Conclusion, 273

THE EGS CODE SYSTEM

12. Structure and Operation of the EGS4 Code System 287 Walter R. Nelson and David W. O. Rogers 12.1 Introduction, 287 12.1.1 History Prior to EGS3, 287 12.1.2 The Development of EGS3, 288 12.1.3 EGS4 - A Code Greatly Influenced by Medical Physics, 289 12.2 General Description of EGS4 (and PEGS4), 289 12.2.1 PEGS4 as a Development Tool, 290 12.2.2 PEGS4 as a Preprocessor for EGS4, 290 12.2.3 General Implementation of EGS4, 290 12.2.4 Mortran3 Macros and EGS User Codes, 291 12.3 Some Benchmark Comparisons, 293 12.3.1 Conversion Efficiency of Lead for 30-200 MeV Photons, 293 12.3.2 Large, Modularized NaI(TI) Detector Experiment, 295 12.3.3 Longitudinal and Radial Showers in Water and Aluminum at 1 GeV, 296 12.3.4 Track-Length Calculations, 298 12.4 Summary of EGS4 Capabilities and Features, 299 12.5 EGS4 Graphics Capabilities, 302

13. Experimental Benchmarks of EGS 307 David W. O. Rogers and Alex F. Bielajew 13.1 Introduction, 307 13.2 Detector Response Functions, 308 13.2.1 Photon Spectrometers, 308 13.2.2 Electron Detectors, 310 13.3 Calculated Ion Chamber Response, 311 13.4 Depth-Dose Curves, 311 13.4.1 Photon Depth-Dose Curves, 311 13.4.2 Electron Depth-Dose Curves, 314 13.5 Bremsstrahlung Production, 315 13.6 Conclusion, 317

14. A Comparison of EGS and ETRAN 323 David W. O. Rogers and Alex F. Bielajew 14.1 Introduction, 323 14.2 Class I vs Class II Algorithms, 324 14.3 Differences in Multiple Scattering, 328 14.4 Electron Depth-Dose Curves, 333 14.5 Low-Energy Treatment and Termination of Histories, 335 14.6 Step Sizes and Boundary Crossing, 337 xiv Contents

14.7 Sampling Procedures, 338 14.8 Timing, 338 14.9 Miscellaneous, 340

LOW-ENERGY MONTE CARLO

15. Low-Energy Monte Carlo and W-Values 345 B. Grosswendt 15.1 Introduction, 345 15.2 Low-Energy Electron Monte Carlo Transport Model, 348 15.3 Input Cross Section, 350 15.4 Results Concerning Ionization Yields, 352 15.5 Conclusion, 356

16. Electron Track Simulation for Microdosimetry 361 Akira Ito 16.1 Introduction, 361 16.2 Outline of the Electron Track Simulation 362 16.3 Evaluation of the Electron Cross Section, 362 16.4 Description of an Electron Track Simulation Monte Carlo Program (ETRACK), 366 16.5 Results of Electron Track Simulation, 368 16.6 Basic Physical Quantities Derived from Electron Track Structure, 368 16.7 Patterns and the Proximity Function in Cell Nucleus, 372 16.8 Calculation of the DSB Probability of DNA, 375 16.9 The DSB Probability and RBE, 377 16.10 Concluding Remarks, 378 16.11 The Use of the Physical Random Number Generator, MIKY, 378

GENERAL TECHNIQUES

17. Geometry Methods and Packages 385 Walter R. Nelson and Theodore M. Jenkins 17.1 Mathematical Considerations, 385 17.1.1 Intersection of a Vector with a Plane Surface, 385 17.1.2 The PLANE1 Algorithm Available in EGS4, 386 17.1.3 Intersection of a Vector with a Cylindrical Surface, 386 17.1.4 The CYLNDR Algorithm Available in EGS4, 389 17.2 Geometry Considerations in the EGS4 Code System, 389 17.2.1 The EGS4 User Code Concept, 389 17.2.2 Specifications for (and an Example of) HOWFAR, 390 17.2.2 Auxiliary Geometry Subprograms Available with EGS4, 393 17.2.4 Mortran3 and Macro Forms of the Geometry Routines, 395 17.2.5 Other EGS4-Related Geometry Packages, 396 17.3 Combinatorial Geometry, 397 17.3.1 Constructing Bodies Using Combinatorial Geometry, 400 Contents xv

17.3.2 An Example of a Complex 1'10RSE-CG Geometry, 402 17.4 Geometry Packages in ETRAN, ITS and FLUKA, 402 17.4.1 ETRAN, 402 17.4.2 ITS: The Integrated TIGER Series, 404 17.4.3 The FL UKA Hadronic Cascade Code, 404

18. Variance-Reduction Techniques 407 Alex F. Bielajew and David W. O. Rogers 18.1 Introduction, 407 18.1.1 Variance Reduction or Efficiency Increase?, 407 18.2 Electron-Specific Methods, 408 18.2.1 Geometry Interrogation Reduction, 408 18.2.2 Discard Within a Zone, 409 18.2.3 PRESTA!,409 18.2.4 Range Rejection, 410 18.3 Photon-Specific Methods, 411 18.3.1 Interaction Forcing, 411 18.3.2 Exponential Transform, Russian Roulette, and Particle Splitting, 412 18.4 Other Tricks, 415 18.4.1 Sectioned Problems, Use of Pre-Computed Results, 415 18.4.2 Geometry-Equivalence Theorem, 416 18.4.3 Use of Geometry Symmetry, 417

19. Electron Transport .In -E and -B Fields 421 Alex F. Bielajew 19.1 Introduction, 421 19.2 Equations of Motion in a Vacuum, 422 19.2.1 Special Cases: -E = Constant, --B = OJ B = Constant, -E = 0, 423 19.3 Transport in a Medium, 424 19.4 Application to Monte Carlo - Benchmarks, 427

APPLICATIONS

20. Electron Pencil-Beam Calculations 437 Pedro Andreo 20.1 Introduction, 437 20.2 Point-Monodirectional Beams, 437 20.3 Computational Details, 438 20.4 Monte Carlo Codes for Pencil-Beam Calculations, 440 20.5 Applications, 441 20.5.1 Absorbed-Dose Distributions, 441 20.5.2 Energy Distributions, L144 20.5.3 Pencil Beams as "Benchmarks" for Treatment-Planning Algorithms, 450 XVI Contents

21. Monte Carlo Simulation of Radiation Treatment Machine Heads 453 Radhe Mohan 21.1 Introduction, 453 21.2 Monte Carlo Simulations of Linear Accelerator Heads, 455 21.2.1 Electron Contamination, 459 21.3 Simulation of 60Co Teletherapy Heads, 461 21.3.1 Electron Contamination, 467 21.4 Summary, 467

22. Positron Emission Tomography Applications of EGS 469 A. Del Guerra and Walter R. Nelson 22.1 Principles of Positron Emission Tomography, 469 22.2 Physical Processes in PET, 469 22.2.1 Positron Emitters, 469 22.2.2 Positron Range, 470 22.2.3 Positron Annihilation, 471 22.2.4 Scatter in Tissue, 471 22.2.5 Interaction Within the Detector, 471 22.3 The PET Camera, 471 22.3.1 Scintillator Multicrystal Detector, 471 22.3.2 Gas Detector, 473 22.4 Use of Monte Carlo Codes in Tomograph Design, 473 22.5 An Application: Use of EGS4 for the HISPET Design, 474 22.5.1 The Converter Efficiency Code (UCCELL), 475 22.5.2 Evaluation of the HIS PET Performance (UCPET), 477 22.5.3 Image Reconstruction from EGS4-Simulated Data Output, 481 22.6 Summary, 482

23. Stopping-Power Ratios for Dosimetry 485 Pedro Andreo 23.1 Introduction, 485 23.2 Fundamentals of Stopping-Power Ratios, 485 23.3 The Need for Transport Calculations to Derive Electron Spectra, 488 23.4 Monte Carlo Calculations of Electron Spectra, 489 23.4.1 The Technique of Transport Down to the Monte Carlo Cutoff Plus a CSDA Calculation, 491 23.5 Stopping-Power Ratios for Electron Beams, 493 23.6 Stopping-Power Ratios for Photon Beams, 496

24. Photon Monte Carlo Transport in Radiation Protection 503 B. Grosswendt 24.1 Introduction, 503 24.2 The Anthropomorphic Phantom, 504 Contents XVII

24.3 The Monte Carlo Photon Transport Model, 506 24.3.1 Photon Interaction Model, 507 24.3.2 Interaction Site Model, 508 24.3.3 Bookkeeping Model, 509 24.3.4 The Input Data, 510 24.4 Results Concerning the MIRD Phantom, 510 24.5 Operational Radiation Protection Quantities, 511 24.6 ICRU-Sphere Quantities, 512 24.7 Dose Distribution Geometry, 513 24.8 Special Calculation Techniques, 514 24.9 Results Concerning the ICRU Sphere, 515 24.10 Influence of Electron Transport, 518 24.11 Conclusion, 518

25. Simulation of Dosimeter Response and Interface Effects 523 A. E. Nahum 25.1 Introduction, 523 25.2 An Interface Benchmark, 524 25.3 Electron Steplength Variation, 526 25.4 Ion-Chamber Response, 527 25.4.1 Introduction, 527 25.4.2 In-Air KERMA Calibration in GOCo Radiation, 527 25.4.3 Other Ion-Chamber Simulations, 532 25.5 Dose Distributions at Interfaces, 535 25.5.1 A Benchmarking Situation, 535 25.5.2 Interface Simulations Involving LiF, 536 25.5.3 Aluminium/Gold,539 ·25.5.4 Electron Beams, 540 25.6 Summary and Conclusions, 542

26. Dose Calculations for Radiation Treatment Planning 549 Radhe Mohan 26.1 Introduction, 549 26.2 Conventional Methods of Dose Calculations, 550 26.2.1 Equivalent-Pathlength Methods, 551 26.2.2 Scatter-Integration Models, 551 26.2.3 Electron Beams, 552 26.3 Pencil-Beam-Convolution Method of Dose Calculation, 552 26.4 Examples, 556 26.5 "Differential Pencil-Beam" and "Dose-Spread-Array" Models, 558 26.5.1 Characteristics of Differential Pencil Beams, 559 26.5.2 Dose Computations with Differential Pencil Beams, 563 26.5.3 Examples, 564 26.5.4 Dose-Spread-Array Model, 566 26.5.5 Electron Beams, 569 26.6 Summary, 570 XVIII Contents

27. Three-Dimensional Dose Calculations for Total Body Irradiation 573 Akira Ito 27.1 Introduction, 573 27.2 Photon-Transport Monte Carlo Model, 574 27.3 60Co Gamma-Ray Pencil-Beam Calculation, 578 27.4 Calculation of Tissue Air Ratio (TAR) for 60Co Gamma Rays, 582 27.5 Calculation of Three-Dimensional Dose Distributions in Patients, 585 27.6 Variance-Reduction Techniques, 588 27.7 Three-Dimensional Dose Distribution in a Patient for TBI, 594

28. High-Energy Physics Applications of EGS 599 A. Del Guerra and Walter R. Nelson 28.1 Introduction, 599 28.2 The EGS Code in Electromagnetic Calorimetry, 600 28.2.1 The Electromagnetic Cascade Shower, 600 28.2.2 Electromagnetic Calorimeters, 601 28.2.3 EGS4 Simulation of EM Calorimeters in General, 603 28.2.4 EGS4 Design of a Lead-Glass Drift Calorimeter, 607 28.3 Coupling EGS with Hadronic Cascade Programs, 613 28.3.1 Hadron Calorimetry, 613 28.3.2 Photohadron Production with FLUKA87/EGS4, 614 28.4 Accelerator Design Applications, 614 28.4.1 Positron Target Design, 615 28.4.2 Heating of Beam Pipes and Other Components, 617 28.4.3 Synchrotron Radiation, 617 28.5 Simulation of a Hydrogen Bubble Chamber, 618

Index 623 Introduction and Fundamentals 1. Overview of Photon and Electron Monte Carlo

A. E. Nahum

Joint Department of Physics Institute of Cancer Research and Royal Marsden Hospital Sutton, Surrey, SM2 5PT, U.K.

1.1 INTRODUCTION This is a personal perspective by one who has applied both his own code and one of the standard codes to medical radiation dosimetry problems for about fifteen years. The more general, excellent reviews of the subject by Raeside1 and Turner et al 2 complement what I have to say.

1.2 SOME HISTORY Three ingredients seeded the development of the simulation of radiation transport by the Monte Carlo method. One of them was the development of quantum theory which furnished us with cross-section data for the interaction of radiation with matter. An• other was the seeming intractability of the problem of multiply scattered radiation, in particular photons and neutrons. Finally, automatic calculating machines were devel• oped:

"The procedure used was a simple graphical and mechanical one. The distance into lead was broken into intervals of one-fifth of a radiation length (ab~ut one mm). The electrons or photons were followed through successive intervals and their fate in passing through a given interval was decided by spinning a wheel of chance; the fate being read from one of a family of curves drawn on a cylinder. .. A word about the wheel of chance: The cylinder, 4 in. outside diameter by 12 in. long, is driven by a high speed motor geared down by a ratio of 20 to 1. The motor armature is heavier than the cylinder and determines where the cylinder stops. The motor was observed to stop at random and, in so far as the cylinder is concerned, its randomness is multiplied by the gear ratio ... JJ

The extract is taken from R. R. Wilson, "Monte Carlo Study of Shower Produc• tion", published in the Physical Review in 19523 • Now you know why I didn't give the digital computer as my final ingredient. Good ideas spread fast. Already in 1949, only a few years after von Neumann had given the method the code name of "Monte Carlo"\ a symposium entitled "Monte Carlo Method" was held. A further quote5 from an IBM Computation Seminar, also in 1949, can serve to debunk any remaining notions any of us may have about the vintage of our research tool:

3 4 A. E. Nahum

"But it is worth noting that the Monte Carlo method is not at all novel to statis• ticians ... For more than 50 years when statisticians have been confronted with a difficult problem ...... they have resorted to what they have sometimes called "model sampling". The process consists of setting up some sort of urn model or system, or drawings from a table of random numbers, whereby the statistic whose distribution is sought can be observed over and over again and the distribution estimated empirically. .. Many other examples can be found by leafing through the pages of Biometrika and the other statistical journals".

A symposiums in 1954 already contained papers by Martin Berger on gamma-ray diffusion, Herman Kahn on sampling techniques, and one by Hayward and Hubbell in which the case histories of no less than 67 I-MeV photons were followed! The very useful bibliography lists no less than 13 papers, with abstracts, on radiation-transport simulation including three papers by R. R. Wilson that tackled electron simulation. The appearance the following year of an article "The Monte Carlo Method" in Scientific American" was further proof that the subject had come of age. Indeed, as a young postgraduate student, I seized upon this article and tackled the Buffon needle problem as a gentle introduction to Monte Carlo.

1.3 PHOTONS. ELECTRONS AND MEDICAL PHYSICS

The simulation of photon transport in matter was essentially solved when Kahn6 showed us how to sample from the Klein-Nishina cross section (assuming, of course, that random-number generation is not a problem). Photons undergo on average a mod• est number of catastrophic interactions, and hence the simulation can be done in an entirely analogue fashion. Furthermore, the cross-section data needed for most applica• tions is known to a high degree of accuracy; one can get a long way with Klein-Nishina. The simulation of electron transport requires a different approach. Berger and Wang, in Chapter 2 , estimate the number of elastic collisions an electron makes in the course of slowing down; the figure is of the order of 10" for a I-MeV electron slowing down to 1 keV. Fortunately, the subject of multiple scattering had received the attention of several prominent theoreticians and the "condensed" random-walk method was born of necessity. Sidei et al 1 were among the first to do this. They used the Moliere multiple• scattering theory together with the total stopping power. One hundred 2-MeV electron histories in aluminium were constructed, with 16 multiple scatterings per history, re• ducing the energy by about 20% in each step so as not to violate the Moliere formula. Results were given for the transmission as a function of thickness of the aluminium layer with remarkably small uncertainties. Much ingenuity was displayed in re-using the same set of 100 histories. The authors concluded, amongst other things, that the efficiency of an ionization chamber for various gamma-ray energies easily could be ob• tained if an automatic computer were used. That particular prediction was just a little on the optimistic side!

Martin Berger taught many of us how to do electron Monte Carlo in his 1963 review8 • All the techniques currently used are described in this pioneering article. By the end of the 60's, most research centres had access to reasonably powerful mainframes. Why then wasn't this course with its heavy bias towards medical radiation physics and electron transport held fifteen to twenty years ago? An analysis of papers exploiting Monte Carlo radiation transport published since the early 70's in the journals Physics in Medicine and Biology and Medical Physics can give us some clues (see Fig. 1.1). 1. Overview of Photon and Electron Monte Carlo 5

ui >- .c a. 16 , , , U Q) ~+ + ~ If) Q) + 12 I- "0 + 0 ai u ~ Q) + - If) 0.: • • • OJ 0 C 8 l- ~ - 80 I , If) ... c + • CD 0- + 0> 0 c a. ++ • 'in u, 4 r- • + , . - 40 ::J ~ + c + Q) 0 e Q) -~ Q) 1 1 a. .0 0 0 E 1970 1975 1980 1985 OJ z YEAR Figure 1.1. a) Left ordinate (+): total number of papers (incl. notes, etc.) involving simulation of radiation transport in the journals Physics in Medicine and Biology and Medical Physics. b) Right ordinate (.): percentage of the above articles in which an in-house Monte Carlo code is used.

In-house development of coupled photon-electron codes requires a lot of effort even if the geometry is simple and there is only one medium involved9• When neither of these simplifications applies, the task is beyond the resources of most research departments. However, the availability of general purpose codes has now made such development completely unnecessary. This book contains no less than 9 chapters* emphasizing this very point in the case of the ETRAN, ITS and EGS4 code systems. Another factor of which we are all aware is the widespread availability of minicomputers such as the VAX. For radiotherapy physics research groups, for example, a VAX has become virtually mandatory. In the early 70's, my professor was literally buying CPU time on IBM and ICL mainframes so that I could finish my calculations and write up my thesis. Money that otherwise would have been spent on apparatus bought me 100 hours or so of CPU time. 2000 primary electron histories were all I could "afford"; it was fortunate that the variance on the absorbed dose in thin slabs for incident electrons was sufficiently small. Had the radiation been 20-MeV photons, I'd simply have run out of money. My colleagues, Dave Rogers and Alex Bielajew, had to re-educate me in the use of CPU time. At the start of our collaboration, I thought an hour was a long time; a few months later, the idea of three VAX-ll/780 computers not going all weekend was too awful to contemplate.

There is a third factor that I suspect has contributed to the current high level of Monte Carlo interest in medical radiation physics. Radiotherapy demands more accurate dosimetry than virtually any other field. This demand has intensified with the advent of the CT (computer tomography) scanner. We now have detailed knowledge of the geometry of the body we are irradiating. It was inevitable that this would lead to refinements in the determination of absorbed-dose distributions in patients. Chapters 21 and 27 directly address this problem, to which I will return below.

* Chapters 7 through 14, and 28 . 6 A. E. Nahum

1.4 INTERESTING ELECTRONS

In many cases, even when the incident radiation is a photon beam, the problem be• ing attacked requires that the finite ranges of the secondary electrons be taken into account; in other words, we usually cannot avoid electron transport. The code builder is faced with certain choices as to how to model the physics of electron interactions. Multiple scattering can be done U:sing the simple Gaussian theory as in the HarderlO and Nahum9 codes, or using the more accurate Moliere formalism as in EGS411 and Andreo programsl2 , or that due to Goudsmit and Saunderson, as in ETRAN and its descendants (see Chapters 2 and 7). Energy-loss straggling can be included explicitly using, for example, the Landau theory, as in ETRAN, or simulated implicitly through discrete ,-ray and bremsstrahlung production down to some cutoff, typically 10 keY, as in the codes of Andreol2, Nahum9 and EGS411. Andreo and Brahmel3 have shown how a restricted straggling distribution can be combined with the second of the above two approaches. Some scheme is needed for deciding when an electron is next "multiply" scattered. Rogersl4 implemented a constant fractional continuous energy loss, ESTEPE, in the EGS4 Code System; all the other codes mentioned already incorporate this fea• ture. The list could be lengthened. The point is that there is no one single correct way of "mixing the electron simulation cake". Further, when the recipe is decided, the usersave to choose values for parameters which have no counterpart in photon simula• tion: ESTEPE or its equivalent, the maximum fractional "continuous" energy loss during the step; AE (EGS4 terminologyll), the energy below which 8-ray losses are treated as "continuous", and so on. The physicist's approach then must be to assure that the desired result, be it absorbed dose, an energy distribution, an angular distribution, etc., does not depend on the particular values of ESTEPE etc. selected. In certain cases, this can involve a great deal of work, though often the code architect has chosen default values that "usually work" for the range of applications envisaged. All this, though potentially a source of problems, makes electron transport simulation interesting.

The entire question of step-size effects receives a great deal of attention in this volume (see Chapters 5, 6 and 25). I wish to show just two examples. Figure 1.2, taken from Rogersl 4, compares angular distributions of 100-keV incident electrons emerging from a 0.004-cm slab of water (~ 1/3rd range), using EGS4 with two very different step-size choices, and using CYLTRAN. There is a significant narrowing of the EGS4 distribution when a maximum 1% continuous energy loss per step is allowed instead of the default step-size values. In this case, there are very few discrete events, so the agreement with CYLTRAN remains excellent despite the fact that the Moliere multiple• scattering formalism is used in EGS4, and the Goudsmit-Saunderson formalism is used in CYLTRAN (with lO-keV kinetic energy cutoffs). This demonstrates very clearly that, at least using EGS4, one needs to choose ESTEPE with care, the default being unsatisfactory.

The second example (Fig. 1.3) is of a 20-MeV beam incident on water; the quantity scored is the absorbed dose in 0.5-1.0 cm slabs, and the geometry is broad beam. The transport cutoff ECUT (total energy) has been varied from an extremely low value, 10 ke V (kinetic energy), to an unreasonably high value, 2.5 MeV (kinetic energy), with very little influence on the result (there appears to be no statistically significant variation until ECUT is greater than 1.5 MeV. This must be trying to tell us that electrons with energies as low as 2.5 MeV have reached a state of near total diffusion; their residual energy is not deposited in any preferred direction. 1. Overview of Photon and Electron Monte Carlo 7

100keV e on O.004cm H20

I III +' c:: 10-1 III "C ·rl U c:: ...... c:: ·M D ...... I QJ

o 20 40 60 60 100 ANGLE (degrees) Figure 1.2. A comparison of the angular distributions calculated for 100-keV electrons passing through a 0.004-cm slab of water (from Rogers14).

4r------l E '"u >- C) 2 '0... 3 -Ecur-700 keV w uz UJ ~ J l4. cECUr-1.5 MeV "•W U) o CI __ Ecur-3. 0 MeV CI W §! o U) III < o 4 6 e 10 DEPTH (em) Figure 1.3 Variation in the depth-dose curve as a function of ECUT for a 20- MeV broad parallel beam of electrons incident normally on a slab of water (AE=521 keV (total energy) for all cases) (from Rogers and Bielajew15).

This example illustrates how, in certain circumstances, one can get away with "un• reasonable" values of certain parameters. Indeed, nowhere in the work of Andreo12 or Nahum9 were step-size effects found to be a problem. This was partly due to the energy range studied (5-50 MeV), partly due to following Berger8 and choosing a value for the equivalent of ESTEPE equal to ~ 0.04, and partly due to the particular geometry. Depth-dose distributions at such energies in broad beam geometry are relatively insen• sitive to the value of ESTEPE. In a sense, Andreo and I were fortunate. Had either of us attempted to score the energy deposited in a small air cavity, however, then things would have been very different, as Bielajew and Rogers (Chapter 5) and I (Chapter 25) make clear. A natural development is to design an electron algorithm that chooses 8 A. E. Nahum its own parameters depending on the geometry. In the EGS4 context, the PRESTA algorithm (see Chapter 5) represents a major advance in this direction.

Let us assume, however, that artefacts due to step-size effects and the like are not present. Can we say that we have got the physics of electron transport right? Firstly, do the results of the different codes agree with each other, and secondly, do any of them agree with experiment? Again, part of this book is devoted to answering these questions (Chapters 6, 8, 13 and 14 in particular). A few examples here will suffice. Figure 1.4, taken from a recent paper comparing certain features of EGS4 and ETRAN16, shows significant differences between the predictions of these two codes for the depth-dose distribution for 20-MeV electrons in water (divergent-beam geometry). The experimental measurements favour EGS4 over the first half of the electron range. In fact, Rogers and Bielajewl 4, showed that the version of ETRAN used contained a minor error in its use of the Landau energy-loss straggling; this has now been corrected (Chapter 7). The question that immediately arises is, "how clean was the experimental geometry?" The authors suggest that the "remaining discrepancies" (between EGS4 and experiment) are "related to the non-monoenergetic nature of the experimental beam". It is shown in Chapter 13 that these remaining differences disappear when the geometry is modelled more faithfully.

N e ~ >- 2<:> , 3 -0 UJ HISTOGRAM - CYLT RAN u z UJ ~ 2 ...J DIAMONDS - EGS ...... LL UJ en 0 STARS - EXPERIMENT 0 0 UJ CD II: 0 en CD < 0 0 2 e 8 10 12 DEPT H (em)

Figure 1.4. Comparison of a measured central axis depth-dose curve in water (stars) with those calculated using EGS4 or CYLTRAN (ETRAN) for a point source of monoenergetic 20-MeV electrons 100 em from the phantom surface. The mean and most probable energies of the experimental data were estimated to be 19.84 and 20.49 MeV, respectively (from Rogers and Bielajew16).

The geometry is extremely "clean" in my other example, taken from a compre• hensive report on experimental benchmarking by Lockwood et aP7. The geometry was broad beam; there was no window and no air as the experiments were done in the accelerator vacuum. The source was highly monoenergetic with a voltage ripple of less than 100 volts (i.e. 0.01% on 1 MeV) and an angular spread of less than 1 milliradian18. 1. Overview of Photon and Electron Monte Carlo 9

The code used was TIGER (see Chapters 10 and 11), with transport cutoffs of 1.0 keV (kinetic energy) for electrons and 10.0 keV for photons. The electron transport algo• rithm follows ETRAN (see Chapter 7). The measurements were made using a thin-foil calorimetric technique, with the heat-absorbing element being made of the substance under investigation, thus avoiding any stopping-power-ratio conversion (Chapter 23).

CARBON 1.0MeV O· 3.0 C CALORIMElER THICKNESS 2 2 1.561 X...... 10- 9/cm

5 ~ 2.0 ~

." EXPERIMENT "I. lMEORY

O.O'--_-'-_---l-:--_-'-_--L:--_L-_-'-_---l__ -'-_--L_= QO QI Q2 Q3 Q4 QS Q6 Ql Q8 Q9 1.0 FRACTION OF A MEAN RANGE Figure 1.5. Comparison of experimental and theoretical energy-deposition profiles in semi-infinite carbon for 1.0-MeV electrons incident at an angle of 0° (from Lockwood et aliT).

4.0 COPPER 1. 0 MeV O' Cu CALORIMElER THICKNESS 2.194 X 1002 9/cm2 I-i

~- ." EXPERIMENT ~ 1. THEORY ii: ~ z 0 ;:: ~ ~,.. '" ~ 1.0

FRACTION OF A MEAN RANGE

Figure 1.6. Comparison of experimental and theoretical energy-deposition profiles in semi-infinite copper'for 1.0-MeV electrons incident at an angle of 0° (from Lockwood et aliT). 10 A. E. Nahum Figure 1.5 shows very good agreement for I-MeV electrons in carbon, except pos• sibly around the depth of maximum dose. Figure 1.6, also for I-MeV electrons, but in copper, demonstrates outstanding agreement at depths greater than 1/3rd of the range, but reveals a serious discrepancy in the dose maximum region. It must be pointed out, however, that these TIGER calculations include the error in ETRAN referred to above.

From the above examples, it would appear that it has yet to be shown conclusively that any code gets the central axis depth-dose distribution absolutely right over a wide range of energies and media. However, any simulation is only as good as the cross• section data fed into it. For years, Berger and Seltzer have provided us with electron stopping-power data, which are described in Chapters 3 and 4. There are areas of uncertainty in the data, especially where collision stopping powers are concerned. It will be fascinating to see if discrepancies between Monte Carlo and experiment can be used to decide between different sets of data, for example, between different evaluations of the density effect, or alternative values of the mean excitation energy, I.

1.5 THE ULTIMATE (RADIOTHERAPY) PROBLEM

Thus far little has been said about the limitations of Monte Carlo imposed by the finite speed of computers. The problems that have been tackled to date have, rather obviously, been the feasible ones. Are they, however, the most important ones? In a perfect world, what would we really like to be able to'do? Again my own dream simulation is taken from the world of radiotherapy. For the unfamiliar, the prime physical goal in external beam radiotherapy is to determine the distribution of absorbed dose at all positions in the patient. Figure 1.7 can serve to illustrate the extraordinary complexity of the clinical situation. Monte Carlo has already played a significant role towards achieving the above-mentioned goal. The precision of the absolute determination of the absorbed dose to water in a water phantom using an air-kerma calibrated ionization chamber has been significantly improved due to Monte Carlo derived stopping-power ratios for both photon and electron beams (see Chapter 23) and precise Awall values obtained from simulations of ion chamber response in a 6OCO beam (see Chapter 25). The passage of the radiation from the vacuum window through any target, filter, collimators, scattering foils and air is the object of considerable Monte Carlo interest at present (Radhe Mohan covers this in Chapter 21).

Both machine-head simulations and stopping-power ratios are one-off computations; a lot of computing time can th'us be expended on them, if necessary. However, dose distributions in patients are required continually for many different field arrangements, beam energies, areas of the body, for the extremely heterogeneous nature of the popula• tion. Roughly one in eight of us will find ourselves undergoing radiotherapy on current trends, barring miracle cures for cancer. Current analytical methods of correcting the dose distributions measured in water for body inhomogeneities and surface curvature are, and always will be, approximate. Also in this area, Monte Carlo has played a significant role, as Chapters 20 and 26 explain. However, there is no doubt that if we could "Monte-Carlo" the patient in order to eliminate all the approximations involved in the various analytical methods, then we would be doing this, at least for a selected number of patients. Ito describes his work on this subject in Chapter 27.

Recently, I have carried out a feasibility study. The goal I set was to determine the dose to ±2% (one S.D.) in an elementary water cube of side 0.5 cm (0.125 cms volume). The geometry was, in principle, that illustrated in Fig. 1.8, except that there was a vacuum between the point source and the phantom surface, 100 cm away. 1. Overview of Photon and Electron Monte Carlo 11

Accelerator Quadrupoles Bending Magnet I===:::=e--:I

Vacuum Window Initial Elc<:: tron Beam -

o ::"'.-:'l ,, -~. ::. :. ~; . Primary Primary - _ _ g{::S1;;:j"~:t:.~{ Scattering Foi l Collimator '£~/ ;\ ", '3~~On dary r ::":"'--~c!f.=~\? Scattering Foil

Ii I ~~ .~

Collimation I

Figure 1.7. The complete geometry for radiotherapy treatment (from N ah um19). 12 A. E. Nahum

A Focus ,1,'/::\ Electron /::\ Beam ,I 1\ Collimator J:,I "' \ . , I ! J.-.!--J. ~i ': ' \

" , I ' I L AI 'II IIy I 'I

/ 'I ~ Phantom ~ ~ ;,- ~ ;,-~

Figure 1.8. The ultimate 3-D Monte Carlo simulation geometry (from Man• fredotti et aI 2o ).

The code used was EGS4 with User Code DOSRZ, for cylindrically symmetric ge• ometry, supplied to me by A. Bielajew. Simulations were carried out for monoenergetic electrons of incident energies 10 and 20 MeV, and for monoenergetic photons of energies 1,5, 10 and 20 MeV. The beam was collimated to a radius of 5.6 cm (equal in area to a "standard" 10 cm by 10 cm square beam) on the circular end-face of a cylinder of water of radius 15 cm and height 20 cm. The dose was scored in elementary cylindrical volumes of height 0.5 cm and radius 0.28 cm (same volume as a 0.5-cm sided cube) on the axis of the beam, at depths from 2.5 to 5.0 cm for the electron beams, and 2.5 cm to 12.0 cm for the photon beams, to correspond approximately to the depths where high precision in dose is required. The electron transport cutoff was set to 0.5 MeV (i.e., ECUT = 1 MeV), at which energy the csda range of an electron is 0.18 cm in water. The data set used had AE = 700 keV; i.e., no delta rays were created with kinetic energy below 189 keV (see Chapter 6). ESTEPE was set at 0.04.

The uncertainty on the dose value is estimated in DOSRZ by dividing the histories into 10 equal batches and then calculating the mean value and the standard deviation on the mean from these 10 dose values. An average uncertainty (1 S.D.) was obtained from the numbers for the individual elementary cylinders. The CPU time for 2% uncertainty was estimated from the relation

(1.1)

where <':1 and <':2 are the uncertainties in N1 and N2 histories, respectively. 1. Overview of Photon and Electron Monte Carlo 13

For 10-MeV electrons, the CPU time on a microVAX II computer to achieve ±2% (1 S.D.) on the dose in the "0.5-cm sided cube", Topu (2%, 0.5 cm), was found to be approximately 50 hours, corresponding to about 9 x 105 primary electron histories. In the case of the photon beams, the results are shown in Fig. 1.9.

5xl07

'" .... If) 400 .... '- ...... :::J ...... 0 (f) .s::. W n ...... ~ 2 - .... -{] 0:: ::> .... 0 0.. "- I- t) (f) I- 200 "- "[SJ :r: " , 5xl06 0 2 5 10 20 PHOTON ENERGY (MeV) Figure 1.9. The CPU time (left ordinate) necesary on a microVAX II to compute the dose to a precision of ±2% (1 S.D.) in an elementary cube of side 0.5 cm at a depth ~ 5 cm in a water phantom of realistic patient dimensions irradiated by a monoenergetic photon beam of radius 5.6 cm at the surface. The right-hand ordinate is the number of primary photon histories required. The precision of the values shown is ~ ± 15%. [EGS4 User Code DOSRZ, with AE = 700 keV (total energy), ESTEPE = 0.04, ECUT = 1.0 MeV (total energy).]

The left-hand ordinate is the quantity, Topu (2%, 0.5 cm), and the right-hand ordinate is the number of primary photon histories required for this precision level. The time per history increased from 0.03 seconds at 1 MeV to 0.10 seconds at 20 MeV, reflecting the increasing importance of electron transport. Tcpu (2%, 0.5 cm) decreases somewhat with increasing photon energy, rather than increasing as might have been expected. As the photon energy increases, the importance of Compton scattering decreases, and the initial kinetic energy of the charged particles is deposited less "locally". Both these effects will tend to decrease the variance per primary history.

No variance reduction has been employed. However, techniques such as range rejection and importance sampling are unlikely to be useful when the dose is required in everyone of the elementary cubes in the irradiated volume (see, however, Chapter 27).

The above results tell us unequivocably that a VAX, in this case a MicroVAX II, does not make such calculations feasible. What are the alternatives? 14 A. E. Nahum

1.6 COMPUTER TECHNOLOGY The cheapness of VAX-type machines has got some of us a long way, but one must put things in perspective. Many workers run their simulations on considerably more powerful machines. Ito chose a mini-supercomputer, the FPS M64/30, and reports a speed-up by a factor of 8 over a VAX-ll/750 with floating point acceleration (Chapter 27). Figure 1.10 (from Ref. 21) and Table 1.1 attempt to summarise the present situation. All the figures are approximate. The RELATIVE SPEED column has been obtained from a comparison given in Dongarra21 on solving a system of linear equations using the LINPACK software in a FORTRAN environment. The relative speeds for running a coupled photon-electron Monte Carlo code will be somewhat different, but of the same order of magnitude.

Can one fully exploit machines such as a CRAY by vectorizing a Monte Carlo code? Martin et a1 22 are the experts here:

"The independence of the particle histories makes it impossible to vectorize the conventional Monte Carlo algorithm. Trying to follow a "vector" of par• ticles would be fruitless because each component of the vector (i.e., each par• ticle) follows a different path since the histories are all different. Hence, the "history-based" algorithm is not vectorizable. However, if instead of following "histories" the algorithm follows "events", then a vectorized algorithm can be formulated. In particular, an event is defined as a portion of a history that begins with the particle emerging at a position r with a velocity v. This parti• cle may have been the result of a scattering collision, it may have been emitted by a source, or it may be sitting on a boundary ready to be transported into the adjacent zone. As the event proceeds, the particle is tracked either to its next collision or to the next boundary or collision site, where the position and velocity of the continuing particle (if any) are known. These can then be used to initiate the next event. For time-dependent Monte Carlo, the event may be terminated by reaching the end of the current time step. Whereas all histo• ries are different, all events are similar. The key to vectorizing Monte Carlo is to construct an "event-based" algorithm and process the particle vector for many events, continually updating the particle vector by eliminating deleted particles and adding new particles until the requisite number of simulations is performed. "

These authors quote a speed-up factor of 4.5 for the vector code VPHOT over an optimized scalar version of the vector code run on a CRAY XMP /2. VPHOT processes 2000 particles simultaneously. These figures were obtained from an inertial confinement fusion plasma problem. One must seriously question whether such a modest speed-up justifies the considerable coding effort involved. Brown and Martin23 report speed-ups of at least 20 to 40 times on a CYBER-205 compared to scalar calculations on a CDC- 7600. They consider that the significant effort for stylized coding and major algorithmic changes is worthwhile. 1. Overview of Photon and Electron Monte Carlo 15

100~~------~~~~------'~ Unit MFlOPS

History of Computer Center. Tokyo Univ. HITAC ,820/20 (1983- ~ 10 1 10 M680H (1986-

main 1 Frame '[-'M280H

(1973-1980)

0.1 (1968-1973)L

by UNPACK (ANL Aug.1986)

O.o1'------...&..--...L..--....L-..&....-....&.---'·O.OI

Figure 1.10. A comparison of the performance of various computers in solving a system of linear equations using the LINPACK software in a FORTRAN environment24•

Table 1.1. The performance of various computers. The number of million floating point operations per second (MFLOPS) and relative speeds were obtained from a benchmarking study by Dongarra21 j the vector capabilities of vector machines were not exploited.

TYPE MACHINE MFLOPS RELATIVE APPROX. TIME FOR PHOTON SPEED BEAM DOSE DISTRIBUTION (Hours) Super-Mini microVAX II 0.13 1.0 500 VAX-ll/780 0.14 1.1 450 VAX 880 0.99 7.6 70 Mainframe IBM 370/195 2.5 19 25 Mini-Super FPS-264 5.6 43 12 Super-Mainframe IBM 3090-200 6.8 52 10 Super CRAY-2 15 115 4.5 16 A. E. Nahum

We Monte Carloists have an insatiable appetite for CPU time. However, we probably do not talk in terms of "70,000" CRAY years, a figure recently quoted by Gaines and Nash25 for the precise prediction of baryon masses by lattice gauge theory. With such ambitions one needs a quantum leap in computing speed. Gaines and Nash advocate very strongly moving away from "big" machines towards grids connecting to• gether large numbers of single board computers (SBCs) based on 32-bit microprocessors. Compare the development at the Edinburgh Regional Computing Centre. A new ma• chine, the Edinburgh Concurrent Supercomputer, will contain over one thousand TSOO floating-point transputers. Each has the power of a large main frame computer giving it a performance of over 1000 million floating point operations per second (1 Giga-flop). My own view is that this is the way to go for Monte Carlo rather than the coding effort required to exploit vector machines. One seeds the transputers with one history each and they all run independently, being fed new histories by the network whenever they are ready. Very little new coding should be involved. One can envisage having one hundred or so microprocessors each with the power of a MicroVAX connected into a suitable network. Gaines and Nash consider that a cost effectiveness better than $500 per VAX-11/7S0 will shortly be possible by exploiting the transputer approach. My own department has recently acquired a transputer development kit consisting of four TSOO's, and a preliminary benchmark against the microVAX II indicates a speed-up by a factor of 2.S per TSOO for a Monte Carlo electron transport code. Thus, we can expect a factor of 11 when all four TSOO's are running the same code. This will enable us to compute patient doses in one-off cases. To perform Monte Carlo dose calculations for every patient will require execution times of the order of an hour or less, which still is not approached even by the CRAY-2 in the table.

1.7 THE APPEAL OF MONTE CARLO

I wish to make a few concluding remarks. The usefulness of Monte Carlo for solving pressing problems in radiation transport is beyond doubt. That is the purpose of the course at Erice and this book. However, I wonder whether that is the whole story. Surely even high-brows such as physicists can't rule out that we sometimes choose a method of tackling a problem because it "appeals" to us. That was certainly true in my case. Wherein then, lies this appeal?

One important ingredient must be the intuition that many of us have that Monte Carlo can, in principle, do anything because it simulates physical reality. It is con• ventional, in general lectures on Monte Carlo, to include an attempt at showing that solving a problem using Monte Carlo is formally equivalent to solving the Boltzmann transport equation. I consider that this is the wrong way of looking at the problem. What could be more fundamental than following the fate of individual particles and then computing averages, etc. Surely the onus is instead on the users of analytical transport equations to prove that their solutions correspond to averages over discrete particle histories. I contend, then, that Monte Carlo simulation does not need formal justification any more than counting particles in an experiment does; all one has to do is to get the "physics" right.

What one is doing with Monte Carlo is performing a mathematical experiment. However, the rules can be changed unlike in "real" experiments. This for me opens up fascinating possibilities. One can label the electrons coming from the side wall of the. detector; I almost feel that this is cheating! One can switch on and off different 1. Overview of Photon and Electron Monte Carlo 17 effects and observe the result. IUy favorite Monte Carlo figure, Fig. 1.10, makes this very point, as Seltzer and Berger understood so very well. There is so much physics in this picture which also is covered by Rogers in Chapter 6. Interestingly, Fig. 1.11 also reveals the minor error in the code used to produce it; this error, and its rectification, have been referred to above. Non-specialists in radiation dosimetry might like to try and figure out what this is. As a teaching aid, both for ourselves and any students of radiation physics, Monte Carlo has enormous possibilities, especially if particle histories are visualized. My last figures (Fig. 1.12a,b), taken from the EGS4 manual 11 , make this point in explaining clearly why 8.5-MeV electrons are much easier to focus through a slit after passing through a scattering foil than are 3.5-MeV electrons.

1.2

C~DA MULllPU SCATT 1.1

1.0 : CSDA. l:: STRAIGHT-AHEAD 0.9 ::/, ...... ~l ~

0.7 , -0'0 0.6 To , 0.5 WATER , To = 30 M.V

0.4 '. '"' 13.1 em

0.3 -

0.2

0.1 - BREMSS1RAHLUNG CONTRIBUTION

0 ...... -...... -...... :.:..:...::.:-~=-=-=---- ._-----...... -~-'------' 0.1 0.2 0.3 0.4 0.5 0.6 0.7 1.3 1.4 z/ro

Figure 1.11. Effect of various approximations on the calculation of electron depth-dose distribution. Results pertain to a broad beam of 3D-MeV electrons perpendicularly incident on a semi-infinite water medium, and are given in terms of scaled, dimensionless quantities (from Seltzer et a1 26 ).

Don't we all like to gamble? Isn't the very unpredictability of Monte Carlo output part of the appeal? Surely I am not the only Monte Carloist who gets excited when plotting a histogram. One watches the rise and fall of the values and tries to imagine a "smooth" curve going through them that confirms one's theory. Some interesting trend is discernible so one immediately runs more histories, dreaming of weekends of uninterrupted CPU time, if only one can wait that long. What analytical method can compare with that? 18 A. E. Nahum

Figure 1.l2a. EGS4 simulation* of 8.5 MeV electrons scattering in a 0.38 mm copper foil (100 events). ·A 2.6 kG magnetic field focuses the the electrons through a lead slit. Solid lines are electrons and dots are photons.

Figure 1.l2b. Same as Fig. 1.12a but energy is 3.5 MeV and magnetic field strength is 1.0 kG.

Acknowledgement: I wish to thank David Rogers for pointing out and helping me rectify an error in the timing benchmarks in Section 1.5, and Akira Ito for supplying Figure 1.10.

* EGS4 User Code UCBEND. 1. Overview of Photon and Electron Monte Carlo 19

REFERENCES 1. D. E. Raeside, "Monte Carlo Principles and Applications", Phys. Med. BioI. 21 (1976) 181. 2. J. E. Turner, H. A. Wright and R. N. Hamm, "Review Article: A Monte Carlo Primer for Health Physicists", Health Phys. 48 (1985) 717. 3. R. R. Wilson, "Monte Carlo Study of Shower Production", Phys. Rev. 86 (1952) 261. 4. D. D. McCracken, "The Monte Carlo Method", Sci. Am. 192 (1955) 90. 5. Symposium on Monte Carlo Methods, edited by H. A. Meyer (John Wiley and Sons, New York, 1954). 6. H. Kahn, "Applications of Monte Carlo", U.S. Atomic Energy Commission report AECU-3259 (1956). 7. T. Sidei, T. Higasimura and K. Kinosita, "Monte-Carlo Calculations of the Mul• tiple Scattering of the Electron", Mem. Fac. Eng. Kyoto Univ. 19 (1957) 220. 8. M. J. Berger, "Monte-Carlo Calculations of the Penetration and Diffusion of Fast Charged Particles", in Methods in Computational Physics, Vol. 1, edited by B. Alder, S. Fernbach and M. Rotenberg (Academic Press, New York, 1963) 135. 9. A. E. Nahum, "Calculations of Electron Flux Spectra in Water Irradiated with Megavoltage Electron and Photon Beams with Applications to Dosimetry" , Doc• toral Thesis, Univ. of Edinburgh (1976); (University Microfilms International, No. 77-70,006). . 10. D. Harder, "Durchgang Scneller Elektronen durch Dicke Materieschichten", Doc• toral Thesis, University of Wiirzburg (1965); also available as Argonne National Laboratory translation ANL-Trans-608 (1967). 11. W. R. Nelson, H. Hirayama, D. W. O. Rogers, "The EGS4 Code System", Stan• ford Linear Accelerator Center report SLAC-265 (1985). 12. P. Andreo, "Aplicacion del Metodo de Monte Carlo a la Penetracion y Dosimetria de Haces de Electrones", Doctoral Thesis, University of Zaragoza (1981). 13. P. Andreo and A. Brahme, "Restricted Energy-Loss Straggling and Multiple Scattering of Electrons in Mixed Monte Carlo Procedures", Radiat. Res. 100 (1984) 16. 14. D. W. O. Rogers, "Low Energy Electron Transport with EGS", Nucl. Instr. Meth. A 227 (1984) 535. 15. D. W. O. Rogers and A. F. Bielajew, "The Use of EGS for Monte Carlo Calcula• tions in Medical Physics", National Research Council of Canada report PXNR- 2692 (1984). 16. D. W. O. Rogers and A. F. Bielajew, "Differences in Electron Depth-Dose Curves Calculated with EGS and ETRAN and Improved Energy-Range Relationships", Med. Phys. 13 (1986) 687. 17. G. J. Lockwood, L. E. Ruggles, G. H. Miller and J. A. Halbleib, "Calorimet• ric Measurement of Electron Energy Deposition in Extended Media-Theory vs Experiment", Sandia National Laboratory report SAND 79-0414 (1980). 18. J. A. Halbleib, private communication (1987). 20 A. E. Nahum

19. A. E. Nahum, "Monte Carlo Electron Transport Simulation II: Application to Dose Planning", in The Computation of Dose Distributions in Electron Beam Radiotherapy, edited by A. E. Nahum, (Umea University, 1985); 319. 20. C. Manfredotti, U. Nastasi, R. Ragona and S. Anglesio, "Comparison of Three• Dimensional Monte Carlo Simulation and the Pencil Beam Algorithm for an Electron Beam from a Linear Accelerator", Nucl. Instr. Meth. A255 (1987) 355. 21. J. J. Dongarra, "Performance of Various Computers using Standard Linear Equa• tions Software in a Fortran Environment", Argonne National Laboratory Tech• nical Memoranundum No. 23, Mathematics and Computer Science Division, (1986). 22. W. R. Martin, P. F. Nowak and J. A. Rathkopf, "Monte Carlo Photon Transport on a Vector Supercomputer", IBM J. Res. and Dev. 30 (1986) 193. 23. F. B. Brown and W. R. Martin, "Monte Carlo Methods for Radiation Transport Analysis on Vector Computers", Prog. in Nucl. Ener. 14 (1984) 269. 24. A. Ito, private communication, based on data in Ref. 21. 25. I. Gaines and T. Nash, "Use of New Computer Technologies in Elementary Par• ticle Physics", in Reviews of Nuclear and Particle Physics, Vol. 37, edited by J. D. Jackson, H. E. Gore and R. F. Schwitters, (Annual Reviews, Inc., Palo Alto, Calif., 1987). 26. S. M. Seltzer, J. H. Hubbell and M. J. Berger, "Some Theoretical Aspects of Electron and Photon Dosimetry", IAEA-SN-222/05 (p. 3-43) in National and International Standardisation of Radiation Dosimetry, Vol.II, IAEA Publication STI/PUB/471 (1978). 2. Multiple-Scattering Angular Deflections and Energy• loss Straggling

Martin J. Berger and Ruqing Wang*

Center for Radiation Research National Bureau of Standards Gaithersburg, Maryland 20899, U.S.A.

2.1 INTRODUCTION When energetic electrons pass through an extended medium, the number of Coulomb interactions with atomic nuclei and orbital electrons is so large that-even with the supercomputers now available-a direct Monte Carlo simulation of all these collisions is impractical. The transport codes of the type discussed in this book, with the exception of Chapter 16, therefore abandon direct simulation of every individual collision, and make use of a "condensed random walk" model in which multiple scattering theories are used to sample angular deflections and energy losses in successive short track seg• ments. The lengths of these segments (also referred to as step-sizes) are chosen to satisfy two conditions: 1) a sufficient number of collisions must occur within each segment, so that the application of the multiple-scattering theories is justified; 2) the cumulative deflections and energy losses in each segment must be small enough so that the con• densed random-walk model provides a sufficiently accurate simulation of electron-track generation, bo.undary crossings, the scoring of energy deposition, etc.

Various types of Monte Carlo models based on this approach have been developed. As demonstrated in Chapters 8, 11 and 13, the predictions from these models are, on the whole, in rather good agreement with a large body of experimental transport data. A case could be made, therefore, that one could let matters rest in regard to cross sections and multiple-scattering distributions, and should concentrate further efforts on refining the transport codes as practical tools, making them easily applicable to complex geometrical configurations of radiation sources and media. This, of course, is the approach that has been used very successfully by the developers of the TIGER and EGS programs.

Nevertheless, it seems worthwhile to review from time to time the infra-structure of the Monte Carlo transport codes, including the pertinent cross sections and multiple• scattering theories. Such a reexamination can indicate reasons for some remaining discrepancies between calculated and experimental transport results; it can provide a posteriori justifications for various assumptions and approximations made in the

* On leave from the Institute of Chemical Physics, Academia Sinica, Dalian, People's Republic of China.

21 22 M. J. Berger and R. Wang

Monte Carlo models; it can make it possible to delineate more precisely the condi• tions of applicability of the models; and it can provide input for a future refinement of the Monte Carlo models.

2.2 ELASTIC·SCATTERING CROSS SECTION Both the EGS and ETRAN codes treat the elastic scattering of electrons by atoms according to a cross section derived by Moliere in a small-angle WKB approximation evaluated using the Thomas-Fermi model of the atom. In the EGS case, the use of such a cross section is implicit in the use of Moliere's multiple-scattering distribution. In ETRAN, the cross section at larger angles is corrected to take into account spin and relativistic effects. In the present section, we explore the use of using more accurate cross sections obtained with partial-wave expansions and based on potentials derived from Hartree-Fock wave functions. The possible improvements can become significant at low energies, especially for high-Z materials.

2.2.1 Factorization There are two important methods by which one can calculate the cross section for the elastic scattering of electrons by atoms with an accuracy better than that provided by the first Born-approximation. Both methods involve the assumption of a static, screened Coulomb potential. One method is to evaluate phase shifts through the nu• merical solution of the Dirac equation and/or the application of the WKB method, and to calculate the cross section from a partial-wave expansion. This requires much computation, because many hundreds of phase shifts may have to be included. The second method, which is computationally easier, involves expressing the cross section as the product of three factors:

(2.1)

The first factor, (TRuth, is the Rutherford cross section for an unscreened Coulomb potential. The second factor, K.an takes into account the screening of the nuclear charge by the orbital electrons, and can be calculated from the scattering theory of Molierel . The third factor, Kr.l, takes into account spin and relativistic effects, and is given by the theory of Mott'.

As was shown by Zeitler and Olsen3, this factorization gives accurate results, pro• vided that screening modifies the Rutherford cross section only at small angles, and spin and relativistic effects are significant only at large angles, so that there is no sig• nificant overlap between the two corrections. According to Zeitler and Olsen, the order of magnitude of the relative error resulting from the factorization is indicated by

2Z4/ 3 (1" + 1) (2.2) q = 1372 1"(1" + 2) ,

where Z is the atomic number and 1" is the kinetic energy in units of the electron rest mass. Table 2.1 shows, for several elements, the electron kinetic energies at which the error estimate q has the values 0.01, 0.02 or 0.05. There is a wide range of energies and Z-values for which the factorization is adequate, and only for high atomic numbers 2. Multiple-Scattering Angular Deflections and Energy-Loss Straggling 23 and low energies is there a requirement for the more costly phase-shift calculations for a screened Coulomb potential.

Table 2.1. Electron energies for which the Zeitler-Olsen error estimate, 'fJ (pertaining to factored elastic-scattering cross sections), has the values 0.01, 0.02 or 0.05.

Kinetic energy (ke V) for which Z 'fJ = 0.01 'fJ = 0.02 'fJ = 0.05 6 30.5 15.1 6.0 13 90.0 43.3 16.9 29 297 135 50.8 47 640 281 101 79 1467 639 217 92 1861 816 274

2.2.2 Spin and Relativity Effects

The spin-relativistic factor J{r.l is equal to the the ratio of the Mott cross section to the Rutherford cross section. This ratio can be computed using Mott's phase-shift formulas for an unscreened Coulomb potential2,4. The Mott cross section, in the form of a Legendre series, must be evaluated numerically which requires only a moderate amount of computation. Tables of J{r.l were published by Doggett and Spencer5.

2.2.3 Screening Effects

An accurate method for calculating the elastic-scattering cross section, that does not depend on the first Born-approximation, was developed by Molierel in a small-angle approximation. Moliere used a screened Coulomb potential derived from the Thomas• Fermi model of the atom, which he represented by the expression

(2.3)

where r A = 0.885 X I'BZ-l / 3 is the atomic radius, rB is the Bohr radius and e is the electronic charge. The other parameters in this formula have the values al = 0.1, a2 = 0.55 , as = 0.35, bl = 6, b2 = 1.2 and bs = 0.3. This potential, which we shall refer to as the Thomas-Fermi potential, has also proved to be very useful in many other applications.

The expression for the Moliere cross section, 0"Mo), is a complicated integral that must be evaluated numerically. Moliere used ingenious approximations, relying in part on interpolations between limiting cases that are amenable to partial analytical eval• uation. These approximations are also described in a review by Scott6 • Zeitler and Olsens,7,8 rederived Moliere's cross-section formula, rearranged it into a form conve• nient for accurate numerical evaluation on a computer, and provided many numerical results. We have repeated and somewhat extended their calculations, thus obtaining a 24 M. J. Berger and R. Wang database from which the Moliere cross section for any element, at all energies above 1 keY, can easily be generated by interpolation.

As shown by Zeitler and Olsen, Moliere's method easily can be combined with the use of Hartree-Fock instead of Thomas-Fermi potentials, provided the potentials are approximated by the sum of three or more Yukawa-typeterms, i.e., an expression similar to Eqn. 2.3, but with more than three terms. Such fitted Hartree-Fock potentials are, in fact, available for selected elements from the work of Cox and Bonham9 and others.

2.2.4 Characteristic Screening Angle

Moliere has shown that in a multiple-scattering theory based on the small-angle approx• imation, the information contained in the screening factor K.cr ( 0) can be expressed in terms of a single quantity, the screening angle, XII' which he defined by the equation

j: -lnXII = }~~ [J K.cr(O) ~ + ~ -In k] (2.4) ° This expression is related to the average of the square of the deflection angle because

(2.5)

Moliere evaluated the screening factor, K.cr(O) by an ingenious combination of an• alytical and numerical procedures, inserted the results into Eqn. 2.4, and thus obtained values of the screening angle XII which he in turn fitted into a simple formula,

2 6.8 X 10-6 Z2/3 2 XII = r(r + 2) (1.13 + 3.76a) , (2.6) where a = Z/137/3. Finally, for the purpose of calculating multiple scattering, the screening factor K.cr itself then can be approximated sufficiently accurately by the expression (1 - cos 0)2 0'" (2.7) K.cr(O) = (1 _ cos 0 + lx~)2 '" (0 2 + X~)2 ' which was used in the multiple-scattering theory of Moliere10, in the moment-method electron-transport theory of Spencerll, and in the ETRAN code.

With the cross-section database now available for the Moliere single-scattering cross section, it is almost as easy to use exact numerical values of K.cr as input for multiple-scattering calculations; however, this makes a significant difference only for very short pathlengths involving a few collisions (plural scattering).

Using precise numerical values of K.cr for the case of a Thomas-Fermi potential, we have computed the screening angle, XII, according to Eqn. 2.4 and have thus obtained a correction factor by which the value of X! from Moliere's approximate formula (Eqn. 2.6) should be multiplied. This factor, given in Table 2.2, is quite close to unity, indicating that Moliere's approximations were remarkably good. The correction only changes 2. Multiple-Scattering Angular Deflections and Energy-Loss Straggling 25

Moliere's multiple-scattering distribution very slightly, and can safely be omitted. We have also evaluated analogous correction factors which can be applied to Eqn. 2.6 to obtain X! with screened Hartree-Fock potentials as parametrized by Cox and Bonham9 • These results are also given in Table 2.2.

Table 2.2 Modification of the Moliere screening angle as a function of the parameter a = Z/137{3. The quantity tabulated is the correction factor by which the value of X! from Eqn. 2.6 should be multiplied.

Thomas- Hartree-Fock Potentials a Fermi H He Be C N 0 Al Cu Ag Potential 0.0 1.037 0.863 1.530 1.130 1.058 1.081 1.116 1.186 1.101 1.183 0.1 1.028 0.856 1.517 1.120 1.048 1.071 1.104 1.175 1.091 1.172 0.2 1.004 0.836 1.482 1.094 1.025 1.047 1.081 1.149 1.066 1.145 0.4 0.950 0.790 1.401 1.034 0.969 0.990 1.022 1.086 1.008 1.083 0.6 0.918 0.764 1.355 1.000 0.939 0.957 0.989 1.050 0.975 1.047 0.8 0.912 0.759 1.246 0.994 0.931 0.951 0.982 1.043 0.969 1.040 1.0 0.918 0.764 1.356 1.001 0.937 0.957 0.989 1.050 1.075 1.047 1.2 0.929 0.773 1.370 1.012 0.947 0.968 1.000 1.062 0.986 1.059 2.4 0.968 0.905 1.427 1.054 0.987 1.008 1.041 1.106 1.027 1.102 4.8 0.982 0.818 1.450 1.070 1.011 1.023 1.057 1.122 1.041 1.117

2.2.5 Calculations by the Partial-Wave Method Many authors have computed differential elastic-scattering cross sections by the partial• wave method. Particularly useful for electron transport calculations are the tables published by Riley, MacCallum and Biggs12 , which include results at 7 energies between 1 and 256 keY. These results are given in tabular form at 33 angles for 24 elements, and in terms of 12-parameter fits in angle for all elements. The numerical procedures used for this cross-section tabulation have been described by Riley13 • A static screened Coulomb potential was used, which was derived from the relativistic wave functions of Ma:nn and Waber14. Exchange effects were neglected, but the resulting error is insignificant at energies above 1 ke V. The neglect of the polarization of the atom, inherent in the use of a static potential, introduced a more important error. Riley estimated that in mercury at 1 ke V, the inclusion of polarization effects would raise the cross section by a factor 3.4 at 0 degrees, but only by 3 percent at 10 degrees. At higher energies, this error is much smaller and confined to angles of a few degrees.

Using Riley's computer code, we have generated a set of elastic-scattering cross sections, at 65 angles and at 11 energies (1,2,4, ... ,1024 keY), for all elements (atomic numbers Z = 1 to 100), using potentials derived from Hartree-Fock wave functions. We have made additional calculations for C, Al and Au at the same energies with Thomas-Fermi and solid-state potentials, and for positrons with Hartree-Fock poten• tials. Additional calculations for hydrogen, oxygen and gold at other energies indicate that, by log-log cubic-spline interpolation on the basic grid of 11 energies, one can ob• tain cross sections at other energies with interpolation errors that are, at most, on the order of 1 percent, and generally much smaller. 26 M. J. Berger and R. Wang

To obtain the screened Coulomb potentials for the Hartree-Fock model, we have used electron-density distributions obtained from the multi-configuration, relativistic Dirac-Fock wave-function program of Desclaux15 • In terms of the electron density, po, the screened Coulomb potential is given by

00 + 41re2 Jxpo(x)dx (2.8)

The densities from Desclaux's code are applicable, strictly speaking, only to a gas of free atoms. We have also used densities appropriate to the solid state, taken from tables given in Ziegler, Biersack and Littmark16, which are based largely on calculations by Moruzzi, Janak and Williams17 according to the muffin-tin model, adapted to the specific lattice structure of the material. In this model, the electron densities (and resulting potentials) are assumed to be spherically symmetric, and are truncated at one-half the distance to the nearest neighbor atom.

2.2.6 Comparisons of Elastic-Scattering Cross Sections

Figure 2.1 compares (curves 3 and 4) elastic-scattering cross sections in gold, factored according to Eqn. 2.1, with those from a partial-wave expansion. Both sets of cross sections were calculated with a Thomas-Fermi potential, and are shown plotted in units of the Rutherford cross section. Also shown are the individual factors K,cr and Kroz. It can be seen that the factorization works quite well at 512 keY, but poorly at 16 keY, as predicted by the criterion of Zeitler and Olsen (Eqn. 2.2).

2 4

o ~ Ir oZ 61 lJ.J en CIl oCIl Ir U 512keV

OL-----~----~----~-- o 50 100 150 ANGLE (degrees) Figure 2.1. Factorization of the elastic-scattering cross section for gold (Thomas• Fermi potential). Curve 1: Screening factor K,cr(B); Curve 2: Spin-relativistic factor Kr.z(B); Curve 3: Product K,.r(B)Kr.z(B) = approximate cross section, in units of the Rutherford cross section; Curve 4: Exact cross section from partial-wave expansion, in units of the Rutherford cross section. 2. Multiple-Scattering Angular Deflections and Energy-Loss Straggling 27

Figure 2.2, also for gold and a Thomas-Fermi potential, gives the ratio of the factored cross section to the exact cross section from a partial-wave expansion, at 7 energies between 1024 to 1 keY. This family of curves indicates how the error inherent in the factorization increases with decreasing electron energy. It should be noted that the factored cross section is quite accurate at small scattering angles regardless of the electron energy.

1.00 1024 keY 512 256 128 0 I-

o ~ ______-L______~ ______-L ____ _ o 50 100 150 ANGLE (degrees)

Figure 2.2. Ratio of the approximate-to-the exact elastic-scattering cross sec• tion for gold (Thomas-Fermi potential). The approximate cross section is factored according to Eqn. 2.1, and the exact cross section is calculated by a partial-wave expansion.

Figure 2.3 shows the elastic-scattering cross sections at 11 energies in gold for electrons and positrons; both sets of cross sections were calculated with a partial-wave expansion and a Hartree-Fock potential. The cross sections for electrons at 1, 2 and 4 keY have pronounced resonances (attributable to virtual bound states). Such resonances are absent in the case of positrons, because the potential in this case is repulsive. 28 M. J. Berger and R. Wang

POSITRONS

1 keV 2 4 8 1 6 -.... 32 (/) 64 ...... C\J0« 1 28 ...... 256 512 z ~ 10-8 '--____....1..- ____-'-- ____--'- ___ 1024 U 0 50 100 150 ~ 10 2

(/) (/) ELECTRONS o ua:: 100 1 ke V 2 4 8 1 6 32 64 128 256

51 2 1 024 1~8~------~------~------~---- o 50 100 5 ANGLE (degrees) Figure 2.3. Cross sections for the elastic scattering of electrons and positrons from gold, calculated by a partial-wave expansion with a Hartree-Fock poten• tial derived from an electron density calculated with Desclaux's code. 2. Multiple-Scattering Angular Deflections and Energy-Loss Straggling 29

Ratios of elastic-scattering cross sections in gold for positrons to those for electrons are given in Table 2.3, at energies of 1024, 64 and 16 keY. Also listed for comparison are the corresponding cross-section ratios calculated with an unscreened Coulomb potential according to Mott's theory, which are considerably different at the lower two energies.

Table 2.3. Ratio of the elastic-scattering cross section for positrons to that of electrons, in gold: (a) From Mott's theory for an unscreened Coulomb potential; (b) From partial-wave expansions with a screened (Hartree-Fock) Coulomb potential.

Angle 1024 keY 64 keY 16 keY (deg) (a) (b) (a) (b) (a) (b) 0 1.000 1.000 1.000 1.000 1.000 1.000 5 0.930 0.940 0.985 0.959 0.996 0.979 10 0.879 0.877 0.969 0.927 0.992 0.956 15 0.813 0.811 0.962 0.898 0.991 0.956 30 0.609 0.606 0.945 0.828 0.965 0.812 60 0.340 0.337 0.759 0.622 1.013 0.688 90 0.233 0.231 0.589 0.453 0.907 0.541 120 0.212 0.208 0.491 0.356 0.658 0.299 150 0.291 0.282 0.435 0.302 0.485 0.174 180 0.489 0.471 0.415 0.284 0.432 0.143

Figure 2.4 compares elastic-scattering cross sections computed with different po• tentials (Thomas-Fermi, Hartree-Fock and solid-state) for carbon at 1 and 8 keY, and for aluminum and gold at 8 keY. The Thomas-Fermi potential gives cross sections that are much too large for deflection angles from zero up to a few degrees, indicating that this potential falls off too slowly at very large distances from the nucleus. As expected, the use of truncated solid-state potentials from the muffin-tin model leads to cross sections which, at very small angles, are somewhat smaller than those obtained with Hartree-Fock potentials.

The differences between the cross sections for various potentials are confined to angles of a few degrees. In electron-transport calculations, these differences will have significant consequences only for plural scattering (involving, say, no more than 10 or 20 successive collisions), but not for multiple scattering. A pertinent quantity for com• paring the influence of different potentials on multiple scattering is the transport cross 11" section, 211" J(1- cos O)O"( 0) sin OdO. The transport cross section, by itself, is sufficient to o calculate the mean cosine of the multiple-scattering deflection (see Section 2.3.4) and the average difference between the depth of penetration and the pathlength traveled by an electron.

The differences between transport cross sections computed with Hartree-Fock and Thomas-Fermi potentials are shown in Table 2.4 for carbon and gold, as functions of the electron energy. The differences are appreciable only at low energies and for high-Z. 30 M. J. Berger and R. Wang

40r------r------~-----, 50 r------,------_,

C, 1 KEV C, 8 KEV 40 30

30

20 20 \\~\ --:;; 10 (\1..... 10 0<[ F \" F····.. ... Ss...... '" .. '"~>,:::., ...... 55 ...... z >-__ ::: .. _ o OL---~--~~===d OL-----~L-==~~ I- 5 10 15 2.5 5.0 U o o lLJ (J)

(J) -. (J) 120 ~ 60 AL, 8 KEV AU, 8 KEV u \;~

80 40 '\F

F 40 -.--.-.--.-~ 20 ······ .... 5.2F\\ ......

O~------~------~ O~------~--~----~ o 2.5 5.0 0 2.5 5.0 ANGLE (degrees) Figure 2.4. Cross sections for elastic scattering of electrons calculated by a partial-wave expansion, using various screened Coulomb potentials. TF: Thomas-Fermi potential; HF: Hartree-Fock potential, derived from electron density calculated from atomic wave functions obtained with Desclaux's code; SS: Solid-state potential, calculated from muffin-tin model electron density given by Ziegler, Biersack and Littmark. 2. Multiple-Scattering Angular Deflections and Energy-Loss Straggling 31

Table 2.4. Percentage amount by which the transport cross section computed with a Thomas-Fermi potential exceeds that computed with a Hartree-Fock potential.

Energy Percent Difference (keV) Carbon Gold 1024 0.01 1.64 512 0.09 1.85 256 0.21 2.03 128 0.24 2.22 64 0.28 2.43 32 0.32 2.72 16 0.39 3.15 8 0.50 3.80 4 0.65 3.93 2 0.85 7.23 1 1.10 13.02

2.2.7 Molecular and Solid-State Effects No rigorous, accurate method appears to be available which could be used routinely to calculate the elastic scattering of electrons by molecules at energies above 1 keY, for a large number of materials. The simplest approximation is to use an independent-atom model, in which the cross section for a molecule is obtained by the linear combination of the cross sections for the constituent atoms. Fink and Kesslerl8 measured scatter• ing from molecular nitrogen and oxygen at 37 keY and 69 keY, and found that the independent-atom approximation is accurate for scattering angles larger than about 2 degrees, but overestimates the cross section by 20 percent or more at smaller angles. Fink and Kessler suggest that the error of the independent-atom approximation is due largely to molecular binding, which changes the spatial distribution of the valence elec• trons, and therefore the screening of the nuclear charges. The independent-atom model also fails to take into account interference effects due to scattering from two or more atoms in a molecule. Moreover, the assumption of a spherically symmetric potential is, in general, no longer to be justified for scattering from molecules.

Even greater complications arise for the elastic scattering of electrons in solids. The crystalline or microcrystalline structure of the material gives rise to a diffraction pattern that is superimposed on the strong forward maximum of the scattering cross section. A clear demonstration of this can be found, for example, in an experiment of Briinger and Menzl9, who measured elastic-scattering cross sections for carbon and germanium at energies from 20 to 60 keY. Their results for germanium at an electron energy of 51.2 keY are shown in Fig. 2.5. The experimental cross section at angles smaller than a few degrees is considerably smaller than that calculated with a Hartree-Fock potential. Furthermore, the cross section exhibits a pronounced modulation (diffraction effect) which is observed not only for crystalline germanium but also for amorphous germa• nium. Similar findings were made by Hilgner and Kessler20 who measured and analyzed the plural scattering of 100-keV electrons in thin films of amorphous Ge, Ni, Ag and Au, and found that the cross sections derived with Hartree-Fock potentials for single atoms predict too high a scattering cross section at small angles in the forward direction. It 32 M. J. Berger and R. Wang appears that even in amorphous materials, there can be a sufficient amount of sttucture (in microcrystalline regions) so that a significant amount of diffraction is superimposed on elastic scattering.

.gl; 10 1

o 0.2 0.4 0.6 !l..!lm U{ -I) >. Figure 2.5. Differential cross section for elastic scattering of 51.2-keV electrons from germanium (Adapted from Brunger and Menz19). The solid curves are experimental. The dashed curves were calculated by Brunger and Menz (1: for an atomic gas; 2: with estimated corrections for interface effects). The points were calculated by present authors for an atomic gas (. : Hartree-Fock potential; *: solid-state potential). Electron wavelength A = 0.0084Ao.

In electron microscopy, the penetration of electrons through very thin films is often of interest, so that the inclusion of diffraction phenomena in transport calculations is important. A great deal of pertinent information can be found in books by Reimer21 and Egerton22 . Transport codes such as ETRAN or EGS are not intended or suit• able for application in the thin-film, plural-scattering regime, where interference effects depending on the structure of the material play an important role. For electron pene• tration through thick targets, one expects the diffraction effects to be less significant, resulting in changes of perhaps the same order of magnitude as those due to differences between Thomas-Fermi and Hartree-Fock potentials.

2.3 CALCULATION OF MULTIPLE-SCATTERING DEFLECTIONS In some Monte Carlo electron-transport codes, for example EGS, angular deflections are sampled from the multiple-scattering distribution of Moliere10. In others, for example ETRAN, the sampling is done from the distribution of Goudsmit and Saunderson23 • Both multiple-scattering distributions have their advantages and disadvantages.

The Moliere distribution is a universal function of a scaled angular variable, which simplifies the random sampling, and makes it easy to select multiple-scattering deflec• tions for randomly selected pathlengths. On the other hand, the Moliere distribution is based on a small-angle approximation, and is intended to be used for multiple-scattering deflections no larger than about 20 degrees. Spin and relativistic effects are not taken 2. Multiple-Scattering Angular Deflections and Energy-Loss Straggling 33 into account. Finally, the applicability of the Moliere distribution is restricted to path• lengths long enough so that they include at least on the order of 100 elastic collisions, a condition which sometimes is difficult to satisfy in condensed random-walk Monte Carlo models.

Random sampling from the Goudsmit-Saunderson distribution is somewhat more complicated, so that it is advantageous to sample deflections from stored multiple• scattering distributions for a pre-selected set of pathlengths, which is the procedure adopted in the ETRAN code. There are, however, several compensating advantages. The Goudsmit-Saunderson distribution can be applied for much smaller pathlengths, and is applicable without restriction at all scattering angles. It can easily be used with a variety of elastic-scattering cross sections calculated with different potentials, with spin and relativistic effects included. It is possible to take into account differences between the scattering of electrons and positrons. These advantages are particularly important at energies below 1 MeV and in high-Z materials.

2.3.1 Moliere Multiple-Scattering Distribution

The multiple-scattering distribution of MolierelO, as a function of the deflection angle B and the pathlength s, is

where 2 f(n)(19) = ~Joo uduJo(19u)e-u2/4(U2ln u )n , (2.10) n. 4 4 o and Jo is a zero-order Bessel function. This distribution is normalized so that the result is unity when it is integrated with respect to 19 from 0 to infinity. Deflections are expressed in terms of a scaled variable,

19 = OIXcVB , (2.11)

where Z2 [ T + 1 ]2 Xc = 0.6009-:4 T(T + 2) s, (2.12) with s in units of g/cm2 • The expansion parameter B is obtained from the solution of the equation B -lnB = In(x~/1.167X!) , (2.13) where Xa is given by Eqn. 2.6.

The error due to the termination of the expansion in Eqn. 2.9 is of the order II B3, and is thus about 1 percent for B = 4.5. Moliere regarded the value 4.5 as the lower limit for the applicability of his distribution.

The functions fen) (19) (with n = 1 and 2) needed In Eqn. 2.9 have been evaluated by Moliere10 and by Bethe24 by a combination of analytical and numerical procedures. We have found it simple to do the integrations in Eqn. 2.10 by straightforward numerical 34 M. J. Berger and R. Wang quadrature for any value of n. Therefore, it would be easy to extend the expansion so as to include terms with n > 2. However, according to Moliere, this would not be sufficient to keep the error acceptably small for B < 4.5 because of other approximations made in the derivation of his distribution.

The Moliere distribution does not include spin-relativistic effects. As pointed out by Bethe24 and demonstrated by Spencer and Blanchard25 , this can be done rather ac• curately by multiplying the Moliere distribution by the single-scattering spin-relativistic factor Krel, provided that the multiple-scattering distribution is predominated by single scattering at the larger angles where K rel deviates significantly from unity. This condi• tion can be satisfied in the multi-MeV energy region, but not below 1 MeV, particularly for high-Z materials.

MolierelO mentioned the possibility of extending his theory to the case of a medium consisting of irregularly distributed crystallites. He suggested that his theory would remain applicable to such a material if the crystallites are small enough so that at most one elastic collision would occur in each, so that the incoherence of the multiple• scattering process remains preserved. It would be necessary, however, to multiply the elastic-scattering cross section by a structure factor that takes into account diffraction effects. Such a method has not yet been implemented, perhaps because of the difficulty of obtaining the appropriate structure factors.

2.3.2 Goudsmit-Saunderson Multiple-Scattering Distribution

The distribution of Goudsmit and Saunderson23 , which is valid for arbitrarily large angles, has the form of a Legendre series:

00 FGs(O,s) = 2:)£+ 1j2)exp(-sGt)Pt(cosO). (2.14) l=O

This distribution is normalized so that the result is unity when it is integrated with respect to 0 from 0 to 11'. The Pt'S are Legendre polynomials, s is the pathlength (in gjcm2 ), and the coefficients Gt are calculated from the equation ,.. Gt = 211":" /[1- Pt(cosO)]o-(O)sinOdO , (2.15) o where N .. is Avogadro's number and A is the atomic weight. The energy loss of the electron along its track can be taken into account in the continuous-slowing-down approximation, by making in Eqn. 2.14, the substitution

(2.16)

where To is the initial electron kinetic energy, Tl is the energy reached after the traversal of a pathlength s, and where S(T)jp is the mass stopping power (in MeV-cm2 jg). 2. Multiple-Scattering Angular Deflections and Energy-Loss Straggling 35

In the ETRAN code, a cross section factored according to Eqn. 2.1 is used, and J{r.1 is fitted by a simple expression such that the integrals in Eqn. 2.15 can be done an• alytically. Recursion relations for the Gt'S thereby are obtained which can be evaluated quickly with little computation. Riley, MacCallum and Biggs12 also provided fitting formulas for their elastic-scattering cross sections which can be used to obtain similar recursion relations for the Gt's.

In the present work, we have not used recursion relations, but instead have relied on direct numerical quadrature to obtain the Gt's. This requires more computation, but nevertheless can be done rather easily and has several advantages. There is no need for developing accurate analytical approximations to the cross sections obtained with various potentials. It possible to apply an angular cutoff to the single-scattering cross sections. For large values of e, numerical integration was found to provide more accurate results than the use of recursion relations because the latter are subject to a buildup of round-off error, and can become unstable.

The number of terms required to make the Goudsmit-Saunderson Legendre series converge increases as the pathlength becomes shorter, particularly for large scattering angles. We have used up to 999 terms for the Goudsmit-Saunderson series, and have found that, for carbon and gold (Hartree-Fock potential), convergence can be achieved for pathlengths as short as 20 mean free paths at 1024 keY, and for pathlengths as low as 6 mean free paths at lower energies.

The convergence of the Goudsmit-Saunderson series in the case of small pathlengths can be improved by separating the contributions to the multiple-scattering distribution from electrons that have had either no collision or exactly one collision. The Goudsmit• Saunderson distribution then takes the form

00 (2.17) + L(e + 1/2)[e-·Gt - e-"'(1 + J-ts - sGt))Pt(cosO) t=o where ,.. J-t = 21r'l J<1(O)sinOdO. (2.18) o

The first term on the right-hand side of Eqn. 2.17, proportional to a delta function, represents unscattered electrons, and the second term, proportional to the scattering cross section, represents singly-scattered electrons.

2.3.3 Contribution of Inelastic Collisions to Multiple Scattering

Bethe24 recommended that in the Moliere distribution, the factor Z2 in Eqn. 2.12 should be replaced by Z(Z + 1) in order to include the effects of inelastic scattering by atomic electrons. Fano26 introduced a further refinement which takes into account the fact that the screening effects for electron-atom and electron-electron collisions are somewhat 36 M. J. Berger and R. Wang different. Fano showed that for electrons, in addition to replacing Z2 by Z(Z + 1), one should change Eqn. 2.13 by adding a term on the right-hand side:

B -lnB = In(x;/1.167X!) + (u a - u;n)/(Z + 1) , (2.19) where (2.20) and where U is a constant for a given material, which must be calculated with the use of incoherent-scattering factors. Fano gave the values -3.6 for H, -4.6 for Li, -5.0 for 0 and -6.3 for Pb. In the ETRAN code and in the present work, a set of u-values for the elements (shown in Table 2.5) has been adopted, which was obtained by interpolation between the values given by Fano.

Table 2.5. Values of the screening parameter U;n adopted in present work.

Atomic Number -U;n Atomic Number -u;n 1 3.6 21 - 24 5.5 2 4.0 25 - 29 5.6 3 4.3 30 - 34 5.7 4 4.6 35 - 40 5.8 5 4.8 41 - 47 5.9 6 5.0 48 - 55 6.0 7 - 11 5.1 56 - 65 6.1 12 - 14 5.2 66 - 76 6.2 15 - 17 5.3 77 - 88 6.3 18 - 20 5.4 89 -100 6.4

The equivalent change for the Goudsmit-Saunderson distribution, first introduced by Spencerll, consists of multiplying the coefficients Gt by

(2.21 ) where

f I = (U;n - ua)/ [( Z + 1) In x!l . (2.22)

With the procedures discussed above, all deflections from inelastic collisions, re• gardless of their magnitudes, are allowed to contribute to the multiple-scattering distri• bution. In a more accurate Monte Carlo model, one would include contributions only from inelastic deflections which are smaller than a specified cutoff value Be, corresponding 2. Multiple-Scattering Angular Deflections and Energy-Loss Straggling 37 to a chosen small energy loss. The coefficients GI. then could be calculated according to the following prescription: ,.. Gt = 211":" {/[1 - Pt(cos 0)]0"(0) sinOdO o I. (2.23) + ~(1 + f./) /[1- PI.(cosO)]O"(O)sinOdO}. o

A multiple-scattering angular distribution for 1024-keV electrons in carbon, evalu• ated with a 12-degree cutoff for the contributions from inelastic deflections, is compared in Fig. 2.6 with distributions obtained in simpler approximations.

Carbon 1024 keV 6

AN GLE (degrees) Figure 2.6. Contribution of inelastic scattering to the multiple-scattering angular distribution in carbon at 1024 keY. Calculated with Thomas-Fermi po• tential, partial-wave expansion and Goudsmit-Saunderson series. Solid curve 1: Contribution of inelastic scattering omitted. Solid curve 2: Contribution of inelastic scattering included through factor (1 + l/Z)(l + f./). Dashed curve: Similar to curve 2, but with 12-degree cutoff for inelastic collisions, as dis• cussed in the text.

2.3.4 Number of Elastic Collisions and Mean Deflection Angle

In the continuous-slowing-down approximation, the number n. of elastic collisions which an electron will undergo while slowing down from an initial energy To to a final energy Tl is given by To p,(T) n. = JS(T)/ pdT, (2.24) Tl where p, is given by Eqn. 2.18. The mean value of the cosine of the multiple-scattering deflection angle is To Gl(T) } (cos 8) = exp{ - JS(T)/ pdT . (2.25) Tl 38 M. J. Berger and R. Wang

This result is obtained by multiplying Eqn 2.14 by cos 0 = Pl(O), integrating with re• spect to 0 (taking advantage of the orthogonality properties of the Legendre polynomials Pt ), and using the substitution (Eqn 2.16) to take into account the energy dependence.

Table 2.6 gives the number n. of elastic collisions made by electrons slowing down to an energy of 1 keY in carbon, aluminum and gold. These results were obtained with cross sections computed with Hartree-Fock potentials. For low initial energies, say below 20 or 50 keY, n. is so small that the use of the condensed random-walk model becomes questionable; it would be more accurate and not much more expensive to follow all individual elastic collisions. If solid-state potentials were used and diffraction effects were included, the number of collisions would be even smaller than shown in Table 2.6.

Table 2.6. Average number of elastic collisions which an electron undergoes while slowing down from energy T to 1 keY. Calculated in the continuous-slowing-down approximation, using cross sections computed with Hartree-Fock potentials.

T Average Number of (keV) Elastic Collisions Carbon Aluminum Gold 1024 9525 14820 27380 512 5240 8468 14880 256 2861 4655 7764 128 1563 2557 3912 64 856 1401 1916 32 468 761 921 16 252 406 441 8 131 207 211 4 63 2 23

Table 2.S gives the average number of collisions, n. and the mean deflection angle (arccos( (cos 0))) for track segments with lengths typically used in the ETRAN code. These results take into account not only elastic collisions, but also inelastic collisions, through the inclusion of a factor (1 + liZ) in the formula for Ct. The correspo!lding distributions of multiple-scattering deflections are shown in Fig. 7.2 of Chapter 7. It can be seen there that in carbon, the distributions are well-behaved down to an energy of'S keY (with a segment length of 4.5 mean free paths), and in gold down to 32 keY (with a segment length of 4.5 mean free paths). At these energies, the angular distributions begin to exhibit slight irregularities due to lack of adequate convergence of the Goudsmit-Saunderson series, and at lower energies, the convergence would be very poor. In the ETRAN code, a crude but serviceable approximation is used to force convergence (see Chapter 7). However, when only a small number of collisions occur per segment, it would be better to abandon the track-segment model, and to sample all individual elastic collisions. 2. Multiple-Scattering Angular Deflections and Energy-Loss Straggling 39

Table 2.7. Values of the Moliere parameter B, scale angle X• ..Jjj and pathlength S, for the multiple-scattering angular distributions shown in Figs. 2.5 and 2.6. All pathlengths correspond to 200 mean free paths for a cross section calculated with the Thomas-Fermi potential. Energy loss is not taken into account.

Carbon Gold

T B X • ..Jjj S B X • ..Jjj S (keV) (deg) mg/cm2 (deg) mg/cm2 1024 6.35 7.3 8.65 5.22 16.5 5.85 512 6.35 11.8 7.22 5.07 26.2 4.82 256 6.27 17.7 5.03 4.87 40.4 3.69 128 6.22 25.8 3.15 4.58 60.6 2.57 64 6.20 37.3 1.82 4.18 89.3 1.69 32 6.17 53.2 0.984 3.69 130.83 1.08 16 6.11 75.4 0.513 3.10 187.1 0.677

Table 2.8. Energy intervals, pathlengths, mean number of collisions and mean multiple• scattering-deflection angles for track segments similar to those used in the ETRAN code. The energy intervals extend from T1 = (T/2)(1 +K + (1-K)/m) to Tz = (T/2)(1 + K - (1- K)/m), with K = 2-1/ 8 = 0.9170 and m = 2 for carbon and 18 for gold. S is the pathlength, n. the average number of elastic collisions, and () Au is the mean deflection angle, arccos ( < cos () » from Eqn. 2.25.

Carbon Gold

T S n. ()Au S n. ()Au (keV) (mg/cm2) (deg) (mg/cm2) (deg) 1024 25.08 335.7 15.0 4.189 108.0 19.7 512 11.79 191.4 17.7 2.954 62.9 22.9 256 4.685 104.4 18.7 0.8694 34.7 24.9 128 1.638 56.8 19.8 0.3212 18.3 25.6 64 0.5205 31.1 20.4 0.1081 9.24 25.2 32 0.1565 17.2 20.8 0.0348 4.52 24.3 16 0.04592 9.60 21.0 8 0.01343 5.43 21.3 4 0.00397 3.11 21.5

2.3.5 Comparison of Multiple-Scattering Distributions

For small multiple-scattering deflections, the distributions of Moliere and Goudsmit• Saunderson are essentially equivalent, i.e., FMo1«())()dO '" FGs(O) sin OdO. For large de• flections, the Goudsmit-Saunderson distribution is more accurate. The relation between the two theories has been analyzed by Bethe24, and by Winterbon27• Bethe showed that 40 M. J. Berger and R. Wang the accuracy of the Moliere distribution for large angles can be improved through mul• tiplication by a factor (0/ sin 0)1/2, so that

FMo1 ( 0)( 0 sin 0) ~ dO '" FGs( 0) sin OdO. (2.26)

In order to indicate the quantitative relationship between the two multiple-scattering theories, a few plots are shown in Figs. 2.7 a-d and 2.8. The quantities plotted are (0 sin 0)1/2 FMo1 ( 0) and sin OFGs( 0).

8 0.8

Carbon Carbon 0.6 6 1024 keV 16 keV

4 0.4

2 0.2 z 0 I- 0 0 :::l (I) 0 5 10 15 20 0 60 120 180 ct: I- 3 0.5 (f) 0 Gold 1024 keV 2 0.3 Gold 16 keV

o L---L_-L-_L----L.:.::::::~:=:ol 0 ~__L _ __'___L-__L_ __'__---1 o 20 40 60 0 60 120 180 ANGLE (degrees)

Figure 2.7. Multiple-scattering angular distributions, for a pathlength of 200 mfp (for a Thomas-Fermi potential). Solid curves: Goudsmit-Saunderson se• ries. Dashed curves: Moliere's multiple-scattering theory.

The Moliere distribution is implicitly based on the use of the Thomas-Fermi po• tential. Therefore, the Goudsmit-Saunderson results chosen for comparison were also evaluated with the Thomas-Fermi potential. The distributions shown were calculated without taking into account energy loss, and pertain to a pathlength equal to 200 mean free paths (values in mg/cm2 are given in Table 2.7). 2. Multiple-Scattering Angular Deflections and Energy-Loss Straggling 41

In carbon, the agreement between the two theories is very close at 1024 keY, and still rather good at 16 keY. For gold, the agreement is fairly good at 1024 keY, but not as close as for carbon. At lower energies in gold, the differences between the two theories are much greater. This is illustrated for 64 keVelectrons in Fig. 2.8 which shows three results: the Moliere distribution, and two curves calculated with Goudsmit• Saunderson series, one based on a factored cross section and the other one based on a cross section from a partial-wave calculation. Neither the Moliere distribution, nor the Goudsmit-Saunderson theory with a factored single-scattering cross section, are adequate in this case. At 16 keY in gold, the disagreement between the Moliere and Goudsmit-Saunderson distributions is much smaller than at 64 keY. Actually both distributions, for a pathlength of 200 mean free paths, become almost isotropic, so that each of the plotted curves is close to sin (I.

0.75

/ ...... Gold,64 keV / -- '\... / '" / / '\ 0.50 z // \, 0 I,' , f- \, ::J I,' , CD \ ' n: I,' , f- I,' \ ' , (f) , \ , 0 0.25 I,' , I,' '\ , I,' l "

0 0 60 120 180 ANGLE (degrees) Figure 2.8. Multiple-scattering angular distributions in gold at 64 keY for a pathlength of 200 mfp's (calculated with a Thomas-Fermi potential). Solid curve: Goudsmit-Saunderson series, with cross section from partial-wave ex• pansion. Short dashes: Goudsmit-Saunderson series, with factored cross sec• tion. Long dashes: Moliere's multiple-scattering distribution.

The differences between the l\loliere and Goudsmit-Saunderson distributions can, to some extent, be related to the values of the Moliere expansion parameter B and scale angle Xcv'B, which are given in Table 2.7. For carbon, the value of B is always safely above the required minimum value of 4.5, the scale angles are smaller than 26 degrees down to 128 keY, and the agreement between the two theories therefore is excellent. At 16 keY, the scale angle is rather large (75 degrees), and the agreement begins to deteriorate. For gold, the values of B are greater than 4.5 above 64 keY, and the scale angle is smaller than 26 degrees above 256 xeV. The discrepancy between the Moliere and Goudsmit-Saunderson distributions at 1024 keY therefore must be ascribed to the omission of spin and relativistic effects in Moliere's theory. 42 M. J. Berger and R. Wang

The difficulties encountered in the application of Moliere's distribution to a high-Z material such as gold are further illustrated in Table 2.9, which gives values of B and the scale angle Xc..fii, for pathlengths corresponding to step-sizes typically used in the ETRAN code. At low energies, it is impossible to have B sufficiently large (greater than 4.5), and at the same time the scale angle Xc..fii sufficiently small (less than, say, 25 degrees).

Table 2.9. Parameters of Moliere multiple-scattering distribution in gold, as functions of pathlength and electron kinetic energy T. Pathlength s is given in units of mg/cm2 and in units of mean free paths (for elastic scattering with Thomas-Fermi potential). Pathlengths are for track segments typically used in the ETRAN code.

T s B Xc..fii (keV) (mg/cm2) mfp (deg) 512 8.20 336 5.72 36.3 4.10 167 4.69 23.6 2.05 84 3.98 15.6

256 3.48 186 4.80 38.9 1.74 93 3.90 24.8 0.87 46 2.92 15.7

128 2.57 200 4.58 60.5 1.28 109 3.66 38.3 0.64 67 2.64 23.0

Figure 2.9 shows angular distributions in carbon for 1024 keY, obtained with a Thomas-Fermi potential and with the Goudsmit-Saunderson series, for a pathlength of 200 mean free paths, with various cutoffs (3, 6 and 12 degrees) applied to the under• lying single-scattering cross section. Distributions of this kind can be used in a Monte Carlo model in which small elastic deflections in a track segment are combined and sampled from a multiple-scattering distribution, whereas large deflections are sampled individually (Andreo and Brahme28).

Finally, Fig. 2.10 compares Goudsmit-Saunderson angular distributions for a path• length of 200 mean free paths in gold for 128-keV electrons (with Thomas-Fermi and Hartree-Fock potentials) and for 128-keV positrons (with a Hartree-Fock potential). The difference between the two distributions for electrons with different potentials is quite small, whereas the difference between the distributions for electrons and positrons is large. 2. Multiple-Scattering Angular Deflections and Energy-Loss Straggling 43

20

Carbon 1024 keV 15 z Q I- ~ m 10 a:: enI- 0 5

5 10 15 20 ANG LE (degrees) Figure 2.9. Multiple-scattering angular distributions calculated with vari• ous cutoffs for underlying single-scattering cross section. Calculated with a Thomas-Fermi potential, a partial-wave expansion, and a Goudsmit-Saunderson series (for a pathlength of 200 mfp's).

I '\ Gold \ 128 keV

z o I• ~ ~ 0.5 a:: I•en o

o~~--~~--~~--~--~~--~~~~~ o 60 120 180 ANGLE (degrees) Figure 2.10. Multiple-scattering angular distributions for electrons and positrons. Calculated with a partial-wave expansion and a doudsmit-Saunderson series. Solid curve: Hartree-Fock potential, electrons. Long dashes: Hartree-Fock potential, positrons. Short dilShes: Thomas-Fermi potential, electrons.

The comparisons shown so far have pertained mainly to multiple-scattering de• flections of moderate size. To illustrate the differences for large deflections, we com• pare in Table 2.10 the probabilities, calculated on various assumptions, that the de- 44 M. J. Berger and R. Wang flection angle in a pathlength of 200 mean free paths will exceed 90 degrees. In the cases examined, the misapplication of the Moliere theory to very large angles leads to an overestimate of large deflections for carbon, and an underestimate for gold. Even though the probabilities for backscattering are small for anyone track segment, they can become appreciable when they are compounded for many successive track segments. The Goudsmit-Saunderson distribution therefore is to be preferred for the calculation of quantities such as backscattering coefficients.

Table 2.10. Probability of a multiple-scattering deflection for electrons and positrons through an angle greater than 90 degrees. Results are for a distance of 200 mean free paths (for the case of a Thomas-Fermi potential). The method of calculation is indicated as follows: Multiple-scattering theory: Single-scattering cross section: M: Moliere (M): Moliere (WKB) GS: Goudsmit-Saunderson (TF): Partial-wave exp., Thomas-Fermi potential (HF): Partial-wave exp., IIartree-Fock potential

Carbon

Energy, keY 1024 512 256 Pathlength, mg/cm2 8.65 7.22 5.03 PARTICLE METHOD PROBABILITY (percent) Electron M(M) 0.050 0.137 0.337 Electron GS(TF) 0.023 0.078 0.239 Electron GS(HF) 0.023 0.078 0.239 Positron GS(HF) 0.021 0.073 0.226 Gold

Energy, keY 1024 512 256 Pathlength, mg/cm2 5.85 4.82 3.69 PARTICLE METHOD PROBABILITY (percent) Electron M(M) 0.346 0.1.09 3.70 Electron GS(M) 0.665 2.23 5.83 Electron GS(TF) 0.668 2.70 7.01 Electron GS(HF) 0.663 2.17 6.86 Positron GS(HF) 0.149 0.632 2.71

2.4 ENERGY-LOSS STRAGGLING

In the ETRAN code, energy losses in track segments are sampled from the straggling distribution of Landau29 , modified to include binding effects according to Blunck and LeisegangSo • This treatment is accurate only when the track segments are sufficiently long. As discussed below, this condition is difficult to satisfy below 1 MeV in high-Z material. 2. Multiple-Scattering Angular Deflections and Energy-Loss Straggling 45

It is possible to treat energy-loss straggling by alternative procedures applicable to arbitrarily short pathlengths. Two steps are involved: (a) the calculation of an energy• loss cross section for single collisions; (b) a calculation of the distribution of the sum of many energy losses in many successive collisions, which can be done by a variety of convolution techniques involving the use of Laplace transforms, numerical integrations, or random sampling. Step (a) is a difficult task which must be done separately for each material by combining all the available experimental and theoretical data. A good example of such a critical data analysis is the recent work of Bichse131 on energy loss straggling in silicon.

2.4.1 Landau's Distribution: Applicability, Refinements

Let 1(6., s )d6. be the probability distribution of an energy loss between 6. and 6. + d6. in a pathlength s. In Landau's theory, it is given by

A(6.,s)d6. = ¢£(>..)d>.. , (2.27) where (h(>") is a universal function of a scaled energy-loss parameter >.., which has been tabulated by Boersch-Supan32 . The relation between the scaled and the actual loss is

2~ mc2(32 ] 6. = ~{>.. + In [ (1 _ (32)12 - (32 + 0.4228 - 8}, (2.28) where 1 is the mean excitation energy and 8 is the density-effect correction (see Chapter 3 for a discussion of these two characteristics of the medium). The quantity

(2.29) in Eqn. 2.28 can be interpreted as the single-scattering energy loss that is exceeded, on the average, only once in a pathlength s.

Landau's theory admits arbitrarily large energy losses in single inelastic collisions, which has the consequence that (/>L(>") does not have a finite mean value. Therefore, it is necessary to truncate the distribution in order to obtain the appropriate mean energy loss per unit pathlength (stopping power). As discussed in Chapter 7, the truncation value depends on the electron energy and pathlength.

The Landau theory takes into account only the mean value, but not the higher moments, of the cross section for small energy transfers, (for which binding effects are important). The Blunck-Leisegang correction takes into account the correct second moment, and consists of convoluting the Landau distribution 1(6., s) with a Gaussian distribution with a zero mean value and with the appropriate variance 0'2. The choice of 0'2 is difficult because the cross sections for small energy losses are not known sufficiently accurately. Various procedures for estimating 0'2 can be found in the literature. Blunck and WestphaP3 gave a particularly simple prescription which is based on the Thomas• Fermi model of the atom: (2.30) where ~ is the mean energy loss in pathlength s , and where Q is a constant set equal to 10 eV. As discussed in Chapter 7, application of the Blunck-Westphal prescription 46 M. J. Berger and R. Wang to the energy-loss straggling of electrons gives results for the most-probable energy loss, and for the full-width at half-maximum of the energy-loss distribution, which are in good agreement with a large body of experimental data.

A condition for the applicability of Landau's theory is that the quantitye (defined by Eqn. 2.29), which is proportional to the pathlength s, must be large compared to the mean excitation energy I. In an analysis of the approximations involved in the derivation of Landau's distribution, Chechin and Ermilova34 have shown that the relative error 7]BL of the Landau theory (with the Blunck-Leisegang correction) depends as follows on the ratio U I: e( le)3]-1/2 7]BL '" [lO I 1 + 10 I . (2.31)

This error pertains to the Laplace transform of the straggling distribution as evaluated by the method of Landau.

For e/ I equal to 1, 4, 10, 40 and 100, the magnitude of 7]BL is 0.3, 0.1, 0.04 and 0.005, respectively. In a Monte Carlo model, the length of succcessive track segments therefore should be such that the value of U I is equal to 10 or greater. This requires track segments longer than desirable for the sampling of multiple-scattering deflections. A double-segmentation scheme therefore is used in ETRAN such that several very short sub-segments used for sampling angular deflections are combined into a single segment that is used for sampling energy losses. This scheme was illustrated earlier in Table 2.5 for carbon and gold, and is discussed more generally in Chapter 7.

Table 2.11 shows the average reduction of the electron energy for pathlengths such that U I = 10. In carbon, the reduction is acceptably small, amounting to no more than 6 percent even for an initial electron energy as low as 200 keV. In gold, however, the reduction exceeds 10 percent below 1 MeV and 20 percent below 500 keV. Such large energy changes per segment are undesirable in the modeling of tracks, so that it may be preferable to compromise by chosing a smaller ratio, such as U I = 4.

The energy spectrum of electrons is determined by pathlength straggling as well as energy-loss straggling. By pathlength straggling, we mean the fluctutations of the pathlengths traveled by electrons to the point of interest, taking into account the detours resulting from multiple-scattering deflections. The lower the energy and the higher the atomic number of the material, the greater is the relative importance of pathlength straggling. Therefore, for conditions in which the errors in the treatment of energy-loss straggling are greatest, energy-loss straggling is relatively unimportant compared to pathlength straggling. 2. Multiple-Scattering Angular Deflections and Energy-Loss Straggling 47

Table 2.11. Average energy loss of an electron in a pathlength such that the quantity e; I has the value 4 or 10. Calculated with I =78 eV for C and with I = 790 eV for Au. The Chechin-Ermilova error estimate '1lBL is 0.035 for e/ I = 10, and 0.095 for e/ I = 4.

Carbon Gold Gold e; 1= 10 e; 1= 10 e; 1= 4 Energy Energy Loss (MeV) (Percent of Initial Energy) 100 0.02 0.2 0.07 10 0.2 1.5 0.6 5 0.3 2.8 1.1 2 0.8 6.3 2.5 1 1.5 10.9 4.4 0.8 1.8 13.2 5.3 0.6 2.3 16.8 6.7 0.5 2.7 19.5 7.8 0.4 3.3 23.5 9.4 0.3 4.3 29.9 11.9 0.2 6.1 49.6 19.9 0.1 11.2 74.1 29.6

2.4.2 More Elaborate Treatment of Straggling Straggling theories, based on the calculation of an energy-loss cross section and a sub• sequent statistical treatment of multiple collisions, have been developed for various materials, e.g., for aluminum by Bichsel and Saxon35 and by Perez et a1 36, for various gases by Chechin and Ermilova34, for argon by Talman37, and for silicon by Bichse131.

In the calculation of energy-loss cross sections, often it is useful to separate inelas• tic collisions into ·soft collisions (with large impact parameters and small momentum transfers and energy losses) and hard collisions (with small impact parameters and large momentum transfers and energy losses). It is possible to derive information on soft collisions from cross sections for the interactions of photons with matter, expressed in terms of oscillator strengths (which are proportional to photoelectric cross sections), or dielectric response functions (which can be obtained from optical data such as the complex-valued index of refraction). In the Weizsacker-Williams method of virtual quanta, the perturbing field of the incident electron is replaced by an equivalent pulse of radiation, analyzed into a frequency spectrum of virtual photons. The ionization process then is calculated as a photoelectric process for these virtual photons, on the assumption that the effects of the various frequency components add incoherently. The condition for the applicability of the Weizsacker-Williams method are essentially the same as those for the applicability of the first Born-approximation.

2.4.3 Energy-Loss Straggling in Water Cross-section database. We shall now discuss a rough-and-ready method for estimating the required energy- 48 M. J. Berger and R. Wang loss cross sections for water. Input data for water vapor are used because they are more readily available than those for liquid water. In applications in radiological physics and dosimetry, of course one is interested in liquid water as a material representative of tissue. However, at the energies from 10 keY to 1 MeV considered here, the differences between the energy-loss cross sections for vapor and liquid are expected to be small. For example, the estimated mean excitation energy is 75 e V for liquid water compared to 71.6 eV for water vapor.

We have used experimental total ionization cross sections from Schutten et a1 38 at energies from 20 eV to 20 keY, K-shell ionization cross sections from Glupe and Mehlhorn39 from 539.7 eV (threshold) to 1300 eV, excitation cross sections obtained by Olivero, Stagat and Green40 from the analysis of experimental data and by theoretical estimates at energies up to 1000 eV, and experimental cross sections for dissociative excitations measured by Beenacker et al 41 at energies up to 1000 eV. To extend these cross sections to higher energies, we have made use of the result26 that at sufficiently high energies, where the first Born-approximation is valid, the product of (32 and the cross section (for ionization or excitation) depends linearly on the logarithm of the electron kinetic energy. More precisely,

(2.32) where aB is the Bohr radius, and p and q are constants independent of the electron energy. For water, a so-called Fano plot, i.e., a plot according to Eqn. 2.32, indicates that linearity (with constant values of p and q) holds above 500 eV. The fitting of the experimental cross section above 500 eV leads to the following values for p and q:

p= 29.48 and q = 2.84 for the total ionization cross section, p= 12.30 and q = 1.26 for the total excitation cross section, p= 0.222 and q = 0.0346 for the cross section for ionization from the K-shell of oxygen.

We have used these results to obtain inelastic-collision cross sections up to 1 MeV by extrapolation. The reliability of this extrapolation over such a wide energy span, from 1 or even 20 keY up to 1 MeV, might seem questionable. However, as shown below, the reliability of the extrapolation has been confirmed by the good agreement between the stopping powers constructed from the extrapolated cross sections with the stopping powers from the Bethe theory at energies from 10 keY to 1 MeV. Further confirmation is provided by comparisons with calculations of differential ionization cross sections according to the Weiszacker-Williamsmethod (described by Seltzer in Chapter 4). The total ionization cross sections for water vapor obtained by Seltzer in this manner at energies from 1 to 50 keY can be represented accurately by Eqn. 2.32, with parameters p = 28.11 and q = 2.916, and differ from our extrapolated cross sections by 3.3% at 1 keY, and 0.2% at 1 MeV.

In addition to the total ionization cross section, one needs to know the relative probabilities for ionization from the different orbitals of the water molecule. The binding energies of the orbitals, and the respective probabilities, are listed in Table 2.12. For the orbitals with the four lowest binding energies, these probabilities were taken from the results of the calculation by the Weizsacker-Williams method. The relative probability 2. Multiple-Scattering Angular Deflections and Energy-Loss Straggling 49 for ionization from the orbital with a binding energy of 539.7 eV was obtained from experimental results of Glupe and Mehlhorn for ionization from the K-shell of oxygen. Table 2.12 also shows the relative probabilities of various types of excitations, derived from the data given by Olivero, Stagat and Green40, and of Beenacker et a1 41 •

Table 2.12. Estimated relative probabilities of electron-impact excitation and ionization of a water molecule.

Energy, keY 1000 100 10 1 IONIZATION (partial) Relative Probability Orbital Binding Energy (ev)

Ib1 12.6 0.2559 0.2570 0.2587 0.2644 3al 14.7 0.2084 0.2093 0.2112 0.2160

Ib2 18.4 0.1725 0.1733 0.1730 0.1732 2al 32.2 0.0606 0.0609 0.0624 0.0627 01s 539.7 0.0056 0.0049 0.0038 0.0007 IONIZATION (total) 0.7030 0.7054 0.7091 0.7170 EXCITATION 0.2970 0.2946 0.2909 0.2830

The next step is the construction of a differential ionization cross section. If the binding of the atomic electrons could be neglected, one could use the M~ller cross section42 dO') _ 21lT. mc2 [1- bf 1 - b(1 - f) ] ( ------+ +a. (2.33) dT. M j32 T2 f2 (1 - f)2

Here T is the kinetic energy of the primary electron, T. is the kinetic energy transferred to a secondary electron, and f = T./T; r. is the classical electron radius, and mc2 is the electron rest energy. The parameters a and b are given by

(2.34)

We have adopted the following approximate differential ionization cross section which is a modification of the M~ller cross section and has a functional form convenient for random sampling: dO' ( ) l7;on '"""' / (2.35) dT = I< T,T. T wPi!;. ci· . 1

The sum in Eqn. 2.35 extends over the five orbitals of the water molecule, l7;on is the total ionization cross section, Pi is the relative probability of ionization from the jth orbital, and

1 - bf 1 (2.36) !; = f2 + bi/T )2' O:S; f < fi = "2(1 - Bi/T), 50 M. J. Berger and R. Wang where Bi is the binding energy of the jth orbital. The c;'s are normalization constants defined by

The parameters Ii have been assumed to be 13 eV for all orbitals. This is not a good choice for the oxygen K-shell orbital, but this does not matter much because the contribution to the total ionization from the K shell is quite small.

The modification factor, K(T, Ts), was taken to be

if T. :5 WI; K(T, T.) = { tW2 - T. + Ko(T)(T. - WI )](W2 - WI}-I if WI < T. < W2; (2.38)

Ko(T) if T. 2:: W2 • with Ko(T) chosen so that the differential ionization cross section has the same value as the M¢ller cross section for T. greater than W2 • We have set WI = 80 eV and W 2 = 300 eV. Values of Ko(T) are given in Table 2.13. The cross sections calculated with Eqn. 2.35-Eqn. 2.38 are in fairly good agreement with experimental results for water vapor of Opal, Beaty and Peterson43 at 500 eV and of Bolorizadeh and RuddH at 2 and 3 ke V, and with the Weizsacker-Williams results of Seltzer at energies from 1 to 50 keY.

Table 2.13. Average energy transfers in inelastic collisions of electrons with water molecules. The last column gives values of I(o(T) used in the differential ionization cross section (Eqn. 2.38).

Average Energy Transfer, eV Energy Per Ionization Per Per Inelastic (keV) P.E. K.E. Sum Excitation Collision Ko(T) 1000 20.6 57.1 77.7 12.8 58.4 0.489 100 20.0 50.8 70.8 12.9 53.7 0.615 10 19.2 42.2 61.4 13.1 47.2 0.802 1 16.9 27.7 44.6 13.5 35.8 1.000

Table 2.13 shows the average energy transfers calculated with the adopted cross sections, including the potential energy used to release an electron saven ionization event (an average over the binding energies for the various orbitals, weighted by the pertinent cross sections), the average kinetic energy transferred to a secondary electron (based on the differential ionization cross section), the energy used for excitations, and the average energy transfer, (W), averaged over all possible excitation or ionization processes.

The stopping power is equal to

1 -Scol = jlin . (W) , (2.39) P 2. Multiple-Scattering Angular Deflections and Energy-Loss Straggling 51 where (2.40) is the inelastic-scattering coefficient, i.e., the probability per unit pathlength of an inelastic collision. Stopping powers from Eqn. 2.39 are shown in Table 2.14; they are in good agreement above 10 keV with stopping powers from Bethe's theory (evaluated with a mean excitation energy of 71.6 eV).

Table 2.14. Interaction coefficients for inelastic collisions and stopping powers for elec• trons in water vapor. J1in is the probability of a collision per unit pathlength. (W) is the mean energy loss per inelastic collision. Sao'; p is the stopping power, calculated (a) as the product of /lin and (W); (b) from Bethe's stopping-power formula, using a mean excitation energy I = 71.6 eV.

Energy 106 J1in (W) Saol/ p (keV) (cm2/g) (eV) (MeV-cm2/g)

(a) (b) 1000 0.03298 58.4 1.926 1.881 100 0.07748 53.7 4.161 4.142 10 0.4693 47.4 22.15 22.77 1 3.074 35.5 110.0 122

Monte Carlo calculations of straggling. The cross sections described above have been used in a simple Monte Carlo program to obtain the distribution of energy losses by I-MeV electrons in various pathlengths in water vapor. A numerical convolution technique, of the type described by Bichsel and Saxon35 , could have been used instead, and would have required less computation. However, the Monte Carlo method was used because the work was the initial phase of a project which cannot be done easily by methods other than random sampling, namely, the calculation of energy-deposition distributions in a small region, taking into account the transport of energy by secondary electrons into and out of the region.

The inelastic-scattering coefficient J1in is used to sample the pathlength traveled by the electron to the next inelastic collision. The collision is selected to be either an excitation or an ionization from one of the five orbitals of the water molecule. In the case of an ionization, the kinetic energy given to a secondary electron is sampled according to Eqn. 2.35. The energy of the electron is reduced either by the threshold energy for the excitation event, or by the sum of the orbital binding energy and the kinetic energy transferred to the secondary electron.

The energy-loss-straggling distributions obtained with the Monte Carlo program are shown in Fig. 2.lla-d, and are compared with distributions from the Landau theory with a Blunck-Leisegang binding correction. The agreement is quite close for e/ 1= 50, somewhat less close but still good for e/ 1= 10, and poor for e; I = 1, in agreement with the predictions of Chechin and Ermilova. The distribution for a I-micron pathlength, shown in Fig.2.lld, is pertinent to microdosimetry and has a complicated structure, reflecting the irregularies of the underlying energy-loss cross section. 52 M. J. Berger and R. Wang

0.04 (a) 0.2 ( b) Water Water 1 MeV 1 MeV 372 p..m 74.4 p..m 0.02 0.1 ...-. T

>Q) ~ 0L.....,ij~..L.-J....,....L...... L...:r::=:l:::::::t:==~ 0L.....-~~...L...----l_:.:::r==~.....1 30 60 90 120 150 0 10 20 30 1.5 15~--~~~---.--~-. - (c) ( d) - Water Water 10 1.0 t MeV t MeV 7.44 p..m lfLm 0.5 5

o l...... <~~...L...-_~J.:::::::=d 0 LL-__...L.- __-'--~ _ __' o 2 3 0 0.1 0.2 0.3 t::,. (keV)

Figure 2.11. Energy-loss distributions for I-MeV electrons in water. The his• tograms are Monte Carlo results obtained by simulation of all individual inelas• tic collisions. The curves are from the straggling theory of Landau combined with the binding correction of Blunck and Westphal. Pathlengths are indicated in microns for a unit-density medium, and also in mean free paths (mfp). (a) 372 microns (1230 mfp), ell = 50; (b) 74.4 microns (246 mfp), ell = 10; (c) 7.55 microns (24.6 mfp), ell = 1; (d) 1 micron (3.3 mfp), ell = 0.134.

Energy-loss distributions with cutoffs. In a detailed treatment of energy-loss straggling, such as that sketched above for wa• ter, it is easy to include only individual energy losses smaller than some chosen cutoff value. We have done a few Monte Carlo calculations with an energy-loss cutoff equal to the quantity e defined by Eqn. 2.29. The results for two different pathlengths, shown in Fig. 2.12a and b, indicate that the stra.ggling distributions with a cutoff applied bear strong resemblances to Gaussians. This is in conformity with the predictions of the straggling theories of Williams45 and Vavilov46• However, Vavilov's theory would indicate that a somewhat larger cutoff energy is needed to achieve a truly Gaussian distribution. 2. Multiple-Scattering Angular Deflections and Energy-Loss Straggling 53

In any case, the option is available of using a modified track-segment Monte Carlo model in which small energy losses below the selected cutoff are combined and sampled from a Gaussian distribution, whereas large losses are sampled individually. A scheme of this kind has been used successfully in an electron transport calculation by Andreo and Brahme28• However, these authors neglected binding effects when calculating the variance of the Gaussian distribution.

To be consistent, one should apply a cutoff not only to energy losses, but simul• taneously also to angular deflections from inelastic collisions, as already discussed in Section 2.3.3. If the cutoff energy W. is large compared to the binding energy, the re• lation between W. and the deflection angle cutoff (). can be calculated from the M~ller. cross section: • 2 () W./T (2.41) S111 • = 1 + (W./T)(T /2mc2) . For example, with a 12-degree angular cutoff for 1024 keY electrons in carbon (whose effect is shown in Fig. 2.6), the corresponding energy-loss cutoff is 46.3 keY.

0.4

(0) 3 (b) ~ I > .:.::'" 2 0.2

o L-.-J;:_L...... __ L...... __~_---.J 0 L-.~"---_L-___L-=_--...l 5 10 15 0 0.4 0.8 1.2 6. (keV) Figure 2.12. Energy-loss distributions for I-MeV electrons in water, with a cutoff on energy losses in individual inelastic collisions. The cutoff was chosen to be equal to the parameter e, so that on the average, one collision with an energy loss greater than the cutoff value occurs in the given pathlength: (a) Pathlength 74.4 microns, cutoff 716 eV. (b) Pathlength 7.44 microns, cutoff 71.6 eV.

Acknowledgement: This work was supported by the Office of Health and Environmental Research, U.S. Department of Energy. We are grateful to Dr. M. E. Riley for putting at our disposal his computer code for calculating elastic-scattering cross sections. 54 M. J. Berger and R. Wang

REFERENCES 1. G. Moliere, "Theorie der Streuung schneller geladener Teilchen I: Einzelstreuung am abgeschirmten Coulomb-Feld", Z. Naturforsch. 2a (1947) 133. 2. N. F. Mott, "The Scattering of Fast Electrons by Atomic Nuclei", Proc. Roy. Soc. A124 (1929) 426. 3. E. Zeitler and H. Olsen, "Screening Effects in Elastic Electron Scattering" , Phys. Rev. 136A (1964) 1546. 4. J. W. Motz, H. Olsen and H. W. Koch, "Electron Scattering without Atomic or Nuclear Excitation", Rev. Mod. Phys. 36 (1964) 881; formula 1A-109(b). 5. J. A. Doggett and 1. V. Spencer, "Elastic Scattering of Electrons and Positrons by Point Nuclei", Phys. Rev. 103 (1956) 1597. 6. W. T. Scott, "The Theory of Small-Angle Multiple Scattering of Fast Charged Particles", Rev. Mod. Phys. 35 (1963) 231. 7. E. Zeitler and H. Olsen, "Elastic Scattering of Electrons and Positrons by Screened Nuclei", Z. Naturforsch. 21a (1966) 1321. 8. E. Zeitler and H. Olsen, "Complex Scattering Amplitudes in Elastic Electron Scattering", Phys. Rev. 162 (1967) 1439. 9. H.1. Cox and R. A. Bonham, "Elastic Electron Scattering Amplitudes for Neu• tral Atoms Calculated Using the Partial Wave Method at 10,40,70, and 100 keY for Z = 1 to Z = 54", J. Chern. Phys. 47 (1967) 2599. 10. G. Moliere, "Theorie der Streuung schneller geladener Teilchen II: Mehrfach- und Vielfachstreuung", Z. Naturforsch. 3a (1948) 78. 11. 1. V. Spencer, "Theory of Electron Penetration", Phys. Rev. 98 (1955) 1597. 12. M. E. Riley, C. J. MacCallum and F. Biggs, "Theoretical Electron-Atom Elastic Scattering Cross Sections. Selected Elements, 1 keY to 256 keV", Atom. Data and Nucl. Data Tables 15 (1975) 443. 13. M.E. Riley, "Relativistic, Elastic Electron Scattering from Atoms at Energies Greater than 1 keV", Sandia Laboratories Report SLA-74-0107 (1974). 14. J.B. Mann and J.T. Waber, "Self-Consistent Relativistic Dirac-Hartree-Fock Cal• culations of Lanthanide Atoms", At. Data 5 (1973) 201. 15. J. P. Desclaux, "A Multiconfiguration Relativistic Dirac-Fock Program", Compo Phys. Commun. 9 (1975) 31. 16. J. F. Ziegler, J. P. Biersack and U. Littmark, The Stopping and Range of Ions in Solids (Pergamon Press, New York, 1985). 17. V. 1. Moruzzi, J. F. Janak and A. R. Williams, Calculated Electronic Properties of Metals, (Pergamon Press, New York, 1978). 18. M. Fink and J. Kessler, "Absolute Measurements of Elastic Cross Section for Small-Angle Scattering of Electrons from N2 and O2'', J. Chern. Phys. 47 (1967) 1780. 19. W. Brunger and W. Menz, "Wirkungsquerschnitte fur elastische und inelastische Elektronenstreuung an amorphen C- und Ge-Schichten", Z. Phys. 184 (1965) 271. 20. W. Hilgner and J. Kessler, "Mehrfachstreuung mittelschneller Elektronen m Folien", Z. Phys. 187 (1965) 119. 21. 1. Reimer, Transmission Electron Microscopy, (Springer, Berlin, 1984). 22. R. F. Egerton, Electron Energy-Loss Spectroscopy, (Plenum, New York, 1986). 2. Multiple-Scattering Angular Deflections and Energy-Loss Straggling 55

23. S. Goudsmit and J. 1. Saunderson, "Multiple Scattering of Electrons", Phys. Rev. 57 (1940) 24. 24. H. A. Bethe, "Moliere's Theory of Multiple Scattering", Phys. Rev. 89 (1953) 1256. 25. 1. V. Spencer and C. H. Blanchard, "Multiple Scattering of Relativistic Elec• trons", Phys. Rev. 93 (1954) 114. 26. U. Fano, "Inelastic Collisions and the Moliere Theory of Multiple Scattering", Phys. Rev. 93 (1954) 117. 27. K. B. Winterbon, "Finite-Angle Multiple Scattering", Nucl. Instr. Meth. B21 (1987) 1. 28. P. Andreo and A. Brahme, "Restricted Energy Loss Straggling and Multiple Scattering of Electrons in Mixed Monte-Carlo Procedures", Radiat. Res. 100 (1984) 16. 29. L. Landau, "On the Energy Loss of Fast Particles by Ionization", J. Phys. USSR 8 (1944) 201. 30. O. Blunck and S. Leisegang, "Zum Energieverlust schneller Elektronen in dunnen Schichten", Z. Phys. 128 (1950) 500. 31. H. Bichsel, "Energy Loss and Ionization Spectra of Fast Charged Particles Travers• ing Thin Silicon Detectors", private communication (submitted to Rev. Mod. Phys., 1987). 32. W. Boersch-Supan, "On the Evaluation of the Function

<7+;00 (;l(A) = ~ J exp(uln u + Au)du 21rl u-ioo

for Real Values of ,\ " , J. Res. NBS 65B (1961) 245. 33. O. Blunck and K. Westphal, "Zum Energieverlust energiereicher Elektronen in dunnen Schichten", Z. Phys. 130 (1951) 641. 34. V. A. Chechin and V. C. Ermilova, "The Ionization-Loss Distribution at Very Small Absorber Thickness", Nucl. Instr. Meth. 136 (1976) 551. 35. H. Bichsel and R. P. Saxon, "Comparison of Calculational Methods for Straggling in Thin Absorbers", Phys. Rev. A 11 (1975) 1286. 36. J. Ph. Perez, J. Sevelyand B. Jouffrey, "Straggling of Fast Electrons in Aluminum Foils Observed in High-Voltage Electron Microscopy", Phys. Rev. A 16 (1977) 106l. 37. R. Talman, "On the Statistics of Particle Identification Using Ionization", Nucl. Instr. Meth. 159 (1979) 189. 38. J. Schutten, F. J. de Heer, H. R. Moustafa, A. J. H. Boerboom and J. Kistemaker, "Gross and Partial Ionization Cross Sections for Electrons in Water Vapor in the Energy Range 0.1 - 20 keV", J. Chern. Phys. 44 (1966) 3924. 39. G. Glupe and W. Mehlhorn, "Absolute Electron Impact Ionization Cross Sections of N, 0 and Ne", J. Phys. (Paris) 4 (1971) 40. 40. J. J. Olivero, R. W. Stagat and A. E. S. Green, "Electron Deposition in Water Vapor with Atmospheric Applications", J. Geophys. Res. 77 (1972) 4797. 41. C. I. M. Beenakker, F. J. De Heer, H. B. Krop and G. R. Mohlman, "Dissociative Excitation of Water by Electron Impact", Chern. Phys. 6 (1974) 445. 56 M. J. Berger and R. Wang

42. C. MllSller, "Zur Theorie des Durchgangs schneller Elektronen durch Materie", Ann. Phys. 14 (1932) 568. 43. C. B. Opal, E. C. Beaty and W. K. Peterson, "Tables of Secondary-Electron• Production Cross Sections", Atom. Data 4 (1972) 209. 44. M. A. Bolorizadeh and M. E. Rudd, "Angular and Energy Dependence of Cross Sections for Ejection of Electrons from Water Vapor", Phys. Rev. A 33 (1986) 882. 45. E. J. Williams, "The Straggling of Beta Particles", Proc. Roy. Soc. A125 (1929) 420. 46. P. V. Vavilov, "Ionization Losses of High-Energy Heavy Particles", JETP 5 (1957) 749. 3. Electron Stopping Powers for Transport Calculations

Martin J. Berger

Center for Radiation Research National Bureau of Standards Gaithersburg, Maryland 20899, U.S.A.

3.1 INTRODUCTION The reliability of radiation transport calculations depends on the accuracy of the input cross sections. Therefore, it is essential to review and update the cross sections from time to time. Even though the main interest of my group at NBS is in transport calculations and their applications, we find ourselves spending almost as much time on the analysis and preparation of cross sections as on the development of transport codes. Stopping powers, photon attenuation coefficients, bremsstrahlung cross sections, and elastic-scattering cross sections in recent years have claimed our attention. (see, for example, Chapters 2 and 4)*. This chapter deals with electron stopping powers (with emphasis on collision stopping powers), and reviews the state of the art as reflected by Report 37 of the International Commission on Radiation Units and Measurements2•

3.2 DEFINITION OF STOPPING POWER

The electron stopping power represents the mean energy loss per unit pathlength, and is defined ast - dE = N W du dW (3.1) dx J dW' where du / dW is the cross section for inelastic scattering resulting in an energy loss W, and N is the number of scattering targets per unit volume. When du / dW pertains

* Photon cross sections will not be covered. A database and computer program are available 1 for use on a personal computer to calculate cross sections for coherent and incoherent scattering, photoelectric absorption and pair production, as well as attenuation coefficients. The program produces such outputs for any element, compound or mixture, and provides cross-section tables as well as computer-usable data files useful in the preparation of input for transport calculations. The quantity -dE/ dx should not be interpreted as a mathematical derivative, but as the ratio of the mean energy loss dE to the pathlength dx, for a pathlength such that: (1) dx is long enough so that many inelastic collisions occur in dx; (2) dx is short enough so that a further reduction of dx will not significantly change the value of -dE/dx. Only in the continuous-slowing-down approximation, to be discussed, is -dE/ dx treated as a derivative.

57 58 M. J. Berger to collisions with atomic electrons resulting in ionization and excitation, -dEI dx is called the collision stopping power. When do-IdW pertains to collisions with atoms resulting in the emission of bremsstrahlung photons, -dEI dx is called the radiative stopping power. When the pathlength increment dx is expressed in mass units (i.e., in g/cm2 ), the stopping power, in the terminology of the ICRU, is called the mass stopping power. The collision and radiative mass stopping powers are denoted as Scol(T)1 p and Srad(T)1 p, respectively, with T the kinetic energy of the electron and p the density of the medium.

The collision stopping power predominates at low and intermediate energies, and the radiative stopping power at high energies. Collision and radiative stopping powers are equal at an energy that depends on the atomic number of the medium. For example, this so-called critical energy is close to 90 MeV for water or tissue, and 10 MeV for lead.

Figure 3.1 shows the radiative stopping power for gold, illustrating the considerable changes in recommended values that have occured over the last 25 years. Radiative stopping powers will not be discussed further here because they are included in Chapter 4 on bremsstrahlung cross sections.

z.' 79

15

1"9- :;;: 10 o 0

o 0

5 • TSENG - PRATT X BETHE-HEITLER. OLSEN. MAXIMON o NAS - NRC 1133

O~-U~~~llll~~~~~~~-LLU~~~~ 10-3 10-2 10-1 100 10' 102 103 T (MeV) Figure 3.1. The circles (0) indicate the values adopted in a tabulation in 1964 (Berger and Seltzer3,4), based on Bethe-Heitler theory with empirical corrections from Koch and Motz5• The crosses (x) above 50 Me V are from the high-energy theory of Bethe and Heitler, Olsen and Maximon as summa• rized by Koch and Motz. The asterisks (*) below 2 MeV are based on the calculations of bremsstrahlung cross sections in Tseng and Pratt6 and Pratt et al 7,s. The solid curve is based on a least-squares cubic-spline interpola• tion, and represents the radiative stopping powers which have been adopted in ICRU Report 372 •

3.3 CONTINUOUS-SLOWING-DOWN APPROXIMATION

The energy loss per unit pathlength fluctuates strongly about the average values given by the stopping powers. In fact, it is possible (but very rare) for an electron to lose its 3. Electron Stopping Powers for Transport Calculations 59 entire energy in a single collision with an orbital electron of the target atom (or half its energy, if one adopts the standard convention that the faster of the two electrons after the collision is the primary electron). One therefore might question the utility of stopping powers. However, even when a transport calculation is designed to treat energy-loss fluctuations as completely as possible, stopping power values still are needed. The simulation of straggling in a Monte Carlo calculation, for example, must be done so that mean energy loss per unit pathlength is equal to the stopping power.

The use of the continuous-slowing-down approximation (csda) simplifies and speeds up transport calculations. In this approximation, the statistical fluctuations of energy losses in successive inelastic collisions are disregarded, and electrons are assumed to lose energy along their tracks in a deterministic, continuous manner, at a rate given by the stopping power. The transport of electrons in bulk matter is influenced not only by energy-loss straggling, but also by pathlength straggling due to multiple-scattering deflections. The use of csda introduces only a small error into transport calculations under conditions where the effects of energy-loss straggling are small compared to the effects of pathlength straggling. For example, as shown in Fig. 3.2, the error incurred in a calculation of depth-dose curves in carbon or lead is quite small at 1 MeV, but is much more noticeable at 10 MeV where multiple scattering is less important.

4 C,IMeV Pb,l MeV

CSOA

z/ro

Figure 3.2 Calculated depth-dose curves in carbon and lead using ETRAN (1969 version), for electron beams incident perpendicularly on a semi-infinite mediurri. csda: Electron energy loss treated in the continuous-slowing-down approximation. S': Electron energy-loss straggling included (bremsstrahlung photons assumed to escape from the medium). S: Electron energy-loss strag• gling included (complete electron-bremsstrahlung cascade treated, including deposi tion of energy by secondary electrons from photon interactions). 60 M. J. Berger

A problem for which csda is inadequate is illustrated in Fig. 3.3, which pertains to electron flux spectra at two depths in a water medium irradiated by a 20-MeV electron beam.

1.0 f\C5DA I \ I \ I I \ DEPTH ~ I \ 5.88- 5.97cm ~ I , I , ! / 5TRAG a: I J: 0.1 /

0.0 I L-.::::I=::!:::::=L.....l...... L..l_L...L.Jc.lL-L-lL.-J_LL...L....J 2 4 6 8 10 12 14 16 T (MeV) Figure 3.3 Spectra of primary electrons at two depths in water. Calculated with ETRAN for 20-MeV incident electrons, either in the continuous-slowing• down approximation (csda), or with inclusion of energy-loss straggling.

The total stopping power (sum of the collision and radiative stopping powers) can be used to evaluate the csda range. This quantity is obtained by integrating the recipro• cal of the total stopping power with respect to energy. The csda range is approximately equal to the total pathlength which a primary electron travels in the course of slowing down to rest. Csda ranges are usually included in stopping-power tables because they are easy to calculate, and provide at least an approximate description of the penetra• tion of electrons. For better accuracy, csda ranges of course must be replaced by other quantities, such as practical ranges derived from experimental or theoretical depth-dose or transmission curves. Nevertheless, csda ranges are useful as scaling parameters. By expressing target dimensions in units of the csda range at the energy of the incident electron beam, it is possible to reduce and almost eliminate the explicit dependence of such quantities as transmission coefficients or depth-dose curves on the beam energy.

The csda implies that electrons lose their energy gradually, in infinitesimal amounts, whereas the losses actually occur in finite chunks which must be at least as large as the various ionization or excitation thresholds. For example, a minimum energy of 12.6 eV is needed to eject an electron from the least tightly bound orbital of a water molecule. It can be estimated that the average energy loss per ionization in water is approximately 80 eV at 1 MeV, 45 eV at 1 keY and 33 eV at 0.1 keY, taking into account the kinetic energy given to secondary electrons. These numbers suggest that the assumption of a gradual energy loss is realistic for I-MeV electrons, and is acceptable perhaps down to 1 keY. It is unrealistic, however, for electrons with energies of 200 eV or lower which can lose all their energy in a small number of collisions. At such low energies, stopping powers are mainly useful as summary descriptions of energy-loss cross sections. 3. Electron Stopping Powers for Transport Calculations 61

3.4 STOPPING-POWER FORMULAS AND TABLES

Stopping power and range tables for energies above 10 keV were published by Nelms9 ,lO, Berger and Seltzer·, Pages et al ll , and in Report 37 of the International Commission on Radiation Units and Measurements2• The tables of Nelms contained collision stopping powers and ranges. All later tables have also included radiative stopping powers.

In all of these tabulations, electron collision stopping powers were obtained with Bethe's stopping-power formula12,13. There are two quantities in this formula, the mean excitation energy of the medium, and the density-effect correction (which was absent in Bethe's original formulation) whose evaluation is not trivial and requires the use of experimental input data.

The formula for the mass collision stopping power for electrons with kinetic energy T in a medium with density p is

where (3.3)

When Eqn. 3.2 is applied to positrons, F must be replaced by

F+(T) = 2ln 2 - ((32/12)[23 + 14/(T + 2) + lO/(T + 2)2 + 4/(T + 2)3]. (3.4)

The various symbols in Eqn. 3.2 through Eqn. 3.4 have the following meaning: r. is the classical electron radius (2.8179 X 10-13 cm), me2 is the rest energy of the electron (0.511 MeV), (3 is the electron velocity in units of the velocity of light, Na is Avogadro's number (6.022045 x 1023 mol-I), Z is the atomic number and A the atomic weight of the target material, T is the kinetic energy in units of me2, I is the mean excitation energy, and 8 is the density-effect correction.

The differences between electron and positron stopping powers arise from two causes. First, large energy losses are treated according to the MS'lller cross section for electrons, but according to the Bhabha cross section (without exchange) for positrons. Second, on account of the indistinguishability of the incident and atomic target electron, an electron, by convention, is only allowed to lose at most half its energy in a single collision, whereas no such restriction applies to a positron. Typical positron/electron collision stopping-power and csda range ratios are given in Table 3.1. 62 M. J. Berger

Table 3.1. Ratio of positron to electron collision stopping power and csda range.

WATER LEAD Electron (+)/(-) (+)/(-) (+)/(-) (+)/(-) Energy Stopping Power csda Range Stopping Power csda Range (MeV) Ratio Ratio Ratio Ratio 0.01 1.101 0.895 1.192 0.757 0.1 1.039 0.948 1.022 0.927 1 0.979 1.005 0.972 1.054 10 0.973 1.027 0.968 1.122 100 0.975 1.020 0.973 1.107

The mean excitation energy, I, is a constant which characterizes in summary form the stopping properties of the medium. The density-effect correction, 8, takes into account the reduction of the stopping power due to the polarization of the medium by the incident particle. It depends mainly on I and the density of the medium.

Except in the case of the simplest atomic gases, mean excitation energies cannot yet be calculated with sufficient accuracy from first principles, but must be extracted from experimental data. The changes of stopping-power values from one compilation to the next have resulted mainly from the use of different values of mean exCitation energies.

The applicability of Bethe's stopping-power formula is limited by the condition that the velocity of the incident electron should be much larger than the velocities of the atomic electrons with which it collides. At energies above 1 keY, this condition is satisfied for the outer-shell electrons in all atoms. It is not satisfied, however, for inner-shell electrons, particularly K-shell electrons which have mean kinetic energies ranging from 0.44 keY in carbon to 2.17 keY in aluminum to 112 keY in gold. For high-Z elements, the inner-shell electrons constitute only a small fraction of all orbital electrons, which reduces the overall uncertainty of the collision stopping power.

Starting with the work of Nelms9,10, most stopping-power tabulatipns for electrons have been limited to energies above 10 keY. For protons and alpha particles, correction terms for Bethe's formula have been developed which extend its applicability down to velocities that are no longer large compared to the velocities of the atomic electrons. These are called shell corrections because they are applied separately to atomic shell energies. The application of shell corrections for protons to electrons of the same velocity suggests itself, but has no sound theoretical basis.

The uncertainties of the stopping powers tabulated in ICRU Report 37 were es• timated to be 2-3 percent for low-Z materials, and 5-10 percent for high-Z material between 100 and 10 keY. Auxiliary stopping power tables of lesser accuracy for low-Z materials (such as water, air, plastics) are given down to 1 keY, with estimated uncer• tainties at 1 keVon the order of 10 percent.

In ICRU Report 37, mean excitation energies and density-effect corrections are presented for 78 materials of interest in dosimetry. Recommended mean excitation 3. Electron Stopping Powers for Transport Calculations 63 energies, obtained on the same basis, are given in Seltzer and Berger14 for 278 materials. A compilation by Sternheimer, Berger and Seltzer15 also gives the mean excitation energies for these 278 materials, as well as the density-effect corrections in terms of an accurate approximation formula.

The mean excitation energy enters logarithmically into the stopping-power formula, so that the relative uncertainty of the collision stopping power is several times smaller than the relative uncertainty of I. The density effect also depends on the mean excita• tion energy, so that the stopping power actually depends on the quantity 2 log I + 8(1) which, in the limit of very high energies, approaches a constant value (depending only on the density of the medium). The variation of the stopping power with I therefore vanishes at high energies, and is greatest at low energies where 8 is negligibly small. The percent change of the stopping power resulting from a 10 percent change of I is indicated in Table 3.2 for water, air and lead.

Table 3.2. Percent decrease of the collision stopping power for electrons resulting from a 10 percent increase of the mean excitation energy.

Energy Percent Decrease (MeV) Water Air Lead 0.01 2.0 2.0 3.8 0.1 1.4 1.4 2.1 1 0.6 1.0 1.2 10 0.2 0.8 0.5 100 0.01 0.2 0.1

3.5 MEAN EXCITATION ENERGIES 3.5.1 I-Values from Stopping-Power Data.

Historically, the most widely used method of determining I-values has been to extract them from measured proton or alpha-particle stopping powers or ranges. The stopping• power formula for heavy charged particles is similar to Eqn. 3.2; however, the auxiliary function F, which represents close collisions with large energy transfers, is different. Furthermore, within the square brackets on the right-hand side of Eqn. 3.2, three addi• tional correction terms are included for heavy charged particles. The most important of these is the shell correction, needed when the velocity of the incident particle is not large compared to the velocities of the atomic electrons. In addition, there are the so-called z3 and z4 corrections (also named Barkas and Bloch corrections after their originators) which take into account departures from the first Born-approximation.

Accurate shell corrections for the K and L shells have been calculated by Walske16•17, Bichsel18 and others. For elements of moderate and high atomic numbers, many more atomic shells must be included, and a rigorous theoretical treatment for all shells is not yet available. Bonderup19 used the statistical electron-gas model of Lindhard20 to compute stopping powers, and extracted shell corrections (summed over all shells) from his results. Bichsel has extended Walske's K- and L-shell corrections to higher shells by a semi-empirical scaling procedure described in ICRU Report 37; in effect, he estimated 64 M. J. Berger

I-values as well as shell corrections simultaneously by fitting measured stopping powers and ranges for protons and alpha particles. Bichsel's shell corrections are quite close to those of Bonderup for elements with atomic numbers Z < 50, but are smaller for Z > 50.

One of the standard sources of I-values is the tabulation of proton stopping powers by Andersen and Ziegler21. These authors relied on Bonderup's shell corrections when choosing I-values, whereas the ICRU Committee also gave great weight to Bichsel's shell corrections. As a result, the I-values adopted in ICRU Report 37 are somewhat larger for high-Z elements than those recommended by Andersen and Ziegler, for example I = 823 eV rather than 759 eV for lead. The choices made by the ICRU Committee were also influenced by the fact that range measurements of Barkas and von Friesen22 for 750-MeV protons, under conditions where the shell corrections are insignificant, support the higher I-values. This is illustrated in Fig. 3.4 for lead. Additional measurements of proton stopping powers or ranges at energies of several hundred MeV, where the shell corrections (and all other corrections) are small, would be useful for resolving this question.

I I

850 r- -

BARKAS & I VON FRIESEN T :;., 800 r- -

H S.RE:S~ II II ANDERSEN ! OBTAINED USING SHELL CORRECTIONS OF: 750 i- • BICHSEL - ! ! HI! o BONDERUP

700~--~~~~~I----~~~~~I--~L-L-LJ I 10 100 500 T (MeV) Figure 3.4 Determination of mean excitation energy of lead from proton stop• ping power measurements of S¢rensen and Andersen23 and range measure• ments of Barkas and von Friesen22 •

3.5.2 I-values from Photon Cross Sections Knowledge of photon cross sections can be brought to bear on the determination of I-values. For gases, the mean excitation energy can be expressed as

00 00 InI = 1dEdf InEdEII dEdf dE, (3.5) o 0 where df IdE is the density of optical dipole oscillator strength per unit of excita• tion above the ground state, and is proportional to the photoelectric cross section. I-values for many molecular gases have been determined from photoelectric data by 3. Electron Stopping Powers for Transport Calculations 65

Zeiss et al24•25 • For example, the value I = 71.6 eV for water vapor was obtained in this way.

For materials in the condensed phase, the alternative formula for the mean.excita• tion energy is

00 InI = [211T(hwp)2] Jhwlm[-l/ f (w)] In(hw)hdw, (3.6) o where 1 hwp = 28.816 (pZ I A) "2 (3.7) is the plasma energy (in eV), and feW) is the complex-valued dielectric response func• tion, which is related to the real and imaginary parts of the complex index of refrac• tion. Experimental optical data have been used by Shiles et al 26 to obtain the value 1= 166 eV for aluminum, and by Ashley27 to obtain I = 75.4 eV for water. The ICRU Committee considered the I-values derived from oscillator strengths and dielectric re• sponse functions to be as accurate as the values derived from the best stopping-power measurements.

3.5.3 Survey of Mean Excitation Energies for Elements Figure 3.5 shows I-values for all elements, derived in the critical data analysis by the ICRU Committee, in the form of a plot of liZ VB Z. Values determined from experi• mental data have error bars; the others were obtained by interpolation. Except for the smallest atomic numbers, I (in units of eV) is approximately equal to 10Z. However, 1/Z varies irregularly as a function of Z, which is attributed to atomic shell struc• ture. The non-smooth Z-dependence can be predicted rather accurately on the basis of Lindhard's electron-gas model28 •

22 ----r--

Gases Solids • Experimen to I - Interpolated

§ ;:; 16 ..... H

12

20 40 60 80 100 ATOMIC NUMBER Z Figure 3.5 Variation of mean excitation energy with atomic number (ICRU Re• port 372 ). 66 M. J. Berger

Mean excitation energies depend on the physical state of aggregation of the mate• rial. No dependable theory is yet available for predicting this dependence quantitatively. Semi-empirical estimates are given in Table 3.3 for a few elements. The I-values for atomic gases used in these comparisons are theoretical.

Table 3.3. Dependence of mean excitation energies on the state of aggregation of the medium. (From ICRU Report 372)

Mean Excitation Energy, eV Element Atomic Molecular Liquid Solid Gas Gas H 15.0 19.2 21.8 C 60.0 78 N 76.9 82.0 90.5 0 93.5 95.0 104.3 Al 124 166 Si 132 173 Ti 182 233 Fe 226 286 eu 274 322 Ge 292 350

The differences between I-values recommended by different authors are small, and have resulted not only from gradual improvements of the experimental input data, but "also from different subjective judgements. The changes of the adopted I-values for nitrogen and lead in the course of time are illustrated in Fig. 3.6.

3.5.4 Mean Excitation Energies for Compounds

The number of compounds of interest is much greater than the number of those for which experimental values of the mean excitation are available. Collision stopping powers for compounds therefore are often approximated by the linear combination of stopping powers for the atomic constituents (Bragg additivity):

(3.8)

where Wi is the fraction by weight and [~Scol(T) Lis the mass collision stopping power of the j'th constituent. The corresponding relation for the mean excitation energy is

L-i Wi(Zi/A;) In Ii InI = . (3.9) L-i wi(Zi/Ai) 3. Electron Stopping Powers for Transport Calculations 67

MEAN EXCITATION ENERGY FOR Nz GAS 90

85

ICRU

80

BICHSEL 8JCH5EL

> 75 I ~ 1960 1965 1970 1975 1980 1985

H MEAN EXCITATION ENERGY FOR LEAD

850

Figure 3.6 Comparison of mean excitation energies for molecular nitrogen and for lead adopted in various compilations: NAS-NRC3, Janni29 , Bichsel3o, Turner et a1 3t, Bichsel32, Andersen and Ziegler21 , Ziegler33, Ahlen34 and ICRU2 .

The error incurred by the use of Eqn. 3.9 can be reduced by adopting I-values for the atomic constituents that differ from the I-values for elements, and which, at least approximately, take into account the chemical environment and the state of aggregation of the medium.

Experimentally based I-values were available to the ICRU Committee for 13 molec• ular gases (with 2-percent uncertainties), for 27 liquids (with 3-percent uncertainties), and for 14 solids (with uncertainties ranging from 3 to 8 percent and, in one case (A-150 plastic), 16 percent). It was found that these I-values can accurately be represented by using Eqn. 3.9 provided one uses for the atomic constituents the I-values listed in Table 3.4. The fitting errors in most cases are smaller than, or comparable with, the uncertainties of the experimental I-values.

The assignment scheme in Table 3.4 distinguishes between compounds in the gas and condensed phase, but does not include explicitly the chemical environment, i.e., the types of chemical bonds involved. In future tabulations based on more abundant input data, it would be desirable to remedy this defect and take into account the underlying chemical physics. Such a scheme in fact W8,$ proposed by Thompson35 to explain 1- values which he deduced from his accurate measurements of partial ranges of 340-MeV protons in 27 liquids. An updated version of Thompson's assignment scheme is given in Table 3.5. 68 M. J. Berger

Table 3.4. Mean excitation energies recommended in ICRU Report 37 for atomic constituents of compounds.

Assignment Rule 1 (Gas) Assignment Rule 2 (Liquid or Solid) Constituent I (eV) Constituent I (eV) H 19.2 H 19.2 C 70 C 81 N 82 N 78 0 97 0 106 Cl 180 Others: 1.13 x I-value for element in condensed phase

Table 3.5. Mean excitation energies for atomic constituents of liquids. This is an updated version of an assignment scheme proposed by Thompson35 (from ICRU Report 37 2).

Constituent Type of Bonding I (eV) H saturated 19.0 ± 0.8 unsaturated 16.0 ± 0.8 C saturated 81.1 ± 2.5 unsaturated 79.8 ± 2.3 highly chlorinated 69.0 ± 3.7 N amines, nitrates 105.7 ± 10.6 in rings 81.9 ± 7.0 0 -0- 104.6 ± 9.2 0= 94.4 ± 4.9 Cl all 179.7± 11.9

The assignment scheme in Table 3.4 includes the recommendation to use, for the constituents of solids (other than H, e, N, F and el), I-values which are 13 percent larger than the I-values for the elemental substance in the condensed phase. This was done to obtain good fits to the experimentally well-established I-values of three materials (aluminum oxide, silicon dioxide and photographic emulsion). The validity of extending this prescription to other materials with high-Z constituents is not obvious, and should be tested when relevant experimental I-values for other compounds become available. Not only a better understanding of phase and chemical-binding effects, but also information of much practical use could be obtained through careful stopping-power experiments on compounds used as radiation detectors, e.g., lithium fluoride, calcium fluoride, or sodium iodide.

For the 48 compounds included in leRU Report 37, the uncertainties of the mean excitation energies are estimated to be 2 to 3 percent for 25 cases, 5 to 10 percent in 18 cases, and 10 to 15 percent in 5 cases. 3. Electron Stopping Powers for Transport Calculations 69

3.6 DENSITY-EFFECT CORRECTION The density-effect correction, that is, the reduction of the stopping power due to the polarization of the medium, can be computed from knowledge of the dielectric response function of the medium:

00 6 = [2/1I"(fiwp)2] Jfiw 1m [-1/ f(W)] In(1 + 12/w2) fi dw - (11" /2)l2(1 - ,82), (3.10) o where fiwp is the plasma energy defined by Eqn. 3.7, and l is a root of the equation

1 - ,82f(il) = O. (3.11)

Inokuti and SmithS6 analyzed optical data for aluminum, which are especially abundant, and derived a dielectric response function which they applied to the cal• culation of the density-effect correction. A similar calculation for water, also based on optical data but requiring more theoretical extrapolation, was carried out by Ashley37 for water.

For most materials, the available optical data are scarce, or have not yet been critically analyzed, so that input for density-effect calculations is lacking. However, a satisfactory alternative is provided by the theory of Sternheimer38• This theory is based on a simple model for the dielectric response function, and requires as input the mean excitation energy energy, the density, and the plasma energy of the material, as well as a set of atomic binding energies. The key to the success of Sternheimer's model consists of multiplying all atomic binding energies by a common factor so that they are consistent with the experimentally determined mean excitation energy. In Sternheimer's method, the analog of Eqn. 3.10 is

(3.12) where l is a root of the equation

N ..!.. -1 = L in (3.13) ,82 n=1 (PSt En/fiwp)2 + in'

The oscillator strengths in are set equal to shell occupation numbers. In ICRU Report 37,the atomic binding energies En were taken from tables of Carlson39• For conductors, the binding energy for the outermost shell (n = N) is set equal to zero. The parameter On = 2/3 for n -:f; Nj ON = 1 for conductors and 2/3 for non-conductors. The atomic binding energies En are replaced by pstEn , where the adjustment factor PSt then is chosen so that N Lin In{(PStEn/fiwp)2 + Onin} = 2 In (I /fiwp). (3.14) n=l Stopping powers for aluminum and water, evaluated with the density-effect cor• rection, calculated according to Sternheimer's method, are compared in Fig. 3.7 with 70 M. J. Berger results obtained with the inherently more accurate density-effect corrections of Inokuti and Smith36 for aluminum, and of Ashleys7 for water. At all energies, the differences are smaller than 0.5 percent for water, and smaller than 0.2 percent for aluminum. One might expect, although of course further confirmation is desirable, that the use of Sternheimer's method for other materials would introduce uncertainties for collision stopping powers no greater than 0.5 percent.

0.5 "0 en" z IJJ (!) Z eX 0 J: U I- Z IJJ Ua:: a.IJJ -0.5 0.1 10 100 T (MeV) Figure 3.7 Change of collision stopping powers when the density-effect cor• rection of Ashleys7, or the correction of Inokuti and SmithS6, are replaced by corrections calculated according to Sternheimer's method (ICRU Report 372 ).

The direct output of a density-effect calculation by Sternheimer's method is a set of numerical values of 6 at various energies. However, Sternheimer has devised a simple, accurate fitting formula:

4.6052X + C ,X>XI 4.6052X + a(XI - x)m + C , Xo < X < Xl 6(X) = j . (3.15) o for non - conducting materials ,X

6(Xo)102(X-Xo) for conducting materials ,X

In this equation, X =10g10( r( r + 2))1/2, r is the electron kinetic energy in units of the rest mass, and C = -2ln(I/,hwp ) -1 . (3.16) The parameters Xo, Xl, a, m, and C have been determined for 278 materials, and are listed in Sternheimer, Berger and Seltzerl6.

An unresolved question is how one should modify the theory of the density effect for inhomogeneous materials. A case in point is graphite, which is of interest because of its common use in ionization chambers and calorimeters. Graphite is a porous material consisting of loosely packed crystallites with a density of 2.265 g/cms. The average bulk density is much lower, ranging from 1.5 to 1.9 g/cms, depending on the method of manufacture. As demonstrated in Table 3.6, a change in the assumed density from 2.265 to 1.7 g/cms can lower the computed collision stopping power by as much as 1 percent. In ICRU Report 37, there is a very tentative recommendation to calculate 6 with the bulk density. However, this question is one which should be resolved by further theoretical or experimental work. Similar considerations also arise for other 3. Electron Stopping Powers for Transport Calculations 71 inhomogeneous materials of interest in dosimetry, for example, photographic emulsion, or A-150 tissue-equivalent plastic.

3.7 COMPARISONS WITH EXPERIMENTS The tabulation of electron stopping powers is predominantly a theoretical matter, with some empirical input. Stopping-power measurements are perhaps more difficult for electrons than for heavy particles because of the wiggliness of the electron tracks due to multiple-scattering deflections. Perhaps experimenters have been discouraged from making measurements because, unlike for heavy particles, the electron stopping powers in general cannot be used directly in simple formulas, but must be used in more elaborate transport calculations which include multiple-scattering effects. Another complication for electrons is the long tail of the energy-loss distribution over which the energy loss must be averaged. Very rare large losses can make a significant contribution to the elec• tron stopping power. Most experimenters have preferred to concentrate their efforts on determining the most-probable energy loss and the width of the energy-loss distribution (for a review of these quantities, see Chapter 7).

Measured electron stopping powers from work done in recent years are shown in Fig. 3.8 for aluminum, copper, silver, and gold. The quantity plotted is proportional to the stopping power multiplied by (32 and, in the first Born-approximation, is expected to be a linear function of the logarithm of the electron energy. Comparisons are made in Fig. 3.8 with the Bethe theory, and for aluminum at low energies also with other theories designed to work at low energies. The agreement between theory and experiment is not particularly close. The experimental evidence is inconclusive, providing little guidance how the results from the Bethe theory should be modified at low energies.

Table 3.6. Density-effect correction 0 and mass collision stopping power SeQl in graphite.

DENSITY EFFECT Electron CORRECTION Energy Density Density Percent (MeV) 1.7 g/cm3 2.265 g/cm3 Difference 0.1 0.0376 0.0474 24.1 1 0.709 0.803 13.3 10 3.26 3.45 5.8 100 7.42 7.70 3.8 COLLISION STOPPING POWER Percent (MeV-cm2 /g) Difference 0.1 3.674 3.671 -0.1 1 1.617 1.609 -0.5 10 1.745 1.730 -0.9 100 1.950 1.928 -1.1 72 M. J. Berger

T/(MeV) 0.001 0.01 0.1

16 (a) -. 0 AI u (f) 12 -IQ... ~ Y ~IN ? NGI... 8 N c..> => E I:: (\J 4 "" N C!l 0 -12 -10 -8 -6 -4 -2 0 2

140 .02 0.05 0.1 0.2

12

-. 10 0 u (f) -IQ... 8 12 ~ ~IN NGI... 10 10 N c..> => E I:: 8 12 (\J

N C!l 10

8

6 -6 -5 -4 -3 -2 -I In [yo2 (yo + 2)] -13 2

Figure 3.8 Comparison of calculated and experimental electron collision pow• ers (from ICRU Report 37). (a) Stopping power for aluminum. Theoretical curves: 1, Ashley et a1 40; 2, Ashley et a1 41 ; 3, Ashley et a1 42 ; 4, Bethe theory. Experimental points: circles, Hubbell and Birkhoff43 ; triangles, Pugachev and Volkov44; squares, Ishigure et a1 45 • (b) Stopping powers for copper, silver and gold. Theoretical curves: Bethe theory. Experimental points: circles, Hubbell and Birkhoff42 ; triangles, Pugachev and Volkov44• 3. Electron Stopping Powers for Transport Calculations 73

3.8 STOPPING-POWER RATIOS In ionization chamber dosimetry, the absorbed dose in a medium is deduced from the measured ionization in the gas in the chamber by using the Bragg-Gray relation

D". = J, . W . P . S';" (3.17) e where D". is the absorbed dose in the medium, J, is the charge liberated per unit mass of gas, W is the average amount of energy needed per ionization, e is the charge of the electron, and p is a perturbation factor. S,: is the stopping-power ratio (see Chapter 23), i.e., the stopping powerfor the medium (averaged over the electron track length spectrum at the point of measurement) divided by the stopping power for the gas (averaged over the same spectrum). By an extension ofthe cavity ionization theory of Spencer and Attix·6 , as formulated by Nahum·7, the stopping-power ratio at depth z in a medium irradiated by a beam of electrons of energy T can be calculated from the expression

To J y(T, z)[L(T, ~)l".dT + y(~, z) ~ [Scol(~)l". S';"(z, To) = =A~~o------­ (3.18) J y(T,z)[L(T, ~)l,dT + y(~, z) ~ [Scol(~)l, A

Here, y(T, z) is the tracklength of electrons with energies T at depth z, L(T,~) is the restricted collision stopping power and Scol(T) the total collision stopping power. L(T,~) includes only energy transfers smaller than a cutoff value~. The magnitude of ~ is chosen to be equal to the energy of an electron which has sufficient range just to cross the cavity. The restricted stopping power is calculated by an equation similar to Eqn. 3.2 but with F replaced by a more complicated expression.

Equation 3.18 has been used to evaluate medium-to-air stopping-power ratios for water, graphite (densities of 2.256 and 1.7 g/cm3), polystyrene and PMMA (lucite), irradiated by broad, parallel, monoenergetic beams of electrons with energies from 1 MeV to 60 MeV. The tracklength distributions, y(T, z) in the respective media were calculated with the ETRAN code, and the restricted stopping powers were calculated according to the prescription in ICRU Report 37. Various values of ~ have been used, ranging from 2 to 50 keV. The stopping-power ratios are rather insensitive to the value of ~, which is fortunate because in the Spencer-Attix theory the appropriate cutoff is not very clearly defined. The results for the most common situation, ~ = 10 keV, have been published as part of a Protocol prepared by Task Group 21 of the American Association of Physicists in Medicine·s.

A similar evaluation of stopping-power ratios was done entirely independently by Nahum, who used his own Monte Carlo code to obtain electron tracklength distributions as input for Eqn. 3.18. His results for water were reported in Report 35 on Radiation Dosimetry of the ICRU·9. When the stopping powers used in Nahum's calculation are the same as those used in ICRD Report 37 (in regard to mean excitation energy and the density-effect correction), Nahum's stopping-power ratios are in close agreement with those given in the Protocol of Task Group 21 of the AAPM.

The calculated stopping-power ratios from Eqn. 3.18 can be compared with experi• mental values obtained with the Bragg-Gray relation from measured values for D". and 74 M. J. Berger

JII • Figure 3.9, taken from an interesting review by Nahum50, shows that the closeness of the agreement between calculated and measured stopping-power ratio depends on the density-effect correction used. With the density-effect correction of Sternheimer and Peierls51 , the agreement is close in water at all depths at 10 and 20 MeV, and at depths greater than 8 cm at 30 MeV. With the density-effect correction of Ashleys7, the calcu• lated results are lower than the experimental results by as much as one percent, except for depths less than 8 cm at 30 MeV. This is puzzling because the Sternheimer-Peierls density-effect correction is based on an approximation scheme which is inherently less accurate than the direct application of Sternheimer's method which, in turn, is less accurate than Ashley's method based on the use of an experimental dielectric response function.

10 MeV

1.10

1.05

0 i Cf) Expt* 1.00 Svensson 1971 Mattson 1982

Theory -- Berger and Seltzer 1981 (Sternheimer and Peiers ~) ---- Berger 1981 (Ashley~)

o 5 10 15 DEPTH IN WATER/Cern)

*Sosed on Em G = 352 10-6 rn2 kg-I G{I

Figure 3.9 Comparison of calculated and measured stopping-power ratios in water (from Nahum50). The calculated results were obtained with the density-effect correction of AshleyS7( dashed curves) and of Sternheimer and Peierls51(solid curves).

Discrepancies between theory and experiment could arise from a variety of causes. Perhaps the most serious difficulty arises from the fact that in the calculation, the electron beam is assumed to be strictly monoenergetic and monodirectional, which is not the case for clinical radiotherapy beams whose quality is affected by their passage through beam-spreading foils, collimators and air. Another error is introduced by the neglect in the calculation of the perturbation of the electron spectrum at the point of measurement due to the presence of a cavity. Furthermore, the interpretation of the experimental results is complicated by the fact that the extracted experimental stopping-power ratios depend on the assumed W-value for air, and on the yield of the ferrous sulfate, which must be known to determine the absorbed dose. It is usually assumed, for example, that the W-value is strictly energy-independent, with the same value as that measured for 60 CO gamma rays. This assumption is open to question, and there is some evidence to the contrary. 3. Electron Stopping Powers for Transport Calculations 75

3.9 STOPPING POWERS AT LOW ENERGIES

For most of the transport calculations discussed in this volume, the limitation of stopping-power tables to energies above 10 keY (or 1 keY for low-Z materials) is not a serious hindrance. Often it is a good approximation to assume that the electron effec• tively stops and deposits its energy on the spot when it reaches an energy of 10 keVor, more accurately, that the electron deposits all of its energy uniformly over a (roughly estimated) residual range. For some applications such as microprobe analysis or low• energy radionuclide dosimetry, it would be desirable to improve the stopping-power information at energies below 10 keY. This also is important for radiobiological mod• eling involving track-structure calculations which often are extended down to energies close to the ionization thresholds.

During the past ten years, considerable effort has gone into the evaluation of elec• tron stopping powers from 10 keY down to 100 eV, or even lower. Low-energy stopping powers often have been obtained as by-products of the construction of sets of inelas• tic scattering cross sections used for transport calculations. These data depend on a complicated mix of theory and experimental data. Among the prominent contributors to this work are A. E. S. Green and collaborators who concentrated on electron energy losses in low-Z gases, Waibel and Grosswendt at PTB, and Paretzke and collaborators at GSF in Germany who also worked on gases, and a group at the Oak Ridge National Laboratory including Ritchie, Ashley, Tung and others, who calculated stopping powers and more detailed energy-loss cross sections for many solids. Illustrative plots of stQP• ping power vs energy are shown in Fig. 3.10 for water, and in Fig. 3.11 for aluminum and gold. References to the literature, and additional plots of low-energy stopping powers, can be found in ICRU Report 37.

300 WATER VAPOR

~I-ti -I Q.. I

Figure 3.10 Comparison of stopping powers for water from various calcula• tions. Dashed curves 3 and 7 are from Bethe's theory, evaluated with with mean excitation energies of 71.6 eV for water vapor and 75 eV for liquid wa• ter, respectively. Curve 1: Paretzke and Berger52; Curve 2: Green (private communication); Curve 4: Kutcher and Green53; Curve 5: Ritchie et a1 54 ; Curve 6: Ashley27. 76 M. J. Berger

The uncertainties in the available low-energy stopping powers are not well known. The theory as developed by the Oak Ridge group makes implicit use of the first Born• approximation, which ceases to be valid below a few hundred eV in low-Z materials. At energies around 100 eV, at which the stopping power typically peaks, the uncertainties may well be 15-20 percent, or greater. More measurements extending down to low energies, similar to those pioneered by Cole55 , or by other methods, would be valuable.

102

,--, t}I ---E 0 0> 1 ~Q) 10 ~ 102

"0 Au ~ WIuu )( 4 ;"------1Cl.. 1/ ...... --I 10 1 I I~ I 5 I

10°u-~~~~~--~~~~~~~~wu~ 10 1

Figure 3.11 Comparison of stopping powers for aluminum and gold from several calculations. Curve 1: Ashley et aZ4.° j Curve 2: Ashley et al 41 j Curve 4: Ashley et al4.o. Dashed curves 3 and 5 are from Bethe's theory (from ICRU Report 372).

3.10 CONCLUDING REMARKS

New information pertinent to the evaluation of electron stopping powers accumulates rather slowly, and the lifespan of tables is perhaps 10 to 15 years. No substantial new body of data, or new theory, is available at present which would strongly suggest a change of the stopping powers and ranges tabulated in ICRU Report 37 in 1984. A significant reduction of the uncertainties may have to await the availability of improved experimental techniques or calculational methods. 3. Electron Stopping Powers for Transport Calculations 77

At present, the uncertainties in the stopping powers at energies from 10 to 100 keY are estimated to be 2 to 3 percent for low-Z materials, and 5 to 10 percent for high-Z materials. Above 100 keY, the uncertainties are estimated to be 1 to 2 percent. For many applications in transport calculations, this level of accuracy probably is sufficient.

The stopping-power ratios required for ionization-to-dose conversions have esti• mated uncertainties of 1/2 to 1 percent, greater than is acceptable for precision dosime• try as practiced in national standards laboratories. To reduce these uncertainties, it would be desirable (and probably feasible with presently available Monte Carlo codes) to estimate the effect of the perturbation of the electron flux due to the presence of the ionization chamber.

In order to help settle the question of mean excitation energies, particularly for high-Z materials, new proton stopping-power measurements at energies of several hun• dred MeV would be helpful. The determination of mean excitation energies for com• pounds should be put on a sounder basis than the crude, semi-empirical rules adopted in ICRU Report 37.

The evaluation of the density-effect correction appears to be in a satisfactory state for most but not all practical purposes. An unsolved problem of interest is the evaluation of the density-effect correction in inhomogeneous materials, especially graphite because of its wide use in measurement instruments. In addition to the appropriate theory, direct stopping-power measurements would be valuable, in graphite as well as in other materials used as radiation detectors, such as lithium fluoride, calcium fluoride.

It would be useful to extend the analysis of dielectric response functions for solids to a larger set of materials, using the same approach successfully used for aluminum and silicon. This would provide the basis for improved calculations of energy-loss straggling and low-energy electron stopping powers.

The evaluation of electron stopping powers and more detailed energy-loss cross sections at energies below 10 keY is still an unfinished business. Measurements at energies down to 100 eV, if feasible, would provide valuable checks on the cross sections presently used for radiobiological modeling, as well as guidance for theoretical work.

Acknowledgement: This work was supported by the Office of Health and Environmental Research, U.S. Department of Energy. 78 M. J. Berger

REFERENCES 1. M. J. Berger and J. H .Hubbell, "XCOM: Photon Cross Sections on a Personal Computer", National Bureau of Standards report NBSIR 87-3597 (1987). 2. International Commission on Radiation Units and Measurements (ICRU), "Stop• ping Powers for Electrons and Positrons", ICRU Report 37 (1984). 3. M. J. Berger and S. M. Seltzer, "Tables of Energy Losses and Ranges of Electrons and Positrons", in Studies in Penetration of Charged Particles in Matter, Publi• cation 1133, (National Academy of Science, National Research Council, 1964). 4. M. J. Berger and S. M. Seltzer, "Tables of Energy Losses and Ranges of Electrons and Positrons", NASA Publication SP-3012 (1964). 5. H. W. Koch and J. W. Motz, "Bremsstrahlung Cross-Section Formulas and Re• lated Data", Rev. Mod. Phys. 41 (1959) 920. 6. H. K. Tseng and R. H. Pratt, "Exact Screened Calculations of Atomic-Field Bremsstrahlung", Phys. Rev. A3 (1971) 100. 7. R. H. Pratt, H. K. Tseng, C. M. Lee, 1. Kissel, C. MacCallum and M. Riley, "Bremsstrahlung Energy Spectra from Electrons of Kinetic Energy 1 keY ::; T ::; 2000 keY Incident on Neutral Atoms 2 ::; Z ::; 92", Atom. Data and Nucl. Data Tables 20 (1977) 175. 8. R. H. Pratt, H. K. Tseng, C. M. Lee, 1. Kissel, C. MacCallum and M. Riley, "Bremsstrahlung Energy Spectra from Electrons of Kinetic Energy 1 keY ::; T ::; 2000 keY Incident on Neutral Atoms 2 ::; Z ::; 92", Atom. Data and Nucl. Data Tables 26 (1981) 477. 9. A. T. Nelms, "Energy Loss and Range of Electrons and Positrons", National Bureau of Standards Circular 577 (1956). 10. A. T. Nelms, "Energy Loss and Range of Electrons and Positrons", Supplement to National Bureau of Standards Circular 577 (1958). 11. 1. Pages, E. Bertel, H. Joffre, and 1. Sklavenitis, "Energy Loss, Range and Bremsstrahlung Yield for lO-keV to 100-MeV Electrons in Various Elements and Chemical Compounds", Atom. Data 4 (1972) 1. 12. H. A. Bethe, "Zur Theorie des Durchgangs schneller Korpuskularstrahlen durch Materie", Ann. Phys. 5 (1930) 325. 13. H. A. Bethe, "Bremsformel fur Elektronen relativischer Geschwindigkeit", Z. Phys. 76 (1932) 293. 14. S. M. Seltzer and M. J. Berger, "Evaluation of the Collision Stopping Power of Elements and Compounds for Electrons and Positrons", Int. J. Appl. Radiat. and Isot. 33 (1982) 1189. 15. R. M. Sternheimer, M. J. Berger and S. M. Seltzer, "Density Effect for the Ionization Loss of Charged Particles in Various Substances", Atom. Data and Nucl. Data Tables 30 (1984) 261. 16. M. C. Walske, "The Stopping Power of K Electrons", Phys. Rev. 88 (1952) 1283. 17. M. C. Walske, "Stopping Power of L-electrons", Phys.Rev. 101 (1956) 940. 18. H. Bichsel, "L-shell Correction in Stopping Power", Univ. of S. California report USC-136120 (1967). 19. E. Bonderup, "Stopping of Swift Protons Evaluated from Statistical Atomic Model", K. Dan. Vidensk. Selsk. Mat.-Fys. Medd. 35 (No.17) (1967). 3. Electron Stopping Powers for Transport Calculations 79

20. J. Lindhard, "On the Properties of a Gas of Charged Particles", K. Dan. Vidensk. Selsk. Mat.-Fys. Medd. 28 (No.1) (1954). 21. H. H. Andersen and J. F. Ziegler, "Hydrogen: Stopping Powers and Ranges in All Elements", Vo1.3, in The Stopping and Ranges of Ions in Matter, (Pergamon Press, New York, 1977). 22. W. H. Barkas and S. von Friesen, "High-velocity Range and Energyloss Mea• surements in AI, Cu, Pb, U and Emulsion", Nuovo Cimento (Suppl.) 19 (1961) 41. 23. H. S(2Srensen and H. H. Andersen, "Stopping Power of AI, Cu, Ag, Au, Pb, and U for 5-18-MeV Protons and Deuterons", Phys. Rev. B8 (1973) 1854. 24. G. D. Zeiss, W. J. Meath, J. C. F. MacDonald and D. J. Dawson, "Accurate Evaluation of Stopping and Straggling Mean Excitation Energies for N, 0, H2, N2 , O2 , NO, NHs, H20 and N20 Using Dipole Oscillator Strength Distributions. A test of the Validity of Bragg's Rule", Radiat. Res. 70 (1977) 284. 25. G. D. Zeiss, W. J. Meath, J. C. F. MacDonald and D. J. Dawson, "Dipole Oscillator Strength Distributions, Sums and Some Related Properties for Li, N, 0, H2, N2, O2, NH 3, H20, NO and N20", Can. J. Phys. 55 (1977) 2080. 26. E. Shiles, T. Sasaki, M. Inokuti and D. Smith, "Self-consistency and Sum-rule Tests in the Kramers-Kronig Analysis of Optical Data: Applications to Alu• minum", Phys. Rev. B22 (1980) 1612. 27. J. C. Ashley, "Stopping Power of Liquid Water for Low-energy Electrons", Ra• diat. Res. 89 (1982) 25. 28. W. K. Chu and D. Powers, "Calculation of Mean Excitation Energy for All Elements", Phys. Lett. 40A (1972) 23. 29. J. F. Janni, "Proton Range-Energy Tables, 1-keV-I0-GeV", Atom. Data and Nucl. Data Tables 27 (1982) 147. 30. H. Bichsel, "Charged Particle Interactions", in Radiation Dosimetry, Vol.1 (2nd Ed.), edited by F. H. Attix and W. C. Roesch, (Academic Press, New York, 1968) 157. 31. J. E. Turner, P. D. Roecklein and R. B. Vora, "Mean Excitation Energies for Chemical Elements", Health Phys. 18 (1970) 159. 32. H. Bichsel, "Passage of Charged Particles Through Matter", in American Insti• tute of Physics Handbook (3rd Ed.), edited by D. E. Gray, (McGraw-Hill, New York, 1972) 8. 33. J. F. Ziegler, "Handbook of Stopping Cross Sectios for Energetic Ions in All Elements", Vol. 5, in The Stopping and Ranges of Ions in Matter, (Pergamon Press, New York, 1980). 34. S. P. Ahlen, "Theoretical and Experimental Aspects of the Energy Loss of Rel• ativistic Heavily Ionizing Particles", Rev. Mod. Phys. 52 (1980) 121. 35. T. J. Thompson, "Effect of Chemical Structure on Stopping Powers for High• energy Protons", Lawrence Berkeley Laboratory report UCRL-1910 (1952). 36. M. Inokuti and D. Y. Smith, "Fermi Density Effect on the Stopping Power of Metallic Aluminum", Phys. Rev. B25 (1982) 61. 37. J. C. Ashley, "Density Effect in Liquid Water", Radiat. Res. 89 (1982) 32. 38. R. M. Sternheimer, "The Density Effect for the Ionization Loss in Various Ma• terials", Phys. Rev. 88 (1952) 851. 80 M. J. Berger

39. T. A. Carlson, Photoelectron and Auger Spectroscopy, (Plenum Press, New York and London, 1975). 40. J. C. Ashley, C. J. Tung and R. H. Ritchie, "Electron Inelastic Mean Free Paths and Energy Losses in Solids, 1. Aluminum Metal", Surface Sci. 81 (1979) 409. 41. J. C. Ashley, C. J. Tung, R. H. Ritchie and V. E. Anderson, "Calculations of Mean Free Paths and Stopping Powers of Low Energy Electrons « 10 keY) in Solids Using a Statistical Model", IEEE Trans. NucI. Sci NS-23 (1976) 1833. 42. J. C. Ashley, C. J. Tung and R. H. Ritchie, "Electron Interaction Cross Sections in Al and A1203; Calculations of Mean Free Paths, Stopping Powers and Electron Slowing-down Spectra", IEEE Trans. NucI. Sci. NS-22 (1975) 2533. 43. H. H. Hubbell and R. D. Birkhoff, "Calorimetric Measurements of Electron Stop• ping Power of Aluminum and Copper Between 11 and 127 keV", Phys. Rev. A26 (1982) 2460. 44. A. T. Pugachev and Yu. A. Volkov, "Determination of the Energy Losses Suffered by Electrons on Passage Through Thin Films", Fiz. Verd. Ela 21 (1979) 2637; Sov. Phys.-Solid State 21 (1979) 1517. 45. N. Ishigure, C. Mori and T. Watanabe, "Electron Stopping Power in Aluminum in the Energy Region from 2 to 10.9 keV", J. Phys. Soc. Japan 44 (1978) 973. 46. L. V. Spencer and F. H. Attix, "A Theory of Cavity Ionization", Radiat. Res. 3 (1955) 239. 47. A. E. Nahum, "Water/Air Mass Stopping Power Ratios for Megavoltage Photon and Electron Beams" , Phys. Med. BioI. 23 (1978) 24. 48. Task Group 21, Radiation Therapy Committee, American Association of Physi• cists in Medicine (AAPM), "A Protocol for the Determination of Absorbed Dose from High Energy Photon and Electron Beams", Med. Phys. 10 (1983) 741. 49. International Commission on Radiation Units and Measurements (ICRU), "Ra• diation Dosimetry: Electron Beams with Energies between 1 and 50 MeV", ICRU Report 35 (1984). 50. A. E. Nahum, "Stopping Powers and Dosimetry", National Research Council of Canada report PXNR-2653 (1983). 51. R. M. Sternheimer and R. F. Peierls, "General Expression for the Density Effect for the Ionization Loss of Charged Particles", Phys. Rev. B26 (1971) 3681. 52. H. G. Paretzke and M. J. Berger, "Stopping Power and Energy Degradation for Electrons in Water Vapor", in Sixth Symposium on Microdosimetry, Vol. II, (Harwood Academic Publishers, New York, 1978) 747. 53. G. J. Kutcher and A. E. S. Green, "A Model for Energy Deposition in Liquid Water", Radiat. Res. 67 (1976) 408. 54. R. H. Ritchie, R. N. Hamm, J. E. Turner and H. A. Wright, "The Interaction of Swift Electrons with Liquid Water", in Sixth Symposium on Microdosimetry, Vol.I, (Harwood Academic Press, New York, 1978) 345. 55. A. Cole, "Absorption of 20-eV to 50,000-eV Electron Beams in Air and Plastic", Radiat. Res. 38 (1969) 7. 4. Cross Sections for Bremsstrahlung Production and Electron-Impact Ionization

Stephen M. Seltzer

Center for Radiation Research National Bureau of Standards Gaithersburg, Maryland 20899, U.S.A.

4.1 INTRODUCTION

Electron transport codes require extensive information on the cross sections that govern electron interactions with the atoms that make up the medium. These processes include bremsstrahlung production in the atomic field, excitation and ionization of atomic elec• trons, and elastic scattering by screened atomic nuclei. These fundamental processes are of basic interest in many fields, but their inclusion in general purpose Monte Carlo transport codes imposes the requirement that reasonably accurate cross-section data be available over a very wide range of energies and for virtually any material. In this chap• ter, we discuss two of these processes: bremsstrahlung production and electron-impact ionization. Both of these interactions result in the production of secondary radiations that can be important in radiation transport calculations. Other processes are covered elsewhere in this volume.

Section 4.2 outlines the recent work1 on the development of a comprehensive set of bremsstrahlung cross sections, differential in emitted photon energy. This set covers electrons with energies from 1 keY to 10 GeV incident on neutral atoms with atomic number Z = 1 to 100, and was prepared through the synthesis of various theoretical results that go beyond Bethe-Heitler, Born-approximation results. Explicit account is taken of the contributions from interactions both with the screened atomic nucleus and with the orbital electrons. Included also in this Section is a brief summary of electron-positron differences.

In Section 4.3, we describe a calculation of the shell-by-shell electron-impact ion• ization cross section. The calculation is based on a Weizsacker-Williams treatment, and shows generally good agreement with measured data.

4.2 BREMSSTRAHLUNG PRODUCTION

The singly differential bremsstrahlung cross section da / dk depends on the energy k of the emitted photon, the kinetic energy Tl of the incident electron, and the atomic

81 82 S. M. Seltzer number Z of the target atom. (The energy of the outgoing electron is T2 = Tl - k.) For a neutral atom, the cross section can be written as the sum of two terms,

( 4.1) where dun/ dk represents the bremsstrahlung produced in the field of the screened atomic nucleus, and Z(du./dk) represents the bremsstrahlung produced in the field of the Z atomic electrons. It is convenient to rewrite Eqn. 4.1 as

du ( 7]) dUn (4.2) dk = 1 + Z dk' where 7] is the cross-section ratio

(4.3)

A substantial body of theory on the bremsstrahlung cross section has been devel• oped in various approximations, and with various limitations and regions of applica• bility. The analytical theories available through 1959 can be found in the still largely current review of Koch and Motz2 • Recent reviews by Pratt3 present further discussion on low-energy theories. None of the analytical theories by themselves are adequate to describe accurately the bremsstrahlung cross section over a wide range of conditions. Thus, for the electron-nucleus cross section dun/ dk, Koch and Motz recommended a set of formulas, Coulomb-correction factors, and empirical corrections to be used according to the values of initial and final electron energies involved. We used somewhat different prescriptions in later Monte Carlo calculations·, in conjunction with updated empiri• cal correction factors. As was the usual practice, the electron-electron bremsstrahlung contribution was included by simply assuming TJ = 1 in Eqn. 4.2.

With the availability of accurate results by Pratt et al 5 who obtained the electron• nucleus cross section for 1 keV $ Tl $ 2 MeV on the basis of numerical phase-shift calculations, it became no longer necessary to rely on approximate cross-section formulas and poorly known empirical corrections at these energies. For high-energy electrons, the analytical electron-nucleus bremsstrahlung theory of Davies, Bethe, Maximon and Olsen6 (DBMO) is available. Their results, derived in the high-energy approximation in which Tl and T2 are large compared to the rest mass of the electron, go beyond the Born approximation through their use of Sommerfeld-Maue Coulomb wave functions. Their cross section is written in terms of the Bethe-Heitler, Born-approximation formula, including screening effects, plus a Coulomb-correction term. The DBMO results are considered accurate for incident electron energies Tl greater than about 50 MeV, except near the high-frequency limit (k = Tt} of the emitted photon spectrum. One can interpolate rather well between the high-energy DBMO predictions and the numerical Pratt et al data at the lower energies. It was on this basis, along with refinements in the evaluation of the high-energy results and a more accurate evaluation of the electron• electron cross section, that the new bremsstrahlung dataset was prepared. Full details can be found in Ref. 1; here we give a brief outline of the procedure. 4. Cross Sections for Bremsstrahlung Production and Electron-Impact Ionization 83

4.2.1 Electron-Nucleus Bremsstrahlung

Numerical partial-wave results for TJ < 2 MeV. Tseng and Pratt 7 developed a relativistic, partial-wave, multi pole, independent-particle calculation in which the bremsstrahlung process is treated as a single electron transi• tion in a self-consistent central potential representing the screened Coulomb field of the atomic nucleus. The code was used to calculate the electron-nucleus cross section for selected Z, Tl and klTl values, and these results were supplemented by calculations8 for k = a using a low-energy theorem which connects the low-frequency limit of the bremsstrahlung spectrum with the electron elastic-scattering cross section. Pratt et al 5 expanded this somewhat sparse database by interpolation to provide complete coverage for 2 :::; Z :::; 92, for a :::; klTl :::; 1, and for 1 keV :::; Tl :::; 2 MeV. We have used linear extrapolation to extend their results to the cases Z = 1 and Z = 93 to 100.

High-energy analytical results for TJ > 50 MeV. The cross section was evaluated using a combination of results from bremsstrahlung theory, dC1n 40: r; Z2 {un.cr 8 8} dk = k XBorn + scr.en + Coul , ( 4.4) where XB~::': is the Bethe-Heitler, Born-approximation result for an unscreened nucleus, derived with no energy approximations (formula 3BN in Ref. 2), and where 8.creen is a screening correction, and 8coul is a Coulomb correction. In Eqn. 4.4, 0: is the fine structure constant, and r. the classical electron radius.

The screening correction was calculated on the basis of the Bethe, Born-approximation high-energy formula for a screened nucleus (formula 3BSb in Ref. 2). 8screen was ob• tained as the difference between results evaluated (i) with Hartree-Fock atomic form factors from Hubbell et a/ 9 , and (ii) with the form factor set equal to zero (no screen• ing).

A Coulomb correction was devised which combines (a) knowledge of the high• frequency limit, or tip value, of the cross section (corresponding to k = T l ); (b) the Elwert factorlO in the region near the tip; and (c) the Coulomb correction from DBMO theory away from the tip. The result can be written as

(4.5) where JE is the Elwert factor, and 8g~Jt° is the DBMO Coulomb correction. In Eqn. 4.5, i3l and i32 are the velocities of the incoming and outgoing electrons in units of the velocity of light. The function W(T2) varies smoothly from a to 1 with increasing T2 (see Eq. 15 in Ref. 1) so as to switch on the DBMO Coulomb correction as the Elwert term (curly brackets in Eqn. 4.5) goes to zero. The exponential factor in Eqn. 4.5 ensures that the cross section goes to the adopted value for the high• frequency limit which is incorporated in the parameter ( (see Eq. 13 in Ref. 1). The result of this combination is illustrated in Fig. 4.1 where the percent contribu• tion of the final Coulomb correction to dC1n ldk is plotted as a function of T2 • For large values of the outgoing electron energy, the curves show the reduction of the cross section predicted by DBMO theory. As T2 is reduced below a few MeV, the Coulomb correction changes sign, and the cross section is raised to the correct tip 84 S. M. Seltzer value. This dependence is similar to the dependence of the Coulomb correction on the incident photon energy given by 0verb\'l11 for the related process of pair production.

o2

tw a: a: o u III g:2 :Jo U >• III o2 i= :J IIIa: f- o2 U f- 2 w U a: w c..

Figure 4.1. Coulomb correction expressed as a fraction of the electron-nucleus bremsstrahlung cross section. Results are given as a function of the kinetic en• ergy T2 of the outgoing electron; they apply for initial electron kinetic energies Tl > 50 MeV, and are nearly independent of T1•

The Born-approximation cross section vanishes at the high-frequency limit. The product fEXB~-:; gives tip values correct to the first order in o:Z. More accurate tip values, in the high-energy limit, /31 = 1, were obtained from the theory of Jabbur and Prattl2 . Their numerical results, judged to be most complete, are shown in Fig. 4.2, where they are compared to the other available predictions. To obtain tip values at lower energies, the Jabbur-Pratt high-energy limit results were connected to the tip values from the tables of Pratt et at for Tl $ 2 MeV, as is shown in Fig. 4.3a. In Fig. 4.3b, the resultant tip values are compared, as a function of electron energy T1, with results obtained from cross-section measurements. 4. Cross Sections for Bremsstrahlung Production and Electron-Impact Ionization 85

1.6

1.4

1.2 f,... 1.0 ....gl =l!l 0.8 -_JAr:~~tR~8ArrTT ~IN 0.6 ---JArf~lt~g::TT --- DECK, MULLIN AND HAMMER ----- ELWERT x BETHE-HEITLER

20 30 40 50 60 70 80 90 100 ATOMIC NUMBER Z Figure 4.2. High-frequency limit (tip) of the bremsstrahlung cross section in the limit!3I = 1. Curves of the scaled tip value, (!3U Z2) kda / dk for k = TI , vs. atomic number Z, are from various sources: numerical results of Jabbur and Pratt (points and solid curve); analytical results of Jabbur and Pratt (dot-dash curve); analytical results of Deck, Mullin, and HammerI3 (long-dash curve); and the product of the Elwert factor and the Bethe-Heitler, Born-approximation cross section (short-dash curve).

T1, MeV 3CX1 20 10 5 2

o JABBUR AND PRATI o PRATI, etal

13 6

Figure 4.3a. High-frequency limit (tip) of the differential bremsstrahlung cross section as a function of incident electron energy TI . The scaled tip value, (!3UZ 2) kda/dk for k = Tl, is plotted as a function of the variable (1 - !3;)1/2 / !3;. The corresponding electron kinetic energy TI is given on the upper scale. The squares, for !3I = 1, are from the theory of Jabbur and Pratt (see solid curve in Fig. 4.2): the circles for TI ~ 2 MeV are from the calculations of Pratt et a15 • ; the curves represent least-squares cubic-spline fits to the points. 86 S. M. Seltzer

6

5 Z = 13 4

3

..c 2 E. ..:- II 1 "'" .g"O°1"'" 0 "'" ~IN 6

5 I I' II Z= 79 4 "" "" 3 ""\'~ '\ '-. 2 ...... 1'-...:------e_·_e

T,. MeV Figure 4.3b. High-frequency limit (tip) of the differential bremsstrahlung cross section as a function of incident electron energy Tl . Comparison of calculated and experimental results. The points were obtained by extrapolating to k = Tl the experimental bremsstrahlung cross sections shown in Figs. 4.Sc, 4.Se and 4.10. The solid curves are the values adopted in the present work. The dashed curves were obtained by combining the energy dependence of the photoelectric• effect cross section of Gavrila14 with the analytical theory of the tip cross section obtained in the limit (31 = 1 by Jabbur and Pratt and by Deck et al. (dot-dash curve).

Intermediate-energy results for 2 < Tl < 50 MeV. The interpolation across the gap region 2 MeV < Tl < 50 MeV was carried out by fitting least-squares cubic splinesls to the cross sections at lower and higher energies. The results are illustrated in Fig. 4.4a for C and Fig. 4.4b for Au. It would appear that the curves in the gap region are rather tightly constrained by the values of the cross section for Tl below 2 MeV and above 50 MeV, and could differ little from those shown. Tests involving. different choices of the interpolation variables gave results that differed by less than 2%. In addition, a direct check on the accuracy of the interpo• lation procedure can be made by comparing the interpolated results with those from the exploratory partial-wave numerical calculations of Tseng and Pratt16 for 5 and 10- MeV electrons in Al and U. These comparisons are shown in Fig. 4.5, and confirm the reliability of the interpolation. 4. Cross Sections for Bremsstrahlung Production and Electron-Impact Ionization 87

T, . MeV

Figure 4.4a. Interpolation of scaled electron-nucleus bremsstrahlung cross sections for carbon. Curves of (J3i / Z2) kdO' / dk vs. the incident electron kinetic energy Tl are plotted for various values of k/T1 , where k is the energy of the emitted photon. Points and solid curves below 2 MeV are from the phase-shift calculations of Pratt et al. The solid curves above 50 MeV are based on our evaluation of the high-energy theory. The dashed curves in the shaded area between 2 and 50 MeV are interpolated. The lightly shaded area indicates the region in which the cross sections (with the modified Coulomb correction) adopted here differs by > 3% from DBMO cross sections.

"E

Tlo MeV

Figure 4.4b. Interpolation of scaled electron-nucleus bremsstrahlung cross sections for gold (see Fig. 4.4a). 88 S. M. Seltzer

15,.....----,-.--.....---,-...,---,--r--.,.-,..--, 14

T,-5MeV

~ 9 .fl~ 8 ~~ "'" 7 ~I'N 6 5 4 3 2

o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 kiT, kIT, Figure 4.5 Comparison of differential electron-nucleus bremsstrahlung cross sections, interpolated as indicated in Figs. 4.4a and 4.4b, with the results of phase-shift calculations by Tseng and Pratt (points). The error bars indicate an estimated 5% uncertainty.

, .c \ E \ , "I:" 9 \ ~"C ,, Z = 79 , , --- BETHE·HEITLER THEORY z , , -- PRESENT WORK o 7 ... , , w~ III 6 '... , T=10 MeV III , IIIo , a: 5 U " <.!l ~ 4 \ ...J \ ::c \ ~ 3 \ \ !Ii \ III ~ 2 \ w \ a: \ !II \ \ \

01 2 3 4 5 6 7 8 9 10 BREMSSTRAHLUNG PHOTON ENERGY. MeV Figure 4.6. Comparison of new bremsstrahlung cross sections in gold with those from Bethe-Heitler theory (formula 3BS in Ref. 2). 4. Cross Sections for Bremsstrahlung Production and Electron-Impact Ionization 89 Fig. 4.6 shows the present electron-nucleus cross section for Au at 5 and 10 MeV and the corresponding results from Bethe-Heitler theory (formula 3BS in Ref. 2). The comparison indicates differences on the order of 10% (and larger near the tip). 4.2.2 Electron-Electron Bremsstrahlung

At very high energies, the electron-electron and electron-nucleus cross sections are nearly identical for an un screened target of unit charge17. This is the basis for the appro~­ imation 7] = 1 that is often used in Eqn. 4.2. There are a number of differences between the electron-electron and electron-nucleus cross sections which cause 7] to be other than unity: (1) the screening in the field of the atomic electrons is different from that in the field of the nucleus; (2) the electron-electron cross section goes to zero at the high-frequency limit due to the Coulomb repulsion between the electrons; (3) in electron-electron interactions, there is a component of low-energy bremsstrahlung pro• duced by the recoiling electron; (4) there is a kinematic upper limit on the photon energy emitted in electron-electron interactions, which varies from kmao; = T1 for /31 = 1 to kmao; = Tt/2 for /31 = 0; (5) the electron-electron cross section tends to vanish at low incident-electron energies as a consequence of the lack of a dipole moment for the electron-electron system.

These features have been taken into account III our procedure. The electron• electron cross section was evaluated according to

dO"e 40;1'; { r 8.} dk = ~ J e-.XHaug + 8cre.n , ( 4.6) where XHaug is Haug's18 Born-approximation result for an unscreened free target elec• tron, including recoil effects and derived with no energy approximations, and where f.-. is the Coulomb correction of Maxon and Corman 19, and 8:creen is a screening correction.

The screening correction was evaluated on the basis of the high-energy theory of Wheeler and Lamb20, in analogy with the nuclear screening correction. 8;creen was obtained as the difference between results evaluated (i) with the incoherent scattering factors from Hubbell et a1 9 , and (ii) with the incoherent scattering factor set equal to unity (no screening). Combining this screening correction with the Haug cross section is justified at high energies because screening mainly affects small momentum transfers which are adequately described by the Bethe-Heitler recoil distribution used in the Wheeler-Lamb treatment, and exchange corrections and the radiation emitted by the recoiling electron is associated with large momentum transfers. At low energies, the errors incurred by using a screening correction derived in the high-energy approximation are greatly mitigated by the fact that when such errors become large, the contribution of electron-electron bremsstrahlung to the total cross section is very small.

The results for the electron-electron cross section are shown for selected atoms in Fig. 4.7 in terms of the quantity 7]. At high energies, 7] is larger than unity due to the fact that the atomic electrons are, on the average, less effectively screened than is the atomic nucleus. At lower energies, 7] becomes smaller than unity over much of the spectrum. 90 S. M. Seltzer

1.4rr--....------,--.----.----.

1.4rr--,--....---.----,.---.

11

Figure 4.7. Quantity"l which relates the electron-electron to the electron• nucleus bremsstrahlung cross section. The total bremsstrahlung cross section, differential in emitted photon energy, is obtained by multiplying the electron• nucleus cross section by the factor (1 + "lIZ).

4.2.3 Comparisons of Calculated and Measured Cross Sections

Data on do"/dk from a number of measurements can be found in the literature. Figs. 4.Sa-e compare measured21 - 26 and calculated results in the energy region of the Pratt et at data (Tl ::; 2 MeV). The agreement is generally within the combined limits of experimental uncertainty and a theoretical uncertainty estimated to be 5 to 10%. The earlier measurements of Motz26 are higher than the theoretical results as well as the other experimental results, probably due to an incomplete background subtraction in the Motz experiment23• 4. Cross Sections for Bremsstrahlung Production and Electron-Impact Ionization 91

Figs. 4.9 - 4.11 present some comparisons at higher energies, mainly in the interpo• lation region, 2 MeV < Tl < 50 MeV, and show good agreement between the calculated and measured data. In addition, the bremsstrahlung cross section was recently studied experimentally by Martins et a1 27 through the analysis of the photodisintegration yield . .

12 (b) 18 (0) AI 2 0 3 10

8 T, = 0.38 MeV 6

.c 12 .c E E .gl~ 12 -BI~ ... , .>t: ';Q:/'N 10 ~/N 8 " ~ •~ 8 '" , 6 T, = 0.18 MeV 6

4 4

2 2

0 0.2 0.4 0.6 0.8 1.0 0 0.2 0.4 0.6 0.8 1.0 kiT, kiT,

Figures ·i.Sa and b. Comparison of calculateu and measured bremsstrahlung spectra in the low-energy region. The solid curves represent the electron• nucleus cross section. The dashed curves include the contribution from electron• electron bremsstrahlung. The points are from the measurements of Aiginger21, Aiginger and Unfried22 , Rester and Dance2S , Rester and Edmonson24, Motz and Placious25 , and Motz26 . a) For I-MeV electrons incident on beryllium. b) For 0.18 and 0.38-MeV electrons incident on aluminum oxide; experimental cross sections have been divided by Z;" = 112.4.

of a 63CU target placed behind thin radiators of Cu, Mo, Ta, and Th and irradiated by electrons with energies Tl = 20 - 60 MeV. They found that the present bremsstr• ahlung cross sections provided a good fit to their data, better than that using either the uncorrected DBMO cross section or the Schiff spectrum which they found to be unacceptable. 92 S. M. Seltzer

12 AI AI

8 8 = 2.5 MeV = 0.5 MeV 6 6 14 14 4 12

.c E -81=E 16 16

14

12 12

10 o o

8 8 T, = 0.05 MeV

6 6

4 4

2 2

o 0.2 0.4 0.6 0.8 1.0 0 0.2 0.4 0.6 0.8 1.0 kiT, kiT,

Figure 4.8c. Comparison of calculated and measured bremsstrahlung spectra in the low-energy region. The solid curves represent the electron-nucleus cross section. The dashed curves include the contribution from electron-electron bremsstrahlung. The points are from the measurements of Aiginger21, Aigin• ger and Unfried22 , Rester and Dance23 , Rester and Edmonson24, Motz and Placious25 , and Motz26Por 0.05 to 2.5-MeV electrons on aluminum. Experi• mental cross sections have been divided by Z;v = 112.4. 4. Cross Sections for Bremsstrahlung Production and Electron-Impact Ionization 93

12 12 Sn 10 10

8 Hj To = 2.5 MeV 8 6 'Tt 6 T1hT 4 p~~ . 4 hp~p 2 ~~. 2

8 0 0 = 1.7 MeV 6 6 .c E 12 12

"C"Cbl~ ~

N_INcc.N

o 0.2 0.4 0.6 0.8 1 .0 o 0.2 0.4 0.6 0.8 1.0 k/TI k/TI Figure 4.Sd. Comparison of calculated and measured bremsstrahlung spectra in the low-energy region. The solid curves represent the electron-nucleus cross section. The dashed curves include the contribution from electron-electron bremsstrahlung. The points are from the measurements of Aiginger21, Aigin• ger and Unfried22 , Rester and Dance23 , Rester and Edmonson2., Motz and Placious25 , and Motz26 • For 0.05 to 2.5-MeV electrons on copper and tin. Experimental cross sections have been divided by Z;" = 112.4. 94 S. M. Seltzer

Au 10 Au 10

T, = 2.5 MeV 8 T, = 0.38 MeV 8

6 6

12 4 12

2 2 T, = 0.2 MeV 8 T, = 1.7 MeV o o

6 6 16 4 4 .c I E 2 2 -81=6 12 T, = 1 MeV o NCQ.N-IN 10 T, = 0.18 MeV 10

8 8

6 6

4 12 12 2 10 10 T, = 0.05 MeV o 8 8

6 6

4 4

20~~~-L~~~L-~-L-L~ 2~L-~-L~ __L-~-L~ __~ 0.2 0.4 0.6 0.8 1 .0 o 0.2 0.4 0.6 0.8 1.0 kiT, kiT, Figure 4.Se. Comparison of calculated and measured bremsstrahlung spectra in the low-energy region. The solid curves represent the electron-nucleus cross section. The dashed curves include the contribution from electron-electron bremsstrahlung. The points are from the measurements of Aiginger21, Aigin• ger and Unfried22 , Rester and Dance23 , Rester and Edmonson24, Motz and Placious25 , and Motz26 . For 0.05 to 2.5- MeV electrons incident on gold. Ex• perimental cross sections have been divided by Z;u = 112.4. 4. Cross Sections for Bremsstrahlung Production and Electron-Impact Ionization 95

14

13 I A: T, = 500 MeV, kiT, = 0.47 I I \ B: T, = 34 MeV, kiT, = 0.51 12 \ \ \ C: T, = 24 MeV, kIT, = 0.73 \ 11 \ , D: T, =16.9 MeV, kiT, =0.85 ,, 10 , \, ' ......

.0 9 ',.... -zr. .... __ E. '-" 8 -'---0. "C"Cbl'>< ...... >< _- _------7 ... ' ...... ~IN 6 ~"------~------­ ----- 5 Q.-___ ~C lr------n------.g..------______D CL 2 4

3 0 10 20 30 40 50 60 70 80 90 100 Z

Figure 4.9. Comparison of the calculated and measured Z-dependence of the bremsstrahlung cross section for high-energy electrons. The solid curves rep• resent the calculated electron-nucleus cross section, and the dashed curves include the contribution from electron-electron bremsstrahlung. The points are from the relative measurements of Brown28 (case A), Barber et al 29 (cases B and C), and LanzI and Hanson30 (case D), and have been normalized to the theoretical curves. The same normalization factor was used for cases Band C in order to preserve the measured relative difference between the 24 and 34-MeV results from the same experiment.

5

4 .c E. 3 -81~ -'" Z =79 ('IT"OQ.N 2 1;=4.54 MeV

0 0.6 0.7 0.8 0.9 1.0 kIT, Figure 4.10. Comparison of calculated and measured bremsstrahlung spectra in the tip region, for 4.54-MeV electrons incident on gold. The curves include the small contribution from electron-electron bremsstrahlung. The points are from the doubly differential measurements by Starfelt and Koch31, converted to the cross section integrated over photon angle by Fano, Koch, and Motz32 • 96 S. M. Seltzer

kiT, 5 1.0 0.95 0.90 0.85

4

Z =90

o.~----~----~~----~----~~----~ ____ ~ 15.0 15.5 16.0 16.5 17.0 17.5 18.0 T,. MeV

Figure 4.11. Comparison of the calculated and measured shapes of the brems• strahlung cross sections in the tip region, for 15 to 18-MeV electrons incident on thorium. The scaled cross section ((3i1Z 2 )kdf7/dk, for a fixed photon en• ergy k = 15.1 MeV, is plotted as a function of the incident electron kinetic energy Tl . The corresponding values of k/Tl are given on the upper scale. The curve is based on the adopted cross sections, including the small contri• bution of electron-electron bremsstrahlung. The points are from the relative measurements of Hall et al 33 and have been normalized to the theoretical curve.

4.2.4 Radiative Stopping Power

The mean energy loss of an electron per unit pathlength due to bremsstrahlung emission is usually written as

--1 (dE)-d = -ANA ar.Z2 2( Tl + me 2) iI>rad, (4.7) p X rad where -1/p(dE/dx)rad is the radiative stopping power, p is the mass density, A is the atomic weight of the medium, NA is Avogadro's constant, me2 is the electron rest energy, and Tl iI>rad = Jk df7dk dk /[ar.Z 2 2 (Tl + me2 )1 , (4.8) o is the dimensionless, scaled, integrated bremsstrahlung energy-loss cross section. In analogy with Eqns. 4.1 through 4.3, iI>rad can be written as the sum of the electron• nucleus and the electron-electron components: 4. Cross Sections for Bremsstrahlung Production and Electron-Impact Ionization 97

;r,. _ ;r,.(n) Z ;r,.(.) 'J!'rad - 'J!'rad + 'J!'rad' (4.9) or (4.10) where _ 4>(') /( 1 ",(n)) ." = rad Z2 'l'rad • (4.11)

Illustrative results for 4>~:~ are plotted in Fig. 4.12, and for 'i7 in Fig. 4.13.

20 20 z =1 l=6

15 15 A,(n) 't'rad 10 10

5

Z'13 l=29

15 15 A,(n) 't'rad 10 10

5 5

20 Z =47 l =79

15 15 A,(n) 't'rad 10 10

5 5

0~~-~3WWI0~-~2~10=-I~~I~~10~~10~~1~0~~10' 0~10~-~~~-~2~1~~~I~I~~10~~~~rw~. T/MeV T/MeV Figure 4.12. Scaled electron-nucleus bremsstrahlung integrated energy-loss cross sections. The points indicate the data from the exact numerical calcula• tions below 2 MeV and from high-energy theory above 50 MeV, upon which our cross sections are based. 98 S. M. Seltzer

1.4

1.2

1.0

0.8 1) 0.6

0.4

0.2

0 10-3 T,. MeV Figure 4.13. The quantity fJ which accounts for the contribution of electron• electron bremsstrahlung to the integrated radiative energy-loss cross section. The total integrated radiative energy-loss cross section is obtained by multi• plying the electron-nucleus result by the factor (1 + fJ/Z).

The total energy-loss cross sections, cI>rad are shown in Fig. 4.14a for C, and in Fig. 4.14b for Au, and can be compared to those used previously in our Monte Carlo calculations4 and to those from our old stopping-power tables!4 which were based on the Koch and Motz prescription.

o N Present Work Previous ETRAN/DATAPAC NAS-NRC 1133/NASA SP 3012

/ / -' -' .,,,,," # ""'- __ ~_ -=.::"" ~ h z 6

Figure 4.14a. Comparison of present radiative energy-loss cross sections for carbon (solid curves) with those from the empirically corrected Bethe-Heitler cross sections used in the previous ETRAN database4 (long dashed curves) and with those from the Koch-Motz2 prescription used in an early tabulation of electron stopping powers34 (short dashed curves). 4. Cross Sections for Bremsstrahlung Production and Electron-Impact Ionization 99

Present Work

Previous ETRAN/DATAPAC

NAS-NRC 1133/NASA SP 3012

..,0 ..o~

z 79

o~~~~~~~~~~~~~~~~~~~~~~~~ 10-3 10-2 10-' 10° 10' 102 103 T,. MeV Figure 4.14b. Comparison of present radiative energy-loss cross sections for gold (solid curves) with those from the empirically corrected Bethe-Heitler cross sections used in the previous ETRAN database4 (long dashed curves) and with those from the Koch-Motz2 prescription used in an early tabulation of electron stopping powers34 (short dashed curves).

The effect of these changes in the cross sections on transport results is indicated in the example shown in Fig. 4.15. Curves are given of the emergent bremsstrahlung yield from 5-MeV electrons perpendicularly incident on uranium slab targets, as a function of target thickness. The yield represents the fraction of the incident electron kinetic energy that emerges as bremsstrahlung in the forward direction (transmitted) or backward direction (reflected). Fig. 4.15 compares the results based on the present cross sections with those based on the two earlier cross-section datasets.

4.2.5 Positron Bremsstrahlung

Feng et al 35 and Kim et al 36 have obtained some results for the positron-nucleus bremsstrahlung cross section for neutral atoms using the Tseng-Pratt partial-wave mul• tipole expansion code. These results are available only for the combination of incident energies Tl = 10 and 500 keY and atomic numbers Z = 2, 8, 13,29,47, and 92, plus Tl = 50 keY in the cases Z = 8 and 92. Figs. 4.16a and 4.16b show the positron-nucleus cross sections and the corresponding electron-nucleus cross sections for Z = 8 and 92, for 10 keY :::; Tl :::; 500 keY. The two spectra become less similar with decreasing Tl and higher Z. These differences reflect the fact that electrons are attracted, while positrons are repulsed, by the nuclear charge. At much higher energies, the differences are expected to be small; in fact they vanish in the DBMO approximation. 100 S. M. Seltzer

SLAB THICKNESS, mm 1.5 0.13 ' ... 0.12 -- TRANSMITIED -'- 0.11 0 ...J 0.10 ',,-"""""" '''' W >= t!l 0.09 Z 1983 ~ ...J ::t: 0.08 - 1970 «ex: I-en 0.07 To ~ 5 MeV en :2 w 0.06 ex: til ______1964 I- 0.05 Z W ~_------1983 t!l _ ------1970 ex: 0.04 w REFLECTED :2 w

0,01

1.0 1.5 2.0 2.5 3.0 SLAB THICKNESS, g/cm2

Figure 4.15. Bremsstrahlung yields for 5-MeV electrons normally incident on uranium slab targets, Results are given for the fraction of the incident en• ergy emerging as bremsstrahlung in the forward direction (transmitted) and in the backward direction (reflected), as a function of the slab thickness. Re• sults are shown based on the use of bremsstrahlung production cross sections from our early stopping-power tables34 (curve labeled 1964), from the previ• ous ETRAN database4 (curve labeled 1970), and from the present data (curve labeled 1983).

These data on the quantitative differences between the positron-nucleus and electron• nucleus differential cross sections are rather fragmentary and do not lead easily to a comprehensive description. We have found36,37 however, that the ratio of the respective integrated energy-loss cross sections is, to a good approximation, a universal function of the variable Ttl Z2. This simple scaling law is shown in Fig. 4.17, and can be used to extend the complete coverage of the electron radiative stopping power to the case of positrons* . An example of such an application is given in Fig. 4.18 for Pb, which includes also results for the bremsstrahlung efficiency (the fraction of the incident ki• netic energy converted to bremsstrahlung photons as the particle slows down to rest, evaluated in the continuous-slowing-down approximation).

* Ref. 37 outlines also an approximate procedure used to estimate the contribution of the radiative energy loss of positrons in the field of the atomic electrons. 4. Cross Sections for Bremsstrahlung Production and Electron-Impact Ionization 101

N r---~--~--~----r---~---r--~----~--~---, Z = 8 e

.0 00 E

o~ __~ ____~ __~ ____~ ____~ __~ ____~ __~ ____-~-~:~~~-~ 0.0 0.2 0.4 0.6 0.8 1.0

Figure 4.16a. Comparison of electron-nucleus (solid curves) and positron• nucleus (dashed curves) bremsstrahlung cross sections for oxygen. These re• sults are from the exact partial-wave calculations of Feng et a1 35 and are shown for incident electron kinetic energies Tl of 10 and 50 keY.

N ~--~----~--~----~------r----~--~----~--~ Z = 92 e

.0 00 E '\ .::<.. '\ ·0

"-b '\ "0 '\ .::<. '\ '\ ~ '\ '"N .... " , , "-.,;: , '-"

'-'-. 50 Tl = 500 keY 10 -. ------o~ __~ ____~ __~ --- ____~_-~-~-~:~~~-~-~~~~ --- __~~~~ __~ 0.0 0.2 0.4 0.6 0.8 1.0 kiT,

Figure 4.16b. Comparison of electron-nucleus (solid curves) and positron• nucleus (dashed curves) bremsstrahlung cross sections for uranium. These results are from the exact partial-wave calculations of Feng et al 35 and are shown for incident electron kinetic energies Tl of 10, 50 and 500 keY. 102 S. M. Seltzer

C! ~

Z I> 2 0 8 x 13 ? 29 + 47 I + -'Ol() 0 92 ~f?ci '- + ~.., .5f? 0&

o o~1-0·-~7~~1~O-~6~~~1~O--5~~~1~O--~4~~1~O--~3~~1~O--~2~~1~O~-~'~~~100 T,/Z2. MeV

Figure 4.17. Ratio of positron-nucleus to electron-nucleus integrated energy• loss cross sections, as a function of Tli Z2. The points are from the results of exact partial-wave calculationss6 •

N o~ ______~~ __~~ __~ __~~~~~ ____~ ____~~~

Bremsstrahlung Efficiency (7.)

z 82 oo

(dE/dx),od (MeV cm2/g)

I o

N oI

T,. MeV

Figure 4.18. Positron (e+) and electron (e-) differences in Pb for the radia• tive stopping power, in MeV cm2 I g, and for the bremsstrahlung efficiency, in percent, as a function of the incident energy Ti. 4. Cross Sections for Bremsstrahlung Production and Electron-Impact Ionization 103

4.3 ELECTRON-IMPACT IONIZATION The mean energy loss due to ionization of the atom by the incident electron is already included in the collision stopping power. However, separate more detailed information on the electron-impact ionization cross section is important for other aspects of electron transport. For example, (a) knowledge of the shell-by-shell total cross section is needed for calculating the probability of emitting x-rays subsequent to inner-shell ionizations, and (b) information on the cross section differential in energy transfer can be important for the accurate descriptjon of energy-loss straggling (see Chapter 2).

We will mal

The calculation is based on the Weizsacker-Williams41,42 method (see also the ear• lier work of Fermi43), in which the cross section, differential in energy transfer E, is written as the sum of two contributions:

( 4.12)

The cross section for close collisions dael dE is described in terms of a collision between two electrons; and the cross section for distant collisions dad/dE is described in terms of the interaction of the equivalent radiation field (virtual photons) with the orbital electrons.

4.3.1 Cross-Section Formulas Close collisions. Close collisions, those which take place at small impact parameters and which are associated with large energy transfers, are assumed to be essentially governed by well• known knock-on electron cross sections. The knock-on cross sections refer to a collision with a free target electron. If each of the target atom orbital electrons are assumed to be initially at rest, the knock-on formula can be used directly. For the jth orbital, one can interpret the energy transferred E as the sum of the orbital binding energy Bj and the kinetic energy W of the ejected electron. Such a procedure ignores the sharing of the momentum transfer among the atomic electrons and nucleus, and assumes that one of the orbital electrons receives all of the momentum transfer. The cross section is then simply the sum of contributions from binary collisions between each orbital electron and the incident electron. In more refined forms of binary-encounter theory, the binding of the orbital electrons is included more fully by taking into account the initial momentum distribution for the target electron. For example, the extension of the Rutherford44 and Mott45 cross sections to the Vriens46 binary-encounter formula, is discussed by Kim4o. 104 S. M. Seltzer

In a similar fashion, we have applied a binary-encounter form of the MlIlller cross section·7 to each orbital according to

du~;) 271" r~ mc2 {Ill ( r ) 2 dE = 131 niPi (W + Bi)2 + (Tl - W)2 + T; r + 1 (4.13) 2r + 1 1 } - (r + 1)2 (W + Bi )(Tl - W) + Gi , where ni is the number of electrons in the orbital, r is the kinetic energy of the incident electron in units of the electron rest energy (r = Tdmc2 ), W is the kinetic energy of the ejected secondary electron, and Bi is the binding energy of the target electron. The slowest electron emerging from the collision is considered to be the secondary, so that 0:::; W:::; (Tl - Bi )/2. In Eqn. 4.13, the factor Pi is a so-called focusing term which we write as (4.14) where Ui is the mean kinetic energy of the target electron in the jth orbital. The factor

SUi [1 1] [ v'Y(y - 1)] Gi = 371" (W + Bi)S + (Tl _ W)S arctan v'Y + (y + 1)2 (4.15) W y=-, Ui is the result of averaging over an isotropic, hydrogenic distribution of orbital electron velocities.

Distant collisions. For the collisions with large impact parameters, one uses the method of virtual quanta. The perturbing fields of the incident electron are replaced by an equivalent pulse of radiation which is analyzed into a frequency spectrum of virtual quanta. The cross sec• tion for ionization then is calculated for the virtual-photon spectrum. Thus, we write the distant-collision cross section for each orbital as

du~;) (il dE = ni J(W + Bi ) UPE(W + Bi), ( 4.16) where uJt1 is the photoionization cross section for the jth orbital (per orbital electron), for an incident photon of energy E = W +Bi. The virtual-photon spectrum, integrated over impact parameters bmin < b < 00, is given by .8

J(E) = :;1~ ({Xmin J(O(Xmin)J(l(Xmin) - X;in [J(f(Xmin) - J(6(Xmin)]}

2 ( 4.17) + {(I - f3f) X'2in [/(f(Xmin) - J(6(Xmin)]}) 4. Cross Sections for Bremsstrahlung Production and Electron-Impact Ionization 105

where (4.18)

](0 and ](1 are the Bessel functions of order 0 and 1, and nc is Planck's constant (divided by 211") times the velocity of light. In Eqn. 4.17, the longitudinal component of the virtual-photon spectrum, given in the second set of curly brackets, tends to vanish at high electron energies (PI -+ 1), where the method of virtual quanta has well-established validity. In using this method at lower energies, we have ignored this component under the assumption that it is included in the close-collision cross section. The low-energy results are further modified due to the fact that we have integrated over impact parameters bmin < b < bmaz, where

1.123 nc PI ( 4.19) bmaz = B j (1 _ Pl)1/2 is a cutoff similar to that used in Bohr's49 classical theory of energy loss. This resultant virtual-photon spectrum is

I(E) = 2~2 El ({Xmin ](O(Xmin)](l(Xmin) - x!.in [](~(Xmin) - ](g(Xmin)]) 1I"}Jl 2 ( 4.17') 2 - {Xmaz](O(xmaz)]((Xmaz) - X~az [](~(xmaz) - ](g(xmaz)]}) where x maz is defined analogously to Xmin in Eqn. 4.18, but with bmaz in place of bmin.

This treatment of the distant collisions ignores the density effect in condensed media. Taking into account the dielectric polarization of the medium leads to a modifi• cation of the virtual-photon spectrum through terms involving Pi times the frequency• dependent dielectric constant, indicating that the correction is important at high en• ergies. The quantitative effect of the density-effect correction on inner-shell ionization cross sections has been estimated by Dangerfield and Spicer50 and by Scofield39 and found to be significant for incident electron energies above a few hundred MeV. For outer shells, the effect sets in at lower incident energies.

4.3.2 Input Data

For atoms, values for the binding energies Bj have been taken from Beardon and Burr5l, supplemented by data from Siegbahn et al 52 and from Moore53 • Values of the mean kinetic energies Uj of the orbital electrons were calculated from the results of MannS"; and his values of (r), the expectation value of the electron radius for the orbital of interest, were used for bmin•

The photoionization cross section for each atomic shell of interest, 0"~1(k), was synthesized using the following sources of data: for photon energies E ~ 1 keV, from Scofield55 ; for 0.1 keV ~ E < 1 keV, from Henry et a1 56; and for E < 0.1 keY, from McGuire57• In their calculations of the photoionization cross section, these authors used slightly different binding energies from each other and from those adopted in the present calculations. To adjust these data to a common choice of binding energy, it was assumed that the cross section is simply a function of the energy excess E - Bj • 106 S. M. Seltzer

In exploratory applications to molecules, photoionization cross sections for molec• ular orbitals were assembled from a linear combination of cross sections for atomic or• bitals, using parentage coefficients derived from Hartree-Fock calculations. Appropriate values for the binding and mean kinetic energies were taken from the available literature.

4.3.3 Illustrative Examples

The first example pertains to the sub-shell total cross section,

( 4.20)

In Fig. 4.19, results from our calculations and those from the rigorous high-energy treatment by Scofield39 are compared to measured cross sections50,58-68 for the ioniza• tion of the K and L shells of Au. Fig. 4.20 gives a similar comparison for the K-shell cross section for Ni and AI. Our results agree quite well with Scofield's results (and the experimental data) at energies above about 20 times the ionization threshold en• ergy. Below 20 Bj , our results appear to be more consistent with the experimental data than are Scofield's. Some discussion of the relationship of Scofield's inner-shell results (and, indirectly, ours) to those from other more approximate theoretical formulations, including Arthurs and Moisewitsch69, Gryzinski70 and Kolbenstvedt71, can be found in Peek72 •

The second example is based on our results for the differential cross section for water vapor. Table 4.1 gives the calculated energy spectrum of secondary electrons produced by an incident 10-keV electron, summed over all orbitals of the water molecule. The results are given relative to the spectrum obtained from the M011er cross section for free target electrons. Important differences occur at ejected energies below about 200 eV, where the M0ller spectrum goes to infinity (as 1/W2) while the present results remain finite. Differences at secondary energies greater than about 4.7 ke V are due to the dropping-out of orbitals that can no longer participate in the present treatment (as indicated by the upper limit of the integral in Eqn. 4.20). Our spectrum agrees quite well with that adopted by Berger and Wang (as discussed in Chapter 2) from their analysis of the available experimental data.

The ionization stopping power can be calculated from our spectra:

(T,-B')/2 ' daU) j (W + B j ) dE dW. (4.21 ) o

Results are given in Table 4.2 for incident energies between 50 and 1 keY. As shown in Table 4.2, if the relatively small contribution due to excitation (obtained from Berger (see Chapter 2)) is added, the total collision stopping power agrees very well with results from the recent evaluation of Bethe theory37. 4. Cross Sections for Bremsstrahlung Production and Electron-Impact Ionization 107

, ...... ----_...... --._.

T. MeV Figure 4.19. Comparison of theoretical electron-impact ionization cross sec• tions for K and L shells in gold with experimental results. The solid curves are from our Weizsacker-Williams calculations, and the broken curves are from Scofield's39 Born-approximation calculations. The points are from various measuremen ts50.58-68 . 108 S. M. Seltzer

z o Ishii et al (1977) o • Hink a Ziegler (1969) • Genz et al (1981) o Li-Schalz et al (1973) " Seif el Nasr et al (1974) • Peckman et al (1947) " Smick a Kirkpatrick (1945) o Jessenberger a Hink (1975)

l- tt u --....Ni---- ~ ;;1 Z 0 0:: I- U W ...J W I 10 T, MeV

Figure 4.20. K-shell electron-impact ionization cross sections in aluminum and nickel. The solid curves are from our Weizsacker-Williams calculations; the broken curve is from Scofield'ss9 Born-approximation calculation for Ni (Scofield did not ca~culate the cross section for AI). The points are from various measurements59,73-79.

Table 4.1

Spectrum of secondary electrons produced by 10-keV electrons in water vapor. The spec• trum, as a function of the kinetic energy W of the secondary electron, is given in terms of the ratio to the Mpller cross section.

W(eV) Ratio W(eV) Ratio W(eV) Ratio 0 0 500 0.962 4800 0.815 10 0.427 1000 0.969 4990 0.612 20 0.896 2000 0.978 4992 0.408 50 1.33 3000 0.990 4993 0.205 100 1.12 4000 1.01 5000 0 200 1.00 4600 1.03 4. Cross Sections for Bremsstrahlung Production and Electron-Impact Ionization 109

Table 4.2

Stopping power of electrons in water vapor. Results, in units of MeV-cm2 /g, are given for the ionization component from the present calculations, and the excitation component the calculations of Berger and Wang (see Chapter 2). The total is compared to results from recent ICRU tables37.

T (keV) Ionization Excitation Total ICRU % Discrepancy 50 6.057 0.484 6.541 6.650 -1.6 40 7.114 0.575 7.689 7.834 -1.9 30 8.802 0.722 9.524 9.726 -2.1 20 11.96 1.00 12.96 13.38 -2.4 15 14.91 1.27 16.18 16.61 -2.6 10 20.34 1.78 22.12 22.77 -2.9 8 24.11 2.15 26.26 27.1 -3.1 6 29.98 2.73 32.71 33.8 -3.2 4 40.54 3.82 44.36 45.9 -3.4 3 49.98 4.84 54.82 56.9 -3.7 2 66.57 6.73 73.30 76.2 -3.8 1.5 80.99 8.49 89.48 93.1 -3.9 1 105.4 11.7 117.1 122 -4.0

Acknowledgement: This work was supported by the Office of Health and Environmental Research, U.S. Department of Energy. 110 S. M. Seltzer REFERENCES 1. S. M. Seltzer and M. J. Berger, "Bremsstrahlung Spectra from Electron Interac• tions with Screened Atomic Nuclei and Orbital Electrons", Nucl. Instr. Meth. B12 (1985) 95; and "Bremsstrahlung Energy Spectra from Electrons with Ki• netic Energy 1 keY - 10 GeV Incident on Screened Nuclei and Orbital Electrons of Neutral Atoms with Z = 1-100", Atom. Data and Nucl. Data Tables 35 (1986) 345. 2. H. W. Koch and J. W. Motz, "Bremsstrahlung Cross-Section Formulas and Re• lated Data", Rev. Mod. Phys. 31 (1959) 920. 3. R. H. Pratt, "Electron Bremsstrahlung Spectrum" , Comments Atom. Mol. Phys. 10 (1981) 121; R. H. Pratt and I. J. Feng, "Electron-Atom Bremsstrahlung" in Atomic Inner-Shell Physics, edited by B. Crasemann, (Plenum, New York, 1985) 533. 4. M. J. Berger and S. M. Seltzer, "Bremsstrahlung and Photoneutrons from Thick Tungsten and Tantalum Targets", Phys. Rev. C2 (1970) 621. 5. R. H. Pratt, H. K. Tseng, C. M. Lee, L. Kissel, C. MacCallum and M. Riley, "Bremsstrahlung Energy Spectra from Electrons of Kinetic Energy 1 keV::; T ::; 2000 keY Incident on Neutral Atoms 2 ::; Z ::; 92", Atom. Data and Nucl. Data Tables 20 (1977) 175; errata in 26 (1981) 477. 6. H. Davies, H. A. Bethe and L. C. Maximon, "Theory of Bremsstrahlung and Pair Production. II. Integral Cross Section for Pair Production", Phys. Rev. 93 (1954) 788; and H. Olsen, "Outgoing and Ingoing Waves in Final States and Bremsstrahlung", Phys. Rev. 99 (1955) 1335. 7. H. K. Tseng and R. H. Pratt, "Exact Screened Calculations of Atomic-Field Bremsstrahlung", Phys. Rev. A3 (1971) 100. 8. H. K. Tseng and R. H. Pratt, "Electron Bremsstrahlung from Neutral Atoms", Phys. Rev. Lett. 33 (1974) 516. 9. J. H. Hubbell, W. J. Veigele, E. A. Briggs, R. T. Brown, D. T. Cromer and R. J. Howerton, "Atomic Form Factors, Incoherent Scattering Functions, and Photon Scattering Cross Sections", J. Phys. Chern. Ref. Data 4 (1975) 471; errata in 6 (1977) 615; and J. H. Hubbell and I. 0verb!,!, "Relativistic Atomic Form Factors and Photon Coherent Scattering Cross Sections", J. Phys. Chern. Ref. Data 8 (1979) 69. 10. G. Elwert, "Verscharfte Berechnung von Intensitiit und Polarisation im Kon• tinuierlichen Rontgenspektrum", Ann. Physik 34 (1939) 178. 11. I. 0verb!,!, "The Coulomb Correction to Electron Pair Production by Intermediate• Energy Photons", Phys. Lett. 71B (1977) 412. 12. R. J. Jabbur and R. H. Pratt, "High-Frequency Region of the Spectrum of Elec• tron and Positron Bremsstrahlung", Phys. Rev. 129 (1963) 184; and "High• Frequency Region of the Spectrum of Electron and Positron Bremsstrahlung. II", Phys. Rev. 133 (1964) B1090. 13. R. T. Deck, C. J. Mullin and C. L. Hammer, "High-Frequency Limit of the Bremsstrahlung Spectrum", Nuovo Cimento 32 (1964) 180. 14. M. Gravila, "Relativistic K-Shell Photoeffect", Phys. Rev. 113 (1959) 514. 15. M. J. D. Powell, "Curve Fitting by Cubic Splines", Atomic Energy Research Establishment (Harwell) report TP 307 (1967). 16. H. K. Tseng and R. H. Pratt, "Electron Bremsstrahlung Energy Spectra Above 2 MeV", Phys. Rev. A19 (1979) 1525. 4. Cross Sections for Bremsstrahlung Production and Electron-Impact Ionization 111

17. J. Joseph and F. Rohrlich, "Pair Production and Bremsstrahlung in the Field of Free and Bound Electrons", Rev. Mod. Phys. 30 (1958) 354. 18. E. Haug, "Bremsstrahlung and Pair Production in the Field of Free Electrons", Z. Naturforsch. 30a (1975) 1099. 19. S. M. Maxon and E. G. Corman, "Electron-Electron Bremsstrahlung from a Quantum Plasma (Z = 1)", Phys. Rev. 163 (1967) 156. 20. J. A. Wheeler and W. E. Lamb, "Influence of Atomic Electrons on Radiation and Pair Production", Phys. Rev. 55 (1939) 858; errata in 101 (1956) 1836. 21. H. Aiginger, "Elektron-Bremsstrahlungswirkungsquerschnitte von 180- und 380- keY Elektronen", Z. Phys. 197 (1966) 8. 22. H. Aiginger and E. Unfried, "Elektron-Bremsstrahlungswechselwirkung", Acta Phys. Austriaca 35 (1972) 331. 23. D. H. Rester and W. E. Dance, "Bremsstrahlung Cross-Section Measurements at Incident Electron Energies of 1.0, 1.7, and 2.5 MeV", Phys. Rev. 161 (1967) 85. 24. D. H. Rester and N. Edmonson, "Bremsstrahlung Cross-Section Measurements at an Incident Electron Energy of 0.2 MeV on AI, Cu, and Au", Phys. Rev. A6 (1972) 1405; and D. H. Rester, personal communication (1967). 25. J. W. Motz and R. C. Placious, "Bremsstrahlung Cross-Section Measurements for 50-keV Electrons", Phys. Rev. 109 (1958) 235. 26. J. W. Motz, "Bremsstrahlung Differential Cross-Section Measurements for 0.5- and 1.0-MeV Electrons", Phys. Rev. 100 (1955) 1560. 27. M. N. Martins, E. Hayward, G. Lamaze, X. K. Muruyama, F. J. Schima and E. Wolynec, "Experimental Test of the Bremsstrahlung Cross Section", Phys. Rev. C30 (1984) 1855. 28. K. 1. Brown, "Z Dependence of Bremsstrahlung for the Case of Complete Screen• ing", Phys. Rev. 103 (1956) 243. 29. W. C. Barber, A. I. Berman, K. 1. Brown and W. D. George, "Z-Dependence of Bremsstrarhlung", Phys. Rev. 99 (1955) 59. 30. 1. H. LanzI and A. O. Hanson, "Z Dependence and Angular Distribution of Bremsstrahlung from 17-MeV Electrons", Phys. Rev. 83 (1951) 959. 31. N. Starfelt and H. W. Koch, "Differential Cross-Section Measurements of Thin• Target Bremsstrahlung Produced by 2.7- to 9.7-MeV Electrons", Phys. Rev. 102 (1956) 1598. 32. U. Fano, H. W. Koch and J. W. Motz, "Evaluation of Bremsstrahlung Cross Sections at the High-Frequency Limit", Phys. Rev. 112 (1958) 1679. 33. H. E. Hall, A. O. Hanson and D. Jamnick, "High-Frequency Limit of 15.1-MeV Bremsstrahlung", Phys. Rev. 129 (1963) 2207. 34. M. J. Berger and S. M. Seltzer, "Tables of Energy Losses and Ranges of Elec• trons and Positrons", National Academy of Sciences (National Research Council, Washington D.C.) Publication 1133 (1964) 205; also National Aeronautics and Space Administration report NASA SP-3012 (1964). 35. I. J. Feng, R. H. Pratt and H. K. Tseng, "Positron Bremsstrahlung", Phys. Rev. A24 (1981) 1358. 36. L. Kim, R. H. Pratt, S. M. Seltzer and M. J. Berger, "Ratio of Positron to Electron Bremsstrahlung Energy Loss: An Approximate Scaling Law", Phys. Rev. A33 (1986) 3002. 112 S. M. Seltzer

37. M. J. Berger and S. M. Seltzer, "Stopping Power and Ranges of Electrons and Positrons (Second Ed.)", National Bureau of Standards report NBSIR 82-2550- A (1983); see also "Stopping Powers for Electrons and Positrons", International Commission on Radiation Units and Measurements (ICRU) Report 37 (1984). 38. C. J. Powell, "Cross Sections for Ionization of Inner-Shell Electrons by Elec• trons", Rev. Mod. Phys. 48 (1976) 33. 39. J. H. Scofield, "K- and L-Shell Ionization of Atoms by Relativistic Electrons", Phys. Rev. A18 (1978) 963. 40. Y. K. Kim, "Energy Distribution of Secondary Electrons. I. Consistency of Experimental Data", Radiat. Res. 61 (1975) 21; and "Theory of Electron• Atom Collisions" in Physics of Ion-Ion and Electron-Ion Collsions, edited by F. Brouillard and J. W. McGowen, (Plenum, New York, 1983) 101. 41. C. F. Weizsacker, "Ausstrahlung bei Stossen sehr schnellen Elektronen", Z. Phys. 88 (1934) 612. 42. E. J. Williams, "Correlation of Certain Collision Problems with Radiation The• ory", Kgl. Danske Videnskab. Mat.-fys. Medd. XIII (1935) 4. 43. E. Fermi, "Uber die Theorie des Stosses zwischen Atomen un elektrisch geldenen Teilchen", Z. Phys. 29 (1924) 315.

44. E. Rutherford, "The Scattering of Q and j3 Particles by Matter and the Structure of the Atom", Philos. Mag. 21 (1911) 669. 45. N. F. Mott, "The Scattering of Fast Electrons by Atomic Nuclei", Proc. Roy. Soc. (London) A124 (1929) 425. 46. 1. Vriens, "Binary-Encounter and Classical Collision Theories", in Case Studies in Atomic Collision Physics I, edited by E. W. McDaniel and M. R. C. McDowell, (North-Holland, Amsterdam, 1969) 335. 47. C. M¢ller, "Zur Theorie des Durchgang schneller Elektronen durch Materie", Ann. Physik. 14 (1932) 568. 48. For example, see J. D. Jackson, Classical Electrodynamics, 2nd Ed., (Wiley, New York, 1975). 49. N. Bohr, "On the Decrease of Velocity of Swiftly Moving Electrified Particles in Passing through Matter", Philos. Mag. 30 (1915) 581. 50. G. R. Dangerfield and B. M. Spicer, "K-Shell Ionization by Relativistic Elec• trons", J. Phys. B8 (1975) 1744. 51. J. A. Beardon and A. F. Burr, "X-Ray Wavelengths and X-Ray Atomic Energy Levels", Rev. Mod. Phys. 31 (1967) 49. 52. K. Siegbahn et al., ESCA-Atomic, Molecular and Solid State Structure Studied by Means of Electron Spectroscopy, (Almqvist and Wiksells, Uppsala, 1967); and K. Siegbahn et aI, ESCA Applied to Free Molecules, (North-Holland, Amster• dam, 1969). 53. C. Moore, "Atomic Energy Levels as Derived from the Analyses of Optical Spec• tra", National Bureau of Standards report NSRDS-NBS 35 (1971). 54. J. B. Mann, "Atomic Structure Calculations. I. Hartree-Fock Energy Results for the Elements Hydrogen to Lawrencium", Los Alamos Scientific Laboratory report LA-3690 (1967); and" Atomic Structure Calculations. II. Hartree-Fock Wavefunctions and Radial Expectation Values: Hydrogen to Lawrencium", LA- 3691 (1968). 55. J. H. Scofield, "Theoretical Photoionization Cross Sections from 1 to 1500 keV", Lawrence Livermore National Laboratory report UCRL-51326 (1973). 4. Cross Sections for Bremsstrahlung Production and Electron-Impact Ionization 113

56. E. M. Henry, C. 1. Bates and W. J. Veigele, "Low-Energy Photoionization", Phys. Rev. A6 (1972) 2131. 57. E. J. McGuire, "Photo-Ionization Cross Sections of the Elements Helium to Xenon", Phys. Rev. 175 (1968) 20. 58. L. M. Middleman, R. 1. Ford and R. Hofstader, "Measurement of Cross Sections for X-Ray Production by High-Energy Electrons", Phys. Rev. A2 (1970) 1429. 59. K. Ishii, M. Kamiya, K. Sera, S. Morita, H. Tawara, M. Oyamada and T. C. Chu, "Inner-Shell Ionization by Ultrarelativistic Electrons", Phys. Rev. A15 (1977) 906. 60. H. H. Hoffmann, H. Genz, W. Low and A. Richter, "z and E Dependence and Scaling Behavior of the K-Shell Ionization Cross Section for Relativistic Electron Impact", Phys. Lett. 65A (1978) 304. 61. K. H. Berkner, S. N. Kaplan and R. V. Pyle, "Cross Sections for K-Shell Ioniza• tion of Pd and Au by 2.5- and 7.1-MeV Electrons", Bull. Am. Phys. Soc. 15 (1970) 786. 62. D. H. Rester and W. E. Dance, "K-Shell Ionization of Ag, Sn and Au from Electron Bombardment", Phys. Rev. 152 (1966) 1. 63. H. Hansen and A. Flammersfeld, "Messung des Wirkungsquerschnitts fur K• Ionisierung durch Stoss Niederenergetishcer Negatonen und Positonen", Nucl. Phys. 79 (1966) 135. 64. J. W. Motz and R. C. Placious, "K-Ionization Cross Sections for Relativistic Electrons", Phys. Rev. 136 (1964) A662. 65. D. V. Davis, V. D. Mistry and C. A. Quarles, "Inner Shell Ionization of Copper, Silver and Gold by Electron Bombardment", Phys. Lett. 35A (1972) 169. 66. S. I. Salem and 1. D. Moreland, "LII and Lm Ionization Cross Sections in Gold at Very Low Energies", Phys. Lett. 37A (1971) 161. 67. M. Green, Ph.D. Dissertation, Univ. of Cambridge, Cambridge, England (1962); data obtained from Ref. 64. 68. Y. K. Park, M. T. Smith and W. Scholz, "Cross Sections for L X-Ray Production and L-Subshell Ionization by MeV Electrons", Phys. Rev. A12 (1975) 1358. 69. A. M. Arthurs and B. 1. Moisewitsch, "The K-Shell Ionization of Atoms by High-Energy Electrons", Proc. Roy. Soc. (London) A247 (1958) 550. 70. M. Gryzinski, "Classical Theory of Atomic Collisions. I. Theory of Inelastic Collisions", Phys. Rev. 138 (1965) A336. 71. H. Kolbenstvedt, "Simple Theory for K-Ionization by Relativistic Electrons", J. Appl. Phys. 38 (1967) 4785. 72. J. M. Peek and J. A. Halbleib, "Improved Atomic Data for Electron-Transport Predictions by the Codes TIGER and TIGERP: I. Inner-Shell Ionization by Electron Collision", Sandia National Laboratories report SAND82-2887 (1983). 73. W. Hink and A. Ziegler, "Der Wirkungsquerschnitt fur die Ionisierung der K• Schale von Aluminium durch Elektronenstoss (3-30 keV)", Z. Phys. 226 (1969) 222. 74. H. Genz, C. Brendel, P. Escwey, U. Kuhn, W. Low, A. Richter, P. Seserko and R. Sauerwein, "Search for the Density Effect in Inner-Shell Ionization by Ultra• Relativistic Electron Impact", report IKDA 81/16, Institute fur Kernphysik, Technische Hochschule Darmstadt, Germany (1981). 114 S. M. Seltzer

75. A. Li-Scholz, R. Colle, I. L. Preiss and W. Scholz, "Cross Sections for K-Shell Ionization by 2-MeV-Electron Impact", Phys. Rev. A 7 (1973) 1957. 76. S. A. H. Seif el Nasr, D. Berenyi, and Gy. Bibok, "Inner Shell Ionization Cross Sections for Relativistic Electrons", Z. Phys. 267 (1974) 169. 77. L. T. Pockman, D. L. Webster, P. Kirkpatrick and K. Harwort, "The Probability of K Ionization of Nickel by Electrons as a Function of Their Energy", Phys. Rev. 71 (1947) 330. 78. A. E. Smick and P. Kirkpatrick, "Absolute K-Ionization Cross Section of the Nickel Atom under Electron Bombardment at 70 kV", Phys. Rev. 67 (1945) 153. 79. J. Jessenberger and W. Hink, "Absolute Electron Impact K-Ionization Cross Sections of Titanium and Nickel (:::; 50 keV)", Z. Phys. A275 (1975) 331. 5. Electron Step-Size Artefacts and PRESTA

Alex F. Bielajew and David W. O. Rogers

Division of Physics National Research Council of Canada Ottawa, Canada KIA OR6

5.1 INTRODUCTION In the first half of this chapter, we shall discuss electron step-size artefacts, the reasons for calculation of spurious results under some circumstances, and simple ways by which these calculational anomalies may be avoided. In the second half, we shall discuss a more sophisticated electron-transport algorithm called PRESTA which, in most cases, solves this problem of step-size dependence.

The chapter will proceed within the context of the EGS4 code!, although the ideas put forth apply to all electron transport codes which use condensed-history methods. Calculations which signal the existence of step-size anomalies will be presented along with the improvements to the electron-transport algorithm which were used to circum• vent the problem.

5.2 ELECTRON STEP-SIZE ARTEFACTS 5.2.1 What is an Electron Step-Size Artefact?

An electron step-size artefact is characterized by the dependence of some calculated result upon arbitrary "non-physics" parameters of the electron transport. This is illus• trated by the example given in Fig. 5.1. In this example, I-MeV electrons were incident normally upon a 3ro/2 thick slab of water. The quantity ro is the range calculated in the continuous-slowing-down approximation (csda). The energy deposited between ro/2 and ro was scored. Several ways of calculating the energy deposition are depicted. The first, (no ESTEPE control - lower dashed line), is the default EGS calculation. In this case, a path-length correction, which includes the effect of electron path curvature for each electron step, was included. The second, (no ESTEPE control - upper dashed line), neglects this correction. Rogers2 added an electron step-size limit, ESTEPE, the maximum allowable fractional kinetic energy loss per electron step to "continuous" energy-loss processes in order to obtain better agreement with experiment in the low• energy region. One notices a dramatic increase in the "with PLC" curve with smaller ESTEPE and a commensurate decrease in the "no PLC" curve. Why should a cal• culated result depend so strongly on an arbitrary parameter such as electron step-size unless some basic constraints of the underlying theory are being violated? What is the role of path-length corrections? Does the electron-transport algorithm have enough

115 116 A. F. Bielajew and D. W. O. Rogers physics content to simulate electon transport accurately? An even more important question is "What is the correct answer?". (If a correct answer is to be obtained for a case that exhibits step-size dependence, it is always found at smaller step-sizes within certain constraints that we shall discuss later.)

EGS (no PLC)

... NILEs..T.E.P..E .. CQNreQL......

8.88 8.05 8.10 8.15 0.20 ESTEPE Figure 5.1. The relative energy deposition from I-MeV electrons incident nor• mallyon a 3ro/2 slab of water. The energy deposited between ro/2 and ro is shown. The upper dashed line is an EGS calculation without the electron step-size shortened by ESTEPE and without path-length corrections (PLC's). The lower dashed line is an EGS calculation without ESTEPE control and in• cluding the default PLC employed by EGS. The solid lines depict calculations with electron step-sizes shortened using ESTEPE control.

As another example of dramatic step-size effects, consider the irradiation geometry depicted in Fig. 5.2. In this case, a I-MeV zero-area beam of electrons was incident on the center of the end of an air tube which was 2 mm in diameter and 20 cm long. The results are plotted in Fig. 5.3. The dose deposited in the air cylinder was scored as a function of SMAX, the maximum geometrical step-length allowed. This parameter was also introduced by Rogers2 in adapting the EGS code to low-energy simulations. The default EGS simulation (equivalent to setting SMAX = 20 cm, the length of the tube) is wrong since most often the electrons only take one step through the tube, as depicted in Fig. 5.4. All the "continuous" energy deposition associated with this step is deposited within the air tube resulting in too high a value being calculated. Reducing SMAX to 10 cm, half the length of the tube, almost halves the energy deposition, as seen in Fig. 5.3. In this case, most of the electrons that are transported 10 cm immediately scatter out of the tube, as depicted in Fig. 5.5. Further reduction of SMAX reduces the energy deposited to the tube as the electron transport simulation becomes more and more accurate. Finally, a flat region of "convergence" is obtained in tl~e vicinity of 0.2 to 1.0 cm, a scale of magnitude comparable to the diameter of the tube. As seen in Fig. 5.6, the small transport steps allow the electron to escape the tube or be transported down it, in accord with the random selection of the multiple-scattering angle for each step. In this region, the transport is being simulated more or less accurately. 5. Electron Step-Size Artefacts and PRESTA 117

20 cm • 1 MeV (~) ______ai_r __~Ol2mm e vacuum Figure 5.2. The irradiation geometry of the "thin tube" simulation. A zero• area beam of I-MeV electrons was incident on the center of the end of a 2-mm diameter, 20-cm long tube of air.

2 EGS (SMAX)

SMAX {em} Figure 5.3. The relative dose deposited in the air cylinder is plotted as a function of SMAX. In the "default EGS" case, the electrons are usually trans• ported the length of the tube resulting in an anonamously high calculated dose to the tube.

At step-sizes in the vicinity of 1 mm and smaller, we observe another artefact in Fig. 5.3. We again notice anonamously high results. The reason for the occurance of this artefact has to do with the minimum step-size that can be accomodated by the multiple• scattering formalism used by the EGS code. (EGS uses the Moliere formalisms,4 as expressed by Bethe5.) At these smaller step-sizes, multiple-scattering formalism should be replaced by a "few-scattering" or "single-scattering" formalism. EGS does not do this, but rather "switches off" the multiple scattering, and no elastic electron-nucleus scattering is modelled. Once more the electrons are transported in straight lines down the length of the tube. Therefore, we must qualify a statement expressed earlier in the chapter. If a correct answer is to be obtained with the EGS code for a case that exhibits a step-size dependence, it is obtained by using small step-sizes with the proviso that the multiple scattering is not "switched off" for a substantial number of the electron transport steps. The various limits on transport step-size will be discussed in more detail later in the chapter. 118 A. F. Bielajew and D. W. O. Rogers

SMA! = 20em

Figure 5.4. In the default EGS calculation the electrons most often travel the length of the tube. Note that the vertical scale in this and the next two figures is greatly exaggerated. The tube is actually 2 mm in diameter and 20 cm long. The x's mark the end-points of each electron step.

SMA! = 10em

Figure 5.5. If SMAX is reduced to 10 cm, half the length of the tube, then the electrons that are transported 10 cm usually scatter out of the tube im• mediately.

SMA! = O.5em

Figure 5.6. When the transport steps are shortened to a length comparable to the diameter of the tube, the electron mayor may not scatter out of the tube due to the probabilistic nature of multiple scattering.

The previous example was contrived to show large changes in calculated results with step-size. It represents the extreme limit of what the EGS code is capable of. As a final example, we show a large step-size dependence for a case where the electrons are aLmost in a state of equilibrium. This is the case of a thick-walled ion chamber exposed to 60Co photons. In Fig. 5.7, we show the variation with ESTEPE of the calculated response of a 0.5-g/cm2 thick carbon-walled ion chamber with a cylindrical air cavity 2 mm in depth and 2 cm in diameter exposed to a monoenergetic beam of 1.25-MeV photons incident normally upon one of the flat ends. The results are normalized to the theoretical predictions of Spencer-Attix theory6 corrected for photon attenuation, photon scatter and electron drift effects7,8. According to the theorem of Fan09 , the electron fluence in the chamber in the vicinity of the cavity is almost unperturbed by the presence of the cavity in this situation. (Strictly speaking, Fano's theorem only applies to density changes in one medium. However, carbon and air are not too dissimilar except for their densities, and Fano's theorem may be applied with negligible error.) The electrons depositing energy in the cavity in this simulation are almost in complete equilibrium. Non-equilibrium effects requiring corrections to Spencer-Attix theory amount only to 5. Electron Step-Size Artefacts and PRESTA 119 a few percent of the total response. Why then should the electron step-size play such a critical role in a simulation where electron transport does not matter a great deal to the physics? We observe, in Fig. 5.7 a step-size variation of about 40% when ESTEPE is changed from 1% to 20%! To answer this question requires some closer examination of the various elements of electron tranport.

1 2

D::>- 1.0 0 ION CHAMBER THEORY ::x::r..:I E-< '-Z 0 0.8 S- EGS ~ ~ -< 0.6 U

0.4 000 0.05 0.10 o 15 0.20 0.25 ESTEPE Figure 5.7. Calculated response of a thick-walled carbon ion chamber (0.5 g/cm2 carbon walls, 2-cm diameter, 2-mm thick cylindrical air cavity), exposed to 1.25-MeV photons incident normally on a flat circular end.

5.2.2 Path-Length Correction

To illustrate the concept of path-length correction, we consider the example of lO-MeV electrons incident normally upon a 1-cm slab of water. The top curve in Fig. 5.S depicts a typical EGS electron transport step through this slab with the EGS system used in its default configuration (no ESTEPE or SMAX control). Note that the electron went through in only one step .. The other curves in Fig. 5.S depict similar histories except that ESTEPE, the maximum fractional continuous energy loss per step, has been ad• justed to 10, 5, 2, or 1 %. As ESTEPE gets smaller and smaller, the electron tracks begin to "look" like real electron tracks, similar to those that one would observe, for example, in bubble chamber photographs. We know that electron steps are curved, as depicted in the previous figures. Must we use exceedingly small electron steps to calculate accu• rately in Monte Carlo simulations? The answer depends upon the application. If one is interested in accurate physical "pictures" of electron tracks, then short step-sizes, consistent with the resolution desired, must be used. However, imagine that we are only interested in calculating the energy deposited in this slab. Then, considering Fig. 5.S with the realization that the energy deposited is proportional to the total curved path, it would be possible to simulate passage through this slab using only one step if one could accurately correct for the actual path-length the electron would have travelled if one had used very small steps. Fig. 5.9 depicts the relationship between the total curved path of a step and its straight-line path in the direction of motion at the start of the step. For a given value of the curved path-length, t, the average straight-line path in the starting direction of motion, s, is given by Eq. 5.1 (which has been attribted to Lewis10 ), s = 1t dt'{cos0(t')) (5.1) 120 A. F. Bielajew and D. W. O. Rogers where 8(t') is the multiple-scattering angle as a function of the actual curved path• length along the path, t', and the average value, 0, is to be computed using the prob• ability distribution of any multiple-scattering theory. Several strategies have been de• veloped for calculating 8 using the Lewis equation. Yangll advocated an expansion of Eqn. 5.1 to second order in 8 and the use of a small-angle multiple-scattering theory to compute the average value. This is the strategy employed in the EGS code where the Fermi-Eyges multiple-scattering theory12 is used to compute the average value. (As mentioned previously, the multiple scattering in EGS is performed using Bethe's formulation of the Moliere theory.) Unfortunately, this approach has been shown to pro• duce path-length corrections, (t - 8)/8, a factor of 2 too high1s.14 . Berger15 advocated the relation, 1 8 = 2* + cos(8(t))], (5.2) and showed that 8 calculated using this equation agrees with that calculated using Eqn. 5.1 in the limit of small angle if the multiple-scattering theory of Goudsmit and Saunderson16.17 is used. Bielajew and Rogers14 expanded Eqn. 5.1 to 4th order in 8 and evaluated the average value using Bethe's version of Moliere's multiple-scattering theory. They showed that this approach and 8 calculated using Eqn. 5.2 agree, even for large average scattering angles of the order of a radian. The proof that the approach of Bielajewand Rogers is valid is given later in the chapter.

NO ESTEPE CONTROL

ESTEPE=10Y.

5Y.

2'!.

1Y.

Figure 5.8. A lO-MeV electron being transported through a 1-cm slab of water as simulated by EGS in its default configuration (no ESTEPE or SMAX control, note that the electron takes only one step to cross the water slab) and with an ESTEPE of 10, 5, 2, and 1%.

t p

s

Figure 5.9. A pictorial representation of the total curved path-length of an electron step, t, and the straight-line path-length, 8, in the direction of motion at the beginning of the step. The average lateral displacement of a transport step, p, is related to t. The displacements 8 and p are mutually orthogonal. 5. Electron Step-Size Artefacts and PRESTA 121

..... t=t I max '•....,'" g 13.2 ...., u s•CL> s• o U .c 10% ~ 13.1 ~------___ r:: .-CL> I .c...., 5% to 0..

113-1 Ie Kinetic energy (MeV) Figure 5.10. The path-length correction ir. water versus kinetic energy for various step-sizes as measured by ESTEPE The line t = t max shows the maximum step-size allowed by the Moliere theory.

The path-length correction can be quite large, as seen in Fig. 5.10, where the path-length correction, (t - 8)/8, in water is plotted versus electron kinetic energy for various step-sizes as measured by ESTEPE. If one wishes to reduce computing time by using large electron steps, then one must correct for path-length curvature. The larger the step, the greater the correction. Greater corrections are needed for lower energies as well. Recall that the path-length correction used by EGS is about a factor of 2 too high. This fact is almost entirely responsible for the step-size artefact seen in Fig. 5.1. Too much curvature correction resulted in too much energy being deposited in the first ro/2 slab and, by conservation of energy, too little in the second. In the "no PLC" case, the opposite prevailed. The failure to account for electron path-length curvature leads to less energy depostion in the upstream slab and too much in the downstream one. As the step-size is reduced, however, the electron tracks are modelled more and more correctly, relying on the simulation of multiple scattering for each of the steps for the development of curvature of the tracks. If one uses a correct path-length correction, such as that proposed by Berger15 or Bielajew and Rogers14, then most of the step-size artefact vanishes, as exhibited in Fig. 5.11. The residual step-size dependence has to do with other neglected features of electron transport that we have yet to discuss.

We now return to the ion chamber simulation and see what effect the use of a cor• rect path-length correction has. Fig. 5.12 shows the improvement of the ion chamber calculation, a reduction in the step-size dependence with use of a proper path-length cor• rection. Yet, there still remains a considerable dependence on ESTEPE. Some physics must be missing from this simulation! 122 A. F. Bielajew and D. W. O. Rogers

3.0

~ r.::l t:: 00 2.5 0 ~ r.::l ~

e:=~ 2.0 "PROPER" PLC r.::l Z r.::l

~ 1.5 § r.::le:= 1.0 0.00 0.05 0.10 0.15 0.20 ESTEPE Figure 5.11. The deep energy-deposition problem described for Fig. 5.1. In this case, a "proper" (i.e., demonstrably correct) path-length correction is used to eliminate most of the step-size dependence.

~ ION CHAMBER THEORY e:= 1.0 ~;;;:;;;:~;;:::::=---....::..::.::..:...... c===-=='::':"::"l o

...... o! 0.8 § :::> S 0.6

0.4 L..J..-'-L..J....L...J-l-.J.....1-L..J.....l....L.L..L-'-L..J.....J.....JL..J....L...J-l-J 0.00 0.05 0.10 0.15 0.20 0.25 ESTEPE Figure 5.12. The ion chamber calculation described for Fig. 5.7 with a reduction of the step-size dependence by the use of a "proper" path-length correction. A significant step-size dependence remains.

5.2.3 Lateral Displacement

Returning to Fig. 5.8, we see that as the step-size is made smaller and the electron histories are simulated with increasing accuracy, not only does the electron path acquire curvatttre, it is also deflected laterally. If one is faced with a simulation in which lateral displacement is important (for example, the air tube of Figs. 5.2 and 5.3), but one wishes to use as few electron steps as possible, one ought to account for the lateral displacement during the course of each electron step. If we use a sufficiently small step-size, this lateral displacement will occur naturally as the multiple-scattering angle selected for each electron step deflects the electron, accomplishing the lateral displacement. We saw that in the example of the air tube, if the step-size was restricted to be of the order of the diameter of the tube, the effects of lateral displacement were incorporated properly 5. Electron Step-Size Artefacts and PRESTA 123 in the simulation. Therefore, if one wishes to use fewer transport steps in a simulation of this nature, a more sophisticated approach is needed.

Fig. 5.9 illustrates the basic concept of the lateral displacement. An average lateral displacement, p, is associated with an electron transport step characterized by the total curved path of the step, t. Berger15 has provided a method that correlates p with t and the multiple-scattering angle, e, for the electron step,

p = ~t sin e(t). (5.3)

This is called the "lateral correlation" because the displacement, p, is correlated to the multiple-scattering angle15 . The proof that this prescription is valid will be given later. Fig. 5.13 shows that this correction is large for large step-sizes and small energies. We shall show, in another section, evidence of the reduction of step-size artefacts through the use of Berger's lateral-correlation algorithm, which we call "lateral displacement."

~ 0.6 Q.. ..., 25Y. ENERGY LOSS/STEP d Q.) S 20'1. Q.) Col -a

Q.) bD ...

5.2.4 Boundary Crossing

A general Monte Carlo method should be able to simulate electron trajectories in com• plex geometries. The condensed-history technique, whether the multiple scattering is simulated through the use of the theories of Fermi-Eyges, Moliere, or Goudsmit• Saunderson, is limited by the fundamental constraints of these theories. These theories are strictly applicable only in infinite or semi-infinite geometries. Some theories (e.g., Fermi-Eyges, Moliere) are applicable only for small average scattering angles as well. It would be far too complex to construct a multiple-scattering theory that applies for all useful geometries. In particular, how should a Monte Carlo electron-transport al• gorithm treat the approach and retreat from arbitrarily shaped boundaries, yet still not violate the basic constraints of the underlying theories? Unless multiple-scattering theories become much more sophisticated, there is only one solution - shorten the 124 A. F. Bielajew and D. W. O. Rogers electron steps in the vicinity of boundaries so that for a majority of electron steps in the simulation, any part of the total curved path is restricted to a single medium. In other words, the underlying theories rely upon the particle transport taking place in an infinite, or semi-infinite medium. Therefore, in the vicinity of a boundary, the electron step should be shortened enough so that the underlying theory is not violated, at least for most of the transport steps. The details of how this boundary crossing is accom• plished is very much code-dependent. However, the above "law" should apply for all condensed-history Monte Carlo methods. The details of how this can be accomplished with the EGS code will be given later.

5.3 PRESTA 5.3.1 The Elements of PRESTA

So far we have discussed electron step-size artefacts and how they can be circumvented by shortening the electron transport step-size. The occurances of artefacts were related to a shortcoming in, or the lack of, a path-length correction, the lack of lateral displace• ment during the course of an electron step, or the abuse of the basic constraints of the multiple-scattering theory in the vicinity of boundaries describing the geometry of the simulation. PRESTA, the farameter Reduced E,lectron-S,tep Transport Algorithml", attempts to address these shortcomings with the EGS code. The general features of PRESTA are applicable to all condensed-history codes. The fine details, only a few of which we shall discuss, are not. Before plunging ourselves into the features of PRESTA, we return to the examples dealt with earlier in the chapter and show how PRESTA han• dles the difficulties.

We have discussed the energy deposition in the middle of three 1'0/2 slabs due to 1- MeV electrons. Recall that in Fig. 5.1, we saw large step-size artefacts produced by the EGS code that could be "healed" by using short step-sizes. Later in Fig. 5.11, we cured most of the problem by using an accurate path-length correction. In Fig. 5.14, we also include lateral displacement and a careful boundary-crossing algorithm, (the remaining components of PRESTA), and all residual variation with step-size disappears.

3.0

A r..:I ~ 00 2.5 0 p., r..:I A

~ 2.0 ~ r..:I Z r..:I ~ 1.5 j FERMI-EYGES-YANG PLe fla 1.0 0.00 0.05 0.10 0.15 0.20 ESTEPE Figure 5.14. The energy deposition to the middle of three 1'0/2 water slabs due to I-MeV electrons. This simulation was discussed previously in Figs. 5.1 and 5.2. When PRESTA is used, all evidence of step-size dependence vanishes. 5. Electron Step-Size Artefacts and PRESTA 125

In the ion chamber simulation of Figs. 5.7 and 5.12, the improvement in the calculated ion-chamber response was quite dramatic but still incomplete. The evidence of step-size dependence was still quite strong. Once PRESTA is used for the simulation, however, the step-size artefact vanishes, as evidenced in Fig. 5.15. It is the inclusion of lateral displacement that is responsible for the remaining improvement in this case.

Finally, the improvement in the air-tube calculation of Fig. 5.3 is shown in Fig. 5.16. In this case, it is the boundary-crossing algorithm that is almost entirely respon• sible for the improvement.

1.2

PRESTA ~ 1.0 * 0 ::=rz:I Eo-< ...... z 0.8 "PROPER" PLC e::0 EGS :s:=> C,.) ~ 0.6

0.4 0.00 0.05 0.10 0.15 0.20 0.25 ESTEPE Figure 5.15. The ion chamber response calculation depicted already in Figs. 5.7 and 5.12. The use of PRESTA virtually eliminates any step-size depen• dence in this calculation.

I 2

PRESTA ~ 1.0 * 0 ::=rz:I Eo-< ...... "PROPER" PLC z 0.8 e::0 EGS :s:=> C,.) ....:I 0.6

0.4 0.00 0.05 0.10 0.15 0.20 0.25 ESTEPE Figure 5.16. The air-tube calculation of Fig. 5.3 is dramtically improved by the use of PRESTA. The label blcmin= 1.989 refers to a parameter that controls the boundary-crossing algorithm. This point is discussed later. 126 A. F. Bielajew and D. W. O. Rogers

Therefore, path-length correction, lateral displacement and a careful boundary• crossing algorithm are essential elements of a general purpose, accurate electron-transport algorithm. It remains to be proven in a more rigourous fashion that these components are physically valid in a more general context than the examples given. Otherwise, the improvements may be fortuitous. To do this requires a brief introduction to the Moliere theory, specifically on the limits on electron step-size demanded by this multiple• scattering formalism.

5.3.2 Constraints of the Moliere Theory

In this section, we briefly discuss the physical constraints of the Moliere multiple• scattering theory. Rather than present many mathematical formulae, we concentrate on graphical representations of the various limits. For further detail, the reader is en• couraged to examine Refs. 1 and 14 for the implementation of the Moliere theory in the EGS code. The original papers are enlightenings.4, and the exposition of Moliere's theory by Bethe5 is a true classic of the scientific literature.

The Moliere theory is constrained by the following limi~s: • The angular deflection is "small". (The Moliere theory is couched in a small• angle approximation.) Effectively, this constraint provides the upper limit on step-size. • The theory is a multiple-scattering theory; that is, many atomic collisions partic• ipate to cause the electron to be deflected. Effectively, this constraint provides the lower limit on step-size. • The theory applies only in infinite or semi-infinite homogeneous media. This con• straint provides the motivation for treating the electron transport very carefully in the vicinity of interfaces. • Energy loss is not built into the theory.

Bethe 5 carefully compared the multiple-scattering theories of Molieres.4 and of Goudsmit-Saunderson16•17. The latter theory does not resort to any small-angle ap• proximation. Bethe showed that the small-angle constraint of the Moliere theory can be expressed as an equation that yields the maximum step-sizel •14• Below this limit, the two theories are fundamentally the same. This upper limit is used by PRESTA. (The default EGS upper limit is actually about 0.8 of Bethe's limit.) Bethe's upper limit is plotted in Fig. 5.17 as the curve labelled t max• Also plotted in this figure is the csda range IS • We note that at larger energies, greater than about 3 MeV in water, the csda range is a more stringent restriction on electron step-size. This means that for .high energies, step-sizes can be quite large, up to the range of the electron. However, one must recall that the Moliere theory does not incorporate energy loss directly. Therefore, if we wish to approach the upper limit on step-size, we must treat the energy-loss part of the problem carefully. This topic will be discussed in a later section.

There is a critical parameter in the Moliere theory no that can be interpreted as the number of atoms that participate in the multiple scattering. Moliere considered his development to be valid for no ::::: 20. It has been found14 that sensible results can be obtained for no ::::: e. The lower limit, no = e, represents the "mathematical" limit below which Moliere's formalism breaks down mathematically. It is interesting that Moliere's theory can be "pushed" into the "few-scattering" regime and still produce reliable answers. We shall return to this point later. The minimum step-size, tmin, 5. Electron Step-Size Artefacts and PRESTA 127 obeying no = e is plotted versus electron kinetic energy in Fig. 5.17 for water. We see in this figure that the minimum and maximum step-sizes are the same at about 230 eV in water. Therefore, this represents the absolute minimum energy for which multiple scattering can be modelled using the Moliere theory. (In this energy region, atomic binding effects begin to play an increasingly important role requiring the use of more sophisticated low-energy theories.) As the energy increases, so does the range over which the Moliere theory is valid. The lower limit reaches an asymptotic bound at about 4 X 10-4 cm, while the upper limit continues upwards monotonically with increasing energy. Thus, for high energy, the applicable range in water extends from about 4 microns to the electron csda range .

..- S Co)

--I-< ...... ,Q) CI:1 10-2 ~ .....~ Q) 10-4 .....N rI.l I ~ ...... ,Q) 10-6 rI.l

10 kinetic energy (MeV) Figure 5.17. The minimum and maximum step-size limits of the Moliere theory, tmin and t max respectively. These limits are for water and the behavior for other materials can be obtained in Ref. 1 and Ref. 14. The dashed curve is the csda range18.

In a previous section, we discussed a type of artefact that can be problematic with the EGS code. That is, if one demands a step-size that is too short, EGS "turns off" the simulation of multiple scattering. We saw a dramatic example of this in Fig. 5.3. Fig. 5.IS compares tmin with step-sizes defined by various values of ESTEPE as calculated for water. Note that if one demands a step-size of 1% ESTEPE, then multiple scattering will not be simulated for electrons with energies less than about 40 keY. To circumvent this problem, PRESTA does not allow the ESTEPE restriction to reduce step-size below tmin·

The answer to the question, "Is the Moliere theory valid between these upper and lower limits?", is a complicated one. The benchmarking of PRESTA can be construed as a verification of the consistency of the Moliere theory. If the Moliere theory contained any intrinsic step-size dependence, then so would the results calculated using PRESTA, barring some highly fortuitous coincidences. In the next few subsections, we examine all the components of PRESTA, trying to omit unnecessary complications. 128 A. F. Bielajew and D. W. O. Rogers

__ 10-3 S --C,) ...... ,""'(1) ~ ~ .....l:::: 10-4 (1) .....N 00 I A...... ,(1) 00 10-5 10-1 kinetic energy (MeV)

Figure 5.18. Electron step-size is plotted versus kinetic energy for various values of ESTEPE and tmin' These curves apply for water. For other media, consult Refs. 1 and 15. If one demands a 0.1% ESTEPE in water, then multiple scattering cannot be modelled using the Moliere theory for electrons below about 500 keY.

5.3.3 PRESTA's Path-Length Correction

In Section 5.2.2, we discussed a new path-length correction. This method used the Lewis formula, Eqn. 5.1, expanded it to 4th order in e, and evaluated the mean values using the Moliere distribution functions14. We have seen impressive reductions in step• size dependences exhibited in Fig. 5.11 and Fig. 5.12. It now remains to prove that this path-length correction is valid in more general applications. To this end, we modify our electron-transport algorithm in the following fashion to conform with all the constraints of the Moliere theory: • Energy loss mechanisms are "switched off", including losses to "continuous" and "discrete" processes. • Bounding surfaces of all kinds are eliminated from the simulations. The transport takes place in an infinite medium. • The step-size constraints of the Moliere theory are obeyed.

We performed the following simulations: An electron was set in motion in a given direction, which defines the z-axis for the problem. A history was defined by having the total curved path, summed over all electron steps, exactly equal to the Moliere upper limit. This was achieved by choosing the step-size to be a divisor of t max. That is, one simulation was done with t = tmax, another with t = t max/2, another with t = tmax/3, ... tmax/ N, where N is an integer. The quantity "scored" was the average displacement along the z-axis, (Z)N, at the end of the history. The sum of the curved paths of the N steps always equals t max. We note that lateral displacements play no role in this simulation because they would average out to zero. If the path-length correction and the Moliere theory are both consistent, then the (Z)N'S should be independent of N, or equivalently, step-size independent. 5. Electron Step-Size Artefacts and PRESTA 129

We show two extreme cases in Figs. 5.19 and 5.20. The former, for 10-MeV elec• trons in water, plots {Z}N versus the inverse number of steps, liN. For contrast, the default path-length correction algorithm of EGS and simulations performed without a path-length correction are shown. Recall that there is no energy loss in these sim• ulations. As an indicator of scale, we have included a line indicating the step-size (measured in liN) equal to the csda range in water. We have seen before that at high energies, above 3 MeV in water, the Moliere upper limit exceeds the csda range. We have also included the ESTEPE=20% line, approximately the default EGS step-size in water. If one used the default EGS simulation, one would make path-length related errors of only a few.percent. The new path-length correction would allow the default upper limit on step-size in EGS to be extended upwards, allowing steps approaching the full csda range, without introducing artefacts! The new path-length correction thus shows a potential of speeding up high-energy simulations! Benchmarks have yet to be performed in this area.

12 1 I I /J. 10 MeV II - ,//N;';:~/ - / ...... ' ... / ...... ' " - .... ~-- - ..At"', .... .IJr" -fr: --Z A 9 - ..,. .... ,.. .. ,. ....: .... " NEW PLC - N ~ ts V N II i··,.················ ...... 8 ~ -6 ~ ~ EGS DEFAULT "PLC- IZl <..> 7 I I I I I 0.0 0.2 0.4 0.6 0.8 1.0 liN Figure 5.19. A test of the step-size dependence of the Moliere theory with the new path-length correction and with other path-length correction methods. This case is for 10-MeV electrons in water.

Fig. 5.20 depicts a similar set of simulations at 10 keV, three orders of magnitude less in energy than the previous example. The ESTEPE=20% line, near the default EGS step-size, is close to the Moliere upper limit~ Path-length corrections are very important here. We also show Moliere's lower limit, the 0 0 = 20 line. It was mentioned previously that Moliere's lower limit was found to be too conservative, and that sensible results could be expected for 0 0 ;?: e. This is shown in Fig. 5.20. The new path-length correction (or the Moliere theory) shows evidence of breakdown only in the vicinity of 0 0 = e. It is more likely, however, that this is a numerical problem as various functions, which become singular near this limit, are difficult to express numerically. Similar tests have been performed for other energies and materials. In all cases, the step-size independence of the path-length correction and the Moliere theory was demonstrated. 130 A. F. Bielajew and D. W. O. Rogers

1.5 I I I 1 1

10 keY ______x NO PLC ------.- 1j; ••••,. •• * ...... -...... ,...... S 1.0 Q -- .... , NEW PLC ..... ""6...... A..,. 1 ~ ""0 " ' .... II '"I>;;! """'...... (S) Z N --- 0.5 " ~ - I- ~ " rn I>;;! " ~ cf' cf' V EGS DEFAULT piC'" " " " " ""'" 0.0 I I 1 0.0 0.2 0.4 0.6 0.8 1.0 liN

Figure 5.20. A test of the step-size dependence of the Moliere theory with the new path-length correction and with other path-length correction methods. This case is for 10·keV electrons in water.

5.3.4 PRESTA's lateral-Displacement Algorithm

In Section 5.2.3, we discussed the importance of lateral displacement for each electron step in certain calculations. Berger's algorithm15, Eqn. 5.3, is used by PRESTA. To test this algorithm, we used a test very similar to that used to prove the viability of the path-length correction of the previous section. Again, we modified our electron• transport algorithm to conform with all the constraints of the Moliere theory. Energy• loss mechanisms were "switched off", all bounding surfaces were eliminated from the simulations to make it seem as if the transport took place in an infinite medium, and the step-size constraints of the Moliere theory were obeyed. We performed the follow• ing simulations: An electron was set in motion in a given direction, which defines the z-axis for the problem. As before, a history was defined by having the total curved path, summed over all electron steps, exactly equal to the Moliere upper limit. The quantity "scored" was the average displacement perpendicular to the z-axis, (r}N, at the end of the history. The sum of the curved paths of the N steps always was equal to t max• Path-length corrections played a minor role in the simulations because the geometric straight-line transport distances were somewhat dependent !lpon the amount of curvature correction applied to the electron steps. However, as shown in the pre• vious section, the path-length correction and the Moliere theory are both consistent. If the lateral-displacement algorithm is also consistent, then the (r) N'S should also be independent of N, or equivalently, step-size independent.

We show one representative case in Fig. 5.21 for 100-keV electrons in water, which depicts (r}N versus the inverse number of steps, 1/N. We also show two other calcula• tions of rN which do not include the lateral-displacement algorithm. One is the default EGS calculation with its default path-length correction, and the other has no path• length correction. The relatively small difference between these two curves indicates that this test depends only weakly upon the path-length correction used. (If the new path-length correction was used without a lateral-displacement algorithm, it would lie somewhere between these two curves.) A great reduction of step-size dependence in 5. Electron Step-Size Artefacts and PRESTA 131 this calculation is demonstrated. Only for the large step-sizes is there any evidence of deviation. This feature has been observed at all energies investigated14. However, we shall see in the next section that the remaining dependence is eliminated when energy loss is incorporated. The "ESTEPE=20%" line shows the approximate step-size used by EGS in its default configuration.

I I I I 1- o PRESTA o

2- -

0.2 0.4 0.6 0.8 1.0 liN Figure 5.21. Step-size independence test of the lateral-displacement algo• rithm for 100-keV electrons in water. We show a calculation of < r >N using the PRESTA algorithm. Also shown are two calculations without the lat• eral displacement algorithm, with and without the default EGS path-length correction. This test depends only weakly upon the path-length correction used.

5.3.5 Accounting for Energy Loss

The underlying Moliere theory does not treat energy loss directly. Actually, it is not too difficult to use the Moliere theory in a more general fashion and incorporate energy loss. One merely has to convert integral equations in the following fashion:

t lEO 1dt'!(t', E(t')) ===} dE'! (t'(E'), E') Ils(E')I, (5.4) o E, where !O is any function of the curved path-length, t, and the energy, E. The function, sO, is the stopping power. The familiar equation relating E and t directly is obtained by making the substitution, !O -t 1 in the above equation. However, such equations prove to be difficult to handle numerically and it is not really necessary. In all the formulae used in regards to multiple scattering and the various elements of PRESTA, an integration over t' is involved. It is then sufficiently accurate to make the approximation that the energy is constant if it is evaluated at the midpoint of the step. In more concrete terms, we approximate,

1t dt'! (t', E(t')) ~ 1t dt'!(t', E), (5.5) 132 A. F. Bielajew and D. W. O. Rogers where E = i[Eo + ts(E)]. Note that this latter equation for E is really an iterative equation, and it has been found that it is sufficient to evaluate it only to first order. That is, we make the approximation that E :=:::! HEo + ts(i[Eo + ts(Eo)])}. Some justification for this treatment can be obtained from the following relation,

(5.6) where E = (Eo + E, )/2, !:l.E = Eo - E" and J"(E) is the second derivative of J with respect to E. Thus, if !:l.E is not large with respect to E, and 1"0 is not too large, the approximation, I:=:::! !:l.EJ(E) is valid.

Further justification may be obtained by viewing the step-size independence of (Z}N and (r}N with energy loss incorporated by the above method i.e., evaluating all energy-related expressions at the mid-point of the step. The results are shown in Figs 5.22 and 5.23. . In each case, the step-size was chosen to be a fixed value of Moliere's upper limit. However, as the particle loses energy, this step-size changes owing to it's inherent energy dependence. In each case, the electron's endpoint energy, at which point the transport was terminated, was chosen to be 1 % of the starting energy. The only exception was 10 keY, where the endpoint energy was 1 keY. We note that both (Z}N and (r}N exhibit step-size independence. Even more remarkable is the fact that the minor step-size dependence exhibited by (r}N, shown in Fig. 5.21, has vanished. This improvement appears to be fortuitous, resulting from cancellations of second-order effects. More research is needed to study the theories concerning lateral displacements.

10 I I I I I

10 MeV (EKCUT=100 keY) 1.0 MeV (dO) (EKCUT=10 keY)"

- 100 keY (xl00) (EKCUT=l keY):

o 0 10 keY (xl000) (EKCUT=l keY) - =- I I I I I

0.0 0.2 0.4 0.6 0.8 1.0 1.2 liN

Figure 5.22. A similar test of the path-length correction as shown in Figs. 5.19 and 5.20 but with energy loss incorporated. Electron histories were terminated when the kinetic energy, EKCUT, reached 1% of the starting energy, except in the 10-keV case where it was 10%. 5. Electron Step-Size Artefacts and PRESTA 133

I 10 I I I 10 MeV (EKCUT=100 keY) =- -:

1.0 MeV (EKCUT=10 keY)

c- -: 100 keY (xlO) (EKCUT=l keY)

10 keY (x100) (EKCUT=l keY)

I I I I I 0.0 0.2 0.4 0.6 0.8 1.0 1.2 liN

Figure 5.23. A similar test of the lateral displacement as shown in Fig. 5.21 but with energy loss incorporated. Electron histories were terminated when the kinetic energy, EKCUT, reached 1% of the starting energy, except in the 10-keV case where it was 10%.

5.3.6 PRESTA's Boundary-Crossing Algorithm

The final element of PRESTA is the boundary-crossing algorithm. This part of the algo• rithm tries to resolve two irreconcilable facts: that electron transport must take place across boundaries of arbitrary shape and orientation, and that the Moliere multiple• scattering theory is invalid in this context.

If computing speed did not matter, the solution would be obvious-use as small a step-size as possible within the constraints of the theory. With this method, a great majority of the transport steps would take place far removed from boundaries, and the underlying theory would only be "abused" for that small minority of steps when the transport takes place in the direct vicinity of boundaries. This would also solve any problems associated with the omission of lateral displacement and path-length correction. However, with the inclusion of a reliable path-length correction and lateral• displacement algorithm, we have seen that we may simulate electron transport with very large steps in infinite media. For computing efficiency, we wish to use these large steps as often as possible.

Consider what happens as a particle approaches a boundary in the PRESTA algorithm. First, we interrogate the geometry routines of the transport code and find out the closest distance to any boundary. As well as any other restrictions on electron step-size, we restrict the electron step-size, (total, including path-length curvature) to the closest distance to any boundary. We choose to restrict the total step-size so that no part of the electron path could occur across any boundaries. We then transport the particle, apply path-length corrections, the lateral-displacement algorithm, and perform any "scoring" we wish to do. \Ve then repeat the process. 134 A. F. Bielajew and D. W. O. Rogers

At some point this process must stop, else we encounter a form of Xeno's paradox. We will never reach the boundary! We choose a minimum step-size which stops this sort of step-size truncation. We call this minimum step-size, t:run. If a particle's step-size is restricted to t:run' we are in the vicinity of a boundary. The particle mayor may not cross it. At this point, to avoid ambiguities, the lateral-displacement algorithm is switched off, whether or not the particle actually crosses the boundary. If we eventually cross the boundary, we transport the particle with the same sort of algorithm. We start with a step t:run. We then let the above algorithm take over. This process is illustrated in Fig. 5.24. This example is for a IO-MeV electron incident normally upon a I-cm slab of water. The first step is t:run in length. Since the position at this point is less than t:nin away from the boundary (owing to path curvature), the next step is length t:nin as well. The next 4 steps are approximately 2t:nin' 4t:nin' 8t:run' and I6t:nin in length, respectively. Finally, the electron begins to "see" the other boundary, shortens its steps accordingly. For example, the total curved path "a" in the figure is associated with the transport step "b". The distance "a" is the distance to the closest boundary.

PRESTA'S boundary crossing algorithm

Figure 5.24. Boundary-crossing algorithm example: A IO-MeV electron enters a I-cm slab of water from the left in the normal direction. The first step is t:run in length. Since the position here is less than t:run away from the boundary, the next step is length t:nin as well. The next 4 steps are approximately 2t:nin, 4t:nin , 8t:run, and I6t:run in length, respectively. Finally, the transport begins to be influenced by the other boundary, and the steps are shortened accordingly. The electron leaves the slab in 3 more steps.

Finally, what choice should be made for t:"in? One could choose t:"in = tmin, the minimum step-size constraint of the Moliere theory. Although this option is available to the PRESTA user, practice has shown it to be too conservative. Larger transport steps may be used in the vicinity of boundaries. The following choice, the default setting for t:"in, has been found to be be a good practical choice, allowing both accurate calculation and computing efficiency: Choose t:run to equal t max for the minimum energy electron in the problem (as set by transport cutoff limits). Then scale the energy-dependent parts of the equation for t:"in accordingly, for higher energy electrons. The reader is referred to Ref. 14 for the mathematical details. As an example, we return to the "air tube" calculation of Fig. 5.16. In that figure, the choice of "blcmin", the variable in PRESTA 5. Electron Step-Size Artefacts and PRESTA 135 which controls the boundary-crossing algorithm and which is closely related to t:nin, was set to l.989. This causes t:nin to be equal to t max for 2-keV electrons. A transport cutoff of 2 keV is appropriate in this simulation because electrons with this energy have a range which is a fraction of the diameter of the' tube. In most practical problems, if one chooses the transport cutoff realistically, PRESTA's default selection for t:nin produces accurate results. Again, the reader is referred to the PRESTA documentation14 for further discussion.

PRESTA, as the name implies, was designed to calculate quickly as well as accu• rately, since it wastes little time taking small transport steps in regions where it has no need to. There is no space to go into further discussion about this although there is a brief discussion in Chapter 24. Again, the reader is referred elsewherel4. Typical timing studies have shown that PRESTA, in its standard configuration, executes as quickly, and sometimes much more quickly, then EGS with ESTEPE set so as to produce ac• curate results. For problems with a fine mesh of boundaries, for example a depth-dose curve with a ro/40 mesh, the timing is about the same. For other problems with few boundaries, the gain in speed is about a factor of 5.

5.3.7 Caveat Emptor

It would leave the reader with a mistaken impression if the chapter was terminated at this point. PRESTA has demonstrated that step-size dependence of calculated results has been eliminated in many cases and that computing time can be econo~ized as well. By understanding the elements of condensed-history electron transport, some problems have been solved. Calculational techniques that isolate the effects of various constituents of the electron-transport algorithm have been developed and used to prove their step-size independence. However, PRESTA is not the final answer because it does not solve all step-size dependence problems, in particular, backscattering. This is demonstrated by the example shown in Fig. 5.25. In this example, l.O-MeV electrons were incident normally on a semi-infinite slab of water. The electron transport was performed in the csda approximation. That is, no 8-rays or bremsstrahlung ,'s were set in motion, and the unrestricted collision stopping power was used. The ratio of backscattered kinetic energy to incident kinetic energy was calculated. The default EGS calculation (with ESTEPE control) is shown to have a large step-size dependence. The PRESTA calculation is much improved but still exhibits some residual dependence on step-size.

In general, problems that depend strongly on backscatter will exhibit a step-size dependence, although the severity is much reduced when one uses PRESTA. We may speculate on the reason for the existence of the remaining step-size dependence. Recall that the path-length correction, which relates the straight-line path length, s, and t, the curved path-length of the transport step, really calculates only an average value. That is, given t, the value of s is predetermined and unique. It is really a distributed quantity and should be correlated to the multiple-scattering angle of the step. In other words, we expect the distribution to be peaked in the backward direction if E> = 7r and peaked in the forward direction if E> = O. To this date, distributions of this sort which are accurate for large-angle scattering are unknown. If they are discovered, they may cure PRESTA's remaining step-size dependence. 136 A. F. Bielajew and D. W. O. Rogers

DEFAULT EGS

1.0 MeV

0.05 0.10 0.15 0.20 ESTEPE Figure 5.25. Fractional energy backscattered from a semi-infinite slab of wa• ter with 1.0-MeV electrons incident normally. The electron transport was performed in the csda approximation. (No a-rays or 'r's were set in motion). The default EGS and PRESTA calculations are contrasted. There is still evidence of step-size dependence in the PRESTA calculation. 5. Electron Step-Size Artefacts and PRESTA 137

REFERENCES 1. W. R. Nelson, H. Hirayama and D. W. O. Rogers, "The EGS4 Code System", Stanford Linear Accelerator Report SLAC-265 (1985). 2. D. W. O. Rogers, "Low Energy Electron Transport With EGS", NucI. Instr. Meth. A227 (1984) 535. 3. G. Z. Moliere, "Theorie der Streuung schneller geladener Teilchen I: Einzelstreu• ung am abgeschirmten Coulomb-Feld", Z. Naturforsch. 2a (1947) 133. 4. G. Z. Moliere, "Theorie der Streuung schneller geladener Teilchen II Mehrfach• und Vielfachstreuung", Z. Naturforsch. 3a (1948) 78. 5. H. A. Bethe, "Molire's Theory of Multiple Scattering", Phys. Rev. 89 (1953) 1256. 6. L. V. Spencer and F. H. Attix, "A Theory of Cavity Ionization", Radiat. Res. 3 (1955) 239. 7. A. F. Bielajew, D. W. O. Rogers, A. E. Nahum, "The Monte Carlo Simulation of Ion Chamber Response to 6OCo - Resolution of Anomalies Associated with Interfaces", Phys. Med. BioI. 30 (1985) 419. 8. D. W. O. Rogers, A. F. Bielajew, A. E. Nahum, "Ion Chamber Response and Awall Correction Factors in a 60Co Beam by Monte Carlo Simulation", Phys. Med. BioI. 30 (1985) 429. 9. U. Fano, "Note on the Bragg-Gray Cavity Principle for Measuring Energy Dis• sipation", Radiat. Res. 1 (1954) 237. 10. II. W. Lewis, "Multiple Scattering in an Infinite Medium", Phys. Rev. 78 (1950) 526. 11. C. M. Yang, "Actual Path Length of Electrons in Foils", Phys. Rev. 84 (1953) 599. 12. L. Eyges, "Multiple Scattering with Energy Loss", Phys. Rev. 74 (1948) 1534. 13. D. F. Hebbard and P. R. Wilson, "The Effect of Multiple Scattering on Electron Energy Loss Distributions", Australian J. Phys. 8 (1955) 90. 14. A. F. Bielajew and D. W. O. Rogers, "PRESTA - The "Parameter Reduced Electron-Step Transport Algorithm" for Electron Monte Carlo Transport", Na• tional Research Council of Canada report No. PIRS-0042j and NucI. Instr. Meth. B18 (1987) 165. 15. M. J. Berger, Methods in Computational Physics, Vol. I, (Academic Press, New York, 1963) p.135. 16. S. Goudsmit and J. L. Saunderson, "Multiple Scattering of Electrons", Phys. Rev. 57 (1940) 24. 17. S. Goudsmit and J. L. Saunderson, "Multiple Scattering of Electrons II", Phys. Rev. 36 (1940) 36. 18. M. J. Berger and S. M. Seltzer, "Stopping Powers and Ranges of Electrons and Positrons", Report NBSIR 82-2250-A (Washington DC: U.S. Dept. of Com• merce) (1983). 6. 20-MeV Electrons on a Slab of Water

David W. O. Rogers and Alex F. Bielajew

Division of Physics National Research Council of Canada Ottawa, Canada KIA OR6

6.1 INTRODUCTION This chapter is meant to review many of the aspects of electron transport calculations by means of a simple example. The particular example is that of 20-MeV electrons incident on a slab of water. This is a case of considerable importance in radiotherapy physics, and thus it has been discussed often in the literaturel - 5 • The results presented here are based on the EGS4 code, but the results obtained with other codes are similar except in those cases which are explicitly discussed.

6.2 A THIN SLAB 6.2.1 The CSDA Calculation Let us start with the simple case of 20-MeV electrons incident on a slab of water which is 0.25-cm thick. This slab is sufficiently thin that most of the electrons are not deflected much and hence have a pathlength of 0.25 cm in the slab. This means that their average energy loss can be calculated as 0.25 cm times the total unrestricted stopping power for 20-MeV electrons on water, i.e., they lose 618 keY (3.1% of their energy) while passing through the slab. If we consider a continuous-slowing-down approximation model (csda) in which the electrons generate no secondary electrons or photons and all their energy is lost by continuous processes, then all the electrons will lose the same amount of energy. However, these electrons will undergo multiple scattering and be deflected somewhat. Fig. 6.1 shows that very few electrons are deflected by more than 10 degrees (for which the pathlength in the slab is increased by at most 1.5%), although one must note that the figure is per steradian and there is very little solid angle near zero degrees.

6.2.2 More Realistic Calculations

Let us now consider a more realistic calculation which includes the creation of secondary knock-on electrons and bremsstrahlung photons. In the EGS system6 , the creation of these particles causes the primary electron to lose energy and, in the case of knock-ons, to be deflected. Fig. 6.1 also shows the angular distributions obtained from a calculation

139 140 D. W. O. Rogers and A. F. Bielajew which considered the deflections caused by creating electrons with energies of 10 keY and above. The effect of the creation of secondary electrons on the angular distribution of the primaries (i.e., the difference between the histogram and the diamonds) can be seen to be very small, even at the very large angles*.

histogram - CSDA

stars - full carn

diamonds - no secondaries

10 angle / degree

Figure 6.1. Angular distribution of 20-MeV electrons after passing through a 0.25-cm thick slab of water. The histogram shows a csda calculation. The symbols are for full calculations; the stars include secondaries, and the dia• monds are just the primary electrons (ECUT = 189 keY (kinetic energy)).

The effects of the creation of secondary particles on the energy lost by the primary electrons is considerably more important. This energy-loss straggling can be handled in two quite different manners, depending on whether, in Berger's terminology8, a Class I or Class II electron-transport algorithm is being used. In Class I algorithms, the energy of the primary electron is not directly affected by the creation of a secondary particle, whereas in a Class II model, the energy of the primary is decreased by the energy of each secondary it creates. Let us first consider Class II algorithms since this is what is used in EGS (ETRAN uses a Class I algorithm for the creation of knock-on electrons). Further, let us consider a model which explicitly considers only the creation of secondary photons. At 20 MeV, 17% of an electron's energy loss is via radiative processes, i.e., the creation of bremsstrahlung photons, and of this, roughly 99% is from creating photons with energies above 200 keY, and well over 80% is from creating photons with energies over 2 MeV. Each time one of these photons is created, the primary electron loses an amount of energy equal to the photon energy. This leads to a distribution of electron energies leaving the water slab. This distribution ranges between the low-energy limit for electron transport (called ECUT in EGS and ETRAN) and the energy of an electron which has gone through the slab without creating any secondaries. This upper energy is given by the product of the pathlength in the slab and the restricted stopping power• i.e., the stopping power which considers those radiative processes giving rise to photons

* There are situations in which the scattering by atomic electrons can influence the overall angular distribution (e.g., low-energy electrons on low-Z elements) and in this case, EGS has a slight problem because it already includes the effects of atomic electrons in its multiple-scattering formalism, and thus does some double counting (see Rogers7and Chapter 14). 6. 20-MeV Electrons on a Slab of Water 141 below some threshold (in EGS this threshold is called AP) plus the collision stopping power. If only radiative processes give rise to discrete losses, then most electrons have this upper energy because they do not create secondary photons. The electrons with energies below that of the peak are those which have lost additional energy by creating a secondary photon. The difference between the energy of the electron and the peak energy is just the energy of the bremsstrahlung photon. Thus there is an energy gap below the maximum energy. The width of this gap is equal to the energy of the lowest photon energy which can be created by an electron, i.e., AP. Fig. 6.2 shows a calculation in which AP=lOO keY, and hence there is a 100-keV gap below the peak. The electron energy in the csda calculation is lower because in this case all the radiative losses are accounted for in the continuous energy-loss part of the calculation. Note however that the mean energy lost by electrons in each case is the same.

1':1 0 b 0:.> 4l'" ...., 1':1 ..,'" 10-' stars - create photons> 100 keY '8 .S ""- diamond - CSDA ;>- 10-2 ..!04'" 0 lC

""- 4l'"

10-·

17.0 17.5 18.0 18.5 19.0 19.5 20.0 electron energy / MeV

Figure 6.2. Energy distribution of 20-MeV electrons after passing through a 0.25-cm slab of water. The calculation shown with the stars explicitly sim• ulated the creation of all bremsstrahlung photons over 100 keY, and treated all collisional energy losses on a csda basis. The calculation shown by the diamond was a true csda calculation, hence all electrons had the same energy. In both cases, the mean energy-loss was 618 keY.

If we now explicitly simulate the creation of knock-on electrons as well as brem• sstrahlung photons, we find that: i) there are more electrons in the tail of the energy distribution, starting at a threshold below the peak corresponding to the minimum• energy knock-on electron being simulated; ii) there are fewer electrons in the peak; and iii) the peak is shifted to higher energy because the restricted stopping power becomes smaller. The electron spectrum below about 10 MeV will not change because, by def• inition, an electron cannot lose more than 1/2 of its energy when creating a knock-on electron, and hence all electrons below this energy must have created bremsstrahlung photons. The exact shape of the energy distribution will depend very much on the threshold energy for production of knock-on electrons (called AE in EGS). As the value of AE is reduced, the distribution becomes more and more realistic, as we can see in Fig. 6.3 where all secondaries with energies above 1 keY have been simulated. This 142 D. W. O. Rogers and A. F. Bielajew takes a very large amount of computing time, and hence other techniques have been developed for calculating energy-loss straggling distributions. As has been discussed in the chapters on ETRAN and ITS, these codes all use the Blunck-Leisegang modi• fication of Landau's energy-loss straggling theory. Figure 6.3 shows that there is very good agreement between the energy-loss straggling distributions calculated with the two techniques except in the energy region from 10 MeV to 18 MeV where there is a small difference which is the result of numerical accuracy problems9 (see a discussion of its correction in Chapter 7 by Seltzer).

stars: EGS4 histogram.: CYLTRAN using L(BL)

<-----.

-----> •

5 9 13 17 18 19 20 electron energy / MeV

Figure 6.3. Calculated energy-loss straggling distributions for 20-MeV elec• trons after passing through a 0.25-cm slab of water. The stars represent results calculated with EGS in which all secondaries with energies above 1 keY were explicitly simulated. The histogram presents results calculated with CYLTRAN (ETRAN) which explicitly simulates all bremsstrahlung pho• tons, but simulates the energy-loss straggling due to knock-on electrons by us• ing the Blunck-Leisegang modification of Landau's theory (from Rogers and Bielajew9 ).

Fortunately, there are many situations in which the details of energy-loss strag• gling are not critical because it is the straggling caused by the creation of high-energy secondaries that affects most transport problems. Hence a Class II algorithm will work quite well with reasonably high values for the thresholds for producing secondaries. For example, EGS will calculate accurately a depth-dose curve for 20-MeV electrons on water when simulating all knock-ons above 500 keY, whereas using the same threshold, the distribution of energies behind a 0.25-cm slab of water would bear little relationship to reality.

One final point before leaving the subject of the thin slab. For radiative processes, we saw that by considering the creation of photons above 200 keY, we were thereby explicitly simulating the radiative energy-loss events accounting for 99% of the radiative energy loss. However, for collisional-energy losses, a much lower fraction of the energy loss is from the creation of high-energy secondaries. For example, at 20 MeV, only 6. 20-MeV Electrons on a Slab of Water 143

40% of an electron's collisional energy loss in water comes from the creation of secon• daries with energies greater than 1 keY, and less than 10% goes to knock-ons above 2 MeV. This is one reason that the condensed-history technique is essential for electron transport calculations.

6.3 A THICK SLAB 6.3.1 Typical Histories

We now turn to 20-MeV electrons incident on a thick slab of water. In Fig. 6.4, several typical histories are shown. In history 1, three bremsstrahlung photons were created above 10 keY, two of which were totally absorbed in the slab. There were also 12 knock-on electrons created above the 189 keY threshold for electron production, only two of which were above the 500 keY threshold for electron transport. In history 2, a high-energy knock-on electron was created, and both it and the primary created bremsstrahlung photons after scattering to large angles due to multiple scattering. Note that the creation of the knock-on did not deflect the primary significantly.

,I ______6'----...... -=r

~ 5 o .,-.----._-_=r•

:ern -' r------_L o 3 Mf" ...... -----'....- 2

depth Figure 6.4. Six typical histories for 20-MeV electrons incident on a thick slab of water. The slab is 14-cm thick. The creation or termination of a particle is shown as a point. The simulation included all knock-on electrons created above 189 keY kinetic energy and bremsstrahlung photons above 10 keY, al• though electron histories were terminated at 500 keY. Solid lines show electron histories; dashed lines are photons. ESTEPE=3% was used.

These not-unusual histories illustrate a variety of important points. The first is that most electron histories generate at least one bremsstrahlung photon, although most of these photons have a very low energy which can be inferred from the frequent collisions they undergo. The second point is that the average angle of the electrons increases with depth, mostly due to multiple scattering. Finally, it is worth noting that the creation of the knock-on electrons has little apparent effect on the direction of the primaries, but does have an effect on the energy of the primary, as we saw in the previous section. 144 D. W. O. Rogers and A. F. Bielajew

6.3.2 Depth-Dose Curves

In this section, we will study the effects on the depth-dose curve of turning on and off various physical processes. Fig. 6.5 presents two csda calculations (i.e., no secondaries are created and energy-loss straggling is not taken into account). For the histogram, no multiple scattering is modelled, and hence there is a large peak at the end of the range of the particles because they all reach the same depth before being terminated and depositing their residual kinetic energy (189 keY in this case). Note that the size of this peak is very much a calculational artefact which depends on how thick the layer is in which the histories terminate. The curve with the stars includes the effect of multiple scattering. As we saw in Fig. 6.4, this leads to a lateral spreading of the electrons which thus shortens the depth of penetration of most electrons, and to an increase in the dose at shallower depths because the fluence has increased. In this case, the depth straggling is caused entirely by the lateral scattering since every electron has travelled the same distance.

"'s 5 (.J I no multiple scatter I>. ... t!l - 4 '0-

with multiple scatter--->

OL--L__ ~ __L--L __ ~ __L-~ __-L __ ~~ __-L~

0.0 0.2 0.4 O.B O.B 1.0 1.2 depth / ro

Figure 6.5. Depth-dose curve for a broad parallel beam of 20-MeV electrons incident on a water slab. The histogram represents a csda calculation in which multiple scattering has been turned off, and the stars show a csda calculation which includes multiple scattering (ECUT = 189 keY (kinetic energy)) (from Roger and BieJajew10). The depth scale is in terms of ro, the csda range for 20-MeV electrons (ro = 9.3 cm).

Fig. 6.6 presents three depth-dose curves calculated with all multiple scattering turned off-i.e., the electrons travelin straight lines (except for some minor deflections when secondary electrons are created). In the cases including energy-loss straggling, a depth straggling is introduced because the actual distance travelled by the electrons varies, depending on how much energy they give up to secondaries. Two features are worth noting. Firstly, the energy-loss straggling induced by the creation of bremsstrah• lung photons plays a significant role despite the fact that far fewer secondary photons are produced than electrons-however they have a larger mean energy. Secondly, the 6. 20-MeV Electrons on a Slab of Water 145 inclusion of secondary electron transport in the calculation leads to a dose buildup re• gion near the surface. Fig. 6.7 presents a combination of the effects in the previous two figures. The extremes of no energy-loss straggling and the full simulation are shown to bracket the results in which energy-loss straggling from either the creation of bremsstr• ahlung or knock-on electrons is included. The bremsstrahlung straggling has more of an effect, especially near the peak of the depth-dose curve.

0.0 0.2 0.4 0.6 0.8 1.0 1.2 depth / :r;,

Figure 6.6. Depth-dose curves for a broad parallel beam of 20-MeV electrons incident on a water slab, but with multiple scattering turned off. The solid histogram calculation includes no straggling, and is the same simulation as given by the histogram in Fig. 6.5. Note the difference caused by the different bin size. The dashed histogram includes energy-loss straggling due to the creation of bremsstrahlung photons with an energy above 10 keY. The curve denoted by the stars includes only that energy-loss straggling induced by the creation of knock-on electrons with an energy above 10 keY (from Rogers and BielajewlO).

6.3.3 Fluence vs Depth

In Fig. 6.4, we saw that multiple scattering caused the mean angle of the electrons relative to the z-axis to increase with depth. This leads to an increase in the particle fluence which is given by the total pathlength per unit volume, or alternatively when scoring the fluence at a plane, by summing the particles crossing the plane weighted by llcos B, where B is the angle of the particle's trajectory with respect to the normal to the plane. It is this increase in the fluence which is responsible for the peak in the depth-dose curve (see Figs. 6.5 and 6.7). The differences in these curves, especially for the total fluence, imply that one must define the lower energy cutoff when thefluence is being given since it is a function of this parameter. It is also clear that the number of photons is very large although the dose delivered by these photons is only a small fraction of the total, as can been seen from the bremsstrahlung tail portion of the full calculation curve in Fig. 6.7. The photon fluence builds up with depth both because the electrons continue to create them, and also because of photon scatter. Once past the electron range and after full photon scatter has been achieved, the fluence drops off due to normal photon attenuation. 146 D. W. O. Rogers and A. F. Bielajew

...... ~ 3 - full cal'n =I\) ;;:::= + - no straggle ~2.., .g diamonds - knock-on straggle '-c::I I\) 1: 1 triangles - brem straggle ..,o ..c

'" 0L--L~L-~~--~~--~~--~~~4F~ 0.0 0.2 0.4 0.6 0.8 1.0 1.2 depth / ro

Figure 6.7. Broad parallel beam of 20-MeV electrons on water with multiple scattering included in all cases, and various amounts of energy-loss straggling included by turning on the creation of secondary photons and electrons above a 10-keV threshold (from Rogers and Bielajew10).

I\) 1.5 5 I\) c.> c.> =I\) =I\) ;;:::= ;;:::= .. 4 .. =I\) =I\) '-c::I 1.0 .'-c::I ....c.> ...c.> ....= 3 ....= ...... I\) I\) c.> c.> 2 =I\) =I\) ;;:::= 0.5 ;;:::= photons ----> =0 1 =0 b ..0 c.> I\) ..d Q" -I\) 0.0 0 0 2 4 6 8 10 12 depth / em Figure 6.S. Fluence vs depth for a broad parallel beam of 20-MeV electrons on water. The histograms (all of which coincide at depths less than 7 cm) show the fluence of primaries for ECUT = 10,200 and 500 keY (kinetic energy). Only at depths past 7 cm do the higher cutoffs lead to a reduction in the primary fluence. The symbols show the total fluence for various ECUT values ( +, 500 keY; 0,200 keY; stars, 10 keY). The smooth curve shows the total photon fluence (right axis). For all calculations, AP=AE=10 keY (kinetic energy) (from Rogers and Bielajew10). 6. 20-MeV Electrons on a Slab of Water 147

The average energy of the photons increases from 175 keY in the first 1 mm, to 300 keY in the next 4 mm bin, to 525 keY between 10 and 15 mm, to nearly 700 kev between 6 and 7 cm, and up to nearly 1 MeV at the back of the 13 cm thick slab. Fig. 6.9 presents the photon spectra in two depth intervals with average energies of 375 and 675 keY. At greater depths, lower energy electrons are creating new photons, and hence the average energy of these newly created photons decreases. However, the observed increase in the average photon energy is due to the well-known beam hardening effect in which the low-energy photons are more highly attenuated. At the same time, the relative number of high-energy photons at the greater depth has decreased because they can no longer be created (the most probable electron energy at that depth is about 7 MeV).

d 0 ----, .!:I '--- c.> '--:_. 6-7 em depth Ql'" '-'-, OJ ,"I .., ~. I_!._., d 0-1 ~'" '---C_--:--- _ _, c.>' l LL .9 .. '--!.-L __: '"j:I., 10-2 :>- '--L ::::il'" 1 ...... til ._\ d ..,0 0 ..d j:I.,

10-'" 10-' 1 10 photon energy / MeV

Figure 6.9. Photon fluence spectra at depths of 0-1 cm and 6-7 cm in a water phantom struck by a 20-MeV electron beam. The average energies of the spectra are 375 and 675 keY.

Fig. 6.10 presents electron fluence spectra at three depths in the water phantom. The basic shape of these spectra reflects the energy-loss straggling of the beam as dis• cussed in section 6.2.2. An important fact to bear in mind is that there is a considerable difference in the most probable and mean energies of the electron spectra at any given depth. For example, the mean energy of the primary spectrum between 3 and 3.5 cm depth is 12.05 MeV, whereas the most probable energy is 14.1 MeV, and the average energy of the total electron spectrum is only 10;4 MeV (recall that these figures apply to those electrons above 10 keVonly).

6.4 Conclusions

While the results of this chapter are for 20-MeV electrons incident on water, the physical processes described playa role in a wide variety of situations, although often in different proportions. 148 D. W. O. Rogers and A. F. Bielajew

Q.) sur ace--> CJ solid - total d Q.) ;::j so:: dashed - primaries -' d Q.) ""'<3 .s 10-1 .;:;-' ;::j ...... t> Q.) 10-2 ...... :::Ii!! Q.) CJ d Q.) ;::j <-- surface so::

10-3

1 10 electron energy / MeV Figure 6.10. Electron fluence spectra at three depths for a broad beam of 20-MeV electrons incident on a water phantom. The surface spectrum is from 0-1 mm depth. The solid lines show the spectrum including secondaries, and the dashed spectra are for the primary electrons only. ECUT = 10 keY (kinetic energy) (from Rogers and BielajewlO). 6. 20-MeV Electrons on a Slab of Water 149 REFERENCES 1. M. J. Berger and S. M. Seltzer, "Calculation of Energy and Charge Deposition and of the Electron Flux in a Water Medium Bombarded with 20 MeV Elec• trons", Ann. N.Y. Acad. Sci. 161 (1969) 8. 2. P. Andreo and A. Brahme, "Mean Energy in Electron Beams", Med. Phys. 8 (1981) 682. 3. P. Andreo and A. Brahme, "Restricted Energy-Loss Straggling and Multiple Scattering of Electrons in Mixed Monte Carlo Procedures", Radiat. Res. 100 (1984) 16. 4. A. E. Nahum, "Calculations of Electron Flux Spectra in Water Irradiated with Megavoltage Electron and Photon Beams with Applications to Dosimetry", Ph.D. Thesis, University of Edinburgh (1975). 5. K. R. Shortt, C. K. Ross, A. F. Bielajew and D. W. O. Rogers, "Electron Beam Dose Distributions Near Standard Inhomogeneities", Phys. Med. BioI. 31 (1986) 235. 6. W. R. Nelson, H. Hirayama and D. W. O. Rogers, "The EGS4 Code System", Stanford Linear Accelerator report SLAC-265 (1985). 7. D. W. O. Rogers, "Low Energy Electron Transport with EGS", Nucl. Instr. Meth. A227 (1984) 535. 8. M. J. Berger, "Monte Carlo Calculation of the Penetration and Diffusion of Fast Charged Particles" in Methods in Computational Physics, Vol. I, edited by B. Adler, S. Fernbach and M. Rotenberg, (Academic Press, New York, 1963); 135. 9. D. W. O. Rogers and A. F. Bielajew, "Differences in Electron Depth-Dose Curves Calculated with EGS and ETRAN and Improved Energy-Range Relationships", Med. Phys. 13 (1986) 687. 10. D. W. O. Rogers and A. F. Bielajew, "Monte Carlo Techniques of Electron and Photon Transport for Radiation Dosimetry" in The Dosimetry of Ionizing Radiation, Vol III, edited by K. R. Kase, B. E. Bjarngard and F. H. Attix, (to be published by Academic Press in 1988). The ETRAN System 7. An Overview of ETRAN Monte Carlo Methods

Stephen M. Seltzer

Center for Radiation Research National Bureau of Standards Gaithersburg, Maryland 20899, U.S.A.

7.1 INTRODUCTION

A series of Monte Carlo codes for the calculation of the transport of electrons and pho• tons through extended media has been developed at the National Bureau of Standards over the past 25 years. These codes have been named ETRAN (for Electron TRANs• port), with the various versions representing mainly refinements, embellishments and different geometrical treatments that share the same basic algorithms for simulating by random sampling the course of electrons and photons as they travel through matter. These algorithms, which taken together have been called the ETRAN model, form the basis also of codes written at other laboratories, such as Sandia's older SANDYL code and their more current series of the TIGER, CYLTRAN, and ACCEPT codes described elsewhere in this volume. In this chapter, the ETRAN methods are described as they are currently being implemented in our codes at NBS.

The ETRAN model pertains, strictly speaking, only to the schematization of the electron random walk that is used in the calculation, and not to the particular choices for sources of cross-section information or for sampling methods. The methods used to generate electron trajectories go back to a paper published in 19631 , and involve the sampling from relevant multiple-scattering distributions. Such a scheme is necessary in order to make the problem tractable at higher energies, but is no longer accurate at low energies where the number of collisions suffered by the electron is too small for valid treatment by multiple-scattering theories (see Chapter 2). This limitation should be kept in mind when using these methods for electron energies below, say, 10-20 keY in low-Z, 50-100 keY in medium-Z, and 100-200 keY in high-Z materials.

The ETRAN code was originally developed as a tool for solving electron-transport problems involving energies up to a few MeV. Later, the production and propagation of secondary bremsstrahlung was added to extend the calculation to higher energies. The result is a calculation which takes into account primary electrons, positrons or photons, and all secondary radiations, including knock-on electrons from electron• impact ionization events, electron bremsstrahlung, Compton electrons, photoelectrons, electron-positron pairs, annihilation radiation, and K-shell characteristic x-rays and

153 154 S. M. Seltzer

Auger electrons resulting from electron and photon ionization events. The code fol• lows all generations of electrons and photons with energies up to 1 GeV (and easily extendable to higher energies) and down to 1 keVin any target material.

In Section 7.2, the ETRAN Monte Carlo methods are briefly described, with some details included on the cross sections and techniques used for electron transport. Section 7.3 outlines the structure of current ETRAN codes, and indicates the various run options and output quantities available. Future improvements are discussed in Section 7.4.

7.2 MONTE CARLO METHODS 7.2.1 Photon Transport

Sampling procedures. Successive photon interactions are sampled individually in direct analogy to the phys• ical process. Compton scattering, photoelectric absorption and pair production are included. The cross sections for these interactions are taken from the work of Hubbe1l2 , and have been organized into a databaseS with which the necessary interpolation and summation can rapidly be performed for any material composed of elements with atomic number Z from 1 to 100. The input cross sections are further expanded by interpola• tion and stored for a judiciously chosen fine grid of photon energies to facilitate a rapid, nearest-energy, table look-up scheme in the photon Monte Carlo sampling.

The photon Monte Carlo procedure is based on conventional techniques". The distance to the next photon interaction is sampled from an exponential distribution governed by the attenuation coefficient; the change in attenuation coefficient as a photon crosses material boundaries is taken into account. The type of interaction is then sampled from the appropriate relative probabilities. The history of each photon is continued from collision to collision until the photon either is absorbed, escapes the target, or its energy falls below a chosen cutoff value.

The occurrence of a Compton interaction is sampled on the basis of total incoherent• scattering cross sections which include binding effects through the use of incoherent• scattering functions. The energy and direction of the photon after a Compton scattering is sampled from the Klein-Nishina cross section5, and the initial energy and direc• tion of the Compton electron is obtained from conservation of energy and momentum. Although the total cross section is normalized to the correct value, the shape of the Compton angular distribution thus pertains to the interaction with a free electron, and ignores low-energy effects due to the binding and pre-interaction motion of the atomic electrons. The effect on some photon transport results of the neglect of binding has been investigated by Williamson et al 6 who suggest that form-factor binding corrections be included in problems involving primary photons with energies below a few hundred keY. The relatively infrequent inner-shell ionizations due to Compton scattering are ignored, and no subsequent emission of characteristic x-rays or Auger electrons is sampled.

In the case of pair production in the field of the nucleus, the distribution for the sharing of available kinetic energy between the electron and positron is taken from Bethe-Heitler results given by Bethe and Ashkin7; and the angles of the produced pair are sampled from a distribution based on the leading term, (1 - f3 cos (J)-2, of high• energy theory, where f3 is the ratio of the electron velocity to the speed of light. The 7. An Overview of ETRAN Monte Carlo Methods 155 same distributions are used in the case of pair production in the field of electrons (triplet production)* .

The energy of electrons produced in photoelectric-absorption events is given simply by the difference between the energy of the photon and the binding energy of the atomic shell involved. The angle with which the photoelectron is produced is sampled from the Fischer distributionS for electron energies below 50 keY and from the Sauter distribution9 for higher energies. The filling of the atomic-shell vacancy left by the ejected photoelectron is partially accounted for by sampling the emission of either a characteristic x-ray or an Auger electron from that shell, with a relative probability given by the fluorescence efficiency. The x-ray or Auger-electron energy is sampled from the appropriate emission spectra. The complete relaxation of the ionized atom (i. e., the decay of successively higher shells by the emission of successively lower energy x-rays and/or electrons) is not presently implemented; the codes are usually run taking into account the K shell only.

On the basis of energy cutoff values and other chosen options, the energy, spatial, and directional coordinates of all secondary electrons with energies greater than a cho• sen cutoff value, are stored and used to initialize later electron Monte Carlo histories. An option is available to select only a specified fraction of the photon-descendant sec• ondary electron histories to be followed. An appropriately large statistical weight is then assigned to these secondary electron histories. Although this procedure will tend to increase the statistical fluctuations in the secondary electron scores, it has been found desirable to reduce the number of time-consuming electron histories in some problems involving large numbers of photon histories.

Angular deflections associated with coherent scattering are presently neglected. This is justified to some extent on the basis of the following argument. As a fraction of the total interaction cross section, the contribution of coherent scattering becomes only as large as the order of 10%, and this occurs at energies (in the vicinity of 100 keY) at which the coherent scattering tends to be concentrated at forward angles, thus mimicking no scattering. At higher and lower energies, the cross section for coherent scattering becomes even smaller in relation to the total interaction cross section. It has been confirmed, for example, that in the calculation of the response of Ge detectors to photons with energies from 10 to 300 keY, the neglect of coherent scattering has no discernable effect 10. However, there are certain low-energy photon problems, such as albedo calculations, where coherent scattering can be the dominant process that leads to the result of interest and must be taken into account.

Scoring. The statistical accuracy of the results for photon internal tracklength distributions (fluence spectra) and for emergent distributions (transmission, reflection, etc.) is im• proved by routinely scoring at the beginning of each free path in the photon history the probability that the photon will cross the scoring boundaries of interest without further

* Because in triplet production the kinetic energy of the recoil electron tends to be concentrated at very low values, there should be little recognizable difference between pair production in the field of an electron and in the field of the nucleus7• The transport of the low-energy recoil electron is ignored. 156 S. M. Seltzer interaction. This technique greatly increases the number of scores per photon history, with only moderate additional computational effort.

7.2.2 Electron Transport

The many individual elastic and inelastic Coulomb interactions that occur during the passage of a high-energy electron through matter are too numerous to simulate di• rectly*. Instead, the electron trajectories are divided into many segments in each of which numerous interactions occur. For each segment, the net angular deflection and the net energy loss are sampled from relevant multiple-scattering distributions. This "condensed-history" or "path-segment" model corresponds to the "Complete-Grouping, Class I" scheme outlined in Ref. 1, and involves the use of a predetermined set of seg• ment pathlengths.

Length of path segments. The pathlengths for the steps in the electron random walk are chosen according to conflicting requirements. On the one hand, the steps should be short enough that (1) most of the steps in the electron history are completely inside the target so that the use of multiple-scattering theories for unbounded media are appropriate; (2) the energy loss is, on the average, small so that the use of one-velocity multiple-scattering distri• butions is justified; and (3) the net angular deflection is, on the average, small so that the spatial displacement for the step can be approximated closely by a straight line (neglect of transverse displacements or wiggliness corrections). Factors (2) and (3) are important also in limiting the ambiguity as to the instantaneous energy, direction and spatial coordinates of the electron along the step, which are needed for sampling the production of secondary radiations. On the other hand, the path segments should be long enough that (1) there are a sufficient number of collisions per step to justify the use of multiple-scattering theories, and (2) the number of steps per history is not too large.

The step pathlengths in ETRAN are chosen on two levels. First, major steps are defined such that, on the average, the kinetic energy T of the electron is reduced by the factor 2-(1/k) per step. That is, the length of the nth step is

Tn 8 n = J[Stot(T')t1dT' = ro(Tn) - ro(Tn+1), (7.1) Tn +1 with - 2-(1/k)T. T.n+1 - n, (7.2)

and where Stot is the total (collision plus radiative) stopping power, and r 0 is the elec• tron range evaluated in the continuous-slowing-down approximation (csda). Electron stopping power and range are discussed by Berger in Chapter 3. With this logarithmic energy grid, the average fractional energy change over the step is constant. For exam• ple, with k = 8 (the typical, or "standard" choice), the energy change along the step is on the average 8.3%. The energy dependence of the various electron interaction cross sections resides primarily in the multiplicative factor (3-2 in the Rutherford scattering

* Single-scattering electron Monte Carlo is feasible at low energies (see Chapter 16), and as mentioned above, may be necessary for accurate results. 7. An Overview of ETRAN Monte Carlo Methods 157 formula. These cross sections are evaluated at the mid-energy for the step, so errors associated with the use of a constant energy are estimated to be much less than 1%.

The logarithmic energy grid has the further advantage that the distribution of net multiple-scattering angular deflections changes very slowly from step to step. This feature is exploited in the scheme for sampling angular deflections. In order to keep the mean deflection angle small, the major energy-loss steps are divided into m equal-length sub-steps, at the end of each of which the angular deflections are sampled. It has been found for many problems that an effective choice for m is such that the mean deflection cosine for each sub-step is no smaller than 0.9. Table 7.1 gives for some representative materials values of the mean deflection cosine as a function of electron energy, for the case of k = 8 and the choice of m as indicated. These choices for m are rather typical; however, other choices, including m-values which vary from step to step, have been used in various calculations. Because the change in the distribution of angular deflections from sub-step to sub-step within any major step is negligible, the multiple-scattering angular distribution, evaluated for a single sub-step (which spans the mid-energy of the major step), is used throughout the major step.

Table 7.1

Typical mean d~flection cosines for steps in the ETRAN random walk. The path length for each step is such that the kinetic energy T of the electron at the beginning of the step is reduced on average by the factor 2-1/ 8 = 0.917; this pathlength is further subdivided into m equal sub-steps.

T Be C Al Cu Ag Pb (MeV) m=2 2 3 7 11 18 64 0.9991 0.9989 0.9990 0.9994 0.9995 0.9996 32 0.9981 0.9975 0.9973 0.9982 0.9985 0.9988 16 0.9961 0.9948 0.9938 0.9952 0.9959 0.9963 8 0.9927 0.9900 0.9875 0.9892 0.9900 0.9904 4 0.9874 0.9824 0.9773 0.9791 0.9795 0.9787 2 0.9798 0.9719 0.9628 0.9645 0.9637 0.9602 1 0.9709 0.9595 0.9454 0.9472 0.9444 0.9370 0.5 0.9622 0.9476 0.9285 0.9307 0.9259 0.9152 0.25 0.9551 0.9380 0.9150 0.9182 0.9116 0.9017 0.0125 0.9502 0.9313 0.9058 0.9107 0.9030 0.9005 0.00625 0.9471 0.9272 0.9006 0.9083 0.8994 0.9106 0.03125 0.9452 0.9249 0.8985 0.9109 0.8996 0.9287 0.015625 0.9441 0.9239 0.8995 0.9193 0.9032 0.9535 0.0078125 0.9436 0.9240 0.9044 0.9342 0.9086 0.9800

If the option is chosen to include energy-loss fluctuations, the energy of the electron does not march down the energy grid. The relevant pre-tabulated distributions are either interpolated in energy or simply sampled at the nearest tabulated energy with, as noted above, very small average error. 158 S. M. Seltzer

Distribution of angular deflections. The net angular deflection from the combined effect of the elastic and inelastic colli• sions in a single sub-step is sampled from the Goudsmit-Saunderson multiple-scattering distributionll• This distribution is no more difficult to evaluate than that from Moliere's multiple-scattering theory12, but has two advantages. Goudsmit-Saunderson theory is not derived in a small-angle approximation, so is exact for any angle, and it can be evaluated with any desired single-scattering cross section.

The Goudsmit-Saunderson distribution can be written as the Legendre series,

(7.3) where +1 Gl = 27rN j[dU(O,T)/drt][1- Pl(cosO)]d(cosO). (7.4) -1

In Eqn. 7.4, du(O, T)/drt is the single-scattering cross section for an electron with kinetic energy T, and N is the number of atoms per unit volume (Avogadro's constant divided by the atomic mass). The variation of du/drt due to energy loss along the pathlength s is taken into account by replacing the exponent sGl in Eqn. 7.3 with J; Gl(s')ds'. The integral J; Gl(s')ds' can be evaluated in the continuous-slowing-down approximation either by numerical integration or, as is presently done for ETRAN, using a simple fitting technique due to Spencer13.

In principle, one should use the most accurate available data for the single-scattering cross section du(O"T)/drt. In practice, an approximation is in most cases quite adequate in which the cross section is evaluated in terms of: (a) the Mottl. cross section which is exact for the unscreened point nucleus, and includes spin and relativistic effects; and (b) the Rutherford15 cross section modified by a screening correction term from the work of Moliere16. In this approximation, the cross section can be written13 as

where Z e is the nuclear charge, and p is the momentum and v the velocity of the incident electron. The screening correction consists of replacing the factor (1 - cos ot2 in the Rutherford cross section by the quantity (1 - cos 0 + 2'f/ t2. From the work of Moliere, the screening angle 'f/ is given by

(7.6) where a is the fine structure constant (1/137), m is the electron rest mass, and c is the velocity of light.

With this cross section, the evaluation of the integrals for the expansion coefficients Gl , given in Eqn. 7.4, has routinely been done analytically through a suitable expansion 7. An Overview of ETRAN Monte Carlo Methods 159 of the Mott cross section in powers* of (1 - cos +2'7) and the use of appropriate recursion relations IS. Early experience indicated that the accumulation of round-off or truncation errors caused difficulty in the evaluation of the Goudsmit-Saunderson series. Through numerical experimentation, it was found that reasonably good accuracy could be maintained and convergence at low energies promoted if the following measures are taken. A delta function component, associated with very small scattering angles (and assigned to 00 ), is separated out. If '" < 10-4, the series is summed to i rnaz = 240, using forward recursion (increasing i). If '" > 10-4 , the series is summed to i rnaz = exp(1.794 - 0.3971n "'), but never less than 10, using backward recursion (decreasing i).

The angular deflections due to inelastic scattering by the orbital electrons are approximately accounted for by replacing Z2 with Z(Z + 1) in Eqn. 7.5. This, however, leads to an increase in the cross section at all angles. To correct for this increase at large angles, the final multiple-scattering distribution is then reduced by the factor Z/(Z + 1) at all angles greater than the kinematical cutoff angle determined for the maximum energy transfer in free-free electron collisions.

These details are given partly to complete the record and partly for background in comparing with improved methods and results. Moliere16 found that the use of the small-angle approximation restricted the region of validity for his screening angle to energies above about 100 Z4/3 eV. The analysis by Zeitler and Olsen17 leads to a somewhat higher estimate, based on consideration of the overlap between the screening and the spin and relativistic corrections. They predict errors in the cross section due to neglect of this overlap to be greater than of the order of 10% at energies below about 270 Z4/3 eV.

There are available much more accurate elastic-scattering cross sections at low energies. Riley18 has developed a full phase-shift calculation for the solution of the Dirac equation for an electron in the static, screened Coulomb potential of the atom. He gives complete results for 9 electron energies from 1 to 256 keY and for selected elements from Z = 2 to 94. Recent calculations19 with his code have extended the coverage to energies from 1 to 1024 keY and for Z = 1 to 100. These cross sections, and the resultant multiple-scattering distributions, are discussed further by Berger and Wang in Chapter 2. Here, we use a few of these results to judge the accuracy of the procedures used in ETRAN. Fig. 7.1 shows results for the transport (or momentum• transfer) cross section

+1 O'I(T) = 271" j[dO'(O,T)/dn](l- cosO) d(cos 0). (7.7) -1

Comparing, in Fig. 7.1, the points from the exact phase-shift calculations and the solid curve obtained using the cross section given by Eqn. 7.5 with the screening angle given by Eqn. 7.6, the estimates given above concerning the validity of using the factored Moliere-Mott cross section are rather well confirmed. Moreover, the results using the Moliere-Mott cross section deteriorate rapidly at lower energies, with the result, for

* In Refs. 1 and 13, the large-angle behavior of the Mott cross section is expressed in terms of half-integral powers, j /2, with j = 1 to 5. Present evaluations are done using integral powers j, with j = 0 to 4, which require the use of somewhat different recursion relations. 160 S. M. Seltzer example, that the transport cross section in gold is too small by a factor of 200 at 1 keY. We have found that agreement can be greatly improved by making a strictly empirical adjustment to Moliere's screening angle, using instead

1 2 1/2 rl = - ( arne ) Z2/3[1.13 + 3. 76( aZ/ f3? (_7_) ], (7.8) 4 0.885p 7 + 1 where 7 is the kinetic energy of the electron in units of its rest energy. Results using the screening angle of Eqn. 7.8 are given by the dashed curves in Fig. 7.1, and are in much better agreement with the transport cross sections from the exact phase-shift calculations.

Figure 7.1. Transport cross sections for elastic scattering of electrons. The quantity given is f3 20"1(T), where f3 is the ratio of the electron velocity to the speed of light, and 0"1 (T) is defined by Eqn. 7.7 in the text. The points are from the results of phase-shift calculations18 of the elastic-scattering cross section dO" / dO for the static, screened Coulomb field of the atom. The curves are results based on the use of the Mott elastic-scattering cross section, modified by a multiplicative screening correction. The solid curves are for the screening correction calculated with the screening parameter given by Moliere16; the dashed curves include a simple, empirical adjustment (Eqn. 7.8) to Moliere's screening parameter to improve agreement with the transport cross sections from the phase-shift calculations.

Although, as pointed out by Moliere16 and by Bethe20 , the transport cross sec• tion has the domin

(0) (b)

2

I 'C 0... 0 - 3 --CD (e) (d) 2

20 40 600 20 40 60 8 (degrees)

Figure 7.2 a-d. Multiple-scattering angular distributions for electrons in car• bon. The quantity given is f(O) = 21l'Sin OAGs(O, s), where AGS is the Goudsmit• Saunderson distribution evaluated for the pathlengths typically used in ETRAN calculations. The histograms give results based on the use of the Moliere ( adjusted)• Mott single-scattering cross sections, in conjunction with the analyticalfrecursive algorithm for the Goudsmit-Saunderson expansion coefficients as used in ETRAN. The curves represent the more accurate results and are based on the use of single-scattering cross sections from phase-shift calculations for the Hartree• Fock atomic potential, in conjunction with a numerical integration procedure for the Goudsmit-Saunderson expansion coefficients (see Chapter 2). The irregularities in the curves for the lowest energies are evidence of lack of con• vergence with 1000 terms. Note that the distributions all have unit area; the differences in area between the curves and histograms is accounted for by a a-function at 0° (i.e., 0°-scattering probability) which is introduced to im• prove convergence in the ETRAN algorithm. (a) Energy: 62.67 to 60.02 keY, pathlength = 3.06 {tm, 0°-probability = 1.09 X 10-9• (b) Energy: 31.34 to 30.01 keY, pathlength = 0.921{tm, 0°-probability = 1.06 X 10-5• (c) Energy: 15.67 to 15.00 keY, pathlength = 0.270 {tm, 0°-probability = 1.61 X 10-3 • (d) Energy: 7.834 to 7.502 keY, pathlength = 0.079 {tm, 0°-probability = 0.0257. 162 S. M. Seltzer

(0 2

I --'t:J 0 ...... 0 -Q) 3 (g) ( h) 2

oL-~~--~-L~c=~ o 20 40 600 20 40 60 8 (degrees)

Fig. 7.2 e-h. Same as Fig. 7.2 a-d but for gold. (e) Energy: 245.97 to 244.79 keY, pathlength = 0.450 p.m, 0°-probability = 3.95 x 10-5 ; (f) Energy: 122.98 to 122.39 keY, pathlength = 0.166 p.m, 0°-probability = 5.14 x 10-3 ; (g) Energy: 61.49 to 61.20 keY, pathlength = 0.056 p.m, 0°-probability = 0.0734; (h) Energy: 30.75 to 30.60 keY, pathlength = 0.018 p.m, 0°-probability = 0.286.

Table 7.2

Comparison of mean deflection cosines due to the multiple elastic scattering of low-energy electrons in typical ETRAN steps. Pathlengths are chosen as explained in Table 7.1. Columns labeled ETRAN contain results from present ETRAN treatment. Results labeled "Exact" are from the more accurate evaluation described in Fig. 7.2.

Carbon (m = 2) Gold (m = 18) T Mean Deflection Cosine Mean Deflection Cosine (keV) ETRAN "Exact" ETRAN "Exact" 256 0.9382 0.9470 0.9014 0.9070 128 0.9314 0.9412 0.8951 0.9020 64 0.9271 0.9374 0.8969 0.9047 32 0.9245 0.9350 0.9032 0.9117 16 0.9228 0.9333 8 0.9215 0.9318 7. An Overview of ETRAN Monte Carlo Methods 163

Note that part of the apparent disagreement is due to the removal of a portion of the area in the peak to a delta function at 0° in the ETRAN procedur~ (the probability distributions given in Fig. 7.2 are all normalized to unit area). The corresponding values of the mean multiple-scattering angular deflection are compared in Table 7.2, and agree rather well. The approximate Moliere(adjusted)-Mott cross section is thus used to provide improvement in the low-energy elastic-scattering treatment until the exact phase-shift results are incorporated into ETRAN (the Riley cross sections are already used in some versions of the Sandia codes).

Collision (ionization and excitation) energy loss. The energy loss ~ due to multiple ionization and excitation collisions in a path of length s is described by the Landau distribution21 • Landau's derivation is based on the following assumptions: (1) The pathlength is not too large, so that the mean to• tal energy loss is small compared to the initial energy of the electron, and therefore a one-velocity treatment is adequate. The pathlength must, however, be large enough for a sufficient number of individual collisions to justify the statistical treatment used. (2) The Rutherford-scattering law describes individual collisions with an energy transfer large compared to the mean excitation energy I of the medium, i.e., the probability per unit pathlength for an energy transfer to is given by the quantity (27rr~mc2 N Z {3-2)C2, where m is the electron mass, r. the classical electron radius, and NZ is the number of atomic electrons per unit volume of the medium. Low-energy transfers are described in terms of the first moment of the energy-loss cross section, available from Bethe theory22. (3) Because of the rapid fall-off of the energy-loss cross section, the maximum energy transfer can be allowed to extend to infinity.

The result is a distribution that can be expressed in terms of a universal function of a single scaled variable, f(~,s)d~ = ¢>()")d)", (7.9) where (7.10)

In Eqn. 7.10, 8 is the density-effect correction to the collision stopping power. The parameter ~ is given by (7.11 ) and coincides with the energy above which an energy transfer occurs on the average of once in the pathlength s (this energy is a key quantity in the earlier energy-loss straggling theories of Bohr23 and of Williams24). It is convenient to re-write Eqn. 7.10 as ~-~ )..=--+v (7.12) ~ where ~ is the mean collision energy loss, and the factor

v = In(T/~) - 0.80907 + [r 2/8 - (2r + 1)ln2J/(r + 1? (7.13) in part discounts the M¢ller25 terms that are included in the expression for the mean 164 S. M. Seltzer energy loss, but not in Landau's theory* . The universal function (>.) is given by

c+ioo (>.) = 2~i J exp[uln(u) + >.u] du, (7.14) c-ioo and has been accurately tabulated by Borsch-Supan27 • From his results, (>.) has a maximum at >'p = -0.2225 and a full-width at half-maximum of 4.019.

Blunck and Leisegang28 included the second moment in the expansion at low en• ergies of the energy-loss cross section used in solving Landau's equation. The corrected distribution is given by the convolution

(7.15)

According to Blunck and WestphaI29, the variance of the Gaussian can, to a good approximation, be written as

(}"2 = 10 eV. Z4/3ii. (7.16)

In ETRAN, the collision energy loss for each major step is sampled from the Landau/ Blunck-Leisegang distribution. The rate of energy loss along the step is assumed to be constant.

The Blunck-Leisegang correction provides for only a partial extension of the region of applicability of Landau's theory. For short pathlengths when the Blunck-Liesegang broadening becomes large, either higher moments in the expansion must be included or the entire process must be considered on a more detailed basis30,31. As ETRAN uses Landau/Blunck-Leisegang theory, it is well to illustrate the quality of the resultant energy-loss distributions by comparing the theoretical predictions with experimental results. In Fig. 7.3, we plot the ratio of calculated-to-measured values for the most probable energy loss ~p, and the full-width at half-maximum, FWHM, as a function of the ratio ~/I. Plotted this way, results tend to form a single curve for electrons with en• ergies from 300 keY to 51 GeV in solid targets31- 38 from Be to Pb (squares and crosses) and gas targets39- 42 from CH4 to Kr (circles). Landau states the condition ~/I » 1 for the applicability of his results. Including the Blunck-Leisegang correction, we find good agreement (within < 10%) for e; I > 4, with the results deteriorating rapidly for e; I < 1. Values of ~/ I for the typical pathlengths used in ETRAN calculations are listed in Table 7.3. Except for the occasional partial step to a boundary, one can thus expect less reliable energy-loss results only at the lower energies in high-Z targets.

* For positrons, Eqn. 7.13 would instead contain corresponding terms from the Bhabha cross section26. 7. An Overview of ETRAN Monte Carlo Methods 165

4

Q. 00 >C 3 0 CD 0 Co

0 4

0 Q. >C .. 3 0 ~ I o 0 3 0 0 l.I.. COCO 0 ...... 2 ~ 0 0 ic~o\oO 0 0 o ..,~oS~ ~ Q)o I 0 3 0 l.I.. 0

0 10- 1 10° 10 1 102 103 (/1 Figure 7.3. Comparison of calculated and measured values of the most proba• ble energy loss, and the full-width half-maximum of the energy-loss distribution for electrons traversing thin layers. The calculated values were obtained from evaluations of Landau/Blunck-Leisegang theory. The measured values are from various experiments involving electrons with energies from 300 keY to 51 GeV in solid targets [Ref. 31-38] from Be to Pb (plotted as squares), and in gas targets [Ref. 39-41] from CH. to Kr (plotted as circles). Bich• sel's results42 for 1-GeV electrons in Si, obtained from carefully constructed single-scattering energy-loss cross sections, are treated as experimental and represented by crosses. The comparisons are presented in terms of the ratio of calculated-to-experimental values, plotted as a function of e/ I, where eis the energy value above which an energy transfer occurs on the average of once in that pathlength, and I is the mean excitation energy for the medium. (a) The most-probable energy loss IIp. (b) The full-width at half-maximum, FWHM.

Bichsel42 has preformed a calculation of the energy-loss distribution of electrons in Si, based on the convolution of single-scattering energy-loss cross sections carefully constructed from the best available experimental and theoretical information. Fig. 7.4 compares results from Bichsel's calculation with those from Landau/Blunck-Leisegang theory evaluated for typical ETRAN steps in Si. The level of agreement in the peak region is consistent with the results of Fig. 7.3, and agreement is good in the essentially single-scattering tail for large energy losses. 166 S. M. Seltzer

0.006 0.04 ( b)

/ 0.03 ~ 0.004 I I , 0.02 I I , I ~ 0.002 "- 0.01 ...... I > Q) 0 0 .x 400 500 600 700 800 40 60 80 100 0.4 .:9 ( c) 0.2 0.3

0.2 0.1 "- 0.1

/

O~~~-L~ __h-~~-L~ 0 I 3 5 7 9 0 2 4 6 8 [0; ( keV)

Figure 7.4. Energy-loss (straggling) distributions for electrons in Si. Results are giv~n in terms of the spectrum of energy loss f(6.) as a function of the energy loss 6.. The pathlengths considered are those corresponding to a mean total energy loss of 8.3% as typically used in ETRAN calculations. The solid curves are from Bichsel's42 detailed analysis of the energy-loss cross section for Sij the dashed curves are the Landau distribution convoluted with the Blunck• Leisegang Gaussian. (a) 10 MeV, pathlength = 1789 /-lm, mean collision loss = 707 keY, ~/ I = 185, Gaussian width parameter = 0.483 ~ (15.4 keY). (b) 1 MeY, pathlength = 232 /-lm, mean collision 10ss= 81.6 keY, OI = 27.0, Gaussian width parameter = 1.12 ~ (5.25 keY). (c) 0.1 MeV, path1ength = 10.9 /-lm, mean collision loss = 8.26 ke V, 0 I = 3.73, Gaussian width parameter = 2.59 ~ (1.67 keY). (d) 0.05 MeV, path1ength = 3.34 /-lm, mean collision loss = 4.03 keY, ~/ I = 2.02, Gaussian width parameter = 3.33 ~ (1.17 keY). 7. An Overview of ETRAN Monte Carlo Methods 167

Because the maximum energy loss is allowed to extend to infinity, there is not a finite mean for the Landau distribution. However, in accordance with Eqn. 7.12, the correct mean collision energy loss can be imposed by truncating the distribution at a ).cut such that :\ = II. Values of )..ut are plotted in Fig. 7.5 for the typical ETRAN pathlengths. Earlier versions of ETRAN made sole use of Borsch-Supan's tabulation of the Landau distribution extending only to ). = 100, so that the correct mean energy loss was obtained for electron energies only in the vicinity of 1 MeV. This failure to

Table 7.3

Values of the ratio ~/ I that govern the validity of energy-loss sampling from the Landau distribution (see text). Values are given for the typical ETRAN pathlength in which the mean total energy loss is 8.3% of the initial kinetic energy T of the electron.

T (MeV) H H2 O C AI Cu Ag Pb 64 7115 1681 1678 593 199 99.8 39.4 32 4023 1096 1084 428 164 87.4 36.5 16 2230 652 641 276 121 69.3 31.1 8 1218 366 359 163 79.7 48.9 23.7 4 662 199 196 92.1 48.3 31.3 16.3 2 359 106 105 51.0 27.8 18.8 10.3 1 194 57.0 56.5 28.1 15.6 10.8 6.18 0.5 104 30.7 30.2 15.3 8.61 6.11 3.61 0.25 55.8 16.7 16.2 8.36 4.72 3.40 2.07 0.125 29.8 9.02 8.75 4.56 2.60 1.90 1.18 0.0625 16.0 4.93 4.77 2.52 1.46 1.07 0.69 0.03125 8.70 2.72 2.64 1.41 0.83 0.62 0.41 0.015625 4.77 1.53 1.48 0.81 0.49 0.37 0.25 0.0078125 2.65 0.87 0.85 0.47 0.29 0.23 0.16

predict the correct mean collision loss was pointed out by Rogers and Bielajew43 in their explanation of the discrepancies between the results of EGS and ETRAN calculations of depth-dose curves for electrons incident on water targets. The collision energy-loss sampling algorithm has since been corrected to extend to the appropriate cutoff, with some small but significant changes in calculated electron transport results. Fig. 7.6 shows, for example, the change in the calculated depth-dose curve for 20-MeV electrons incident on water when the corrected energy-loss sampling procedure is used. These newer ETRAN results are compared with those from the corresponding EGS Monte Carlo calculation43 in Fig. 7.7, which shows that there is now rather good agreement. 168 S. M. Seltzer .. o~~ __~~~~~ __~~~~~~~ __~~~~

.., o

No

Figure 7.5. Cutoff parameter 'xcu' for the Landau energy-loss straggling distribution that provides for the correct mean collision energy loss. The results are given as a function of electron kinetic energy T, and are for a pathlength corre• sponding to a mean total energy loss of 8.3% as typically used in ETRAN calculations.

---- Original energy-loBI , sampling o \ \/ \ \ 10 \ \ ci \ \ \ \ \ \ , ~ >- ~ 0.4 0.6 0.8 1.0 1.2 z/r.

Figure 7.6. The depth-dose distribution for a broad beam of 20-MeV electrons incident perpendicularly on a thick water slab. The results, given in terms of scaled quantities, are from ETRAN calculations using the original algorithm for sampling from the Landau energy-loss straggling distribution (dashed curve) and from the new corrected version (solid curve). 7. An Overview of ETRAN Monte Carlo Methods 169

Bremsstrahlung. The radiative energy loss for the electron is determined by sampling the production of bremsstrahlung photons using a recently developed dataset of bremsstrahlung pro• duction cross sections, differential in emitted photon energy45. These cross sections, discussed more fully elsewhere in this volume (Chapter 4), go beyond the r~sults from Bethe-Heitler theory46 in that they incorporate: (a) the results of numerical phase-shift calculations for nuclear-field bremsstrahlung by Pratt, Tseng et a1 47 at energies below 2 MeV, (b) the analytical high-energy nuclear-field bremsstrahlung theory of Davies et a1 48 , including Coulomb corrections and including screening corrections evaluated

Depth, em Figure 7.7. Comparison of results from ETRAN (curve) and from EGS (histogram) for the depth-dose from a broad beam of 20-MeV electrons incident perpendic• ularlyon a thick slab of water. The results from both Monte Carlo calculations are based on the use of recent values of the electron stopping power44(the EGS results are from Fig. 2 of Ref. 43).

with Hartree-Fock atomic form factors, and (c) the analytical electron-electron brems• strahlung theory of Haug49, combined with screening corrections derived from Hartree• Fock incoherent-scattering factors, for bremsstrahlung in the field of the atomic elec• trons. For each sub-step, bremsstrahlung production is sampled from a Poisson distribu• tion, and the photon energy is subtracted from the energy of the electron. Thus, brems• strahlung emission contributes to the energy-loss straggling of the electrons through the direct single-scattering simulation of the energy-loss process. This is illustrated in Fig. 7.8 which shows the energy-loss distribution, with and without bremsstrahlung losses, for 20-MeV electrons traversing 0.25 cm of water. 170 S. M. Seltzer

0

0 ~

~I :::E~ ~ ... I ~

.., I ~ 0 2 :3 4 5 A. MeV

Figure 7.8. The energy-loss spectrum for 20-MeV electrons traversing 0.25 em of water. The histogram is from an ETRAN Monte Carlo calculation which includes ionization losses (by sampling from Landau/Blunck-Leisegang multiple-scattering theory) and bremsstrahlung losses (by sampling individual production events); the curve gives the results for ionization losses only, from Landau/Blunck-Leisegang theory.

If the sampled photon energy is greater than a chosen cutoff value, the photon history is traced. The starting position for the photon is chosen at random along the sub-step. The intrinsic bremsstrahlung emission angle (relative to the direction of the primary electron) is sampled from an angular distribution derived from a combination of Bethe-Heitler cross sections46, differential in emitted photon energy and angle. The direction of the primary is taken to be that at the beginning or end of the sub-step, depending on which end is closer to the chosen production point.

A weighting scheme allows for the sampling of bremsstrahlung photon histories in excess of the natural production rates so that the statistical fluctuations in the bremsstrahlung scores can be reduced without an increase in the number of more time• consuming electron histories. With this option, an artificially enlarged set of photons is sampled with energies greater than the chosen cutoff value and used only for the gener• ation of photon histories, while the radiative contribution to the energy-loss straggling of the electrons is determined by the sampling of "natural" bremsstrahlung photons. The photon histories are then given appropriately small statistical weights so that the computation remainS' unbiased.

No angular deflection of the primary is included as a result of bremsstrahlung events. Angular deflections associated with the emission of the more probable low• energy photons are assumed to be included in the elastic-scattering distribution. The 7. An Overview of ETRAN Monte Carlo Methods 171 angular deflections due to the emission of high-energy photons are ignored on the as• sumption that such events are relatively rare, and the effect is small compared to that of elastic scattering.

Knock-on electrons. The production of knock-on electrons, whose energies are above a chosen cutoff value, is sampled according to the M!i>ller cross section25 for collisions between free electrons (binding effects are disregarded). The intrinsic direction of the knock-on electron is determined by conservation of energy and momentum, with its production point and the direction of the primary electron selected as in the case of bremsstrahlung photons.

Because the energy loss for the primary electron has been accounted for, on the average, in the Landau/Blunck-Leisegang distribution, no change in the energy of the primary is made as a result of the sampled knock-on production event. This procedure thus neglects the correlation between the energy loss of the primary and the energy of the produced secondary electrons. Such a scheme would be unsuitable for simulating an experiment in which primary and secondary electron energies are measured in coinci• dence, but seems to have little deleterious effects in most other problems. On the other hand, an advantage of this scheme is that complete energy-loss straggling is inherently included through the use of Landau/Blunck-Leisegang theory, regardless of the choice of the minimum energy for the sampling and following of knock-on electron histories.

Characteristic x-rays and Auger electrons. K-shell ionization events in each sub-step are sampled on the basis of Kolbenstvedt's cross section50 (older versions of ETRAN used the Arthurs and Moiseiwitsch cross sectionSl). The fluorescence efficiency then is used to select either the emission of characteristic x-rays or Auger electrons, and the emitted energy is selected from the appropriate discrete spectrum. The emission due to ionization of shells other than the K-shell are not included in the present treatment. The emitted energy is not subtracted from the primary energy because the energy transfer is implicitly included in the evalu• ation of the electron collision loss. As in the case of bremsstrahlung photons, an option is provided to generate an artificially enlarged sample of K-shell ionization events.

Positrons. In present versions of ETRAN, positrons are treated as electrons, except for the isotropic emission of two oppositely-directed annihilation quanta when the positron slows to rest. Most of the necessary cross-section information is available within the code for positrons (collision and radiative stopping powers, and elastic-scattering angular distributions), but separate sampling has not been implemented. When it is implemented, knock-on electron production governed by the Bhabha cross section and annihilation in flight will be added.

Boundary crossing and scoring. There are two types of boundaries considered in ETRAN: (1) interface boundaries defined to separate two different materials and/or for the purposes of scoring the emer• gence of radiation through a surface, and (2) minor boundaries to define smaller volumes within a single-material major volume for the purpose of scoring energy depositIon and internal fluence spectra (tracklength distributions). No special action is taken when an electron trajectory crosses a minor boundary. Track1ength and collision energy loss are assumed to be deposited at a randomly selected position along each sub-step, and are 172 S. M. Seltzer scored for the corresponding minor volume. The initial energy and charge of knock• on electrons that are set in motion are subtracted from the corresponding energy and charge-deposition scores, as these will be re-deposited elsewhere.

In the case of an interface boundary crossing, the following action is taken. Because the multiple-scattering angular distributions are pre-tabulated only for the full sub• steps, the angular deflection for the partial sub-step to the boundary is sampled from a Gaussian approximation to the Goudsmit-Saunderson distribution. The emission of secondary radiations is sampled only for the partial sub-step to the boundary. The collision energy loss is re-sampled from the Landau/Blunck-Leisegang distribution for the partial major step to the boundary, and the previous energy deposition scores are adjusted to reflect the new energy loss. The electron history then proceeds from its location at the interface.

Termination of histories. Electron histories are terminated under a variety of conditions. One energy is selected at run time to be the maximum cutoff value (i.e., no electron history is terminated at a higher energy unless it escapes the target). This cutoff can be used, for example, to ensure that the probability of producing more-penetrating bremsstrahlung photons is adequately taken into account before the electron history is dropped. Another energy is selected to be the minimum cutoff value (i.e., no electron is followed to a lower energy). At energies between these two cutoffs, the history is terminated if the residual range is smaller than the distance to either the nearest interface boundary or the nearest minor boundary, depending on the version used and the option selected. If fluence spectra are being scored, electrons are always followed down to the selected minimum cutoff energy.

The residual energy of the electron history terminated inside the target is accounted for in the energy-deposition scores by assuming that the electron loses its energy at a constant rate as it continues to travel along a straight path. The length of this path is equal to the product of the electron's residual csda range and a detour factor which approximately accounts for the reduction of its penetration due to multiple-scattering detours. As this residual path crosses interface boundaries, the residual range and detour factor are accordingly adjusted. The detour factor is estimated to be the ratio of the practical range (obtained by extrapolating to the axis the descending straight line portion of the depth-dose curve) to the csda range. These ratios, usually those obtained from the low-energy results of Spencer52 for perpendicularly incident electrons, are relatively independent of electron energy and are assumed to be constant. The electron's charge is deposited at the end of this track.

Limitations. Some limitations have been indicated above in the discussions of the various processes as presently treated in ETRAN. A few additional remarks are given here.

The Bethe stopping-power theory used44 , and the Landau energy-loss distribution, are applicable for incident electron velocities considerably greater than that of the or• bital electrons of the target atoms. This condition is fulfilled for the outer-shell atomic electrons for incident electrons above rv 1 keY, but not for the inner-shells in high-Z elements. The overall error at low energies is, however, mitigated by the fact that 7. An Overview of ETRAN Monte Carlo Methods 173 only a small fraction of the atomic electrons are in the inner shells. Estimates in Ref. 44 suggest that the Bethe-theory stopping power in high-Z materials is good to about 20-25% at 10 keV.

The treatment in ETRAN is based on the standard linearized Boltzmann equation, with the assumption that the distribution of scattering centers is random and that the passage of one electron does not perturb that of another. Thus, the calculation ignores quantum-mechanical interference (diffraction) which can occur at low energies even in amorphous materials, channeling associated with the transport of charged particles in regular crystals, and collective effects associated with intense beams.

ETRAN neglects processes that occur when the electron enters the nucleus and interacts with the fields of the nucleons (e.g., elastic scattering from nucleons, electro• disintegration of the nucleus, and production of neutrons, alpha particles, nuclear gamma rays, mesons, etc.). These processes set in at rather high energies (well above 100 MeV) and are usually neglected as well in other electron codes that are run in the GeV region.

7.3 ORGANIZATION AND DESCRIPTION OF THE CODE SYSTEM

ETRAN comprises two catagories of codes. Codes in the first group prepare the rather extensive array of cross-section information for the materials, energy coverage and elec• tron step-sizes desired. The second group include the versions of the Monte Carlo code which handle the various target geometries that have been considered. All of the codes are written in FORTRAN.

7.3.1 Data Preparation

ETRAN requires cross-section information for both photons and electrons. At present, photon interaction cross sections and K-shell x-ray/Auger emission spectra are prepared by a code called COMBIXD, and the electron data are prepared by a code called DAT• APAC. These codes use identical input data, and could be combined for convenience.

The DATAPAC code has been updated as better cross-section information has been developed. The present version differs from previous ones mainly in its use of more current collision-loss stopping powers44 and bremsstrahlung cross sections46 . The specification of the target composition is facilitated by the ability of DATAPAC (and COMBIXD) to read in mixtures of compounds in terms of their chemical formulas. Constituent compounds (and single elements) can be specified in parts by weight or parts by volume (partial pressures). DATAPAC can automatically determine the mean excitation energy I used in the evaluation of the Bethe collision stopping-power formula. It also has a number of override options for the I-value to handle those materials not anticipated by the algorithm developed for the preparation of the ICRU tables44 . The few other input parameters specify the density of the target material, whether the particle is an electron or positron, the energy span of the dataset, the choices of the major step parameter k (Eqn. 7.2) and the sub-step parameter m, the desired treatment of low-energy elastic scattering (e.g., Eqn. 7.6 or 7.8), and the amount of detail in the print-out of the run. The output datasets are stored in computer (or tape) files for use in the Monte Carlo calculations. DATAPAC runs rather quickly, typically about 10 seconds per material on an IBM 3081 K. 174 S. M. Seltzer

7.3.2 Monte Carlo Calculations

Geometry. Versions of ETRAN have been written at NBS to handle a few relatively simple but useful target geometries. ETRAN 16 treats the one-dimensional geometry of homoge• neous slab targets that are unbounded laterally. An important feature of this code is that the calculation can be performed simultaneously for a number (say, up to 10 or 20) of different target thicknesses. Thus, for very little more than the computing time required for the thickest slab, results are obtained also for thinner targets.

The version ETRAN 18 treats the three-dimensional geometry of homogeneous, right-circular cylindrical targets and, in a fashion similar to the slab case, can handle up to 10 different cylinder sizes simultaneously. This version includes extra provisions in calculating the spectrum of absorbed energy. For the largest cylinder, one can define a dead region and a surrounding anti-coincidence region, which has been useful in calculations of the response to gamma rays of detectors with active charged-particle shielding.

In the version ZTRAN53, calculations for slab geometry are extended to hetero• geneous multilayer media. The medium is assumed to consist of several (say, up to 20) adjacent plane-parallel layers, each of which can have a different composition. The layers are assumed to be unbounded laterally, and the treatment is one-dimensional. In this version, calculations are performed separately for each target configuration.

A number of source geometries were added to the codes as the need arose. These include externally incident beams of electrons or photons, internal uniformly emitting volumes, and internal source planes, with variously specified distributions of initial energy and angle.

Input parameters and output quantities. The numbers and kinds of input parameters vary somewhat among the versions, and further depend on various options that are selected at run time. Basically, the codes re• quire parameters to specify: the target dimensions (and compositions for ZTRAN); the energy and angular distributions and spatial extent of the incident or initial source, and the desired number of histories; the separate cutoff energies that govern the termination of electron and photon histories; the spatial boundaries and the energy and angular bins for scoring; and the selection of options that govern energy loss and the production of secondary radiations.

Although there are some differences among the versions, they all calculate a similar list of output quantities. At all major interface boundaries (target/vacuum boundaries in the case of homogeneous targets), the energy and angle of emergent radiation is scored. Output tables for the electrons and photons emerging from each surface include the total number and energy, integrated over angle and spectral energy; the energy spectra, integrated over angle; the angular distributions, integrated over spectral en• ergy; and, at selected boundaries, the distribution differential in both energy and angle. For each target (for each major layer in the ZTRAN code), the spectrum of deposited energy is given. Distributions of interest inside the target (one of the multiple targets selected in the ETRAN 16 and 18 codes) that can be calculated include the fiuence, differential in energy and angle, as a function of depth; the spatial distribution of ab• sorbed dose; and the spatial distribution of deposited charge. For some of the quantities, 7. An Overview of ETRAN Monte Carlo Methods 175 the contributions by primary, knock-on, and photon-descendant electrons are tabulated separately.

By way of example, Fig. 7.9 shows depth-dose curves from ETRAN calculations for a broad beam of 20-MeV electrons incident perpendicularly on a water target. The dashed curve giving results in the continuous-slowing-down, straight-ahead approximation is derived simply from the stopping-power and range relationship. The dashed curve for results in the csda with angular scattering is from an ETRAN calculation which follows the histories of primary electrons only (no knock-on electrons and no bremsstrahlung photons or their secondary electrons). The solid curves are from a calculation which includes energy-loss straggling and which follows the histories of all secondary radia• tions; the contributions from knock-on and from bremsstrahlung-descendant secondary electrons are shown separately. As can be seen, the inclusion of energy-loss straggling has a significant effect on the depth-dose curve, causing it to extend beyond the elec• tron's mean range; and the forward transport of energy by knock-on electrons results in an absorbed dose near the entrance surface that is lower than that predicted by the collision stopping power.

~~~~-r~~~T-~~~~~~~~~~~~~~ \ \ \ CSDA \ with eng scatt " CSDA 'I straight , I, ahead \ " \ " \ " \ I," Cl _\____ 1 I \ ' Straggling \ with ang Bcott and secondaries \, I/") \ ci Knock-ons

ci~~---M------======~=C==~::==~~~~ 0.0 0.2 0.4 0.6 0.8 1.0 1.2 z/ro

Figure 7.9. Absorbed-energy distribution for a broad beam of 20-MeV elec• trons incident perpendicularly on a thick slab of water. Results are given in terms of the dimensionless quantity (ro/To)D, where D(z) is the dose ab• sorbed per unit depth at depth z, ro is the incident electron's mean range and To its kinetic energy, and are plotted as a function of the scaled depth z/ro. Results are shown for three transport-model choices: (a) primary electrons only, in the continuous-slowing-.down approximation (csda) and with no an• gular deflections (straight ahead); (b) primary electrons only, csda and with angular deflections; and (c) energy-loss straggling with angular deflections and the transport and subsequent energy deposition by secondary electrons and bremsstrahlung photons. In cases (a) and (b), all bremsstrahlung was assumed to escape the target; for case (c), the contribution to the depth-dose is shown also separately for the knock-on electron (> 1 keY) and the bremsstrahlung components. 176 S. M. Seltzer

Examples of ZTRAN results are given in Fig. 7.10 which shows the effect on the 400-keV electron depth-dose distribution in nylon due to passage first through the thin window of the accelerator and then through an air gap before the electron beam enters a nylon-film dosimeter stack. A comparison with the corresponding measurement of absorbed dose can be found in the second citation in Ref. 53.

Computer memory and time requirements. The present ETRAN codes are made up of 55 to 60 subroutines which contain a total of about 4000 to 4500 FORTRAN source statements. The codes can be divided into three portions of roughly equal length which handle the data input and preparation, the generation and scoring of histories, and the normalization and output of results. When compiled, the codes require about 400 kbytes (perhaps 200 kbytes with overlay of the three portions) exclusive of the large arrays that contain cross-section information and output scores. The overall memory requirement is dependent, of course, on the choice of dimensions for these arrays. With the ample resources available on current mainframe computers, the codes are presently running with up to 3 to 4 Mbytes of memory, without overlay.

A , 5 ~ Qj 6, N 4 E () ~ :2 ;;i 3 (/) -:r;::l o f= ~~A a: 8 I• fu 2 U Q w -' >- >_-e.--L.---w lEw z ~ w 6 o v~~-L __~ ______c

o 0.02 0.04 0.06 0.08 0.10 0.12 DEPTH IN NYLON, 9 cm-2 Figure 7.10. Effect of thin absorbers on the depth-dose distribution in a nylon target. Curve A pertains to direct irradiation of the nylon by a normally incident broad beam of 400-keV electrons; curve B takes into account the passage of the beam first through a I-mil (25.4 {tm) Ti window; and curve C takes into account first the passage through the window and then through a lO-cm air gap. All curves are from results of ZTRAN calculations.

The computation time depends very much on the problem considered and the choices of run parameters, and so is difficult to characterize. However, the examples given above can be used to indicate some run times. The results with straggling and the transport of secondaries given in Fig. 7.9 is based on a sample of 10,000 incident histories of 20-MeV electrons followed down to 0.625 MeV (unless they can cross the entrance face in which case they are followed down to 1 keY). This run involved about 400,000 major 7. An Overview of ETRAN Monte Carlo Methods 177 steps (with k = 8 in Eqn. 7.2) for the primaries, and the sampling of approximately 6.6 million knock-ons and 36,000 bremsstrahlung-produced secondaries with energies above 1 keY. Of these about 11,000 secondary histories were followed through 86,000 major steps, with the rest handled by the track-end (history-termination) algorithm. The run required approximately 32 minutes on an IBM 3081 K. This long run time was due mainly to the choice of 1 keY for the secondary-electron cutoff energy. Changing this cutoff to 50 keY would result in a run time smaller by approximately a factor of 12, with an insignificant effect on the depth-dose results. The results in Fig. 7.10 for a beam of 400-keV electrons incident on the Ti/air/nylon layered target are based on a sample of 10,000 primary histories and all secondaries with energies greater than 1 keY, followed down to 40 keY (or 1 keY if they can cross an interface boundary). This calculation took only 4.2 minutes on the IBM 3081 K. Estimated on the basis on Dongarra's performance ratingsS4 , these computing times might be multiplied by a factor of approximately 0.5 for the CDC Cyber 875, 0.9 for the CDC Cyber 175, from 2 to 5 for a DEC VAX 8600, and from 10 to 20 for the DEC VAX 11/780 or 11/750 (with floating-point accelerators).

7.4 FUTURE IMPROVEMENTS ETRAN would benefit from a thorough recoding. A new version could be made more efficient, in some respects made more accurate, and in general made much easier to un• derstand and update. Improvements associated with the treatment of photon transport would include the addition of coherent scattering, the addition of form-factor binding corrections to the incoherent-scattering distribution, and the use of updated distribu• tions for the sharing of energy by electron-positron pairs and for the angular distribu• tion of photoelectrons. Using available information on the shell-by-shell photoioniza• tion cross section55, characteristic x-ray and Auger electron emission could be included for shells higher than the K-shell. For positrons, appropriate energy-loss and elastic• scattering distributions can easily be added, as can the sampling from the Bhabha cross section for the production of secondary knock-on electrons and the taking into account of annihilation in flight.

Accurate information is available from the exact phase-shift calculations56 of the angular distributions of bremsstrahlung emitted by electrons with energies below 500 keY. This could perhaps be combined with the results from analytical high-energy bremsstrahlung theory to develop an improved database in a manner similar to that done for the bremsstrahlung energy spectra.

Improved electron-impact ionization cross sections could be incorporated, perhaps on the basis of the Weizsacker-Williams results discussed in Chapter 4. These could be used to extend the sampling of the emission of characteristic x-rays and Auger electrons to shells higher than the K-shell.

Acknowledgement: This work was supported in part by the Office of Health and Envi• ronmental Research, U.S. Department of Energy. 178 S. M. Seltzer

REFERENCES 1. M. J. Berger, "Monte Carlo Calculation of the Penetration and Diffusion of Fast Charged Particles", in Methods in Computational Physics, Vol. 1, edited by B. Alder, S. Fernbach and M. Rotenberg, (Academic Press, New York, 1963) 135. 2. J. H. Hubbell, "Photon Mass Attenuation and Energy-Absorption Coefficients from 1 keY to 20 MeV", Int. J. AppI. Radiat. Isot. 33 (1982) 1269, and references therein. 3. J. H. Hubbell, M. J. Berger and S. M. Seltzer, "X-ray and Gamma-ray Cross Sections and Attenuation Coefficients", National Bureau of Standards Standard Reference Database 8 (1985). 4. U. Fano, 1. V. Spencer and M. J. Berger, "Penetration and Diffusion of X-Rays", in Encyclopedia of Physics, Vol. 38/2, edited by S. Fliigge, (Springer, Berlin, 1959) 660. 5. O. Klein and Y. Nishina, "Uber die Streuung von Strahlung durch freie Elektro• nen nach der neuen relativistischen Quantendynamik von Dirac", Z. Phys. 52 (1929) 853. 6. J. F. Williamson, F. C. Diebel and R. 1. Morin, "The Significance of Electron• Binding Corrections in Monte Carlo Photon Transport Calculations", Phys. Med. BioI. 29 (1984) 1063. 7. H. A Bethe and J. Ashkin, "Passage of Radiations through Matter", in Exper• imental Nuclear Physics, Vol. I, edited by E. Segre, (John Wiley, New York, 1953) 166. 8. F. Fischer, "Beitrage zur Theorie der Absorption von Rontgenstrahlung", Ann. Physik 8 (1931) 821. 9. F. Sauter, "Uber den atomaren Photoeffekt bei grosser Harte der anregenden Strahlung", Ann. Physik 9 (1931) 217. 10. S. M. Seltzer, "Calculated Response of Intrinsic Germanium Detectors to Narrow Beams of Photons with Energies up to '" 300 keV", NucI. Instr. Meth. 188 (1981) 133. 11. S. Goudsmit and J. L. Saunderson, "Multiple Scattering of Electrons", Phys. Rev. 57 (1940) 24. 12. G. Moliere, "Theorie der Streuung schneller geladener Teilchen II: Mehrfach- und Vielfachstreuung", Z. Naturforsch. 3a (1948) 78. 13. 1. V. Spencer, "Theory of Electron Penetration", Phys. Rev. 98 (1955) 1597. 14. N. F. Mott, "The Scattering of Fast Electrons by Atomic Nuclei", Proc. Roy. Soc. (London) A124 (1929) 425; see also J. A. Doggett and 1. V. Spencer, "Elastic Scattering of Electrons and Positrons by Point Nuclei", Phys. Rev. 103 (1956) 1597. 15. E. Rutherford, "The Scattering of a and f3 Particles by Matter and the Structure of the Atom", Philos. Mag. 21 (1911) 669. 16. G. Moliere, "Theorie der Streuung schneller geladener Teilchen I: Einzelstreuung am abgeschirmten Coulomb-Feld", Z. Naturforsch. 2a (1947) 133. 17. E. Zeitler and H. Olsen, "Screening Effects in Elastic Electron Scattering", Phys. Rev. A136 (1964) 1546. 7. An Overview of ETRAN Monte Carlo Methods 179

IS. M. E. Riley, "Relativistic, Elastic Electron Scattering from Atoms at Energies Greater Than 1 keV", Sandia National Laboratories report SLA-74-0107 (1974); and M. E. Riley, C. J. MacCallum and F. Biggs, "Theoretical Electron-Atom Elastic Scattering Cross Sections. Selected Elements, 1 keY to 256 keV", Atom. Data and Nucl. Data Tables 15 (1975) 443. 19. R. Wang, M. J. Berger and S. M. Seltzer, "Calculations of Electron Multiple Scattering", Bull. Amer. Phys. Soc. 32 (1987) 765; see also Chapter 2 of this volume. 20. H. A. Bethe, "Moliere's Theory of Multiple Scattering", Phys. Rev. 89 (1953) 1256. 21. 1. Landau, "On the Energy Loss of Fast Particles by Ionization", J. Phys. (USSR) 8 (1944) 201. 22. M. S. Livingston and H. A. Bethe, "Nuclear Physics. C. Nuclear Dynamics, Experimental", Rev. Mod. Phys. 9 (1937) 282. 23. N. Bohr, "On the Decrease of Velocity of Swiftly Moving Electrified Particles in Passing through Matter", Philos. Mag. 30 (1915) 581. 24. E. J. Williams, "The Straggling of ,a-Particles", Proc. Roy. Soc. (London) 125 (1929) 420. 25. C. M!illler, "Zur Theorie des Durchgang schneller Elektronen durch Materie", Ann. Physik. 14 (1932) 568. 26. H. J. Bhabha, "The Scattering of Positrons by Electrons with Exchange on Dirac's Theory of the Positron", Proc. Roy. Soc. (London) A154 (1936) 195. 27. W. Borsch-Supan, "On the Evaluation of the Function

for Real Values of A. ", J. Res. National Bureau of Standards, 65B (1961) 245. 28. O. Blunck and S. Leisegang, "Zum Energieverlust schneller Elektronen in dunnen Schichten" , Z. Physik 128 (1950) 500. 29. O. Blunck and K. Westphal, "Zum Energieverlust energiereicher Elektronen in dunnen Schichten", Z. Physik 130 (1951) 641. 30. V. A. Chechin and V. C. Ermilova, "The Ionization-Loss Distribution at Very Small Absorber Thickness", Nucl. Instr. Meth. 136 (1976) 551. 31. J. Ph. Perez, J. Sevely and B. Jouffrey, "Straggling of Fast Electrons in Aluminum Foils Observed in High-Voltage Electron Microscopy (0.3 -1.2 MV)", Phys. Rev. A 16 (1977) 1061.

32. W. Paul and H. Reich, "Energieverlust schneller Elektronen in Be, C, H2 0, Fe und Pb", Z. Physik 127 (1950) 429. 33. G. Knop, A. Minton and B. Nellen, "Der Energieverlust von 1 MeV- Elektronen in sehr dunnen Schichten", Z. Physik 165 (1961) 533. 34. D. W. Aitken, W. L. Lakin and H. R. Zulliger, "Energy Loss and Straggling in Silicon by High-Energy Electrons, Positive Pions, and Protons", Phys. Rev. 179 (1969) 393. 35. K. Nagata, T. Doke, J. Kikuchi, N. Hasebe and A. Nakamoto, "Energy Loss and Straggling of High-Energy Electrons in Silicon Detectors", Jap. J. Appl. Phys. 14 (1975) 697. 180 S. M. Seltzer

36. W. Ogle, P. Goldstone, C. Gruhn and C. Maggiore, "Ionization Energy Loss of Relativistic Electrons in Thin Silicon Detectors", Phys. Rev. Lett. 40 (1978) 1242. 37. S. P. Moeller, private communication to H. Bichsel (1982), as reported in Refer• ence 42. 38. S. Hancock, F. James, J. Movchet, P. G. Rancoita and L. VanRossum, "Energy Loss and Energy Straggling of Protons and Pions in the Momentum Range 0.7 to 115 GeV Ie", Phys. Rev. A 28 (1983) 615. 39. D. West, "Measurement of the Energy Loss Distribution for Minimum Ionizing Electrons in a Proportional Counter", Proc. Phys. Soc. (London) A 66 (1953) 306. 40. F. Harris, T. Katsura, S. Parker, V. Z. Peterson, R. W. Ellsworth, G. B. Yodh, W. W. M. Allison, C. B. Brooks, J. H. Cobb and J. H. Mulvey, "The Exper• imental Identification of Individual Particles by the Observation of Transition Radiation in the X-Ray Region", Nucl. Instr. Meth. 107 (1973) 413. 41. N. Hasebe, J. Kikuchi, T. Doke, K. Nagata and A. Nakamoto, "Energy Loss of Relativistic Electrons and Its Fluctuation in Gas Proportional Counters", Nucl. Instr. Meth. 155 (1978) 491. 42. H. Bichsel, "Energy Loss and Ionization Spectra of Fast Charged Particles Travers• ing Thin Silicon Detectors", submitted to Rev. Mod. Phys. 43. D. W. O. Rogers and A. F. Bielajew, "Differences in Electron Depth-Dose Curves Calculated with EGS and ETRAN and Improved Energy-Range Relationships", Med. Phys. 13 (1986) 687. 44. M. J. Berger and S. M. Seltzer, "Stopping Powers and Ranges of Electrons and Positrons (2nd Ed.)", National Bureau of Standards report NBSIR 82-2550-A (1983); see also "Stopping Powers for Electrons and Positrons", International Commission on Radiation Units and Measurements (ICRU) Report 37 (1984). 45. S. M. Seltzer and M. J. Berger, "Bremsstrahlung Spectra from Electron Interac• tions with Screened Atomic Nuclei and Orbital Electrons", Nucl. Instr. Meth. B12 (1985) 95; and "Bremsstrahlung Energy Spectra from Electrons with Ki• netic Energy 1 keY - 10 GeV Incident on Screened Nuclei and Orbital Electrons of Neutral Atoms with Z = 1 - 100 ", Atom. Data and Nucl. Data Tables 35 (1986) 345. 46. See, e.g., W. Koch and J. W. Motz, "Bremsstrahlung Cross-Section Formulas· and Related Data", Rev. Mod. Phys. 31 (1959) 920. 47. R. H. Pratt, H. K. Tseng, C. M. Lee, L. Kissel, C. MacCallum and M. Riley, "Bremsstrahlung Energy Spectra from Electrons of Kinetic Energy 1 keY ~ T ~ 2000 keY Incident on Neutral Atoms 2~ Z ~ 92", Atom. Data and Nucl. Data Tables 20 (1977) 175; errata in 26 (1981) 477. 48. H. Davies, H. A. Bethe and L. C. Maximon, "Theory of Bremsstrahlung and Pair Production. II. Integral Cross Section for Pair Production", Phys. Rev. 93 (1954) 788; and H. Olsen, "Outgoing and Ingoing Waves in Final States and Bremsstrahlung", Phys. Rev. 99 (1955) 1335. 49. E. Haug, "Bremsstrahlung and Pair Production in the Field of Free Electrons" , Z. Naturforsch. 30a (1975) 1099. 50. H. Kolbenstvedt, "Simple Theory for K-Ionization by Relativistic Electrons", J. Appl. Phys. 38 (1967) 4785. 7. An Overview of ETRAN Monte Carlo Methods 181

51. A. M. Arthurs and B. L. Moisewitsch, "The K-Shell Ionization of Atoms by High-Energy Electrons", Proc. Roy. Soc. (London) A247 (1958) 550. 52. L. V. Spencer, "Energy Dissipation by Fast Electrons", National Bureau of Stan• dards Monograph 1 (1959). 53. S. M. Seltzer and M. J. Berger, "Electron and Photon Transport in Multi-Layer Media: Notes on the Monte Carlo Code ZTRAN", National Bureau of Standards report NBSIR 84-2931 (1984); see also Int. J. Appl. Radiat. Isot. 38 (1987) 349. 54. J. J. Dongarra, "Performance of Various Computers Using Standard Linear Equations Software in a FORTRAN Environment", Argonne National Labo• ratory report TM 23 (1986). 55. J. H. Scofield, "Theoretical Photoionization Cross Sections from 1 to 1500 keV", Lawrence Livermore National Laboratory Report UCRL-51326 (1973). . 56. L. Kissel, C. A. Quarles and R. H. Pratt, "Shape Functions for Atomic-Field Bremsstrahlung from Electrons of Kinetic Energy 1-500 keVon Selected Neutral Atoms 1~ Z ~ 92", Atom. Data and Nucl. Data Tables 28 (1983) 381. 8. ETRAN - Experimental Benchmarks

Martin J. Berger

Center for Radiation Research National Bureau of Standards Gaithersburg, Maryland 20899, U.S.A.

8.1 INTRODUCTION The ETRAN transport code is the realization of a Monte Carlo scheme for simulating electron histories carried out according to a "condensed random-walk" model. In this model, the sampling of individual elastic and inelastic collisions is replaced by the sampling of multiple-scattering deflections and energy losses in successive short path segments. An overview of the procedures involved can be found in Chapter 7.

The errors in transport results calculated with the ETRAN code arise from various causes: (a) the uncertainties or incompleteness of the single-scattering cross sections used as input; (b) the approximations inherent in the "condensed random walk" model; (c) the approximations made in the various sampling procedures; and (d) coding errors. Some of these sources of uncertainties are inter-related, which complicates an error analysis. An overall impression of the reliability of the ETRAN code can be obtained most directly by comparisons with experimental transport results.

The history of the "condensed-random-walk" model for electron transport can be traced to various publications in the nineteen fifties (e.g., Hebbard and Wilson1 ; Sidei et a1 2 , Leiss et a1 3 ). The ancestor of the ETRAN code was used for model stud• ies and calculations for a variety of electron transport problems at energies up to a few MeV (Berger4). This work was the basis for the development, in collaboration with S. M. Seltzer, of a series of ETRAN versions with increased capabilities. Photon transport was added, including the production and transport of secondary bremsstrah• lung photons. ETRAN thus acquired the capability of calculating all generations of an electron-photon cascade initially started either by electrons or photons.

The transport of photons is treated in ETRAN in almost direct analogy to the physical processes of scattering and absorption, so that the question of the validity of the Monte Carlo model does not arise. The benchmark comparisons in this paper therefore deal largely, if not entirely, with electron transport.

During the development of ETRAN, numerous comparisons were made with ex• perimental transport results in order to validate the Monte Carlo model and to check the adequacy of the cross-section database. Some of these comparisons were published

183 184 M. J. Berger in papers dealing with applications of ETRAN. Others were recorded only in notes or internal memoranda. The input for the present review consists of the accumulation of these comparisons, together with some new calculations.

The partially retrospective nature of this review raises the difficulty that the input cross sections (such as stopping powers, ranges, bremsstrahlung cross sections) changed gradually in the course of time. At the risk of over-simplification, it can be said that three different databases were used successively. The first database is described in Berger" and included the electron stopping powers described in Berger and SeltzerS, with density-effect corrections from Sternheimer6 and bremsstrahlung cross sections with empirical corrections recommended by Koch and Motz7 • The next database, used in the early seventies, included two changes: (a) stopping powers with the density• effect correction evaluated according to Sternheimer and Peierls8, and (b) the use of bremsstrahlung cross sections with empirical correction factors described in Berger and Seltzer9. In the light of later developments, change (b) was a definite improvement, but (a) was not. The third database, used at present, includes stopping powers and ranges given in ICRU Report 3710, and bremsstrahlung cross sections given in Seltzer and Bergerll.12• The photon cross sections for scattering, photoelectric absorption and pair production have also changed slightly in the course of the years. The most up-to• date values presently used can be obtained, for energies from 1 keY to 100 GeV, from a computer program described in Berger and Hubbell13• For the purpose of the present comparisons which deal mainly with electron transport, the small changes of the photon cross sections are unimportant.

Quantitatively, the changes from one database to the next were actually rather minor. A summary description of these changes can perhaps be made most easily in terms of electron ranges, computed in the continuous-slowing-down approximation (csda). These csda ranges depend both on the collision stopping power (including the density-effect correction) associated with the ionization and excitation of atoms or molecules, and the radiative stopping power associated with the emission of bremsstr• ahlung photons. Table 8.1 shows the percent deviations of the csda ranges in the 1964 and 1971 databases from those in the current (1987) database. These differences in most cases are smaller than 1 or 2 percent. Fig.3.1 in Chapter 3 shows the radiative stopping power for gold, and compares the values in the 1964, 1971 and 1987 databases. The 1964 and 1971 values are based on different empirical corrections; those from 1971 have turned out to be in rather good agreement with the values derived from our later synthesis of calculated radiative stopping powersll.

The 1964 database was used to generate the results given in Figs. 8.4 through 8.8, 8.11, 8.17, 8.19 through 8.22. The 1971 data bases was used for Figs. 8.3, 8.9, 8.10, 8.12, 8.13, 8.16, 8.18 and 8.24 through 8.27. The 1987 database was used for Figs. 8.1, 8.2, 8.14, 8.15 and 8.23.

The results of many transport calculations can be expressed with the target di• mensions (e.g., slab thicknesses) expressed in units of the csda range. Such results can easily be updated through the use of up-to-date range values. The comparisons with experimental results would have to replotted to take such an updating into account. Because of the amount of work involved, this has not been done; this is not serious, because the magnitude of most of the changes would be on the order of 1 to 2 percent. In some test cases, we have recently recalculated various transport results, and find it difficult to detect significant differences because the changes often are no larger than the statistical uncertainties of the Monte Carlo calculations. ETRAN - Experimental Benchmarks 185

Table 8.1. Percent deviations of the csda electron ranges in the 1964 and 1971 databases from those in the 1987 database.

10 MeV 1 MeV 0.1 MeV Material 1964 1971 1964 1971 1964 1971 Be -0.9 -0.4 -1.7 -1.8 -1.0 -0.9 C -1.3 -1.6 -0.8 -1.6 -0.2 -0.2 Al -0.3 0.7 -1.0 -0.7 -0.4 -0.5 Cu -2.9 -1.1 -2.7 -1.9 -0.7 -0.7 W -4.4 0.1 -2.6 -0.4 0.1 -0.1 Au -4.3 0.3 -3.0 -0.5 -0.4 0.0 Polystyrene -1.2 -1.1 -1.0 -1.0 -1.3 -1.2 Lucite -1.2 -1.3 -1.0 -1.4 -1.8 -1.3 Water -1.7 -1.2 -1.6 -1.4 -2.2 -2.2 Sodium Iodide -2.6 -1.5 -2.2 -0.8 -1.1 -0.9

All the results shown are plotted so as to correspond to the case of one electron incident on the target. The comparison between experimental and calculated results are absolute except for the results shown in Figs. 8.14 to 8.19 which pertain to distribu• tions of absorbed dose as a function of depth in a semi-infinite medium (an effectively unbounded medium in the case of Fig. 8.19). In these cases, the experimental curves have been normalized so that the area under them is equal to the area under the cor• responding calculated curve. These areas are equal to the total energy absorbed in the medium, e.g., the incident electron energy minus the (generally very small) amount of energy that escapes in the form of backscattered radiation.

In some of the comparisons, transmission or depth-dose curves have been plotted in a scaled manner, with distances expressed in units of the csda range at the energy of the incident electrons. Table 8.2 lists, for these figures, the range values that were used for the calculations and plots.

Except for the progressive elimination of coding errors, the ETRAN code has been rather stable in recent years. One exception should be noted. In the older versions of ETRAN, the sampling of energy losses from the Landau distribution was not quite correct due to the under sampling of rare, very large energy losses so that the average loss was too small. The error introduced thereby is energy dependent and would be largest for the comparisons involving depth-dose and transmission curves in low-Z materials at energies from 2 to 50 MeV where it could become as large as 5 - 8 percent. A correction for this deficiency has recently been incorporated into ETRAN, and is described in Chapter 7. The results in the present review, which are most strongly affected by this change (Figs. 8.1, 8.2, 8.14, and 8.15), have been recomputed with the latest version of the code. 186 M. J. Berger

Table 8.2. Values of the csda range ro assumed in various calculations. These values are listed in order to facilitate the interpretation of those figures in which distances are expressed in units of roo

Figure Material Energy ro (MeV) (g/cm2) 8.1 C 10.0 5.66 8.2 Al 10.0 5.86 8.3 " 0.05 0.00571 " 0.1 0.0186 " 0.25 0.0819 " 0.5 0.244 " 0.75 0.384 " 1.0 0.551 8.11a Al 20.0 10.54 " 10.0 5.84 " 4.0 2.48 " 3.0 1.86 8.11b Be 10.0 6.26 Cu 10.0 6.00 Ag 10.0 5.93 Pb 10.0 5.88 8.14 Lucite 21.2 10.20 8.15 Water 10.0 4.98 Polystyrene 10.0 5.16 8.17a Polystyrene 3.0 1.54 8.17b Al 3.0 1.58 8.18 Al 0.1 0.0186 Be 0.5 0.215 Al 0.5 0.224 eu 0.5 0.258 8.19 Air 0.054 5.63E-03 " 0.032 2.25E-03 " 0.012 4.00E-04 " 0.005 8.73E-05 ETRAN - Experimental Benchmarks 187

8.2 COMPARISONS

The results presented here are for targets irradiated by monoenergetic electron beams (except for Fig. 8.22 which pertains to an incident gamma-ray beam). The target shapes are simple (slabs, semi-infinite media, or cylinders), and the results pertain either to broad parallel beams, or to narrow pencil beams, for electrons incident perpendicularly on the targets. The t.argets are homogeneous and uniform, except for the multilayer targets in Figs. 8.23 a,b.

The problems for which comparisons with experimental transport data are pre• sented can be classified into the following categories: • Transmission coefficients for slab targets: transmission coefficient Figs. 8.1, 8.2 and 8.3. • Energy spectra of electrons transmitted through slab targets: Figs. 8.4, 8.5, 8.6a,b and 8.7. • Angular distributions of electrons transmitted through slab targets: Figs. 8.7, 8.8 and 8.9. • Reflection coefficients for semi-infinite targets: Fig. 8.1 Oa, b. • Energy spectra and angular distribution of electrons reflected from slab targets: Figs. 8.7 and 8.8. • Charge deposition as a function of depth: Fig. 8.11a,b. • Energy deposition as a function of depth: Figs. 8.12, 8.14, 8.15, 8. 17a,b, 8.18 and 8.19. • Energy deposition as a function of depth and radial distance from incident beam: Figs. 8.13 and 8.16. • Detector response functions (obtained by the convolution of statistical energy depositions with intrinsic noise functions): Figs. 8.20a,b, 8.21, and 8.22. • Energy deposition in composite slabs consisting of adjacent layers of different materials: Fig. 8.23a,b (calculated with the one-dimensional multilayer code ZTRAN, which is a descendant of ETRAN). • Emergence of secondary bremsstrahlung photons from thick targets irradiated by electron beams: Figs. 8.24 to 8.27.

8.3 DISCUSSION The number of Monte Carlo histories sampled were such that the statistical error of the ETRAN results in the comparisons was typically of the order of 1 - 2 percent. The systematic error can easily be several times greater, and is due not only to the uncertainties of the cross sections and of the the approximations inherent in the Monte Carlo model, but is also caused by the differences between the idealized geometric configurations assumed by ETRAN and the necessarily somewhat more complicated configurations in the experiments.

There are a number of approximations in the ETRAN code which in principle are avoidable. Important among these are the neglect of lateral spatial multiple-scattering deflections in track segments (see Chapter 5), and the lack of the appropriate correlation between sampled large-energy losses and the setting in motion of energetic secondary electrons. In the course of developing ETRAN, we experimented with more elaborate models which avoided some of these approximations. Comparisons with experimental 188 M. J. Berger data indicated that we did not thereby obtain significantly better results which would justify the additonal computation required.

We have kept the length of track segments (i. e., the step-sizes) in the Monte Carlo model as small as possible (subject to the restrictions imposed by the multiple-scattering theories used), in order to minimize model-dependent errors. The need to compute a large number of steps per particle history is compensated by the decrease of computation achieved through the avoidance of additional calculations per step required with more elaborate models.

From the many comparisons in this paper, one gets the overall impression that the compromises embodied in the ETRAN Monte Carlo model are satisfactory, and that the predictions made with the ETRAN code are reliable. The simpler the quantity of interest, the smaller are the discepancies between theory and experiment. For example, the discrepancies appear to be smaller for transmission coefficients than for spectra of transmitted radiation, and smaller for depth-dose curves than for absorbed-dose distributions as function of both depth and radial distance from the electron beam.

Nothing has been said here about the predictive power of the ETRAN Monte Carlo model for complex source-target configurations. However, this aspect is covered by J. Halbleib in Chapter 11 on the application sets of the TIGER and ACCEPT codes, which combine the same Monte Carlo model with complex geometry routines. Tp.e evidence presented in Chapter 11 indicates that the agreement with experiment is satisfactory.

The uncertainties of the results obtained with the ETRAN code shown in this chapter are due more to systematic than to statistical errors, and we find it difficult to make quantitative error estimates. In the best circumstances, the uncertainties might be as low as 2 to 3 percent. More commonly for simple quantities, they might be 5 percent, and for complex quantities, 10 to 20 percent. There seem to be few, if any, large discrepancies which would point to major features of the ETRAN code that should be changed. However, the ETRAN code undoubtedly could benefit from further fine tuning, for example in regard to the choice of step sizes in order to achieve the best possible combination of accuracy and computational efficiency.

Understanding of the various systematic uncertainties could be improved through further calculations for benchmark situations involving sensitivity studies with different cross-seCtion sets, as well as calculations with two or more codes based on different Monte Carlo models. The comparisons presented by D. Rogers in Chapter 14 are a good example of this. It also would be desirable to automate the database of experimental transport results so that comparisons with results from the latest version of each code could be made routinely with a minimum of labor.

Acknowledgement: This work was supported by the Office of Health and Environmental Research, U.S. Department of Energy. ETRAN - Experimental Benchmarks 189

1 .1

1.0

0.9

0.8

0.7

0.6 TN 0.5

0.4

0.3

0.2

0 .1

o z/ro Figure 8.1. Number transmission coefficient for IO-MeV electrons incident perpendicularly on a carbon slab target. The curve is calculated; the open points (from Ebert, Lauzon and Lent14), and solid points (from Harder15 ), are experimental.

1.1

1.0

0 .8

0.7

0.6 TN AI. 10 Mev 0.5

0.4

0.3 DEBERT ET AL . 10.2 MeV 0.2 "HARDER & POSCHET. 10.83 MeV

0 .1

o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 zlro Figure 8.2. Number transmission coefficient for IO-MeV electrons incident perpendicularly on an aluminum slab target. The solid curve represents the calculated transmission of primary plus secondary electrons; the corresponding experimental points are from Ebert, Lauzon and Lent14. The dashed curve is from a calculation in which the transmission of a primary electron and one or more associated secondary electrons is treated as a single event; the corresponding experimental points are from Harder and Poschet16. 190 M. J. Berger

1.0

• OUPOuy I'I OI. O.O~~V ~ \ . ~ ••. o.,_ 08

06

0 < •.." ... \ 01

o MIl\.£R -8 HtNORtCKS.O.25 MeV o MIl.I.ER 8 HENDRICK$,O.$ MeV o ~Uil!toll.O,5 M'V 0. SE t.t GE~. :;_25 MeV x 04JPO!JY" 01,0.25 MeV ;I: DUPOU Y 1101,0 ,' M eV

·0

0 .• · 0

0.2

o MIU.ER 8. HENDRICKS, 0.7'5 MeV 9 lIGU elC1I,O.7 MeV o NAKAI'IClI , O.e Mell oMllLEA .& I'1 ENORICKS. to MeV 6 SELIG(R,O.96 MtV ¢ N~I( A I .tOl, 1.0 MeV 0.8 + MII.L.E R, l O M,V

)I OUPOUY " 01 ,to MeV

06

0.4 lO- I MeV

0 .2

• 0

z/ro

Figure 8.3. Comparison of experimental and calculated number transmission coefficients for aluminum (from Seltzer and Berger17). The solid curves are calculated results pertaining to the tranmitted current. The dashed curves are from a calculation in which the transmission of a primary and one or more secondary electrons is scored as a single transmission event. ETRAN - Experimental Benchmarks 191

1.6 T._---""-=:':______------L

IA I I 1.2 I

;;;-'" 1.0 E v C,53.0 MeV > x '" 0.8

0.6 J I 0.4 £ { ------8SW~ ~~------L--- 0.2'--_-'-__-'-_--' __ -'- __'--_-L..!

\.6

I~ I

1.2

~ I(t.TL I N E 1.0 v > AI,53.6MeV x'" 0.8 ~ I 0.6

£ f 0.4 --- 0.2 0 Z. 9!cm 2

Figure 8.4, The most-probable energy loss, (b.T)p, and the full-width at half• maximum, W, of the energy-loss distributions for 53-MeV electrons incident perpendicularly on carbon and aluminum foils and emerging in the forward direction.

MC: Monte Carlo calculation.

L: Landau distribution (Landau18 ). BW: Landau distribution with binding correction according to Blunck and WestphaP9

The points with error bars are experimental results of Theissen and Gudden20 for carbon, and of Breuer21 for aluminum. 192 M. J. Berger

10° ,. Be M "- If>

I 1 >Q) 10- ~ 10 1

ll.

10°

Cu

I 0-2 '--'--'--'---'-----'----'_.1....-...1....--'---"-----' 0.6 1.0 1.4 0.6 1.0 1.4 .6T (MeV)

Figure 8.5. Energy-loss distributions for 15.7-MeV electrons transmitted through foils. Results are for electrons incident perpendicularly on foils of 748.3 mg/cm2 of Be, 858.6 mg/cm2 of AI, 840.0 mg/cm2 of Cu, and 965.8 mg/cm2 of Au. Experimental points are for electrons emerging from foil in the forward direction (at 0 degrees with respect to the incident· beam). The histograms are calculated as an average over the emergent directions between o and 5 degrees.

(i) Open points: Experiment of Goldwasser, Mills and Hanson22 •

(ii) Solid points: Experiment of Hall, Hanson and Jamnik23 •

The experimental distribution (i) is that given in the paper by Goldwasser et al. The later paper by Hall et aI, for the same experimental conditions, reported corrected values of the most-probable energy loss. Distribution (ii) has been obtained by shifting distribution (i) so that the most probable en• ergy loss agrees with that of Hall et aI, leaving the shape of the distribution otherwise the same. ETRAN - Experimental Benchmarks 193

10 AI. 1.51 MeV Cu. 1.54 MeV

10- 2 ,---L..-'-::-'c::---'_.,-l-::-_'---::-':::--' 10-2 ,:-,--,-_:-,::---,_.,,-L::_.l..--'-_...l.-..--'_-'-_.J...... --1._...l.-..--' 0.1 0.3 0.5 0.7 0.1 0.3 0.5 0.7 0.9 1.1 1.3

Cu. 2.64 MeV

.~32mg/cm2 .. 0 0 > o .° 000 • 0 ~ o 0 o

10- 1

10-2 ,::,--,--,-:-,::---,_.,,-L::.I..-.l..---::-'::---'---'-=--'-_.l..--'-_-'---' 0.1 0.3 0.5 0.7 0.9 1.1 1.3

AI. 3.70 MeV CU.3.66 MeV

10-2L:::--"-'--::"-::----l_-=-'=_"---:-L~ 10- 2 ':l-:-.....L-'::'::----'-':"-:--'---:!':::---'---::"::---'----,'----L-c'::---' 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 1.1 1.3 "T. MeV Figure 8.6a. Energy-loss distributions for electrons passing through aluminum and copper foils. The histograms represent Monte Carlo results averaged over emergent directions from 0 to 5 degrees. The points are experimental results of Van Camp and Vanhuyse24• 194 M. J. Berger

"0 203 mg/c'" Ag,2.64 MeV Au,2.63 MeV

Ag,3.68 MeV Au,3.68MeV

1~2.L-~~-a __u- __~~ __~ __~ __~~~~ __~ o 0.2 0.4 0.6 0.8 1.0 o 0.2 0.4 0.6 Il.T,MeV Figure 8.6b. Energy-loss distributions for electrons passing through silver and gold foils. The histograms represent Monte Carlo results averaged over emergent directions from 0 to 5 degrees. The points are experimental results of Van Camp and Vanhuyse24• ETRAN - Experimental Benchmarks 195

Pb Pb

10-4'----'-'---'-_-'---'---'_-'--' 10-3 :---'---:::-::--'~;!;:---'-----:-'::--'--c:' 0.4 0.8 1.2 1.6 0 0.4 08 1.2 1.6 AT,MIN Figure 8.7. Distribution of energy losses of transmitted and reflected electrons emerging from a foil at various angles with respect to the incident beam. An• gles of emergence between 0 and 90 degrees pertain to transmission, and angles between 90 and 180 degrees to reflection. The quantity plotted is the emer• gent current of electrons (per MeV and unit solid angle) corresponding to one incident electron. The results are for 1.75-MeV electrons incident perpendicu• larlJ on the foil. The histograms are calculated. The experimental points are from Frank25, for 544 mg/ cm2 AI, 364 mg/ cm2 eu and 463 mg/ cm2 Pb foils. The experimental data are for emergent angles of 0, 60, 120 and 150 degrees, and the calculated histograms are averages over Monte Carlo results for 0-5, 55-65, 115-125 and 145-155 degrees. 196 M. J. Berger

2.0 0.4

1.6 0.3

1.2

0.2

O.S CU,51.1 mg/cm2

0.1 AI, 545 mg/cm2 0.4

I...... !!! 0.4 en z ~ 0.4 l- 0.3 e.) I.LI ...J I.LI 0.3 I- Z 0.2 I.LI ~ 0.2 Cu,ISOmg/cm2 I.LI ~ 0.1 Cu, 364 mg/cm2 I.LI 0.1 U.o a: I.LIm ~ ~ 0.5 0.4

0.3

0.3

0.2

0.2 Cu,364 mg/cm2

0.1 Pb,463 mg/cm2 0.1

a (DEGREES) a (DEGREES) Figure B.S. Angular distribution of electrons transmitted by or reflected from foil targets. Left panel is for transmission, Right panel for reflection. Inci• dent electron energy 1.75 MeV, incident direction perpendicular to foil. The distributions shown include electrons with energies down to 150 keV. The histograms are calculated. The experimental points are from Frank25 • ETRAN - Experimental Benchmarks 197

ALUMINUM Eo. 1.0 MeV

0,.,0 RESTER 8 DERRICt

e (degrees) Figure 8.9a. Angular distributions of electrons with energies greater than 100 keV transmitted through aluminum targets. Results are for I-MeV elec• trons incident perpendicularly on foils of thickness z. Experimental results are from Rester and Derrickson26

ALUMINUM

Eo' 1.0 MeV

0,., D RESTER a DERRICKSON

w en

1.0

E (MeV) Figure 8.9b. Energy spectra of electrons transmitted through aluminum foils, integrated over all forward directions. Results are for I-MeV electrons incident perpendicularly on foils of thickness z. Experimental results are from Rester and Derrickson26 198 M. J. Berger

Figure 8.10a. Comparison of calculated and experimental number reflection coefficient RN , for electrons incident on aluminum foils of saturation thickness, as a function of the electron energy. Curve is calculated and includes primary and secondary electrons backscattered with energies greater than 1 keV (from Seltzer and Berger17).

0.6

o MILLER & HENDRICKS o NAKAI at 81 • COHEN & KORAL 0.5 • SALDICK & ALLEN .• • VERDIER & ARNAL -MILLER '" • RESTER & DERRICKSON A WRIGHT & TRUMP ...z w 0.4 • FREDERICKSON (; eGLASUNOV & GUGLYA. ii: IL W 8 z 0.3 0

5w ....I IL w It: 0.2

0.1 To=l MeV

20 40 60 80 100 ATOMIC NUMBER. Z Figure 8.10b. Comparison of calculated and experimental number and energy reflection coefficients for 1-MeV electrons incident on foils of saturation thick• ness, as a function of the atomic number of the foil. Curves are calculated and include primary and secondary electrons backscattered with energies greater than 1 keV (from Seltzer and Berger17). ETRAN - Experimental Benchmarks 199

1.6

1.2

~0.8 u Q .?ICI> 0.4

-0.4 -0.4

- 0.80~....L..."--;!0."""2 .<..-.l.~O':-.4,.-'-.L.....o.-,;O-'::.6~'-'--"J0."'"8 -'--'---'--,-'1.0 -0 ·SO:-'---'-'-,0f:.2O-,--.L.....o.--;:O;'-;.4c-'--'-'--"J0.76-'--'---'-c0;;..8;;-'-~71.0

2.4 2.4 ,.-.".-,-r--r-r-r,.,--r-r-'-"--,,,-,-r-,-,-,..--,

2.0 2.0

1.6 1.6

1.2 1.2

0.8 0.8

0.4

o

-0.4 -0.4

-0.8 -0.8

-1.2 ~1.2

o 0.2 0.4 0.6 0.8 1.0 1.2 0 0.2 0.4 0.6 0.8 1.0 1.2 z/ra z/ra

Figure 8.lla. Distribution in depth of the charge deposited by electrons inci• dent perpendicularly on thick aluminum foils with energies of 3, 4, 10, and 20 MeV. The distribution D(z) represents the amount of charge per incident elec• tron that is deposited at depth z per unit layer in the foil. The ordinate is the dimensionless quantity roDc(z)je, where ro is the csda range of the incident electrons, and e is the electronic charge. A negative value of D(z) indicates charge depletion due to the escape of secondary electrons. The foil thicknesses are larger than the electron range. The histograms are calculated. The ex• perimental points are from Tabata, Ito and Okabe21, except for the data for 3-MeV electron incident on aluminum, which are from Gross and Wright28; 200 M. J. Berger

5.6 2.0 4.8 n

4.0

3.2

2.4 Be Cu if 10 MeV 10 MeV .?ICU 1.6 o ------...L.....---"''10 0 0.8 -0.4

-0.8 -0.8 -1.2

- 1.6iO=-...... --':0~.2~'-'";;0"'";.4...... --':0~.6;-'-'--';;"0.';;-8 ...... ,1;';.0;-'-'-';-'1.2 0 0.2 0.4 0.6 0.8 1.0

2.0 2.0

1.6 1.6

1.2

Pb 0.4 10 MeV

o 00

-0.4 -0.4

-0.8 -0.8

-1.2 -1.2

o 0.2 0.4 0.6 0.8 1.0 1.2 0 0.2 0.4 0.6 0.8 1.0

z/ro Figure 8.l1b. Distribution in depth of the charge deposited. Similar to Fig. 8.11a, but for 10-MeV electrons incident perpendicularly on thick foils of Be, Cu, Ag and Pb. ETRAN - Experimental Benchmarks 201

..--.. 10 1 0 C\J~ 10 20 30 40 E ()

>Q) ~

I I- a. w 1 0 I- Z :J ...... Z 0 I- (f) 0 a. 10.1 60 w 0 >- (!) a:: w z W

10.2 0 100 200 300 400

DEPTH z (g/cm2 )

Figure 8.12. Energy deposition per unit depth in a semi-infinite water target irradiated by broad, parallel electron beams incident perpendicularly with energies between 60 and 1000 MeV. The histograms (and the curves in the insert) were calculated with the ETRAN code. Also shown (for a beam energy of 1000 MeV) are experimental results of Crannell et a1 29 , represented by circles and triangles, and results calculated by Nelson30with the EGS code, indicated by crosses, which are plotted at the midpoints of his histogram bins. 202 M. J. Berger

1

rt> ---E 0 10-1 ELECTRONS

:>CD To=1000 MeV ---::E w ::E ::::> ....J 0 > ...... !:::-::=~ z: 320-360 g/crn2 z ::::> ...... , "······x····· z x 0 ...... en ········x······ 0 ll. w 0 >- (!) a:: w z w x

o 10 30 40

Figure 8.13. Radial dependence of energy deposition in a water target irradi• ated by a lOOO-MeV pencil electron beam. The quantity plotted is the energy deposition per unit volume, normalized to one incident electron, averaged over the indicated depth intervals. In order to fit the results into one plot, the curves for different depths are shifted along the abscissa, with the zero radial distance indicated by an arrow. Solid Histograms: calculated with the ETRAN code. Crosses: Calculated by Nelson30with the EGS code. Dotted Histograms: Measured by Cranne1l29 • ETRAN - Experimental Benchmarks 203

LUCITE: 21.2 MeV 1 .1 POINTS: HARDER & SCHULTZ HISTOGRAMS: MONTE CARLO 1.0

0.9

0.8

0.7

~D 0.6 T o 0.5

0.4

0.3

0.2

0.1

o 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 z/ra Figure 8.14. Comparison of experimental and calculated electron depth-dose curves in a Lucite target irradiated by 21.2-MeV electrons. The histogram rep• resents results calculated with the latest version of the ETRAN code, which differ slightly from results of an earlier calculation reported in Caswell and Berger31. The points represent experimental results of Harder and Schulz32 obtained with an air-filled ionization chamber. The air ionization values have been converted to absorbed-dose values for Lucite using calculated Lu• cite/air stopping-power ratios averaged over the electron-flux spectra at vari• ous depths. 204 M. J. Berger

1.2

WATER 0.6 To = 10 MeV

0.4

0.2

~D°L-L-L-~L-~~~~~~-J~~~~~~~~~= To 1.2

0.8 POLYSTYRENE To = 10 MeV 0.6

0.4

0.2

Figure S.15. Comparison of calculated depth-dose curves in water and polystyrene at 10 MeV with experimental results of Brahme, Hulten and Svensson33• The histograms are broad beam results recently recalculated with the present ver• sion of the ETRAN code. The solid curves represent experimental results obtained without a beam spreading foil. The dashed curves represent the ex• trapolation of Brahme et al to the case of a monoenergetic, broad, parallel beam. ETRAN - Experimental Benchmarks 205

LUCITE. 10 MeV BEAM RADIUS R=1.25 mm

10-3OL---'----:O-':.O-::5 ---L---:O~.1-::0--'-O:-'.1~5:--L.-::-O.=20:--.L.--:::O-!:.25;-' p/ro

Figure 8.16. Comparison of calculated and measured radial absorbed-dose dis• tributions in a Lucite target irradiated by a narrow beam of lO-MeV electrons (beam radius 1.25 mm). The experimental points are from Lillicrap, Wilson and Boag34• 206 M. J. Berger

3.0 MeV '"c: POLYSTYRENE o POLYSTYRENE :;; 3 • MYLAR o • NYLON ~ u u .;;;E o "0 U 2 .~ o ! ~ ..Too E u > u :E

~ c O~ __~ __~~ __~ __~~~ ____~ __~~~~~~ __~ o U

Figure 8.17a. Comparison of calculated and measured depth-dose curves from 3-MeV electrons in polystyrene. The histograms were calculated with the ETRAN code. The curves are from a moment-method calculation of Spencer35 done in the continuous-slowing-down approximation. The points represent measurements with radiochromic-dye film dosimeters by McLaugh• lin and Hussmann36 .

3.0 MeV ALUMINUM o DYE FILM • ION CHAMBER !!. u .;;;E o "0

c o~o--~~~~--~--~~--~--~~--~~~~~~~.o z,

Figure 8.17b. Comparison of calculated and measured depth-dose curves from 3-MeV electrons in aluminum. The histograms were calculated with the ETRAN code. The curves are from a moment-method calculation of Spencer35 done in the continuous-slowing-down approximation. The points represent measurements with radio chromic-dye film dosimeters by McLaugh• lin and Hussmann36 . ETRAN - Experimental Benchmarks 207

2.0r---r-r---r-r---r-.----.-.----.---, AIGINGER 8 • GONAUSER, O.SMeV o Huffman et ai, 0.104 MeV a FRANTZ, 0.S7MeV I.S

1.0

AI Eo" O.IMeV 0.5

AIGINGER 8 • GONAUSER, 0.5 MeV • AIGINGER 8 2.5 GONAUSER, 0.5 Me " TRUMP el ai, O.S to(Z) a FRANTZ, O.SIS " TRUMP etal,O.5MeV • AGU et ai, O.S

2.0

AI Cu Eo" 0.5 MeV Eo" O.S MeV

0.5

Figure S.lS. Comparison of calculated and experimental depth-dose curves in various materials (from Seltzer and Berger17). The results for 100-keV electrons in aluminum are for an effectively unbounded medium, with a plane• perpendicular source at z = O. The other results are for bounded semi-infinite media. The experimental distributions have been normalized so that the total energy deposited in the target is equal to the calculated value. 208 M. J. Berger

l.6

l.2

0.8

..-.. 0.4 E N /. ~ 0 "

81~ .-~,-~-.-.-.--~.-,-.-~-.-. 1.6

1.2

0.8

0.4

o - ·0.2 Zm Iro Figure 8.19. Comparison of calculated and experimental depth-dose distribu• tions in an unbounded air medium, for plane-perpendicular sources of 54, 32, 12 and 5-keV electrons (from Berger, Seltzer and Maeda37). The histograms were calculated with the ETRAN code. The dashed curve, for 32 keV, is from a moments-method calculation by Spencer35 using the continuous-slowing-down approximation. The solid points represent results from a csda Monte Carlo calculation in which all individual elastic scatterings were sampled. The curves for 5, 12 and 54 keY, and the circles (0) for 32 keY are from an experiment by Griin38• ETRAN - Experimental Benchmarks 209

10T---;----,--r--~___r---.

Z 11 6 1Jol m l • 1051'm '/'• • 0.027 '/'.' 0 .046

10' t-

-

-, 10 - ~ 1 1 1 I

..c: z t. 191-"m Z/'• • 0 .084 ~ a:

Z 0 I- U Z ::::0 U.

W (f) Z 0 a. (f) w a:

• • ,0001'''' l' 30001'"' ./r•• 0.437 z/'o. 1. 31

0.8 I. RELATIVE PULSE HEIGHT h ho Figure S.20a. Response of silicon detectors to I-MeV electrons (from Berger et aI39 ). The points represent measured response functions. The histograms are calculated response functions, obtained by folding a statistical energy• deposition distribution with an experimental Gaussian noise function. The response functions are plotted vs. the relative pulse height hjho , where ho is the most probable pulse height corresponding to the absorption of the entire electron energy in the detector. 210 M. J. Berger

z.611'm z.1051'm z/'.= 0.179 u ••. 0.309

z o I• () Z ::J IJ.. W (/) Z o a... (/) w a::

Figure 8.20b. Response of silicon detectors to O.25-MeV electrons (from Berger et aI39 ). The points represent measured response functions. The histograms are calculated response functions, obtained by folding a statistical energy• deposition distribution with an experimental Gaussian noise function. The response functions are plotted vs. the relative pulse height hjho , where ho is the most probable pulse height corresponding to the absorption of the entire electron energy in the detector. ETRAN - Experimental Benchmarks 211

0.4 To=15MeV NARROW ELECTRON BEAM 5"-5" Na! DETECTOR

>CD ::E

- ~ 0.2 cr

h (MeV) Figure 8.21. Response of a 5" x 5" NaI detector to I5-MeV electrons (from Berger and Seltzer40). The points (0) are from an experiment by Koch and Wyckoff41• The calculated response function was obtained by convoluting an energy-deposition distribution (calculated with the ETRAN code) and a Gaussian representing a 12-percent intrinsic resolution .

.... 10-1 I

>Q) ::::!: .s:::.o ~ c::: 10-2 EO=3.I3 MeV POINT ISOTROPIC SOURCE 3"x3" NaI DETECTOR

Figure 8.22. Response of a 3" x 3" NaI detector to 3.13-MeV gamma rays (from Berger and Seltzer40). The points are from an experiment of Heath42 with a sulfur-37 gamma-ray source at a distance of 10 em from the detector. 212 M. J. Berger

I I I () I I Q) a; Be I Au I Be I I I CI 3 I I '"E () o~~

I I I I I I I I

0.1 0.2 0.3 0.4 0.5 DEPTH, 9 cm-2 Figure 8.23a. Comparison of calculated and experimental absorbed-dose dis• tributions from I-MeV electrons in composite Be/Au/Be slab targets. The points (0) are from calorimeter measurements by Lockwood et a 1"3 • The cal• culated curves were obtained with latest version of ZTRAN code, and differ slightly from earlier results given in Seltzer and Berger44•

4 I AI : Au : Al () Q) I I 0; q I Cl3 N E ()

~-0 >Q) a: 1:; ::! 2.2 ~-w 0 i= ~ iii 0 :r; n. w 0 > Cl a: zw w

0 .1 0.2 0.4 0.5 DEPTH, 9 cm-2 Figure 8.23b. Comparison of calculated and experimental absorbed-dose dis• tributions from I-MeV electrons in composite AI/ Au/ Al targets. The points (0) are from calorimeter measurements by Lockwood et a143• The calculated curves were obtained with latest version of ZTRAN code and differ slightly from earlier results given in Seltzer and Berger44• ETRAN - Experimental Benchmarks 213

• 0.025 : •

~ . .8 0.015 ~ . ::;: . Q) ...c: ~ N • 0.005

o 90 8, degrees Figure 8.24. Angular distribution of the bremsstrahlung intensity I from a thick tungsten target. Results are for lO-MeV electrons incident perpendicu• larlyon a 8.3-g/cm2 target. The experimental points are from Jupiter, Hatcher and Hansen45 • (From Berger and Seltzer9).

W TO=IO MeV

,~ '"

'>CP ::;: ~ 10-2 8=0·

1O-3 1,----:-':---:-'-:----='":---:-'-::--...,-I:,....J o 0.2 0.4 0.6 0.8 1.0 0 0.2 0.4 0.6 klTo Figure 8.25. Bremsstrahlung spectra from thick tungsten target, emitted at angles of 0 and 12 degrees with respect to the incident electron beam. klTo is the ratio of the photon energy to the kinetic energy of the incident electron. Calculated results are for a lO-MeV electron beam incident perpendicularly on a 5.8 g/cm2 target. Experimental points are from an experiment of Starfelt and Koch46 with 9.66-MeV electrons. (From Berger and Seltzer9 ). 214 M. J. Berger

10'

T =10.0 MeV

o 0' DELL, et. al. - THIS CALCULATION

Figure 8.26. Spectra of bremsstrahlung emitted in forward direction. Calcu• lated results are for 20.9-MeV and 10-MeV electron beams incident perpen• dicularlyon a thick target. The experimental points are from O'Dell et a147 and pertain to photons emerging, in a small angular region around 0 degrees, from a composite target consisting of 0.49 g/cm2 of tungsten followed by 0.245 g/cm2 of gold. The histograms represent calculated spectra, summed over the same angular region, for a 9.735 g/cm2 tungsten target. (From Berger and Seltzer9). ETRAN - Experimental Benchmarks 215

Fe To=2MeV

~ 10-2 e u "* 10-3 'v; ~ 10-4 ::;; u: ..

10-£

0.1 0.01 0.1 k, MeV Figure 8.27a Spectra of bremsstrahlung from thick foils, emitted at various angles with respect to the direction of the incident electrons. (From Berger and Seltzer39). Results for 2-MeV electrons incident perpendicularly on 1.74 g/cm2 of aluminum, 1.30 g/cm2 of iron, and 1.62 g/cm2 of gold. Experimental points are from Dance and Baggerly48; histograms are calculated. 216 M. J. Berger

10 20 30 40 50 60 70

~ .. ..

,. ec: u CD q;

Sn To=IOOkeV

10- 4 '--______"--_-----J

e= 1100

Au To=IOOkeV

10-4 LO--'-2.L0--'-....I.40-'-....l6-=-0-"--"SLO""--cILOO::-'-I...l.20.,-'--'140· 0 20 40 60 SO 100 120 140 k,keV Figure 8.27b Spectra of bremsstrahlung from thick foils, emitted at various angles with respect to the direction of the incident electrons. (From Berger and SeltzerS9). Results for 50-keV and lOO-keV electrons incident perpendicularly on 3.8 mg/cm2 of aluminum, 21.6 mg/cm2 of tin, and 19.3 mg/cm2 of gold. Experimental points are from Placious49; histograms are calculated. ETRAN - Experimental Benchmarks 217

REFERENCES 1. D. V. Hebbard and P. R. Wilson. "The Effect of Multiple Scattering on Electron Energy Loss Distributions". Austral. J. Phys. 8 (1955) 90. 2. T. Sidei, T. Higasimura and K. Konosita, "Monte Carlo Calculation of the Mul• tiple Scattering of the Electron", Memoirs of the Faculty of Engineering, Kyoto University, 19 (No. II) (1957). 3. J. E. Leiss, S. Penner and C. S. Robinson, "Range Straggling of High Energy Electrons in Carbon", Phys. Rev. 107 (1957) 1544. 4. M. J. Berger, "Monte Carlo Calculation of the Penetration and Diffusion of Fast Charged Particles", in Methods in Computational Physics, Vol. 1, (Academic Press, New York, 1963) 135. 5. M. J. Berger and S. M. Seltzer, "Tables of Energy Losses and Ranges of Electrons and Positrons", NASA Special Publication SP-3012 (1964). 6. R. M. Sternheimer, "The Density Effect for the Ionization Loss in Various Ma• terials", Phys. Rev. 88 (1952) 85. 7. H. W. Koch and J. W. Motz, "Bremsstrahlung Cross-Section Formulas and Re• lated Data", Rev. Mod. Phys. 31 (1959) 920. 8. R. M. Sternheimer and R. F. Peierls, "General Expression for the Density Effect for the Ionization Loss of Charged Particles", Phys. Rev. B3 (1971) 3681. 9. M. J. Berger and S. M. Seltzer, "Bremsstrahlung and Photoneutrons from Thick Tungsten and Tantalum Targets", Phys. Rev. C2 (1970) 621. 10. International Commission on Radiation Units and Measurements (ICRU), "Stop• ping Powers for Electrons and Positrons", ICRU Report 37 (1984). 11. S. M. Seltzer and M. J. Berger, "Bremsstrahlung Spectra from Electron Inter• actions with Screened Nuclei and Orbital Electrons", Nucl. Instr. Meth. B12 (1985) 95. 12. S. M. Seltzer and M. J. Berger, "Bremsstrahlung Energy Spectra from Electrons with Kinetic Energy 1 keY - 10 GeV Incident on Screened Nuclei and Orbital Electrons of Neutral Atoms with Z = 1 - 100", Atom. Data and Nucl. Data Tables 35 (1986) 345. 13. M. J. Berger and J. H. Hubbell, "XCOM: Photon Cross Sections on a Personal Computer", National Bureau of Standards report NBSIR 87-3597 (1987). 14. P. J. Ebert, A. F. Lauzon and E. M. Lent, "Transmission and Backscattering of 4.0- to 12.0-MeV Electrons", Phys. Rev. 183 (1961) 422. 15. D. Harder, Habilitationsschrift, U.of Wurzburg (1965). 16. D. Harder and G. Poschet, "Transmission und Reichweite schneller Elektronen im Energiebereich 4 bis 20 MeV", Phys. Lett. 24B (1967) 519. 17. S. M. Seltzer and M. J. Berger, "Transmission and Reflection of Electrons by Foils", Nucl. Instr. Meth. 119 (1974) 157. 18. 1. Landau, "On the Energy Loss of Fast Particles by Ionization", J. Phys. USSR 8 (1944) 201. 19. O. Blunck and K. Westphal, "Zum Energieverlust energiereicher Elektronen in dunnen Schichten", Z. Phys. 130 (1951) 641. 20. H. Theissen and F. Gudden, "Energieverlust von 53-MeV-Elektronen in Graphit", Z. Phys. 191 (1966) 395. 21. H. Breuer, "Energieverlust von Elektronen in Aluminium im Energiebereich 20 bis 60 MeV", Z. Phys. 180 (1964) 209. 218 M. J. Berger

22. E. L. Goldwasser, S. E. Mills and A. O. Hanson, "Ionization Loss and Straggling of Fast Electrons", Phys. Rev. 88 (1952) 1137. 23. H. E. Hall, A. O. Hansen and D. Jamnik, "Most Probable Energy Loss of Fast Electron", Phys. Rev. 115 (1959) 633. 24. K. J. Van Camp and V. J. Vanhyse, "Thick-Target Energy Loss Distributions of Electrons", Z. Phys. 211 (1968) 152. 25. H. Frank, "Zur Vielfachstreuung und Riickdiffusion schneller Elektronen nach Durchgang durch dicke Schichten", Z. Naturforsch. 14a (1959) 247. 26. D. H. Rester and J. H. Derrickson, "Electron Transmission Measurements of AI, Si, and Au Targets at Electron Bombarding Energies of 1.0 and 2.5 MeV", J. AppI. Phys. 42 (1971) 714. 27. T. Tabata, R. Ito and S. Okabe, Annual Report of the Radiatiation Center of Osaka Prefecture 9 (1968) 34; also private communication. 28. B. Gross and K. A. Wright, "Charge Distribution and Range Effects Produced by 3-MeV Electrons in Plexiglass and Aluminum", Phys. Rev. 114 (1959) 72. 29. C. J. Crannell, H. Crannell and H. D. Zeman, "Electron-Induced Cascade Show• ers in Water and Aluminum", Phys. Rev. 184 (1969) 426. 30. W. R. Nelson, private communication (1978). 31. R. S. Caswell and M. J. Berger, "Theoretical Aspects of Radiation Dosimetry", U.S. Atomic Energy Commission Publication LA-5180-C (1973) 60. 32. D. Harder and H. J. Schultz, Paper given at European Congress of Radiology, Amsterdam (1971); also private communication. 33. A. Brahme, G. Hulten and H. Svensson, "Electron Depth Absorbed Dose Distri• bution for a 10 MeV Clinical Mictrotron", Phys. Med. BioI. 20 (1975) 39. 34. S. C. Lillicrap, P. Wilson and J. W. Boag, "Dose Distributions in High Energy Electron Beams: Production of Broad Beam Distributions from Narrow Beam Data", Phys. Med. BioI. 20 (1975) 30. 35. L. V. Spencer, "Energy Dissipation by Fast Electrons", National Bureau of Stan• dards Monograph 1 (1959). 36. W. L. McLaughlin and E. K. Hussmann, "The Measurement of Electron and Gamma-ray Dose Distributions in Various Media", in Large Radiation Sources for Industrial Processes, Int. Atom. Energy Agency Publication IAEA SM- 123/43 (1969). 37. M. J. Berger, S. M. Seltzer and K. Maeda, "Energy Deposition by Auroral Elec• trons in the Atmosphere", J. Atmos. Terr. Phys. 32 (1970) 1015. 38. A. E. Griin, "Lumineszenz-photmetrische Messungen der Energieabsorption im Strahlungsfeld von Elektronenquellen", Z. Naturforsch. 12a (1957) 89. 39. M. J. Berger and S. M. Seltzer, "Penetration of Electrons and Associated Brem• sstrahlung Through Aluminum Targets" , in Protection Against Space Radiation, NASA Publication SP-169 (1968) 285. 40. M. J. Berger and S. M. Seltzer, "Response Functions for Sodium Iodide Scintil• lation Detectors", Nucl. Instr. Meth. 104 (1972) 317. 41. H. W. Koch and J. M. Wyckoff, "Response of a Sodium Iodide Scintillation Spectrometer to 10- to 20-Million-Electron-Volt Electrons and X-Rays", National Bureau of Standards J.Res. 56 (1956) 319. 42. R. L. Heath, "Scintillation Spectrometry Gamma-Ray Spectrum Catalog", U.S. AEC Publications IDO-16880-1 and 2 (1964). ETRAN - Experimental Benchmarks 219

43. G. J. Lockwood, 1. E. Ruggles, G. H. Miller and J. A. Halbleib, "Calorimet• ric Measurement of Electron Energy Deposition in Extended Media-Theory vs Experiment", Sandia Laboratories report SAND 79-0414 (1980). 44. S. M. Seltzer ad M. J. Berger, "Energy Deposition by Electron, Bremsstrahlung and Co Gamma-Ray Beams in Multi-Laayer Media", Int. J. Appl. Radiat. Isot. 38 (1987) 349. 45. C. P. Jupiter, J. R. Hatcher and N. E. Hansen, Bull. Am. Phys. Soc. 9 (1964) 338. 46. N. Starfelt and H. W. Koch, "Differential Bremsstrahlung Measurements of Thin• Target Bremsstrahlung", Phys. Rev. 102 (1956) 1598. 47. A. A. O'Dell, C. W. Sandifer, R. B. Knowlen and D. George, "Measurement of Absolute Thick-Target Bremsstrahlung Spectra", Nucl. Instr. Meth. 61 (1968) 340. 48. W. E. Dance and 1. 1. Baggerly, Ling-Temco-Vaught Res. Center report 0 - 71000/5R-12 (1965). 49. R. Placious, "Dependence of 50- and 100-keV Bremsstrahlung on Target Thick• ness, Atomic Number and Geometric Factors", J. Appl. Phys. 38 (1967) 2030. 9. Applications of ETRAN Monte Carlo Codes

Stephen M. Seltzer

Center for Radiation Research National Bureau of Standards Gaithersburg, Maryland 20899, U.S.A

9.1 INTRODUCTION

Of the many possible applications of electron-photon Monte Carlo calculations to prob• lems in radiation physics, a significant subset can be carried out in rather simple ge• ometry, such as slabs or cylinders composed of a single, or perhaps a few, materials. Consideration of simple geometries tends to aid our understanding of the transport re• sults, and often facilitates the development of calculated datasets which can be applied to the solution of a general set of problems without further recourse to the Monte Carlo calculations. In this paper we give a few examples of such problems, which have been investigated through ETRAN Monte Carlo calculations.

9.2 RESPONSE OF PHOTON DETECTORS FOR SPECTROMETRY

In this section, some problems are described that involve the calculation of the response functions of N aI and Ge detectors which are used to measure photon spectra. The conversion of a measured pulse-height distribution into the true incident gamma-ray spectrum requires accurate and detailed information on the response of the detector to monoenergetic photons. The limited number of suitable monoenergetic gamma-ray sources and the complex structure of the response function itself makes it extremely difficult to develop the necessary body of data solely on an experimental basis.

We can write the response function R( E, h) as the convolution of two distributions:

E R(E, h) = JD(E, E) G(E, h) dE , (9.1) o where R(E, h) is the probability per unit pulse height that a photon incident with energy E will produce a pulse of height h, D(E, E) is the probability per unit deposited energy that the incident photon deposits energy E in the detector, and G( E, h) is the probability per unit pulse height that the deposition of energy E gives rise to a pulse of height h. The detector resolution function G( E, h), which depends on the efficiency and

221 222 S. M. Seltzer statistics of signal collection, is usually assumed to be a Gaussian whose full-width at half-maximum as a function of f is best determined experimentally. *

The energy-deposition spectrum D(E, f) can be determined rather accurately by Monte Carlo calculations. In general, this spectrum has the form of a line spectrum plus a continuum C(E, f):

D(E, f) = C(E, f) + Po 8(f - E) + Pl 8(f - E + me2 ) (9.2) + P2 8(f - E + 2me2) + Pz 8(f - E + Ez ).

In Eqn. 9.2, 8 is the Dirac delta function, Po is the probability that the photon energy E is completely absorbed (i.e., the area under the total absorption peak or the pho• topeak efficiency), and Pz is the probability that the full energy is deposited, except for the escape of a fluorescence x-ray of energy Ez (fluorescence escape peak) produced in photoelectric-absorption events. For incident photon energies above", 1 Me V where pair production can occur, Pl and P2 are the probabilities that the entire energy is deposited, except for the amounts me2 and 2mc2 carried out of the detector by one or two unscattered annihilation quanta (single-annihilation and double-annihilation es• cape peaks, respectively). Depending on detector dimensions and photon energy, these components can depend not only on the escape of the photon scattered in the detector, but also on the escape of secondary electrons and their bremsstrahlung.

The detection efficiency ." is the probability that the incident photon will have at least one interaction in the detector leading to the deposition of energy, and thus is equal to the integral of D(E, f) over all f (or, equivalently, the integral of R(E, h) over all h). However, the evaluation of the detection efficiency can be accomplished independently through the solution of the geometrical problem without knowledge of the energy-deposition spectrum. This can be used to advantage because the detection efficiency is a good scaling parameter: dividing D(E, f) by." reduces its dependence on detector dimensions and on the photon angle of incidence.

9.2.1 Nal Detectors The Apollo gamma-ray spectrometer. A program was undertaken in the early 1970's to characterize rather completely the response functions for a 3" (diam.) X 3" (height) cylindrical N aI detectorl. The goal> of this work was to provide the means to generate an extensive, reliable library of monoen• ergetic response functions for use in the pulse-height deconvolution algorithm developed by Trombka2, which was used in the analysis of gamma-ray measurements made on the Apollo flights 15 and 16. These experiments had two purposes: those from lunar orbit measured the cosmic-ray induced and natural emission of gamma-ray lines to provide information for the elemental analysis and geochemical mapping of the Moon's surface; and the measurements during trans-Earth flight were of the diffuse cosmic gamma-ray spectrum whose shape has significant astrophysical implications.

The validity of our ETRAN Monte Carlo calculations was tested through compar• isons of calculated and experimental response functions for various sized detectors at

* The detector resolution usually is expressed as the quantity FWHM/ f, for some reference value of the absorbed energy, €. The resolution is, in general, a function of f. 9. Applications of ETRAN Monte Carlo Codes 223 photon energies up to 20 MeV. Some typical comparisons are presented by Berger in Chapter 8. Systematic calculations of the energy-deposition spectrum were then done for the bare 3"x3" NaI crystal (with no surrounding material) exposed to broad parallel beams of gamma rays incident along the cylinder axis with energies from 0.1 to 20 MeV, based on sample sizes of 50,000 incident photons. The calculated components of the energy-deposition spectrum were refined into a scaled, smooth dataset from which one could easily interpolate to any incident photon energy and, through the evaluation of Eqn. 9.1, generate the complete response function.

Fig. 9.1 shows the resultant peak probabilities divided by the detection efficiency for the 3" X 3" detector. Corresponding results calculated without taking into account the escape of secondary electrons and bremsstrahlung also are shown, and indicate the rather significant effect above about 3 MeV of including electron transport. Be• cause the dimensions of the actual flight detector were somewhat different, the dataset was recalculated for a 2.75" x 2.75" crystal. Fig. 9.2 shows a family of complete re• sponse functions generated for this detector, assuming a resolution of 7.5% for the 137Cs 662-keV gamma ray.

10-1

>• I• :::::; co Ci§ 10-2 o c:: Co.

10-3

10-4L--L~LUllU~-L~UW-i~~~ 0.1 10 100 E ,MeV Figure 9.1. Total absorption peak (Po), single- and double-annihilation radi• ation escape peaks (PI and P2 ) and iodine K-shell fluorescence escape peak (P.,), for a 3" x 3" NaI detector. All quantities are divided by the detector efficiency "I. Dashed curves are calculated disregarding the escape of brems• strahlung and secondary charged particles.

In the analysis of the flight data, a number of background effects had to be sub• tracted before the measured data could be converted to true incident photon spec• tra. This was especially important in the analysis of the diffuse cosmic gamma-ray measurements in which about 80% of the measured counts were from unwanted back• ground sources. The various background components and, in some cases, the Monte Carlo calculations done to study them, are discussed in Trombka et a1 3 • One of the larger background components, and one that is relevant to this discussion of response• function calculations, is the pulse-height distribution from the decay of radioactivity 224 S. M. Seltzer that is induced in the central crystal due to bombardment by the protons, cosmic rays, and secondary neutrons present in the space environment.

12.000 TI:I 2'0. 000 IIIE". 11.5.0 1Ii£y.lCI'IAM'N[l 7.s Z IliESOI.UTIOJot

r 10 1~r-flL~~~~~~~~~~~------~ I- en z w I• Z w 100 ~~~"" > ~ --l W a::

E -2MeV

100 200 300 400 CHANNEL

Figure 9.2. Family of response functions for a 2.75" X 2.75" Nal detector, with an assumed resolution of 7.5% at 662 keY. Curves are normalized such that the area under each is equal to 45 times the detection efficiency. Results pertain to the case of a broad parallel beam incident on the flat end face of the detector.

The theoretical estimation of this background component (see Dyer et at") consists of two parts: the calculation of the rate of isotope production and decay in the detector (i.e., the source function), and the calculation of the pulse-height distribution due to the decay of the internal radioactive products. The second part was done by adding to the ETRAN code the sampling from the complete decay scheme of radioactive atoms uniformly distributed over the detector volumeS. Some typical results for the response of the Apollo detector are shown in Fig. 9.3, Fig. 9.4 and Fig. 9.5 .. The decay of 22Na (primarily by f3+ to an excited state of 22Ne which then de-excites by gamma-ray emis• sion) gives rise to the pulse-height distribution shown in Fig. 9.3. The periodic nature of the distribution is a result of the shift of the f3+ spectrum by combinations of the total absorption of the 1.275-MeV gamma ray and the 0.511-MeV photons produced in the annihilation of the f3+ particles. The gamma ray appears as a discrete line only due to total-absorption events following the decay to 22Ne by electron capture (about 9.5% of the decays). Figure 9.4 shows the pulse-height distribution for the decay of 1241 (77.3% by electron capture, 22.7% by 3 f3+ groups, and by 30 gamma rays in the de-excitation of 12"Te). The prominent peaks are due to combinations of the total absorption of the K-shell binding energy and of the three most-probable emitted gamma rays. Figure 9.5 shows results for 123mTe which decays by two successive transitions: an 88-keV transi• tion, preferentially by internal-conversion electron emission, and a 159-keV transition, preferentially by gamma-ray emission. Notable features here are the Compton shoulder due to incomplete absorption of the 159-keV gamma ray, and the small satellite peak due to the escape of the K x-ray associated with the internal conversion of a K-shell electron. 9. Applications of ETRAN Monte Carlo Codes 225

Y 2mc 2 ! I mTY , 2mc 2+y

>Q) :;: I :2 Ci"

22Na 10-2

10 -3 '--=-'-::----:"-:---::-'-::--::-'-:--c'::-__:_'::--=-'-:--....,.-L:__-::-'-::--::-'-:----::-'::---::-L--="-::---:-':~7 o 1.6 1.8 3.0 h, MeV Figure 9.3. Pulse-height distribution produced by 22Na uniformly distributed within a 7-cm x 7-cm NaI detector, with a resolution of 7.5% at 662 keY. The energies of annihilation quanta (mc2) and the 1.275-MeV gamma ray (,), involved in the decay of 22Na, are indicated. The results are normalized to one decay.

10

124,

10-3 ':---=,O----:"-:---:"7-~:__:-'::----:-'::---:-'-::--~__:_'::--::':----::--::---::'_:_-::'::--::'::--=' o 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 h, MeV

Figure 9.4. Pulse-height distribution produced by 1241 uniformly distributed within a 7-cm x 7-cm Nal detector, with a resolution of 7.5% at 662 keY. The peaks indicated correspond to the absorption of the K-shell binding energy (B: 0.032 MeV) and some high-intensity gamma rays (,1: 0.603 MeV, /2: 0.723 MeV, /3: 1.691 MeV). The results are normalized to one decay. 226 S. M. Seltzer

h, MeV Figure 9.5. Pulse-height distribution produced by 123mTe uniformly distributed within a 7-cm X 7-cm NaI detector, with a resolution of 7.5% at 662 keV. The energy of the K x-ray (x: 0.028 MeV) and the transition energies (-Yl: 0.088 MeV, ')'2: 0.159 MeV) are indicated. The results are normalized to one decay.

Measurement of the NBS 6OCo calibration beam spectrum. The spectrum of the collimated 60CO gamma-ray beam used for instrument calibration at the National Bureau of Standards was measured using a 5" (diam) x4" NaI detector6. The detector was collimated to admit a beam 3.8 cm in diameter (see Fig. 9.6). The measured pulse-height distributions were unfolded using Trombka's numerical least• square matrix-inversion technique. The response functions for monoenergetic photons were generated through ETRAN calculations, in the manner described for the 3" x 3" and the 2.75" x 2.75" detectors.

The photofraction, Pol"" is given in Fig. 9.7 for the 5" x 4" detector. The dashed curve gives results obtained by neglecting the escape of unscattered iodine x-rays, so that the difference between the curves represents the fluorescence escape fraction Pz/",. Complete response functions, including the Compton continuum, are shown in Fig. 9.8, taking into account the detector resolution of 11% for the 137CS 662-keV line.

Typical results for the unfolded true spectrum of scattered gamma rays in the beam are shown in Fig. 9.9. These results are for the case of the source collimator set to produce a field of 25 cm x 25 cm at a distance of 80 cm, and represent the scattered spectrum measured at a source-to-detector distance of ",,1 meter. The contribution from photons scattered to energies 'between 0.1 and 1 Mev is 19% of the total number of incident photons. The shape of scattered spectrum above 1 Me V could not reliably be deduced from the unfolded data; instead, the area under the curve in this region (",,5%) was determined, and the spectrum was represented by flat distributions for the scattered component from each of the 1.17 and 1.33-MeV lines. 9. Applications of ETRAN Monte Carlo Codes 227

6OCo pe llets

-108 em

Nol (Ttl

Figure 9.6. Experimental setup for the 60Co beam spectral measurements. The source-capsule insert shows the activated 60Co pellets backed by a tungsten plug; the capsule is of stainless steel.

0.9 z CI i= uc a:::... 0.8 CI t- CI ::c a..

0.7

0.1 1.0 E • MeV

Figure 9.7. Calculated photofractions for a 5" (diam.) x 4" (height) NaI de• tector. Results are for a 3.8-cm diameter beam of photons. Dashed curve was obtained by disregarding escape of iodine fluorescence x-rays. 228 S. M. Seltzer

10'

RESP~NSE FUNCT I ~NS 4.00·XS.OO· CYLINORICAL NAI CRYSTAL 0.1 0.~50.2 7.J KEY/CHANNEL ri ~ 0.4 lo.eX RESOLUTION 10'

r ~ ~lOI

~ Z

W > ~~\~/ ~tOO \ -' w 0:: \ V \ \

10-I v \ 10-. \

CHANNEL

Figure 9.8. Calculated response functions for a 5" X 4" NaI detector, for a beam diameter of 3.8 cm and for photon energies from 0.1 to 1.5 MeV. The resolution is 11% for the 662-keV 137Cs line, and the channel width is 7.1 keV. The area under each curve is equal to 7.1 times the detection efficiency.

(f)z .0 of• I a. tJ;j I ----- I .... 5% 19.0 ± 1.7% I I I

o 0.4 0.8 1.2 PHOTON ENERGY (MeV) Figure 9.9. Spectrum of scattered photons from one of the NBS collimated 60CO instrument calibration sources. Results pertain to a 31-cm x 31-cm field at a source-to-detector distance of 1 meter. Results indicate that 76% of the photons reaching the detector are unscattered, 19% are scattered to energies between 0.1 and 1 MeV, and 5% are scattered to energies above 1 MeV. 9. Applications of ETRAN Monte Carlo Codes 229

9.2.2 High-Purity Ge Detectors

Spectral measurement of the NBS x-ray calibration beams. Because of their good energy resolution, small high-purity Ge detectors (with volumes from about 0.1 to 30 cmS) are frequently used in the spectrometry of x-rays and gamma rays up to a few hundred keY. For example, measurements have been made at a num• ber of laboratories of the photon spectra produced by x-ray generators that are used in the calibration of radiation detectors or are used in diagnostic radiology. The response function depends, of course, on the detector dimensions. There appeared to be no stan• dard size for the Ge detector (as, e.g., the 3" X 3" NaI detector), but rather a very large variety of available sizes for which information on the response function would be useful. The computational burden for treating a large number of detector sizes is eased by the fact that electron transport is unimportant at energies below a few hundred keY so that the problem involves only photon scattering, and by the fact that the ETRAN code can be run for a number of cylindrical detectors simultaneously. In this way, Monte Carlo calculations were done of the spectra of energy deposited by photons incident, with energies up to 300 keY, on ten typical sizes of Ge detectors. The assumed measurement geometry is shown in Fig. 9.lD, where the pinhole collimator is introduced to reduce the high intensity of photons generated by x-ray tubes to manageable counting rates, and defines pencil-beam geometry for the calculations.

PIN-HOLE COLLIMATOR

WINDOW

CYLINDRICAL SENSITIVE VOLUME

BACKING MATERIAL 0"/////,1;//////,1//M Figure 9.10. Irradiation and detector geometry assumed in the response• function calculations. The intrinsic Ge detector is assumed to be a right circular cylinder, with a pencil beam of photons incident along the cylinder aXIS.

With the resultant base of data at hand, it was found possible to generalize accu• rately the description of the complete response function (photopeak and K x-ray escape efficiencies, and Compton escape and backscatter continua) through analytical formulas developed on the basis of a single-scatter modeF. These formulas can then be used for any detector size and photon energy in our range of interest. Typical results are shown in Figs. 9.11 through 9.14.

Fig. 9.11 compares measured and calculated values of the fluorescence escape probabilities P." for the K" and K,s lines in Ge. A comparison of experimental and 230 S. M. Seltzer calculated photopeak efficiencies Po is shown in Fig. 9.12. The agreement is to within 3- 5%, including some measurements made in broad-beam irradiation geometry. At photon energies below 60 keY, even the smallest detector considered is effectively an infinite slab, and the photopeak efficiency (aside from window attenuation) is independent of detector dimensions and simply the complement of the fluorescence escape probability. At higher energies, the escape of Compton-scattered photons significantly reduces the photopeak efficiency.

0.2

0.1 ..5 ;li 0.Q6 0 IE ~ i3 I!l 0.ll2 ~ 0.01

20 30

INCIDENT PHOTON ENERGY E. keV Figure 9.11. Probability of K x-ray escape from Ge detectors. The results, normalized to one incident photon, give the probability that the energy ab• sorbed is the incident energy minus the energy of either the K.. or the K" x-ray of Ge (EK.. = 9.88 keY and EK, =10.99 keY). The calculated results (curves) are compared with the measured points of Christenson8 •

13 mm x 215 mm2

10 mm x 100 mm!

4mmx3lmm!

INCIDENT PHOTON ENERGY E. keY Figure 9.12. Comparison of experimental and calculated photopeak efficiencies. The curves are from the calculations for a narrow incident beam, and are given for different sizes of detectors as a function of incident photon energy E. The open circles are from Seelentag et al 9 for broad-beam irradiation. The solid squares are from the data of FewelFD for narrow-beam irradiation. The trian• gles are from the data of Soares and Ehrlichll: solid triangles for narrow-beam irradiation, and open triangles for broad-beam irradiation; the dashed curve is calculated assuming a Ge dead layer of 10.9 I'm on the front face of their detector. The normalization of the experimental data is discussed in Ref. 7. 9. Applications of ETRAN Monte Carlo Codes 231

Compton continua, C(E, f), are shown in Fig. 9.13 for 200-keV photons incident on various size detectors. The Monte Carlo results are indicated by the points plotted at the middle of the histogram bins, and the solid lines are from the formulas. Fig. 9.14 shows continua from the Monte Carlo calculations and from the evaluations of the formulas which include the effects of backscattering of the photons from material behind the sensitive detector volume.

~.O 4.5

4,0 7.2 mm It 105 mm 2 3.5 3,0 .- ' . > ::;'" ::; :l 0: t- V I>J 0- (/) Z Q ~, O t- v; 4,5 0 0- I>J 4.0 0 I 3.5 .'. ~ <::> 0:: 3.0 I>J Z I>J 2,5 2,0

1.5 1.0 0.5

0 0

DEPOSITED ENERGY €. keV

Figure 9.13. Comparison of our Monte Carlo and analytical r~sults for the Compton-continuum portion, C (E, f), of the energy-deposition spectrum. The results, normalized to one incident photon, are for a Ge detector with no backing material. The points (x) are the Monte Carlo histogram results plotted at the mid-points of the energy bins, and the curves are from the formulas. The results are fOf an incident photon energy E = 200 keY, for the various detector sizes indicated in the figure.

To use these response functions in the unfolding of x-ray tube spectra, a line• subtraction and backward-stripping algorithm was developed. An example of the results is given in Fig. 9.15 for the case of a W-anode x-ray generator (",0. 7-mm Al inherent filtration plus 8-mm Al added filtration) operated at 149 kVp. Both the measured pulse-height distribution (obtained by FewelllO with a collimated 4-mm X 30-mm2 Ge detector) and the derived photon spectrum are shown. It is interesting to note that the unfolded incident characteristic lines (shown as vertical lines in Fig. 9.15) indicate that the anode contains ",9% Re (see Table 6 in Ref. 7), a figure consistent with information later obtained from the manufacturer of the x-ray tube. 232 S. M. Seltzer

., > ::;:'". ::;: :;) 0:: 200 keY I- Uw Q. Vl Z 0 i= iii I 0 Q. W Cl, >- 0 a:: zw w

100 keY

o 10 20 30 40 50 60 70 0 90 100 DEPOSITED ENERGY E. keY

Figure 9.14. Comparison of our Monte Carlo and analytical results for the continuum portion, C(E, f) of the spectrum of energy deposited in a Ge de• tector with a thick backing of Ge. The results, normalized to one incident photon, are for a 4.1 cmm x 31.7-mm2 detector, and are shown for incident photon energies of 300, 200, and 100 keY. The points are the Monte Carlo histogram results plotted at the mid-points of the energy bins, and the curves are from the formulas. The shaded portion indicates the contribution from the photons backscattered into the detector. 9. Applications of ETRAN Monte Carlo Codes 233

2.0 i ~ ~ o~~ ::;) ~ II: of <9 t; 1.5 6'''1> w Cl. ~ (J) z 0 I- 0 0 J: 1.0 "Is> Cl. I- ~ Z 00 w 000 CI 0 U 0 0 0 ~ 0 0.5 000

o 50 100 150 INCIDENT PHOTON ENERGY E, keY Figure 9.15. Incident photon spectrum derived from a measured pulse-height distribution. The curve gives the unfolded photon spectrum (characteristic x-ray lines are indicated by vertical lines). For comparison, selected points from the measured pulse-height distribution are also plotted.

Q8

~ 0.6 II: f- <.> w "- VI Z 00.4 f- o J: a.

0.2

Figure 9.16. NBS x-ray spectra. Results are given for the beams designated by the old codes HFI and MFK. 234 S. M. Seltzer

The spectra of the NBS calibration x-ray beams were measured with a 4-mm x 30-mm2 Ge detector directly behind a pinhole collimator, at a distance of 400 cm from the x-ray tube. Typical results of the unfolded spectra are shown in Fig. 9.16, for the x-ray machine operated at 200 kVp. The moderately filtered x-ray spectrum (beam code MFK) pertains to a beam with added filtration of 0.50-mm Cu and 3.49-mm AI, plus the inherent filtration of ",1.5-mm AI. The contribution of characteristic x-ray lines to this spectrum is 3.6% for the W KG2 line (57.982 keY), 6.9% for W KG' (59.318 keY) and 2.7% for W KP1,a (67.152 keY). For the heavily-filtered HFI beam (0.77-mm Pb, 4.16-mm Sn, 0.60-mm Cu, and 2.47-mm Al plus 1.5-mm Al inherent), the W and Pb line intensities are all less than a few tenths of a percent. The spectral charac• terization of these beams had previously been done only in terms of half-value layers from attenuation measurements at 50 cm with ionization chambers. Table 9.1 compares the measured half-value layers with those calculated from the unfolded spectra, for the 200-kVp beams. Agreement is quite good; further comparisons, for beams from 20 to 250 kVp, can be found in Seltzer et al 12 •

Table 9.1

Comparison of half-value layers from attenuation measurements with those calculated from the unfolded photon spectra, for the NBS calibration x-ray beams produced at 200 kVp.

Old Half-Value Layer Homogeneity

Beam Measured4 Calculated' CoefficientC Code Cu Al Cu Al Meas. Calc. (mm) (mm) (mm) (mm) HFI 4.09 19.6 4.04 19.3 - 0.97 MFK 1.24 13.2 1.25 13.3 0.92 0.89

a) From attenuation measurements made at a distance of 50 cm. b) From spectra obtained from measurements at a distance of 400 cm. e) Ratio of the first-to-second Al half-value layers.

NASA's Gamma-Ray Remote Sensing Spectrometer. A 5.5 X 5.5 cm high-purity Ge detector has been selected in the design of the Gamma• Ray Remote Sensing Spectrometer to be flown on NASA's Mars Observer Mission. Extending the techniques used on the Apollo flights, this experiment will measure from orbit the gamma-ray line emission induced in the Martian surface by incident cosmic rays, solar protons and secondary neutrons (as well as that due to natural radioactivity), for the compositional analysis and mapping of the Martian surface. Because the high resolution of the Ge detector will separate rather well the peaks associated with the line emissions, the analysis requires knowledge mainly of the delta-function components of the detector response (total-absorption and the single- and double-annihilation escape peaks). These have been calculated with ETRAN for broad beams of gamma rays in• cident with energies up to 20 MeVls. Calculations done as a function of incident angle confirm that, although the absolute peak efficiencies vary by as much as 50%, if they are divided by the detection efficiency then the resultant peak fractions are virtually independent of the incident angle. This also was found earlier for the symmetric 3" X 3" NaI detector. The relative peak probabilities for the Ge detector are shown Fig. 9.17. 9. Applications of ETRAN Monte Carlo Codes 235

o o~~~ __~ __~~~ ______~~~~~ __~~~~~~~

5.5 x 5.5 em Ge

1 >.0 ~ .-' :0 o .t:Jo L. 0.. u II) ON 0 1 1110

.., 1 o 10° Photon Energy, MeV Figure 9.17. Peak probabilities for total absorption, Po, single-annihilation escape, PI, and double-annihilation escape, P2 , for the 5.5-cm x 5.5-cm Ge detector irradiated by broad beams of gamma rays. The results have been scaled by the detection efficiency 1] to remove dependence on the angle of incidence.

9.3 SPACE SHIELDING CALCULATIONS

For purposes of assessing either the reliability of electronic components or the radiolog• ical safety of on-board personnel, it is important to have the ability to predict absorbed doses to material within spacecraft. In some regions of the space radiation environment, the dose produced by incident electrons and their secondary bremsstrahlung represents a significant hazard. The latter can pose a particularly troublesome threat because of the ability of bremsstrahlung photons to penetrate deep into the target. Elcctron• photon Monte Carlo calculations, necessary for accurate estimates of these radiation doses, are capable of including complete geometrical detail of a spacecraft and its con• tents. However, such a complicated calculation is very expensive, particularly for the dose in a relatively small interior volume, and the results for a specific configuration are unlikely to be applicable to another case of interest. An alternative approach is to develop data in somewhat simple, but more generally applicable, geometry in which the accuracy of the transport results can be maintained at modest effort. Thus, in order to provide an efficient utility for dose estimates, ETRAN Monte Carlo calcula• tions were done for the one-dimensional distribution in depth of the absorbed dose, and of the forward- and backward-directed energy-degradation flux spectra, produced by isotropic incident fluxes of monoenergetic electrons (with energies up to 20 MeV) and their secondary bremsstrahlung in semi-infinite aluminum slabs. With these data, it was possible to generate a database of dose kernels for use in an algorithm which quickly interpolates and integrates for any spectrum of incident electrons likely to be encountered, and whose output is given separately for electrons and bremsstrahlung in terms of dose to AI, H20, Si and Si02 , at points in semi-infinite, as well as behind finite-thickness, Al slabs. Further details can be found in Seltzer14, which describes also the computer code, named SHIELDOSE, that uses the database. 236 S. M. Seltzer

Although the assumption that spacecraft are composed mostly of Al is adequate at least in mission planning and design stages, there has been a fair amount of ambi• guity as to how slab results are applied to three-dimensional objects irradiated from all directions. It turns out that the assumption in space-shielding problems that the incident electron flux is isotropic, at least in a time-averaged sense, leads to very useful relations between the doses in slab targets and those in solid spheres and hollow spheri• cal shells, which are more realistic configurations in this application. These connections were explored by Seltzer15 •

The geometry considered is indicated in Fig. 9.18, where the quantities relevant to this presentation are the outer radius R of the sphere or shell, and t which is either the radial depth from the surface of the sphere or the radial thickness of the spherical shell. It is then assumed that there exists a distribution that gives the energy deposited per unit pathlength in the target as if the penetrating radiation travels in a straight line. For protons which do travel nearly in straight lines, this distribution is simply the stopping power. For electrons and bremsstrahlung which do not have straight trajectories, we can nevertheless define the distribution implicitly in terms of the one-dimensional slab depth dose.

a. Sphere

b. Shell

Figure 9.18. Schematic for the calculation of the dose (a) at radius T (depth t) in a solid sphere with outer radius Rj and (b) at the inside surface of a hollow spherical shell with inner radius T, outer radius R, and radial thickness t.

The results (see Ref. 15 for formulae) are that one can estimate the dose as a function of t within either spheres or shells in terms of a simple integral over the slab dose. For the case t = R (i.e., the center of a sphere), the relationship reduces to one involving a derivative of the slab dose with respect to depth. The dose at centers of spheres (as a function of sphere radius R = z) is shown in Fig. 9.19 in terms of its ratio 9. Applications of ETRAN Monte Carlo Codes 237 to twice* the corresponding slab dose at depth z. Results are shown for an exponential spectrum of incident electrons with an e-folding energy of a = 0.5 MeV, and for a fission electron spectrum. For the fission spectrum, there is good agreement with the corresponding ratios obtained by Jordan16, who calculated the electron dose (but not the bremsstrahlung tail) explicitly for both geometries using an adjoint Monte Carlo code. To pin down the accuracy of the approximate transformation of doses in slabs to spheres for the bremsstrahlung component, calculations for 2-MeV electrons incident on an Al sphere of radius R = 20 g/cm2 were done using the ACCEPT code. The results, as a function of depth in the sphere, are compared to those from the transformed SHIELDOSE dose kernel in Fig. 9.20. Agreement is reasonably good, to within about 20%, which is nearly within the statistical uncertainties of the ACCEPT results. It should be pointed out here that the ACCEPT calculation required approximately one hour on a Cyber 205 super computer in order to get sufficient scores in the small volumes near the center of the sphere, while the equivalent accuracy can be obtained from a slab calculation requiring less than about 4 minutes.

7r_---.---,----,----,----~--_r----r_--_r----r_--~

6

5

18 ~ 4 d

3

2

100 200 300 400 500 600 700 800 900 1000 Z, mils Figure 9.19. Ratio of the electron dose D. at centers of aluminum spheres (radius z) to twice the dose Doc in a semi-infinite aluminum medium (at depth z). The results are plotted out to large z where the bremsstrahlung dominates. Solid curves are from the transformation 'of our slab results, given both for incident electrons with a fission spectrum and for electrons with an exponential spectrum characterized by an e-folding energy a = 0.5 MeV. Dashed curve is the fission-spectrum results of Jordan16 who considered explicitly the two geometries. Note that 1 mil = 0.001"; and 1 mil Al = 6.86 mg/cm2•

Fig. 9.21 gives an example of the application of these methods to dose estimates for a geosynchronous orbit with an altitude of 35790 km, an inclination of 00 , and a parking latitude of 160 0 W. For the same geosynchronous spectrum, Fig. 9.22 illustrates the relationship to the slab case of the dose at points in a sphere or at the inside surface of a shell, both with outer radius R = 25 g/cm 2•

* The factor of two comes about because we assume that in space the point would be sandwiched between two slabs of thickness z, irradiated from both sides. 238 S. M. Seltzer

.I. .I.

Figure 9.20. The spatial distribution of absorbed dose from 2-MeV electrons incident isotropically on an Al sphere with a radius of 20 g/cm 2• The results of ACCEPT Monte Carlo calculations for the sphere (histogram) are compared to the results obtained by converting the slab depth-dose distribution to apply to the sphere (curve).

In a study17 of the effectiveness of multilayer shields to reduce the electron brems• strahlung dose within a spacecraft, a series of ACCEPT calculations were done of the energy deposited in a central water sphere surrounded by a shell comprised of an outer Al layer and an inner Pb layer, as indicated in Fig. 9.23. In order to see to what extent the transformation of slab results could be extended to this more complicated case, cal• culations with the one-dimensional ZTRAN code (a multi-slab ETRAN version) were done for the AI/Pb layers followed by a semi-infinite water slab. The dose in the water slab was converted to apply to the layered-shell/central-sphere configuration (see Ref. 15 for details). The results are compared to those from the ACCEPT calculation in Table 9.2 for the geosynchronous spectrum and in Table 9.3 for monoenergetic incident electrons. The discrepancies indicate an accuracy for the transformation procedure of 10-20% in this geometry also, a portion of which is due to the statistical uncertainties in the Monte Carlo results. The transformation procedure is, however, computationally more efficient. In the ACCEPT runs, only from about 1/6 to 1/4 of the bremsstrahlung photons emerging from the shield cross the air region to enter the central water sphere and contribute to the scoring. The corresponding slab calculation, however, makes use of all of the emergent photon histories and requires less time to handle the geometry. Thus, the equivalent information was obtained in an estimated 1/10 of the computer time. Note, from Table 9.2, that the converted slab results correctly predict the reduc• tion by a factor of three in the bremsstrahlung dose, due to the replacment of the inner 1.5 g/cm2 of the Al shield by a more absorbing layer of Pb with the same mass thick• ness. In addition to attenuating the bremsstrahlung with Pb, it seemed worthwhile also to consider reducing the bremsstrahlung yield by replacing an outer portion of the Al with a layer of lower-Z polyethylene. A follow-on ZTRAN calculation (requiring only 5 minutes on an IBM 3081) shows that a CH2(1.0 g/cm2)J AI(0.5 g/cm2)/Pb(1.5 g/cm2) shield results in a reduction of another factor of three, for. an overall reduction of the bremsstrahlung dose by a factor of 10 with no increase in the mass thickness of the shield. 9. Applications of ETRAN Monte Carlo Codes 239

1016

'i I; 107 'i :: ::E N I E u 6 -0 -~ 10 ~ (/) '0 '"uz ....0 '" '---' ~ z 105 0 W t (f) ..'" 0 ..'" 0 z '"0 4 U 0:: 10 !!: 10' <[ W

>-I W Z 103 0

BREMSSTRAHLUNG SOLA~;;Or(iN----__ D. 2 BREMSSTRAHLUNG ---D • 10 .... _-• SOLAR-- P:C;;:~jN---_ D", ------D .. 10 1 o 2 4 6 8 10 z (g/cm2 ) Figure 9.21. Depth-dose distributions in aluminum targets for the radiation encountered during one year in a geosynchronous orbit. The orbit parameters are given in the text; the incident electron-fluence spectrum is given in the inset. Results are given both for the dose Doo at depth z in a semi-infinite medium, and for the dose D. at the center of a sphere of radius z. The solar proton dose, for one anomalously large event, is given by the dashed curves.

Table 9.2

Comparison of direct Monte Carlo results (ACCEPT) with those obtained by converting slab results to apply to spherical geometry (from ZTRAN), for the electron flux expected in a geosynchronous orbit (160° W parking latitude, 0° inclination, 35790-km altitude). Results give average dose in the central 40-cm diameter water sphere and are given in units of mrad/day.

Al Pb ACCEPT From % (g/cm2) (g/cm2) ZTRAN Discrepancy 3.0 0.0 223 259 16.1 1.5 1.5 76.6 83.7 9.3 240 S. M. Seltzer

/, / / / 3 / / R=25 g/cm2 ,/ / / 3 / ~ ..!!! / en / 0 .... ",,/ N 2 ...... SHELL , ...... ' a:: ~ -- ~ -- Q) .s::. -- Co _...... ---- en --- 0

o 5 10 15 20 25 t, g/cm2

Figure 9.22. Ratio of the dose from electrons and secondary bremsstrahlung in spheres to twice the corresponding dose in a semi-infinite slab. Results per• tain to aluminum targets irradiated by an electron flux expected in a geosyn• chronous orbit, and are given at points inside a solid sphere (solid curve) and at points on the inside surface of a hollow spherical shell (dashed curve), both with an outer radius of 25 g/cm 2•

LEAD

1 40cm IOOcm

WATER (TISSUE-EQUIVALENT) Figure 9.23. Schematization of spacecraft for the multilayer shielding calcu• lations. 9. Applications of ETRAN Monte Carlo Codes 241

Table 9.3 Comparison of direct Monte Carlo results (ACCEPT) with those obtained by converting slab results to apply to spherical geometry (from ZTRAN), for monoenergetic electrons incident on the target of Fig 9.23. Results pertain to the average energy deposited in the central 40-cm diameter water sphere, per unit isotropic fluence incident on the AI/Pb layered shield, and are given in units of MeV-cm2 electron-I.

Electron Al Pb ACCEPT From % Energy (g/cm2) (g/cm2) ZTRAN Discrepancy 6 MeV 3.0 0.0 186 132 -29.0 2.0 1.0 120 126 5.0 1.0 2.0 167 179 7.2 0.1 2.9 288 328 13.9 2 MeV 3.0 0.0 9.97 11.4 14.3 1.5 1.5 6.12 7.05 15.2 1.25 1.75 6.01 6.84 13.8 0.7 2.3 5.76 6.34 10.1 0.1 2.9 15: 19.3 28.7 0.5 MeV 3.0 0.0 0.369 0.431 16.8 1.5 1.5 0.0690 0.0755 9.4 0.3 2.7 0.0416 0.0429 3.1 0.1 2.9 0.0403 0.0431 -13.3

9.4 BREMSSTRAHLUNG BEAMS FOR RADIATION PROCESSING

Radiation processing and sterilization with direct electron beams and 60Co sources are done in many industries. Bremsstrahlung beams produced by high-power electron ac• celerators have been considered for use in radiation processing because such beams have substantial penetrating power without the problems of supply and safety associ• ated with large 60CO sources. Calculations using the ZTRAN code were done for this applicationI8 assuming a simple, but realistic, multilayer target configuration. The tar• get, shown in Fig. 9.24, consists first of a tungsten converter plate whose thickness is chosen to produce approximately the maximum transmitted bremsstrahlung yield for the incident electron energy considered. The converter is backed by a Cu/water/Cu cooling channel, which also serves to stop any transmitted electrons, after which there is a small air gap, and finally a thick water phantom which represents the irradiated product.

The absorbed dose in the phantom was obtained by integrating over the prod• uct of the energy, the mass energy absorption coefficient, and the flux spectra of the bremsstrahlung photons, which were calculated as a function of depth in the water. The results are given in Fig. 9.25 for incident electron beam energies of 2, 5 and 10 MeV. The solid curves were obtained using the improved bremsstrahlung production cross sectionsI9 which incorporate the exact numerical results of Pratt et a1 20 (also see Chapter 4); and the dashed curves are based on the use of the earlier ETRAN dataset of empirically corrected Bethe-Heitler cross sections21. 242 S. M. Seltzer

BEAM STOP AIR GAP Ir------~A~ ______~ 0.03 g/cm2 -- -

INCIDENT __ w Cu WATER Cu WATER 2 2 2 ELECTRONS CONVERTER 1.25 g/cm 0.65 g/cm 1.25 g/cm PHANTOM - -- v I COOLING CHANNEL Figure 9.24. Schematic of water-cooled bremsstrahlung converter and water phantom arrangement for bremsstrahlung dose calculations.

'"', ...... 2 MeV, 0.54 g/em2 W ' ...... 10 -4 0!----L---1"=0-....l...-~20:-----1..--=3'=-O-..L.--4Q"::---=t...... :::::...,J50. DEPTH IN WATER, em Figure 9.25. Depth-dose distribution in the water phantom due to bremsstrahlung produced in the converter assembly of Fig. 9.24 by broad beams of normally incident 10, 5, and 2-MeV electrons. Results are given in terms of absorbed dose per electron incident on the converter. The solid curves are from ZTRAN calculations using the updated bremsstrahlung production cross sections, and the dashed curves are based on the use of the older empirically corrected Bethe-Heitler cross sections.

9.5 LIQUID-SCINTILLATION COUNTING OF BETA EMITTERS

Liquid-scintillation counting is an important tool in radionuclide metrology. Dissolv• ing the radionuclide in a liquid scintillator provides essentially 47r counting geometry. However, the energy deposited in the detector is affected by energy lost in or reflected 9. Applications of ETRAN Monte Carlo Codes 243 from the container walls*. To determine the so-called wall correction to the energy deposited by beta particles in the scintillator, we have done a series of ETRAN Monte Carlo calculations for this problem.

Typically, the radionuclide is dissolved in a toluene-based liquid scintillator con• tained in a small cylindrical Pyrex vial. The dimensions assumed for this study (and used at NBS) are for a cylindrical Pyrex container with walls 1.5-mm thick, an inside diameter of 2.5 cm, and an inside height of either 2.04 cm (IO-ml vial) or 3.06 cm (15-ml vial). The calculations were done for isotropic emitters, uniformly distributed throughout the toluene. Results were calculated for monoenergetic emitted electrons, with energies up to 3 MeV, so that the results could later be folded over a number of beta spectra of interest. The objective of the calculation was to determine the fraction of the emitted energy that is deposited in the toluene.

Because the cylindrical version of ETRAN handles only a single material, the calculation was done in stages. Basically, this amounted to calculating the energy deposited in the scintillating cylinder surrounded by toluene walls, and then correcting the energy backscattered from the walls using ratios determined from one-dimensional ZTRAN calculations when toluene walls are replaced by Pyrex walls. The final results are shown in Fig. 9.26 for the two cylinders considered.

~ Cl ~t------__ l-""v; ~O! ",,0 Cl (; ffilX! ZO "" 0'" .... ~o

If) 0"',0=---';2-~""""'''''''''''''''''','''0:--';''' ---~"'-...... ~,o:;:o-- ...... - ...... ~,0' ELECTRON ENERGY, MeV Figure 9.26. Wall correction factors for {3- counting by liquid-scintillation techniques. The curves give the fraction of energy emitted by monoenergetic electrons that is deposited in the toluene-based scintillator, taking into account the energy lost in the 1.5-mm thick walls of the cylindrical Pyrex container. Results pertain to an inside diameter of 2.5 cm, and are given for inside heights H of 2.04 cm (lO-ml vial) and 3.06 cm (15-ml vial).

As one would expect, as the electron energy and its range increases, the electron is likely to lose a larger part of its energy in or beyond the walls, so that the fraction of

* The counting process is further complicated by the nonlinear conversion of de• posited energy to light photons, incomplete collection of the emitted light through the container walls, and the statistics of the phototube response. These are not considered here. 244 S. M. Seltzer the source energy deposited in the scintillator becomes smaller. Although these trends are easy to predict, accurate quantitative results require such Monte Carlo calculations.

To apply these results, beta spectra were calculated using the LOGFT program and the current Evaluated Nuclear Structure Data File from Brookhaven National Laboratory22 for beta-emitting nuclides. Folding these spectra over the wall correc• tion factors from Fig 9.26 gives the distorted or detected spectra. Shown in Table 9.4 are some results for pure beta emitters summarized in terms of the fraction of emitted energy deposited in the detector.

Table 9.4

Wall correction factors for {3- emitters uniformly distributed in a liquid scintillator con• tained in a 0.15-cm thick Pyrex-walled cylindrical vial with an inside diameter of 2.5 cm. Results, for inside cylinder heights H of 2.04 cm (10 ml) and 3.06 cm (15 ml), are given for the fraction of emitted energy deposited in the toluene-based scintillator. Listed in the table are the half-life, the endpoint of the beta spectra, Emaz , and the average emitted energy, E atJ , for the nuclides.

Emitter Half-Life Emaz(MeV) EatJ(MeV) Fraction of EatJ Deposited H=2.04 cm H=3.06 cm 32p 14.3 d 1.71 0.695 0.876 0.889 89Sr 50.5 d 1.49 0.581 0.892 0.904 90Sr 28.5 y 0.546 0.196 0.971 0.974 90y 64.1 h 2.28 0.935 0.766 0.784 144Pr 14.6 d 0.932 0.314 0.946 0.951

Acknowledgement: This work was supported in part by the Office of Health and Envi• ronmental Research, U.S. Department of Energy. 9. Applications of ETRAN Monte Carlo Codes 245

REFERENCES 1. M. J. Berger and S. M. Seltzer, "Response Functions for Sodium Iodide Scintil• lation Detectors", Nucl. Instr. Meth. 104 (1972) 317. 2. J. I. Trombka and R. 1. Schmadebeck, "A Method for the Analysis of Pulse• Height Spectra Containing Gain-Shift and Zero-Shift Compensation", Nucl. In• str. Meth. 62 (1968) 253. 3. J. I. Trombka, C. S. Dyer, L. G. Evans, M. J. Bielefeld, S. M. Seltzer and A. E. Metzger, "Reanalysis of the Apollo Cosmic-Ray Spectrum in the 0.3 to 10 MeV Energy Region", Astrophys. J. 212 (1977) 925. 4. C. S. Dyer, J. I. Trombka and S. M. Seltzer, "Nuclear Data for Assessment of Activation of Scintillator Materials During Spaceflight", in Nuclear Cross Sec• tions and Technology, National Bureau of Standards Special Publ. 425, Vol. II (1975) 480; also "Calculation of Radioactivity Induced in Scintillator Materials During Spaceflight", Trans. Am. Nucl. Soc. 27 (1977) 195. 5. S. M. Seltzer, "The Response of Scintillation Detectors to Internally Induced Radioactivity", Nucl. Instr. Meth. 127 (1975) 293. 6. M. Ehrlich, S. M. Seltzer, M. J. Bielefeld, J. I. Trombka, "Spectrometry of a 60Co Gamma-Ray Beam Used for Instrument Calibration", Metrologia 12 (1976) 169. 7. S. M. Seltzer, "Calculated Response of Intrinsic Germanium Detectors to Narrow

Beams of Photons with Energies up to rv 300 keV", Nucl. Instr. Meth. 188 (1981) 133. 8. 1. H. Christenson, "Comparison between Experimental and Calculated Relative Escape Peak Intensities for an Intrinsic Ge Detector in the Energy Region 11 - 25 keV", X-ray Spectrom. 8 (1979) 146. 9. W. W. Seelentag and W. Panzer, "Stripping of X-Ray Bremsstrahlung Spectra up to 300 kVp on a Desk Type Computer", Phys. Med. BioI. 24 (1979) 767. 10. T. R. Fewell, personal communication (1977). 11. C. Soares and M. Ehrlich, personal communication (1980). 12. S. M. Seltzer, C. Soares and M. Ehrlich, "Characterization of Radiation Beams: Bremsstrahlung Photon Beams", in Quality Assurance for Measurements of Ion• izing Radiation, edited by E. H. Eisenhower, National Bureau of Standards report NUREG/CR-3775 (1984), B-1. 13. S. M. Seltzer, "Calculated Response of a 5.5 X 5.5 cm High-Purity Ge Detector to Gamma Rays with Energies up to 20 MeV", National Bureau of Standards report NBSIR 87-3548 (1987). 14. S. M. Seltzer, "Electron, Electron Bremsstrahlung and Proton Depth-Dose Data for Space-Shielding Applications", IEEE Trans. Nucl. Sci. NS-26 (1979) 4896; and "SHIELDOSE: A Computer Code for Space-Shielding Radiation Dose Cal• culations", National Bureau of Standards Technical Note 1116 (1980). 15. S. M. Seltzer, "Conversion of Depth-Dose Distributions from Slab to Spherical Geometries for Space-Shielding Applications", IEEE Trans. Nucl. Sci. NS-33 (1986) 1292. 16. T. M. Jordan, "Electron Dose Attenuation Kernals for Slab and Spherical Ge• ometries", Air Force Weapons Laboratory report AWRL-TR-81-43 (1981). 246 S. M. Seltzer

17. G. Barnea, S. M. Seltzer and M. J. Berger, "Transport of Electrons and Asso• ciated Bremsstrahlung Through a Composite Aluminum-Lead Shield, With Ap• plications to Spacecraft Shielding", National Bureau of Standards report NBSIR 86-3429 (1986); and G. Barnea, M. J. Berger and S. M. Seltzer, "Optimization Study of Electron-Bremsstrahlung Shielding for Manned Spacecraft", J. Space• craft and Rockets 24 (1987) 158. 18. S. M. Seltzer, J. P. Farrell and J. Silverman, "Bremsstrahlung Beams from High-Power Electron Accelerators for Use in Radiation Processing", IEEE Trans. Nucl. Sci. NS-30 (1983) 1629; and J. P. Farrell, S. M. Seltzer and J. Silverman, "Bremsstrahlung Generators for Radiation Processing", Radiat. Phys. Chem. 22 (1983) 469. 19. S. M. Seltzer and M. J. Berger, "Bremsstrahlung Spectra from Electron Interac• tions with Screened Atomic Nuclei and Orbital Electrons", Nucl. Instr. Meth. B12 (1985) 95; and "Bremsstrahlung Energy Spectra from Electrons with Ki• netic Energy 1 keY - 10 GeV Incident on Screened Nuclei and Orbital Electrons of Neutral Atoms with Z = 1 - 100", Atom. Data and Nucl. Data Tables 35 (1986) 345. 20. R. H. Pratt, H. K. Tseng, C. M. Lee, L. Kissel, C. MacCallum and M. Riley, "Bremsstrahlung Energy Spectra from Electrons of Kinetic Energy 1 keV:5 T :5 2000 keY Incident on Neutral Atoms 2:5 Z :5 92", Atom. Data and Nucl. Data Tables 20 (1977) 175; errata in 26 (1981) 477. 21. M. J. Berger and S. M. Seltzer, "Bremsstrahlung and Photoneutrons from Thick Tungsten and Tantalum Targets", Phys. Rev. C 2 (1970) 621. 22. J. K. Tuli, "Evaluated Nuclear Structure Data File, A Manual for Preparation of Data Sets", Brookhaven National Laboratory report BNL-NCS-51655 (1983). The Integrated TIGER Series 10. Structure and Operation of the ITS Code System

J. Halbleib

Sandia National Laboratories Box 5800 Albuquerque, New Mexico 87185, U.S.A.

10.1 INTRODUCTION The TIGER series of time-independent coupled electron-photon Monte Carlo transport codes is a group of multimaterial and multidimensional codes designed to provide a state-of-the-art description of the production and transport of the electron-photon cas• cade by combining microscopic photon transport with a macroscopic random walk1 for electron transport. Major contributors to its evolution are listed in Table 10.1.

Table 10.1 Contributors

Contributor Contributor Tom Mehlhorn, SNL Kathy Hiebert-Dodd, SNL Ron Kensek, SNL Lee Haggmark, SNL Jim Morel, SNL/LANL Joe Mack, LANL Walt Vandevender, SNL Grady Hughes, LANL Mel Scott, SNL Hsiao-Hua Hsu, LANL

We are primarily code users rather than code developers, and have borrowed freely from existing work wherever possible. Nevertheless, our efforts have resulted in various software packages for describing the production and transport of the electron-photon cascade that we ourselves found sufficiently useful to warrant dissemination through the Radiation Shielding Information Center (RSIC) at Oak Ridge National Laboratory. The ITS system2 (Integrated TIGER Series) represents the organization and integra• tion of this combined software, along with much additional capability from previously unreleased work, into a single convenient package of exceptional user friendliness and portability. Emphasis is on simplicity and flexibility of application without sacrificing the rigor or sophistication of the physical model.

10.2 HISTORY OF THE TIGER SERIES

Table 10.2 chronicles the development of the TIGER series, beginning with the EZTRAN3 and EZTRAN24 codes in the early 1970's. These were basically user oriented versions of

249 250 J. Halbleib the ETRAN codes5 • Because of their restriction to a single homogeneous material, their use in support of many of our laboratory's engineering applications, such as those in Table 10.3, was severely limited. Overcoming this limitation was the original motivation for the development of the TIGER series.

Table 10.2 Chronology of TIGER Series Oevelopment

Code Oate Released Oimension EZTRAN Sep 71 Yes 1-0 EZTRAN2 Oct 73 Yes 2-0/3-0(a) TIGER Mar 74 Yes 1-0 CYLTRAN Mar 75 Yes 2-0/3-0(a) CYLTRANM Jun 77 No 2-0/3-0(a) TIGERP May 78 Yes 1-0 SPHERE Jun 78 Yes 1-0 ACCEPT May 79 Yes 3-0 SPHEM Jul79 No 1-0/3-0(a) CYLTRANP Late 81 No 2-0/3-0(a) ACCEPTM Late 81 No 3-0

(a) The first dimension refers to the material geometry while the second dimension refers to the description of the particle trajectories.

Table 10.3 Local Needs

Flash X-Ray Source Oevelopment Reactor Safety Radiography and X-ray Cinematography High-Pressure, High-Temperature EOS Research Electron Beam Fusion (ICF) Underground Testing Astrophysics' Radiation Effects in Electronics Electron Beam Pumped Gas Lasers Electron Beam Propagation Tokamak Electron Effects Radioactive Waste Transportation Radiation Preservation of Food Radiation Treatment of Sewage Military Projects 10. Structure and Operation of the ITS Code System 251

The TIGER series is based primarily on the ETRAN model. TIGER6, CYLTRAN7, and ACCEPT8 are the base codes of the series and differ primarily in their dimensional• ity and geometric modeling. TIGER is a one-dimensional multilayer code. CYLTRAN employs a fully three-dimensional description of particle trajectories within a cylindrical material geometry, and quite naturally finds application in problems involving electron or photon beam sources. ACCEPT is a general three-dimensional transport code that uses the combinatorial geometry scheme developed at MAGI9.10

These base codes were primarily designed for transport at energies that are large compared to the binding energies of the atomic electrons. Fluorescence and Auger processes are only allowed for the K-shell of the highest atomic-number element in a given material. For some applications, it is desirable to have a more detailed model of the low-energy transport. In the TIGERpll and CYLTRANp12 codes, we include the more elaborate ionization/relaxation model from the SANDYL code13 .

In CYLTRANMu, we combine the collisional transport of CYLTRAN with transport in macroscopic electric and magnetic fields of arbitrary spatial dependence using a Runga-Kutta-Fehlberg algorithm15.16 to integrate the Lorentz force equations. An im• portant modification of this algorithm16 made possible the development of the AC• CEPTM code11 code which combines the collisional transport of the ACCEPT code with macroscopic field transport. SPHERE18 and SPHEM19 are two special purpose codes that are restricted to multiple concentric spherical shells.

Other substantial improvements along the way were the implementation of im• proved low-energy elastic scattering cross sections20 , the implementation of a univer• sal fit of the density-effect correction, to the electronic stopping power21, the addi• tion of a subprogram for the automatic generation of the necessary photon cross• section data22- 24 , and the addition of a capability for plotting multidimensional material geometries and particle trajectories in regions of macroscopic fields.

Though some of the codes in Table 10.2 eventually became obsolete, we still found ourselves maintaining eight separate code packages. Maintaining multiple-code pack• ages became quite burdensome for us as well as for users of the codes. Important modifications were not implemented in a timely fashion, and development of the var• ious codes became uneven such that each code had unique features that had not yet been implemented in the others.

In order to remedy this situation, we have developed the ITS system whose member codes supercede all other versions of the TIGER series codes.

10.3 STRUCTURE OF THE ITS CODE SYSTEM

The ITS system consists of four essential elements: 1. The electron-photon cross-section data file. 2. The source file for the cross-section program library. 3. The source file for the Monte Carlo program library. 4. UPEMVS-the machine-portable emulator of the CDC UPDATE processor26 .

Element (1) is discussed in detail elsewhere2.5•20 and will receive only minimal attention here. 252 J. Halbleib

10.3.1 The Source Files

The two source files were obtained by integrating the eight codes of Table 10.4 in such a way as to minimize the repetition of coding that is common to two or more of these codes. This process led quite naturally to the development of a new code, ACCEPTP. Each of the eight member codes will run on any of four machines - CRAY, CDC, VAX or IBM. Although the source files contain explicit logic only for these four machines, the use of ANSI Standard FORTRAN 77 should facilitate installation on other machines as well. Indeed, users have reported execution on UNIVAC, HP, AMDAHL, ELEXSI, NAS, APOLLO, ALLIANT, and MULTIFLO/TRACE hardware.

Table lOA ITS Member Codes

Standard Enhanced Ionization/Relaxation Macroscopic Fields Codes (P Codes) (M Codes) TIGER TIGERP CYLTRAN CYLTRANP CYLTRANM ACCEPT ACCEPTP ACCEPTM

10.3.2 The UPEML Processor

Figure 10.1 is an example of the kinds of structures one would expect to encounter in a source file. The lines beginning with asterisks (in column 1) are creation-run directives that are recognized by the UPEML processor, and the remaining lines are FORTRAN coding. Although the UPEML processor has many capabilities, it has two primary functions: 1. Creation run. 2. Correction run.

A creation run is performed once and for all and produces a binary program library from a source file. The program library assigns unique identifiers to each line of the source file using the names defined in the DECK and COMDECK directives. For example, Fig. 10.2 is the program library corresponding to the source file in Fig. 10.1.

Once a program library for a given source file is installed via a creation run, day• to-day operation proceeds through various correction runs which select and modify those portions of the program library that are to be compiled for execution. This is accomplished with various correction-run directives such as DEFINE, INSERT, and DELETE. For example, the line *DEFINE, OPT1 in a correction run for the program library in Fig. 10.2, would cause the upper equation for Y(I) to be sent to the compiler. In the Monte Carlo program library, OPT1 and OPT2 might correspond to TIGER and CYLTRAN, respectively - or VAX and CRAY, respectively. 10. Structure and Operation of the ITS Code System 253

------An Example Source File ------*COI!DECK, COMl COMMON /BLOCK1/ X(100), Y(100) *DECK,MAIN PROGRAM SAKPLEl *CALL,COMl DIMENSION Z(100) X(l)=O.O DO 100 I = 1, 100 I *IF DEFINE, OPT1 I :~;D~;FI~:::T: X(I)+FUNC1(I) Y(I) - X(I)+COS(X(I» I *ENDIF

I **C::AL:LK,,~C~O~M:l~· "',... ," I FUNCTION FUNC1(J)

FUNCl = 1.0+(TAN(X(J»**2 I RETURN END I ------Figure 10.1. Structure of an Example Source File.

I ------I ------Structure of a Program Library I *COI!DECK,COMl COMl 00001 COMMON /BLOCK1/ X(100), Y(100) COMl 00002 *DECK,MAIN MAIN 00001 PROGRAM SAKPLEl MAIN 00002 *CALL,COMl MAIN 00003 DIMENSION Z(100) MAIN 00004 X(1)=O.O MAIN 00005 DO 100 I = 1, 100 MAIN 00006 *IF DEFINE,OPTl MAIN 00007 Y(I) - X(I)+FUNC1(I) MAIN 00008 *ENDIF MAIN 00009 *IF DEFINE,OPT2 MAIN 00010 Y(I) - X(I)+COS(X(I» MAIN 00011 *ENDIF MAIN 00012 XCI) - X(I)+0.025 MAIN 00013 100 CONTINUE MAIN 00014 END MAIN 00015 *DECK,FUNCl FUNCl 00001 FUNCTION FUNC1(J) FUNC1 00002 *CALL,COMl FUNCl 00003 FUNC1 = 1.0+(TAN(X(J»**2 FUNC1 00004 RETURN FUNCl 00005 END FUNCl 00006

Figure 10.2. Structure of the Program Library corresponding to the source file in Fig. 10.1. 254 J. Halbleib

10.4 OPERATION OF THE ITS CODE SYSTEM

For the purposes of this section, it is assumed that the program libraries for each of the two source files of the ITS system have been created via UPEML creation runs. The most general run of an ITS code then involves four major job steps: 1. Select a cross-section generating code. 2. Execute the cross-section generating code. 3. Select a Monte Carlo code. 4. Execute the Monte Carlo code.

Steps (1) and (3) are correction runs that use the UPEML processor. Of course, all steps are not always necessary. If the necessary cross-section set has been saved from a previous run, Steps (1) and (2) are unnecessary. If suitable executable codes have been saved from previous runs, Steps (1) and (3) are unnecessary.

10.4.1 Input

Selection of the code to be run and the machine on which it is to be run is mandatory for the correction runs (Steps 1 and 3 above). These selections are made via the DE• FINE directive, and the choices for the cross-section generator and Monte Carlo codes are shown in Tables 10.5 and 10.6, respectively. The only other mandatory input is the modification to subroutine BFLD of the M codes for evaluating the components of the macroscopic fields at an arbitrary location, which is required in the Monte Carlo correction run (Step 3). It is also possible simultaneously to make other more or less elaborate modifications to codes using other correction-run directives, for which we re• fer the reader to References 3 and 26. For example, the user might want to change the integer parameter that determines the number of allowable input zones, or he might want to insert coding for a source distribution not allowed by the unmodified code.

Table 10.5. Choices for Mandatory Define-Directives in Correction Sets for Selecting a Cross-Section Generating Code.

Machine Selection Code Selection CRAY PCODES CDC VAX IBM

Table 10.6. Choices for Mandatory Define-Directives in Correction Sets for Selecting a Monte Carlo Code.

Machine Selection Code Selection Options CRAY TIGER PCODES CDC CYLTRAN MCODES VAX ACCEPT DOUBLE (required for IBM and VAX) IBM PLOTS 10. Structure and Operation of the ITS Code System 255

The scheme for execution input (Steps 2 and 4 above) employed in the older versions of the member codes of ITS has been completely discarded in favor of a much improved method. This very simple and user-friendly procedure is free-format, and is based on a set of descriptive keywords. Column counting is no longer necessary; numerical data are merely separated by one or more blank spaces. The input file is much more eye• readable. The primary keywords are order independent. Input is minimized by making maximum use of defaults (e.g., if the input file for the cross-section generator is empty, l.O-MeV cross sections for aluminum are calculated). The keyword scheme permits the selection of options via standard input that formerly could only be activated via the CDC UPDATE processor or its equivalent. More internal processing makes input rules less rigid. Finally, extensive internal error checking is designed to detect input errors at the earliest possible moment, thus minimizing the need for time consuming debugs and tracebacks.

Table 10.7 Summary of Keywords Used for Input in the Execution of a Cross-Section Generating Code

KEYWORD DEFAULT ENERGY 1.0 MeV MATERIAL AL GAS normal state (pure elements only) liquid/solid (user defined materials) DENSITY normal density (g/cm3) (pure elements only) DENSITY-RATIO 1.0 SUBSTEP internal TITLE no title STEP 8 PRINT-ALL only range tables will be printed

Tables 10.7 and 10.8 list the primary keywords used for execution input to the cross-section generating program and Monte Carlo program, respectively, along with their default values and associated secondary keywords. In order to facilitate use of the Monte Carlo keywords, they have been organized into the five functional categories shown in the table. A discussion of the precise meaning and syntax of the keyword input, along with examples, can be found in Reference 3. 256 J. Halbleib

Table 10.8 Summary of Keywords Used for Input in the Execution of a Monte Carlo Code

KEYWORD DEFAULT

•••• GEOMETRy •••• GEOMETRY required •••• SOURCE •••• ELECTRONS or electron source PHOTONS ~NERGY or 1.0 KeV monoenergetic SPECTRUM POSITION point source DIRECTION monodirectional source CUTOFFS electrons: 5% of maximum energy photons: 0.01 MeV •••• OUTPUT OPTIONS •••• ELECTRON-ESCAPE off ELECTRON-FLUX off PHOTON-ESCAPE off PHOTON-FLUX off PULSE-HEIGHT off •••• OTHER COMMONLY USED OPTIONS •••• TITLE blank title HISTORIES 1000 histories PLOTS [a] no plots TRAP-ELECTRONS no electron trapping check SCALE-BREMS natural bremsstrahlung production SCALE-IMPACT natural electron impact ionization RESTART no restart •••• OTHER RARELY USED OPTIONS •••• ECHO off BATCHES 10 batches DUMP off PRINT-ALL print only final batch output RANDOM-NUMBER machine def aul t NO-SEe-ELECTRONS off NO-SEC-PHOTONS off NO-KNOCKONS off NO-STRAGGLING off NEW-DATA-SET 1 run SIMPLE-BREMS off'

[a] CYLTRANM and ACCEPT codes

Table 10.9 is an input stream for running the TIGER code with all four major job steps. The portions of the input stream required for each of these tasks are separated from one another by "{ eor }"s. Note the use of the commenting feature-asterisk in column one-in the Monte Carlo execution input. Explicit keywords are often used where they are not required (i.e., they specify default values) in order to make the input streams more readable. 10. Structure and Operation of the ITS Code System 257

Table 10.9 Sample Input for a Full Four-Step Run of the TIGER Code

Sample Input of TIGER Code.

F,P *!DENT,DEFINE *DEFINE, CRAY {eor} MATERIAL TA MATERIAL AL TITLE 1.0 MEV CROSS SECTIONS FOR TA AND AL ENERGY 1.0 {eor} F,P *!DENT ,DEFINE *DEFINE,CRAY *DEFINE,TIGER {eor} ECHO 1 TITLE .•• 1.0 MEV TA/AL TEST PROBLEM .************************GEOMETRY ******************************** * MAT NZONE THIK ECUT PTCZ GEOMETRY 2 1 2 0.007 2 2 0.05 ************ ••• **********SOURCE ******.************************* ELECTRONS ENERGY 1.0 CUTOFFS 0.05 0.001 * DEFAULT DIRECTION DIRECTION 0.0 **************.*.* ••* •••• OUTPUT OPTIONS ••••••••••••••••••* ••••• ELECTRON-ESCAPE NBINE 2 NBINT 2 PHOTON-ESCAPE NBINE 2 NBINT 2 ELECTRON-FLUX 1 4 NBINE 2 PHOTON-FLUX 1 4 NBINE 2 ** ••••••••••••••••••••*.*OTHER OPTIONS *.*•• **.**.* •••* ••**.** HISTORIES 10000 BATCHES 10 * ••• X-RAY PRODUCTION SCALING ISCALE-BREMS 500. I .SCALE-BREMS 5.0

While TIGER is the simplest of the ITS member codes, ACCEPTM is the most complex. Table 10.10 is an input stream for running the ACCEPTM code with all four major job steps. Examples of FORTRAN 77 PARAMETER modification and modification of Subroutine BFLD (uniform 10-kG field in the Z direction and no electric field) are included in the input to the Monte Carlo correction run. The lines beginning with "*/" are comments that are ignored by the UPDATE processor. This example also shows how the plot option is activated and used in the ACCEPT codes. 258 J. Halbleib

Table 10.10 Sample Input for a Full Four-Step Run of the ACCEPTM Code I ------F,P !*IDENT,DEFIRE *DEFIRE,CRAY {eor} MATERIAL TA MATERIAL AL TITLE I 1.0 MEY CROSS SECTIONS FOR TA AND AL ENERGY 1.0 !{eor} F,P *IDERT ,DEFINE I*DEFINE,CRAY *DEFIRE,ACCEPT !*DEFINE,MCODES *DEFINE,PLOTS *INSERT,BFLD.56 I EZ - 0.0 I BX - 0.0 BY" 0.0 I BZ = 1.0 RETURII I*/ THE FOLLOWING EXAMPLES OF PARAllETER MODIFICATION ARE USUALLY OPTIONAL, */ BUT MAY SIGNIFICANTLY REDUCE COMPUTER COSTS BY REDUCING MEMORY */ REQUIREMENTS. *IDERT,PMOD *DELETE,PARAMS.5 PARAllETER ( IRMT-2, HELEM-1, IRlUX-64, NSURY-2775, {eor} ECHO 1 TITLE ••• 1. 0 MEY TA/ AL TEST PROBLEM ••• ** •••••••••••••••••••• GEOMETRY •••••••••••••••••••••••••••••••• GEOMETRY RCC 0.00 0.00 0.00 0.00 0.00 -0.007 10.0 RCC 0.00 0.00 0.00 0.00 0.00 5.00 10.0 RCC 0.00 0.00 5.00 0.00 0.00 0.05 10.0 SPH 0.00 0.00 0.00 12.0 ERD I ZX1 +1 ZX2 +3 ZX3 +2 I ZX4 +4 -1 -2 -3 I END * MAT IFLD ECUT PTCZ 1 0 2 0 I o 1 o 0 •• ** ••••••••••••••••••••••••••••••••••••••••••••••••••••••• ** •••• 10. Structure and Operation of the ITS Code System 259

I ------Sample input for ACCEPTK (continued)

******** ••••••••••••••••• SOURCE •••••••••••••••••••••••••••••••• ELECTRONS ENERGY 1.0 I CUTOFFS 0.05 0.001 POSITION 0.0 0.0 -0.007 RADIUS 2.50 I • DEFAULT DIRECTION DIRECTION 0.0 0.0 I••••••••••••••••••••••••• OUTPUT OPTIONS •••••••••••••••••••••••• ELECTRON-ESCAPE lIBlRE 2 I lIBINT 4 PHOTON-ESCAPE I lIBlRE 2 lIBINT 4 ELECTRON-FLUX 1 2 I NBINE 2 PHOTON-FLUX 1 2 NBlRE 2 I••••••••••••••••••••••••• OTHER OPTIONS ••••••••••••••••••••••• HISTORIES 10000 IBATCHES 10 •.•• X-RAY PRODUCTION SCALING SCALE-BREMS 500. I PLOTS 3 -15.0 15.0 -15.0 15.0 180. 90. I -15.0 15.0 -15.0 15.0 0.0 0.0 -15.0 15.0 -15.0 15.0 180. 30. I ------

10.4.2 Output

In addition to certain diagnostic information, the default output consists of: a.) Energy and number escape fractions (leakage) for electrons, unscattered photons and scattered photons. b.) Charge and energy deposition profiles.

These data are sufficient to confirm the general partitioning and conservation of charge and energy.

In addition to the default output, a number of optional outputs may be selected through the use of the appropriate keywords. These are: a.) Escape fractions that are differential in energy for both electrons and scattered photons. b.) Escape fractions that are differential in angle for both electrons and scattered photons. c.) Coupled energy and angular distributions of escaping electrons and scattered photons. d.) Volume-averaged energy distributions of electron and photon scalar fluxes for selected regions of the problem geometry. e.) Pseudo-pulse-height distributions for selected regions of the problem geometry• for example, those regions corresponding to active detector elements. 260 J. Halbleib

The more sophisticated user will be able to modify existing outputs and add addi• tional ones through the correction-run procedures described in References 3 and 26.

Except for the initial diagnostic table which contains accounting information on the various particle types, every output quantity is followed by a one- or two-digit integer that is an estimate of the one-sigma statistical uncertainty of that quantity.

10.5 CONCLUDING REMARKS The ITS system represents the culmination of more than a decade and a half of effort involving contributions by many different people. Nevertheless, we do not expect and it was never intended that this software should remain static. Indeed, Version 2.0 was released by RSIC in July 1987, and Version 2.1, which implements the corrected algorithm for sampling electron energy-loss straggling as described by Seltzer in Chapter 7, was released in February, 1988. In addition to many improvements in the source files, including new options, the original UPEML processor has been replaced by a new one with much increased capability25. We are already working on a number of important new capabilities to be included in Version 3.0. We anticipate and encourage feedback from the user community that will suggest further modifications to enhance the utility and accuracy of the system.

Finally, although we expect that continuous-energy Monte Carlo methods (ETRAN, ITS, EGS27, SANDYL, BETA 28, MCNPE29.30 ) will continue to play an important part in our work for the forseeable future, in our primary role as users we anticipate and welcome powerful alternative techniques that will soon be available. Preliminary results have been obtained from a multi group Monte Carlo method31 which incorporates the CEPX(DATPAC)32,33 multigroup electron-photon cross sections into MCNp29. The re• sulting code will have an immediate adjoint capability, and will be microscopic in the sense that it will not depend on multiple-interaction theories such as multiple-elastic• scattering and energy-loss-straggling theories (similar capabilities are already available in BETA). These same advantages are already being exploited in discrete ordinates applications32,33 which combine the CEPX cross sections with the ONETRAN code34. Moreover, the latter runs in a small fraction of the time required for obtaining the same results from Monte Carlo, and has yielded numerous predictions that are in excellent agreement with Monte Carlo. 10. Structure and Operation of the ITS Code System 261

REFERENCES 1. M. J. Berger, "Monte Carlo Calculation of the Penetration and Diffusion of Fast Charged Particles" in Methods in Computational Physics, Vol. 1, edited by B. Adler, S. Fernbach and M. Rotenberg, (Academic Press, New York, 1963). 2. J. A. Halbleib and T. A. Mehlhorn, "ITS: The Integrated TIGER Series of Cou• pled Electron/Photon Monte Carlo Transport Codes", Nucl. Sci. Eng., 92, No. 2 (1986) 338. 3. J. A. Halbleib and W. H. Vandevender, "EZTRAN-A User-Oriented Version of the ETRAN-15 Electron-Photon Monte Carlo Technique", Sandia National Laboratories report SC-RR-71-0598 (1971). 4. J. A. Halbleib and W. H. Vandevender, "EZTRAN 2: A User-Oriented Version of the ETRAN-18B Electron-Photon Monte Carlo Technique", Sandia National Laboratories report SLA-73-0834 (1973). 5. M. J. Berger and S. M. Seltzer, "ETRAN Monte Carlo Code System for Electron and Photon Transport Through Extended Media", Radiation Shielding Informa• tion Center, Computer Code Collection CCC-107 (1968). 6. J. A. Halbleib and W. H. Vandevender, "TIGER, A One-Dimensional Multilayer Electron/Photon Monte Carlo Transport Code", Nucl. Sci. Eng. 57 (1975) 94. 7. J. A. Halbleib and W. H. Vandevender, "CYLTRAN: A Cylindrical- Geometry Multimaterial Electron/Photon Monte Carlo Transport Code", Nucl. Sci. Eng. 61 (1976) 288. 8. J. A. Halbleib, "ACCEPT: A Three-Dimensional Electron/Photon Monte Carlo Transport Code Using Combinatorial Geometry", Nucl. Sci. Eng. 75 (1980) 200. 9. W. Guber, J. Nagel, R. Goldstein, P. S. Mettelman and M. H. Kalos, "A Geomet• ric Description Technique Suitable for Computer Analysis of Both the Nuclear and Conventional Vulnerability of Armored Military Vehicles, ", Mathematical Applications Group, Inc. report MAGI-6701 (1967). 10. E. A. Straker, W. H. Scott, Jr., and N. R. Byrn, "The MORSE Code with Com• binatorial Geometry", Science Applications, Inc. report SAI-72-511-LJ (DNA 2860T) (1972). 11. J. A. Halbleib and J. E. Morel, "TIGERP, A One-Dimensional Multilayer Elec• tron/Photon Monte Carlo Transport Code with Detailed Modeling of Atomic Shell Ionization and Relaxation", Nucl. Sci. Eng. 70 (1979) 219. 12. J. A. Halbleib and J. E. Morel, Sandia National Laboratories, (unpublished). 13. H. M. Colbert, "SANDYL: A Computer Code for Calculating Combined Photon• Electron Transport in Complex Systems", Sandia National Laboratories report SLL-74-0012 (1973). 14. J. A. Halbleib, Sr., and W. H. Vandevender, "Coupled Electron Photon Colli• sional Transport in Externally Applied Electromagnetic Fields", J. Appl. Phys. 48 (1977) 2312. 15. L. F. Shampine, H. A. Watts and S. Davenport, "Solving Nonstiff Ordinary Differential Equations - The State of the Art", SIAM Rev. 18 (1976) 376. 16. K. 1. Hiebert and 1. F. Shampine, "Implicitly Defined Output Points for Solu• tions of ODEs", Sandia National Laboratories report SAND80-0180 (1980). 262 J. Halbleib

17. J. A. Halbleib, R. Hamil and E. 1. Patterson, "Energy Deposition Model for the Design of REB-Driven, Large-Volume Gas Lasers", Conference Record - Abstracts, IEEE International Conference on Plasma Science, May 18-20, Sante Fe, NM, IEEE Catalogue No. 81CH1640-2 NPS, p. 117. 18. J. A. Halbleib, "SPHERE: A Spherical-Geometry, Mulitmaterial "Electron/Photon Monte Carlo Transport Code", Nucl. Sci. Eng. 66 (1978) 269. 19. P. A. Miller, J. A. Halbleib and J. W. Poukey, "Inverse Ion Diode Experiment", J. Appl. Phys. 52 No.2 (1981) 593. 20. L. G. Haggmark, C. J. MacCallum, and M. E. Riley, "New Scattering Cross Sections for Electron Transport", Trans. Am. Nucl. Soc. 19 (1974) 471. 21. R. M. Sternheimer and R. F. Peierls, "General Expression for the Density Effect for the Ionization Loss of Charged Particles", Phys. Rev. B3 (1971) 3681. 22. F. Biggs and R. Lighthill, "Analytical Approximations for X-Ray Cross Sections II", Sandia National Laboratories report SC-RR-71 0507 (1971). 23. F. Biggs and R. Lighthill, "Analytical Approximations for Total Pair-Production Cross Sections", Sandia National Laboratories report SC-RR-68-619 (1968). 24. J. H. Hubbell, H. A. Gimm and 1. Overbo, "Pair, Triplet and Total Atomic Cross Sections (and Mass Attenuation Coefficients) for 1 MeV - 100 GeV Photons in Elements Z = 1 to 100", J. Phys. Chern. Ref. Data 9 (1980) 1023. 25. T. A. Mehlhorn and M. F. Young, "UPEML Version 2.0: A Machine-Portable CDC Update Emulator", Sandia National Laboratories report SAND87-0679 (1987). 26. "UPDATE, VERSION 1, Reference Manual", Control Data Corporation report 60449900 (Revision 11/23/81). 27. W. R. Nelson, H. Hirayama, and D. W. O. Rogers, "The EGS4 Code System", Stanford Linear Accelerator Center report SLAC-265 (1985). 28. T. M. Jordan, "An Adjoint Charged Particle Transport Method", IEEE Trans. Nucl. Sci. NS-23 (1976) 1857. 29. "MCNP-A General Monte Carlo Code for Neutron and Photon Transport, Ver• sion 3B", edited by J. F. Briesmeister, Los Alamos National Laboratory report LA-7396-M (revised), to be published. 30. R. G. Schrandt and H. G. Hughes, Los Alamos National Laboratory, to be pub• lished. 31. J. E. Morel and W. M. Taylor, Los Alamos National Laboratory, to be published. 32. L. J. Lorence, Jr., W. E. Nelson, and J. E. Morel, "Coupled Electron Photon Transport Using, the Method of Discrete Ordinates", IEEE Trans. Nucl. Sci. NS-32 (1985) 4416. 33. J. E. Morel and L. J. Lorence, Jr., "Recent Developments in Discrete Ordinates Electron Transport", Trans. Am. Nucl. Soc. 52 (1986) 384. 34. T. R. Hill, "ONETRAN: A Discrete Ordinates Finite Element Code for the Solution of the One-Dimensional Multigroup Transport Equation", Los Alamos National Laboratory report LA-5990-MS (1975). 11. Applications of the ITS Codes

J. Halbleib

Sandia National Laboratories Box 5800 Albuquerque, New Mexico 87185, U.S.A.

11.1 INTRODUCTION The structure and operation of the ITS system was discussed in an earlier lecture. Supporting documentation cited in that discussion contains even more information on the subject. Here, we will concentrate on applications of the member codes.

The philosophy behind the system reflects the fact that we are primarily users rather than code developers. Idealistically speaking, one tends to think of accuracy as being the dominant consideration in evaluating code performance. However, from a practical point of view, we find that the capability of applying the system quickly and easily to the solution of a great variety of problems to be of major, if not equal, importance. In the ITS system, we have devoted much effort to minimizing any incom• patibility between these objectives.

We will more or less faithfully follow a certain sequence in presenting the results of what we see as four types of code applications. Applications of a verification nature refer to comparisons of relatively precise measurements with relatively accurate and unambiguous simulations. There is an extensive literature on this type of comparison for the ETRAN system, which is relevant to the ITS system as well. However, this chapter will be confined specifically to verifications of the ITS codes. Corroborative comparisons are those involving experimental and/or code predictions of sufficiently high uncertainties that, taken individually, cannot be used to verify the codes, but, taken as a group, do much to establish credibility for those codes. Predictions refer to applications in which the codes are used primarily as a design tool and for which little or no experimental data exists. Research applications refer to those applications in which the codes have been substantially modified, usually with a view toward a major extension of their capabilities.

Because of space limitations, we will not be able to cover the following examples in any great detail. However, immediately after the discussion of each topic, we have attempted to cite the best available references where more detailed information may be found. The selected graphical information that will be presented in support of this discussion can be found in greater detail in the cited literature. General references may be found in Chapter 10 on the "Structure and Operation of the ITS System", and will not be repeated here.

263 264 J. Halbleib

11.2 VERIFICATION

We begin this review of ITS code applications with those of a verification nature.

11.2.1 Van de Graaff Deposition Profiles

Using a Van de Graaff source, Lockwood et a[1 measured electron energy deposition profiles for a broad matrix of materials, source energies, and source incident angles (in Be through U, at incident energies from 0.3 to 1.0 MeV, with selected data at 50 and 100 keV, and for incident angles from 0 to 60 degrees) and compared the results with predictions of the TIGER code. If there is any tendency, it is for theory to be higher in the peaks and lower in the tails in homogeneous targets, and for some disagreement in low-Z materials near high Z/low Z interfaces. One example, chosen from this series, is the comparison between the TIGER code and measurementsl for I-MeV electrons normally incident on an AI/Au/AI configuration (see Figure 11.1). .

11.2.2 Van de Graaff Electron Backscatter

Lockwood et a1 2 also measured electron backscatter for roughly the same range of materials, but an even greater range of energies and angles, and again compared their results with TIGER code predictions. Agreement was generally very good. However, it was shown that a discrepancy at low energy is at least partially due to breakdown of the Moliere screening approximation used in the Spencer scattering cross section. The use of improved numerical cross sections now employed in the ITS system at energies below 256 keV yielded significantly better agreement; however, predictions are still low in high-Z materials at low energy. A discrepancy in the energy albedo for U at all energies is not understood, but is believed to be an experimental problem because of the good agreement for U02 and other high-Z materials. We include an example from the report by Lockwood2 of a comparison of TIGER and measurements for the electron number backscatter as a function of atomic number at 1 and 0.1 MeV and angles of 0 and 60 degrees (see Figure 11.2).

11.2.3 Van de Graaff Electron Deposition in Film

In order to study the response of critical electronic systems to post-accident environ• ments of nuclear reactors, Buckalew et als developed a magnetic deflection system to raster an electron Van de Graaff beam over large areas. Experimental energy deposition per unit current density in thin film dosimeters as a function of electron kinetic energy from 0.2 to 1.0 MeV was obtained from measured dose, current, accelerator voltage, and mask orifice dimensions. Spread in the downstream radial deposition profile due to scattering of the unrastered beam in a thin Be window was also measured. Comparisons between the measurements and TIGER code predictions show good agreement.

11.2.4 Van de Graaff X-Ray Production and Dosimetry

Sanford et al"·5 used a 750-keV Van de Graaff source to study x-ray production and dosimetry and to compare with Monte Carlo code predictions. In the CYLTRAN code predictions, it was necessary to use a detailed mockup of the converter and its support structure in order to account properly for the production and transport of 11. Applications of the ITS Codes 265 bremsstrahlung by electrons backscattered from the converterfoil and to model carefully the CaF2 thermoluminescent dosimetry (TLD) configurations in order to account for both photon buildup and lack of secondary electron equilibrium. Over a wide range of thicknesses of the Ta converter and x-ray emission angles, the code is found to agree with the measurements. For Ta thickness near those that optimize the radiation output, however, the code overestimates the radiation dose at small angles. Code predictions were also in good agreement with direct measurements of dose enhancement from high-Z (Ta) encapsulation6 • Two examples are given for the TLD x-ray deposition as a function of distance from the beam axis for a Ta converter and as a function of converter thickness for a given angle (see Figure 11.3).

11.2.5 low-Energy Electron Backscatter

Pregenzer7 carried out a series of calculations for electron backscatter at energies of 100 keVand below using the TIGERP code and the improved elastic scattering cross sections now employed in the ITS system, and compared the results with existing exper• imental data. Even with the improved cross sections, backscatter at these low energies is still underpredicted. However, except for the very lowest energies, the discrepancy is less than 20% .

11.2.6 2-D Electron Energy Deposition in Water at Intermediate Energies

Marbach8 compared CYLTRAN results with his measurements of axial and transverse energy-deposition profiles in water that were obtained using electrons from a betatron source. The beam was injected through a Cu exit foil, a W scatterer, and an air space into a water phantom. Good agreement is obtained between predicted shapes of the profiles and ionization chamber measurements made at about 15 and 18 MeV. However, neither the theoretical nor the experimental results are absolute. David Rogers9 has pointed out that for low atomic-number targets in this energy range, the first moment of the straggling distribution is too small (see, however, Chapter 7 for a discussion of a corrected algorithm by Seltzer that has been implemented in Version 2.1 of ITS). In these cases, the first moment is about 5% too low at energies close to the source energies. The inability of the code to account properly for lead and steel structures in the vicinity of the scattering foil was blamed for the poorer agreement at lower source energies. Comparisons of the energy deposition in water as a function of depth and as a function of radius are included for two different electron energies (see Figure 11.4).

11.2.7 High-Energy 2-D Profiles At much higher energies (0.2 to 1.0 GeV), CYLTRAN predictions have been compared with two-dimensional electron energy deposition profiles measured in water, AI, and a multi-material (containing Fe) configurationlO • Also included in the comparisons were the predictions of either the ETRAN or EGS code systems. Agreement among the various theoretical models and the measurements generally is good to very good, with some systematic differences noted. Three examples of this comparison for electron energy deposition in Al are included (see Figure 11.5).

11.2.8 BGO Pulse-Height Distribution

Hsu et aPl used the CYLTRAN code to model the response of a 7.62 cm (diameter) x 7.62 cm (long) bismuth germanate scintillator. They compared the measured pulse 266 J. Halbleib height distribution (PHD) and the predicted spectrum of absorbed energy, or pseudo• PHD, for the 662-ke V gamma rays from 137 Cs. The photopeak and Compton edges agree well. However, the theoretical photopeak is a delta function, whereas the experimental photopeak is broadened due to finite resolution, accounting for the higher theoretical peak-to-valley ratio. The experimental peak at 200 keY is attributed to backscatter from the concrete walls of the room and the photomultiplier tube, which were not included in the calculations. They also compared the measured and calculated absolute photopeak efficiencies for seven monoenergetic gamma ray energies ranging from 165.9 to 2614.6 keY at four source-to-detector distances. The average deviation between calculations and measurements is 3%.

11.2.9 Charge Profiles in Plastic

Frederickson and Woolf12 compared measured charge-deposition profiles in an electri• cally conductive carbon-filled polymer irradiated by monoenergetic electron beams at kinetic energies of 0.4, 1.0, and 1.4 MeV with predictions of the TIGER code. Agree• ment is good, though the Monte Carlo systematically predicts a higher peak deposition and a depth distribution with a narrower width. Charge deposition at these energies is shown in Figure 11.6.

11.3 VERIFICATION/CORROBORATION

Corroborative applications are also supported by experimental data, but the compar• isons are less definitive either because the measurements are not sufficiently precise or because the fidelity of the simulation is not as high as we would like. Taken as a group, however, they can do much to establish the validity and utility of the code system.

We begin this section with the first of several applications involving high-intensity, pulsed, relativistic electron beams (REB's). Because of the large uncertainty in the distributions required to describe the source input for these calculations and the large scatter in the experimental data, the determination of agreement or disagreement be• tween theory and experiment is often ambiguous.

11.3.1 Radial Electron Beam Diode for Gas Laser Excitation

Ramirez and Prestwichl 3,14 used the CYLTRANM code to predict the efficiency and uniformity of energy deposition in their experiments employing a high-intensity, pulsed, radial electron-beam diode operating at a peak energy of 0.75 MeV for gas laser excita• tion. The spectrum of the source electrons was obtained from measured I-V waveforms, and the diode electric field in CYLTRANM was redefined for each source electron to correspond to the sampled kinetic energy. The presence of the field accounted for the reinjection of a portion of the electron energy that was collisionally reflected back into the diode. The measured axial distribution of the source electrons was used in the cal• culations. The deposition in passive radiochromic dosimetry at various radial locations within the cylindrical laser cavity was measured and predicted in 5 gases at pressures up to 3 atmospheres. Good agreement for both the absolute two-dimensional profiles and the total deposition was obtained for all gases except SF6 • The code consistently overperdicts the deposition in the latter gas at high pressure and small radii. This is believed to be caused by the electronegativity of SF6 ; electrons are attached leaving low mobility negative ions with an associated space charge buildup. The greater dis• agreement near the ends of the cavity is probably due to uncertainty in the spatial and 11. Applications of the ITS Codes 267 angular distributions of the source. Two representative examples of the comparison of CYLTRANM and measurements are included for energy deposition in argon (see Figure 11.7).

11.3.2 Helia The Helia accelerator is a 3 MeV, 160 kA, 25ns test bed for the HERMES III accelerator which is projected to be a 20 MeV, 800 kA, 20ns flash x-ray source. Using Helia, Sanford et a/ 1S- 17 compared the measured radiation fields of two coaxial diodes, one with a simple planar anode/converter and one with anode/converter indented to minimize pinching of the electron beam. The stronger pinching of the planar diode leads to focusing near the anode and defocusing at greater distances. Measured results were compared with each other and with the predictions of the CYLTRAN code. Both theory and experiment show that more uniform profiles are available from the indented anode. The CYLTRAN code does a very good job of predicting the shapes of the profiles; however, the absolute predictions are about 25% below the measurements. Agreement is still good, however, because of the 15-20% uncertainties in both theory and experiment. Typical of most applications to high-intensity pulsed sources, the dominant contribution to the theoretical uncertainty is the uncertainty in the source distributions.

11.3.3 Proto II 6-Beam Overlap

Preformed channels were employed to transport six of the 12 electron beams of the Proto II accelerator to a central target. The energy delivered to the cylindrical Al targets was measured as a function of target radii and compared with results obtained from theoretical models employed in inertial-confinement-fusion (ICF) research using electron beamsl8. The CYLTRANM code was also used to model the experiment. The beams were charge and current neutralized by the plasma channels so that the electrons followed trajectories in the net magnetic field resulting from the channel and return currents. The experimental transport efficiencies inferred from x-ray pinhole and thermoluminescent dosimetry data were compared with those predicted by the Monte Carlo calculations. The remarkable agreement does not constitute a definitive verifica• tion of the model, however, since the diagnostics themselves relied on the calculations for their interpretation.

11.3.4 REB/Multiple-Foil Interaction

Plasma channel transport was also used to study enhanced bremsstrahlung conversion from the interaction of a single REB with three Ta foils l9• The experiment was mod• eled with the CYLTRANM code. As the energy of an electron is reduced through in• teractions with the foils, the initial downstream betatron motion changes to upstream gradient-B drift motion. The normalized predicted axial profile in CaF2 TLDs is in good agreement with the measured data. The predictions were normalized to the experiment because of the large uncertainty in channel transport efficiency.

11.3.5 18 Blades A multiple-blade diode has been investigated as a backup design for converting the energy of the 25 TW, 500 kJ, 2 MeV, 20ns electron beam from the Saturn facility into 268 J. Halbleib x-rays (multiple element diodes can minimize beam pinching and the resulting ion cur• rent losses). The diode design attempts to exploit multiple reflexing of beam electrons through the anode/converter between real and virtual cathodes. Sanford et a1 20,21 used the ACCEPT code to model an experiment with a prototype diode on a smaller test accelerator. Instead of the individual blades, the cathode was simulated by a conical surface in the calculations, and reflexing was simulated via localized specular reflection (since there was too much uncertainty about the actual spatial dependence of the electric fields). Absolute Monte Carlo predictions of energy deposition in CaF2 thermolumine• sent dosimetry at a downstream location were compared with the measured deposition as a function of the thickness of the Ta anode/converter. Comparisons were made for three different reflexing assumptions in the theoretical model - no reflexing, reflexing from the virtual cathode only, and reflexing from both the real and virtual cathodes - to see what theoretical assumptions were best supported by experiment. Within the large uncertainties (26% theoretical due primarily to uncertainty in the source distri• butions), the data agree with the second of the above reflexing assumptions, but not with the other two. X-ray pinhole photography provides additional support for the assumption of reflexing from the virtual cathode only.

11.3.6 REB Pumping of Noble-Gas Halide Laser

Hoffman et al 22,23 used a magnetic field axially to inject and confine an electron beam in a cylindrical cavity in order to study REB pumping of noble-gas/halide lasers. Energy transport and radial profiles were measured in an axially movable and total stopping segmented graphite calorimeter. The CYLTRANM code was used to model experi• ments with the calorimeter 75 and 135 cm downstream of the point where the beam was injected into the cavity and confined by a 3-kG magnetic field. Two-dimensional deposition profiles in the gas and radial profiles in the segmented calorimeter were ob• tained in various gases at various pressures. At a position 135 cm from the injection point, normalized CYLTRANM predictions were compared with measured calorimeter profiles in argon at two pressures24• Very good agreement was obtained. A reduction in the energy deposition near the cavity walls was caused by the tendency of a tangential magnetic field to enhance absorption at solid material boundaries. Normalized predic• tions of the total energy deposited in the calorimeter as a function of its distance from the source also agreed reasonably well with experiment.

11.3.7 Gradient-B Drift Transport

The gradient-B drift scheme is being studied as a method of transporting, bunching, and enhancing the x-ray production by pulsed electron beams. As originally proposed, a beam produced in a pinched beam diode is allowed to expand in order to minimize azimuthal motion before injection into a drift region where a current carrying wire gen• erates an azimuthall/r magnetic field that causes the beam electrons to transport via gradient-B drift toward an x-ray converter. Lee et a[25-27 conducted an experiment on the Hydra accelerator in which they used a TLD array to measure the radial profile of the x-ray energy deposition in a plane downstream of the converter. The absolute radial profile measured on a particular shot was compared with absolute profiles cal• culated using the CYLTRANM code. The predicted profiles were obtained assuming gaussian radial beam profiles with various half-widths - the half-width being a major uncertainty in the experiment. Good agreement between measurements and predictions was obtained using a half-width that was consistent with x-ray pinhole photographs and witness-plate damage. Another major uncertainty in the comparison was the injection 11. Applications of the ITS Codes 269 and transport efficiencies used to normalize the calculations. A single figure is included from the comparisons of Lee et al of the absolute radial profiles of x-ray energy depo• sition in TLD's (see Figure 11.8).

11.3.8 Printed Circuit Boards

Beezhold et al 28 combined predictions of the TIGER code with a circuit analysis code to study the IEMP (internal electromagnetic pulse) response of a printed circuit board (PCB) to x-radiation. In particular, they compared absolute predictions and measure• ments of the replacement current in the tinned-Cu conductor as a function of its width. Excessive current could lead to burnout somewhere in devices or interconnects of the PCB. Three predicted x-ray spectra from the PI 737 facility were used as inputs in TIGER code predictions of charge deposition. The TIGER predictions of the trans• fer of charge across the boundaries were then used as inputs to a circuit code. For the unhardened PCBs, there is good agreement for the low-energy spectrum, while the medium- and high-energy predictions appear to be high by about a factor of 2. For the hardened PCBs, there was no systematic disagreement, but there was evidence of a null in the replacement current for certain values of the parameters, making compar• isons more difficult. The measured response of the hardened PCB's is 1 to 2 orders of magnitude less than that of the unhardened PCB's, and the predictions are consistent with this reduction.

11.3.9 Voyager Electron Telescope

Using the CYLTRAN code, design calculations were performed29 for a directional in• terstellar electron spectrometer being flown aboard Voyagers 1 and 2. Basically, the instrument consists of 8 silicon disks surrounded by anticoincidence silicon rings and sep• arated by tantalum absorbers. The coincidence/anticoincidence logic with appropriate trigger levels was added to the existing PHD coding, and the possible trigger/no-trigger combinations of the silicon disks were displayed as 8-digit binary numbers (1 meaning trigger and 0 meaning no trigger). The predictions at 10 MeV were compared with linac measurements at 11 MeV. In the comparisons for the critical combinations-those in• volving a series of successive triggers starting with the initial disk-there is approximate agreement within the large uncertainties, although the predictions are systematically low.

11.3.10 SPEED/Triaxial-Diode Flash X-Ray Source

Hedemann et al 30 used the CYLTRAN code to predict x-ray production and dosimetry from a triaxial diode with a Ta anode/converter. The triaxial diode is the primary diode candidate for the Saturn accelerator. By controlling the balance in the two return current paths of the diode, radial pinching can be prevented, resulting in a large• area, low-voltage, low-impedance flash x-ray source. TLD measurements were made in a downstream planar array, and compared with the absolute predicted radial profiles. The average electron kinetic energy was about 0.6 MeV, and the peak energy 1.2 MeV, but there was much uncertainty in the source distributions needed for the Monte Carlo predictions. Within these large uncertainties, there is good agreement31. The position of the peak energy deposition confirms that there was minimal pinching. 270 J. Halbleib

11.3.11 Inverse Ion Diode

Miller et al 32 designed a novel ion diode intended for ICF applications. The diode is a small metalized thin-walled evacuated glass sphere which is irradiated on opposite sides by two relativistic electron beams that are transported to the sphere via plasma channels. A virtual cathode, formed just inside the glass wall, accelerates ions radially inward, focusing them on a levitated central target where they produce neutrons. The ACCEPTM code was used to study the interaction between the beams and the sphere, and to plot trajectories of the electrons as they reflex through the wall of the sphere. The reflexing is due to the assumed self-magnetic fields of the currents outside the sphere and the simulated virtual-cathode electric field inside the sphere. The predicted reflexing is used as input to a PIC (particle-in-cell) code for predicting the ion current and the voltage at the levitated target. Results are consistent with those inferred from measured neutron yields.

11.4 PREDICTIONS

We turn now to applications of a predictive nature. The codes are used primarily as a design tool, and little or no experimental data exist.

11.4.1 Bremsstrahlung Radiation Environment of PBFA-II

The baseline design of PBFA-II, a 4-MJ accelerator for ICF research, was changed from a 5-MV proton diode to 30-MV lithium-ion diode, necessitating a reevaluation of the bremsstrahlung radiation hazard to both personnel and electrical components caused by electron current losses. The ACCEPT code has been employed by Sweeney et a1 33- 3s to predict x-ray dosimetry throughout the system. While undergoing shakedown testing, little dosimetry was available for comparison with predictions. However, code applica• tions during this period provide a good demonstration of the plotting capability of the ITS system which was used extensively to verify the fidelity of the complex geometri• cal model of the total facility and portions of the surrounding environment, as well as critical subsystems. Code predictions are now being compared with dosimetry data.

11.4.2 PBFA-I, MITL, and Gamma-Ray Telescope Plots

There are other, mostly unpublished, examples of the plotting capability of the ITS system taken from geometrically complex applications of CYLTRANM and the AC• CEPT codes. Examples of applications of the ACCEPT code36•37 are the magnetically insulated transmission lines in the central region of the IS-module PBFA-I accelerator, a more detailed mockup of a single transmission line of that facility, and a design for an intergalactic gamma-ray telescope. The plot utility permits an arbitrary number of parallel projections of the problem geometry in arbitrary directions. For the M codes, an arbitrary number of electron trajectories in regions of nonzero macroscopic fields may be plotted on the final projection.

11.4.3 RAYO: REB-Pumped Gas Laser in Rectangular Geometry

In the case of the ACCEPTM code, one has the option of plotting a sample of the electron trajectories on the final parallel projection. This was a useful feature in the 11. Applications of the ITS Codes 271 application of the code to the design of a large-volume, high-pressure, rectangular ge• ometry gas laser system to be pumped by a large-area, pulsed, REB38. In the model, the beam was injected through a hibachi structure and laterally confined by a uniform magnetic field to prevent loss to the side walls. Gas flow was in a direction normal to one pair of the side walls, with extraction of laser energy in a direction normal to the other pair of side walls. Final projections normal to a side wall from three separate runs at fields of 0, 1, and 3 kG show, via random samples of l.O-MeV electron trajectories in argon (at 2.5 atm), how confinement increases with magnetic field strength. Several features are evident in the trajectory plots-loss to the side walls, loss to the hibachi structure, scattering in the entrance foil, backscatter within the gas, and at least one case of an x-ray-produced secondary electron. The purpose of the field, of course, was to minimize wall loss and to maximize uniformity of deposition. As the field increases from 1 to 3 kG, the argon deposition increases by a factor of 2, and the wall loss de• creases by a factor 5. Using the UPEML correction-run procedure, we were able to reduce the geometry input by hundreds of lines through implementation of automatic subzoning which took advantage of the fourfold symmetry of the geometry (automatic subzoning in the ACCEPT codes is an option in Version 2.0 of ITS). Three projections are included from this study showing how confinement increases with magnetic field strength (see Figure 11.9).

11.4.4 Sector X-Ray Converter with Gradient-B Transport

Lee et a1 39 investigated a novel bremsstrahlung converter concept that couples quite naturally to gradient-B electron transport. The current carrying wire that provides the l/r magnetic field terminates in a pie-shaped planar conducting converter. The field resulting from this current distribution efficiently traps the electrons at the surface of the foil. Plots from ACCEPTM calculations show the interaction of l.O-MeV electrons with a pie-shaped Ta converter foil having a 45-degree half-angle. Electron trajectories are plotted in projections that show the foil edge-on as well as from the side. The predicted total x-ray extraction efficiency is the highest ever obtained from a l.O-MeV source. A two-dimensional spatial distribution of the x-ray output over the surface of the foil was obtained.

11.4.5 TIGER vs TIGERP Line Radiation

The special capability of the P codes is demonstrated in a comparison of TIGER and TIGERP predictions of the reflected photon spectra for 350-keV electrons normally incident on a 1/3 range Ta converter40• The TIGER code predicts only the average K-fluorescent energy. The TIGERP code, on the other hand, predicts the intensities of eleven lines down to the average M-fluorescent energy.

11.4.6 Falcon

Falcon is a project for developing a nuclear-driven laser. Because of the premium on reactor time, Patterson et al'H have used the CYLTRANM code to investigate the pos• sibility of employing an electron-beam-pumped system to study the laser kinetics. Plots from CYLTRANM calculations showing sample electron trajectories were obtained at three uniform axial magnetic field strengths for the same geometrical configuration. The focused beam is injected through a fast opening valve into a long, narrow (l-atm) argon drift which prevents pressure loss during the beam pulse, followed by the laser 272 J. Halbleib gas, primarily Ne and Xe, which is separated from the argon by a second valve. At 1 kG, the beam quickly scatters into the stainless steel wall. At 10 kG, there is well-defined betatron motion, and a small amount of energy reaches the laser medium, but most of the beam cannot get through the second valve. At 50 kG, the second valve is essentially 100% transmitting. Two-dimensional energy deposition profiles were obtained in both the drift and laser gases. At 10 kG, the second valve would have to be removed in order to achieve a power deposition marginally close to the desired few kW fcm. At 50 kG, the desired power deposition should easily be achieved. Much additional model• ing of this concept has been carried out by Sweeney et a142 , including more complex three-dimensional configurations involving nonuniform injection fields. A single plot of confinement in argon gas is included showing the effects of magnetic field strength (see Figure 11.10).

11.5 RESEARCH There are also a number of areas of research which have received varying degrees of support over the years. In research applications, the codes have been substantially modified, usually with a view toward a major extension of their capabilities. We discuss only those areas of research where the modifications have not been made permanent, and thus are not available to ITS users. The following areas are mentioned without further elaboration: a.) CYLTRANM and a two-dimensional hydrodynamic code have been combined in order to model electron-beam-driven ICF experiments. b.) Researchers at Los Alamos National Laboratory have coupled the ACCEPT code to models for generating and transporting Cerenkov and transition radiation. c.) The TIGER code has been used to study certain aspects of continuous-energy adjoint Monte Carlo. d.) A photon-only version of the ITS system is now operational. e.) A program is now under way to develop an ion version of the ACCEPT code in support of research in nuclear-driven laser systems. f.) A modification permitting positron sources has been tested.

In the remaining paragraphs of this section, additional research applications are discussed in more detail.

11.5.1 Time-Dependent Response of the Atmosphere to X-Ray Energy Deposi• tion A modification allowing time dependence has been used to corroborate a semi-empirical model for generating time-dependent responses of the atmosphere to flash x-ray source distributions resulting from the injection of electron beams with energies up to 10 MeV into high-Z converters43• A comparison of the contributions, as predicted by the time-dependent CYLTRAN code and the semiempirical model, of the direct and scattered x-rays to the energy deposition rate in air resulting from the bremsstrahlung conversion of a time-dependent electron source with a peak energy of 10 MeV shows good agreement44• A comparison of the contributions of direct and scattered x-rays to the energy deposition rate in air is included (see Figure 11.11).

11.5.2 Electric Fields in Materials Because the collisional physics of the ITS system is valid only for isotropic media, 11. Applications of the ITS Codes 273 collisional transport within a macroscopic step of the electron random walk is unaffected by magnetic deflections. Thus, it is relatively easy to combine the effects of collisions and magnetic fields. Electric fields, however, have been restricted to voids because they change electron energy, making ambiguous the choice of the energy at which cross sections are to be evaluated. By adding an additional constraint on the integration of the equations of motion that limits the energy loss to the electric field in a random walk step, the CYLTRANM code has been used to study the phenomenon of runaway electrons in large externally applied electric fields. Trajectory plots vividly show how the transport changes as the magnitude of a uniform axial electric field increases from a large negative value through zero to a large positive value45 • In each plot, a IOO-keV monoenergetic electron source at the center of a 60-cm radius, 60-cm long cylinder of argon at atmospheric pressure is directed along the positive Z axis. The asymmetry for parallel vs anti parallel fields is quite evident. The practical range of the electrons is only 8 cm; however, if the field is strong enough, some of the electrons will gain more energy from the field than they lose from collisions, and "run away". Code results show that the IOO-keV source energy plus the energy absorbed from the field agrees well with the sum of the deposited and escape energies. A realistic treatment of this problem would include nonlinear effects such as the time-dependent radiation-induced conductivity. A final comparison is included showing the potential for runaway electrons directed along the positive axial direction in argon (see Figure 11.12).

11.5.3 Hidden Lines A modification of the plot routines allows the primitive body types of the ACCEPT code combinatorial geometry to be plotted so as to show the hidden lines46• There has also been some effort devoted to perspective plotting as opposed to parallel projections.

11.5.4 Self-Consistent Alfven Problem Finally, in a limited foray into the area of self-consistency, the CYLTRANM code was modified to study the Alfven problem47• The logic normally used for batching, in order to obtain statistical estimates, was modified to serve as an iterative scheme for find• ing the steady-state solution for the initially collisionless transport of a charge-neutral plane-parallel uniform beam of 1.0-MeV electrons in its own self-magnetic field. The total beam current was equal to the Alfven critical current, the approximate maximum current at which the beam will propagate. The CYLTRANM solution for the trajec• tories of the beam electrons, exhibiting the characteristic betatron oscillations, is in excellent agreement with the solution obtained from a PIC code. The two-dimensional axial current density profiles are also in excellent agreement. If a 5 micron Au foil is inserted near an axial location corresponding to a focus in the betatron motion at the iteration where the solution for the collisionless beam has reached its steady state, little change is seen as the calculation is continued. If, however, a thick Au foil is in• serted, the steady-state solution is much more strongly modified. The large collisional backscattering leads to a significant reduction in the net current, and the consequent defocusing of the beam in the vicinity of the foil.

11.6 CONCLUSION We believe that the applications discussed here represent a convincing demonstration of the accuracy and flexibility of the ITS system, and of our success in meeting our stated objectives in the development of the system. 274 J. Halbleib

4.0 I"""':":"'""'...... ---:A":""I------, AI CALORIMETER THICKNESS 5.05 X 10- 3g / cm 2 H 3.0 Au CALORIMETER THICKNESS ENERGY 1.399 X 10-2g/ cm2 DEPOSITION H (MeV / 9 / cm2) o EXPERIMENT 2.0 "'"'L THEORY

1.0

0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 FRACTION OF A MEAN RANGE Figure 1l.l. Comparison of predictions of the TIGER series with the mea• surements of Lockwood et all for the energy deposition of l.O-MeV electrons normally incident on an All Aul Al three-layer configuration.

FRACTION OF FRACTION OF ELECTRONS ELECTRONS BACKSCAnEREO BACKSCAnEREO

0.7 r------"";:::::::=-..., 0.7 _------"IP.'l~ 0.' 0.6

0.5 0.5

0.4 0 .•

0.3 0 . 3

0.2 0.2 • EXPERIMENT 0 .1 0. 1 • EXPERIMENT • THEORY • THEORY 0.0 L.IIu...-I_...... _ ...... _ ...... __ ..... 0 . 0 ....;..J"".-I_ ...... _ ...... _ ...... _ ...... o 10 20 30 40 50 to 70 10 gO 100 o 10 20 30 40 50 60 70 10 10 100 ATOMIC NUMBER (Z) ATOMIC NUMBER IZ) Figure 11.2. Comparison of predictions of the TIGER code with measure• ments of Lockwood et a1 2 for the electron number backscatter as a function of atomic number at incident electron energies of 1.0 (left) and 0.1 (right) MeV and angles of 0 and 60 degrees. 11. Applications of the ITS Codes 275

______

24 23 111 .. m To CONVERTER 22 21 20 19 18 17 ~ ~: ~ 1. f13 ~12 llJl1 g 10

-CVLTAAN ~ EXPERIIiENT

10 20 ao 40 50 RADIUS (em)

Figure 11.3a Comparison of predictions of the CYLTRAN code with the measurements of Sanford et al 4,5 for the TLD x-ray deposition as a function of radius for a 111 micron Ta converter at a source electron energy of 0.75 MeV.

6 53° EMISSION ANGLE

5

...... 4 U II) "a-- I!! 3 ...... ~

W 2 U) 0 Q

-.- EXPERIMENT -<>- CYLTRAN 0 0 300

Figure 11.3b. Comparison of predictions of the CYLTRAN code with the measurements of Sanford et a1 4,s for the TLD x-ray deposition as a function of converter thickness for a 53-degree emission angle at a source electron energy of 0.75 MeV. 276 J. Halbleib

100

BO BO 14.9 MeV 17.B MeV CENTRAL AXIS DEPTH 60 60 DOSE (%) 40 40

• EXPERIMENT 20 • EXPERIMENT 20 MONTE CARLO JL MONTE CARLO

O~~~~~~~~~~ °0~~~2--"-4~~~6--"~~ o 2 4 6 B 9 DEPTH IN WATER (cm) DEPTH IN WATER (cm) Figure llAa Comparison of predictions of the CYLTRAN code with the mea• surements of Marbach8 for electron energy deposition in water as a function of depth at source electron energies of 14.9 and 17.8 MeV.

100 - I I RELA TIVE 80 - I DOSE I ELECTRON ENERGY (%) I I DEPOSITION IN WATER 60 I • EXPERIMENT I SL MONTE CARLO : 40

10 8 6 4 2 0 2 4 6 8 10 DISPLACEMENT FROM CENTRAL AXIS (em)

100 -

80 - RELATIVE DOSE (%) 60 - 17.8 MeV • EXPERIMENT 40 _ Jl.. MONTE CARLO

10 8 6 4 2 0 2 4 6 8 10 DISPLACEMENT FROM CENTRAL AXIS (em) Figure 11.4b. Comparison of predictions of the CYLTRAN code with the measurements of Marbach8 for the electron energy deposition in water as a function of radius at source electron energies of 14.9 and 17.8 MeV. 11. Applications of the ITS Codes 277

asps I (em)

MeV/em3, 100 SOURCE ELECTRON

• 1 -EXPT. (CRANNELL) 10 -CYLTRAN --EGS

5 X 10.2 0 10 20 30 40 50 60 70 80 90 DEPTH (em)

MeV lem 3, SOURCE ELECTRON

3s;.ps;. 4 (cm) 6s;.p s;. 8 (cm)

- EXPT. (CRANNELL) - EXPT. (CRANNELL) - CYLTRAN -CYLTRAN -- EGS --EGS

10-4 0 10 20 30 40 50 60 70 80 90 DEPTH (em)

Figure 11.5 Comparison of predictions of the CYLTRAN eode with mea• surements of Crannell and Cranne1l48 for the electron energy deposition in aluminum as a function of depth in three radial intervals (0-1 em, 3-4 em and 6-8 em) at a souree electron energy of 1.0 GeV. 278 J. Halbleib

0.4 MeV CHARGE DEPOSITION IN PLASTIC

CHARGE FRACTION

0.04

0.02

1.0 MeV CHARGE DEPOSITION IN PLASTIC 0.05 r----~Ir"'--~

0.04

CHARGE 0.03 FRACTION

0.02

0.01

1.4 MeV CHARGE 0.035 ..-----~----. DEPOSITION IN PLASTIC 0.030

0.025

CHARGE FRACTION 0.020

0.015

0.010

0.005

Figure 11.6. Comparison of predictions of the TIGER code with the mea• surements of Frederickson and Woolf12 for the electron charge deposition in a conductive carbon-filled polymer irradiated with 004, 1.0, and lA-MeV beams. The ordinate is the fraction of the incident charge deposited in a thickness of 10.7 mg/cm2• 11. Applications of the ITS Codes 279

• SHOT 1/ U03 80 HISTOGRAMS: MONTE CARLO

X.RAY DOSE 60 (krad CaF2 ) ...------.

40

20

o O~--"""--~--~--"'4 RADIUS (em)

Figure 11.7 Comparison of CYLTRANM predictions of the absolute radial profiles of x-ray energy depostion in TLD's, assuming gaussian electron beam profiles of various half-widths, with the measurements of Lee et aZ 25- 21 for a bremsstrahlung converter driven by a 1.0-MeV electron beam that has been propogated 89 cm via gradient-B drift.

JOULES / GRAM

32 r------~ p - 1 .7 em ARGON (1200 Torr) 24 ENERGY DEPOSITION

JOULES / GRAM 32 r------, p - 1.4 em o +12 24 Z-AXIS (cm)

II

o +12 Z-AXIS (cm)

Figure 11.8. Comparison of CYLTRANM predictions of the absolute axial profiles of electron energy deposition in 1200 Torr argon with the measure• ments of Ramirez and Prestwich14 for a cylindrical gas laser configuration driven by a radial electron beam diode. Solid curves are experimental and histograms are Monte Carlo. 280 J. Halbleib

110

'00 so FIELD DEPENDENCE 80 OF E 2 1. RA YO ELECTRON TRANSPORT >- 00 0 w 50 >- t- •• o" D: 30 2Q

. 0

00 -.. " 0 '00 : ~ '~ -... :~ SO - · ,0 • '/ ~ ~:: ~ 8. - /~/. ... - . ~ ~ 10 >- 6. .. , .... .; ..,· .10.~

2:::,' . .:.~ :- =---....-. ____

00 ~______d

-'0 '--'-~------~--' - 10 to 30 50 10 90 " 0 · '0 10 30 SO 10 &0 1 10

ROT ATED X em ( ) ROTATED X ( em) Figure 11.9 Projections normal to the sidewall of a model of an electron-beam pumped high-pressure gas laser system at fields of 0, 1 and 3 kG showing, via random samples of 1.0-MeV electron trajectories in argon obtained from AC• CEPTM calculations, how confinement increases with magnetic field strength.

FALCON PROJECT ELECTRON BEAM PUMPING

E 2.5 10 kG u V

2.5 50 kG

- 10 42 94 1 46 198 250 Z (em) Figure 11.10. Plots of 2.0-MeV electron trajectories in argon obtained from CYLTRANM calculations of a cylindrical electron-beam pumped gas laser system showing how confinement increases with magnetic field strength. 11. Applications of the ITS Codes 281

DOSE RATE AT Z=500m ANDer = 2.5·

DOSE RATE (rad I 5)

so 100 150 200 250 300 350 400 450 500 RETARDED TIME (ns) Figure 11.11 Comparisons of the contributions, as predicted by a time• dependent version of the CYLTRAN code and a semi-empirical model, of the direct and scattered x-rays to the energy deposition rate in air resulting from the bremsstrahlung conversion of a time-dependent electron source.

E-FIELD SUSTAINED ENERGY DEPOSITION

EZ = -0. MY/cm 10 o ~~o --~~--~o~~~ AXIAL POSITION (em) AXIAL POSITION (em)

EZ=-o·03MV/cm o 30 E50 AXIAL POSITION (em) ~ z o i= en o II. -' ;:;'" II:'"

o 30 -30 o AXIAL POSITION (em) AXIAL POSITION (em) :: I EZ ~ ~~'~~ o 30 ~o 0 ~ AXIAL POSITION (em) AXIAL POSITION (em) Figure 11.12. Predictions from a modified version of the CYLTRANM code of the transport of lOO-keY electrons directed along the positive axial direction in one atmosphere of argon in the presence of parallel and anti-parallel electric fields of different magnitudes showing the potential for runaway electrons. 282 J. Halbleib

REFERENCES 1. G. J. Lockwood, L. E. Ruggles, G. H. Miller and J. A. Halbleib, "Calorimetric Measurement of Electron Energy Deposition in Extended Media - Theory vs Experiment", Sandia National Laboratories report SAND79-0414 (1980). 2. G. J. Lockwood, L. E. Ruggles, G. H. Miller and J. A. Halbleib, "Electron Energy and Charge Albedos - Calorimetric Measurement vs Monte Carlo Theory", Sandia National Laboratories report SAND80-1968 (1981). 3. W. H. Buckalew, G. J. Lockwood, S. M. Luker, 1. E. Ruggles and F. J. Wyant, "Capabilities and Diagnostics of the Sandia Pelletron-Raster System", Sandia National Laboratories report SAND84-0912 (1984); Figs. 9 and 16. 4. T. W. 1. Sanford, J. A. Halbleib and W. Beezhold. "Experimental Check of Bremsstrahlung Dosimetry Predictions for 0.75 MeV Electrons", IEEE Trans. Nucl. Sci. NS-32 No.6 (1985) 4410. 5. T. W. 1. Sanford, J. A. Halbleib, W. Beezhold and 1. J. Lorence, "Experimental Verification of Non-Equilibrated Bremsstrahlung Dosimetry for 0.75 MeV Elec• trons", IEEE Trans. Nucl. Sci. NS-33 No.6 (1986) 1261. 6. J. A. Halbleib, T. W. 1. Sanford and W. Beezhold, "Experimental Verification of Bremsstrahlung Dosimetry Predictions for 0.75 Me V Electrons", Sandia National Laboratories report SAND85-1517 (1986). 7. A.1. Pregenzer, "Monte Carlo Calculations of Low Energy Electron Backscatter Coefficients", Nucl. Instr. Meth. in Phys. Res. B6 (1985) 562. 8. J. R. Marbach, "Optimization Parameters for Field Flatness and Central-Axis• Depth-Dose for Use in Design of Therapy Electron Beam Generators", Ph.D. Dissertation, University of Texas Health Sciences Center at the Houston Gradu• ate School of Biomedical Sciences (December, 1978). 9. D. W. O. Rogers and A. F. Bielajew, "Differences in Electron Depth-Dose Curves Calculated with EGS and ETRAN and Improved Energy-Range Relationships", Med. Phys. 13(5) (1986) 687. 10. T. A. Mehlhorn and J. A. Halbleib, "Monte Carlo Benchmark Calculations of Energy Deposition by Electron/Photon Showers up to 1 GeV", in Advances in Reactor Computations, Vol. 1, Proceedings of a Topical Meeting of the American Nuclear Society, March 28-31, 1983, Salt Lake City, Utah, ISBN 0-89448-111-8 (1983) 608. 11. H. H. Hsu, E. J. Dowdy, G. P. Estes, M. C. Lucas, J. M. Mack and C. E. Moss, "Efficiency of a Bismuth-Germanate Scintillator: Comparison of Monte Carlo Calculations with Measurements" , Los Alamos Scientific Laboratory report LA• UR-83-3004 (1983). 12. A. R. Frederickson and S. Woolf, "Electron Beam Current Penetration in Semi• Infinite Slabs", IEEE Trans. Nucl. Sci. NS-28 No.6 (1981) 4186. 13. J. J. Ramirez and K. R. Prestwich, "Investigations of the Radial-Electron-Beam Diode for Gas Laser Excitation", J. Appl. Phys. 50 No. 7 (1979) 4988. 14. J. J. Ramirez and K. R. Prestwich, "Radial Electron Beam Laser Excitation: The REBEL Report", Sandia National Laboratory report SAND78-2005 (1978). 15. T. W. 1. Sanford, J. A. Halbleib, J. W. Poukey, T. P. Wright, C. E. Heath, P. W. Spence, J. Kishi and J. Fockler, "Passive Control of High-Energy High• Current vII ~ 1", IEEE Catalog No. 86CH2317-6 (1986) 45. 16. T. W. 1. Sanford, J. A. Halbleib, J. W. Poukey, T. P. Wright, C. E. Heath, R. Mock, P. W. Spence, G. Proulx, V. Bailey, J. Fockler and H. Kishi, "Improved 11. Applications of the ITS Codes 283

Bremsstrahlung Radiation Uniformity from an Indented-Anode Diode", Appl. Phys. Lett. 50(13) (1987) 809. 17. T. W. 1. Sanford, J. A. Halbleib, J. W. Poukey and T. P. Wright, "Theoret• ical Motivation for Indented-Anode Diode for HERMES III", Sandia National Laboratories report SAND85-2383 (1987). 18. J. A. Halbleib, P. A. Miller, 1. P. Mix and T. P. Wright, "Overlap of Intense Charged Particle Beams for Inertial Confinement Fusion", Nature 286 No. 5771 (1980) 366. 19. J. A. Halbleib, G. J. Lockwood and 1. D. Posey, "Theoretical and Experimental Studies of Multiple Foil Bremsstrahlung Sources", IEEE Trans. Nucl. Sci. NS- 28 No 6 (1981) 4166. 20. T. W. 1. Sanford, J. R. Lee, J. A. Halbleib, J. P. Quintenz, R. S. Coats, W. A. Stygar, R. E. Clark, D. L. Faucett, D. Webb, C. E. Heath, P. W. Spence, J. Kishi, 1. G. Schlitt and D. Morton, "Electron Flow and Impedance of an 18-Blade Frustrum Diode", J. Appl. Phys. 59 (11) (1986) 3868; Fig. 17. 21. T. W. 1. Sanford, J. R. Lee, J. A. Halbleib, J. P. Quintenz, R. S. Coats, W. A. Stygar, R. E. Clark, D. 1. Faucett, D. Webb, C. E. Heath, P. W. Spence, J. Kishi, 1. G. Schlitt and D. Morton, "Electron Flow and Impedance of an 18-Blade Frustrum Diode", Sandia National Laboratories report SAND84-1424 (1985); Fig. 17. 22. J. M. Hoffman, A. K. Hays and G. C. Tisone, "High-Power UV Noble-Gas-Halide Lasers", Appl. Phys. Lett. 28(9) (1976) 538. 23. J. M. Hoffman, E. 1. Patterson and R. A. Gerber, "Energy Extraction from a Large-Volume HF Laser Amplifier", J. Appl. Phys. 50(6) (1979) 3861. 24. J. A. Halbleib, Sr. and W. H. Vandevender, "Coupled Electron Photon Colli• sional Transport in Externally Applied Electromagnetic Fields", J. Appl. Phys. 48(6) (1977) 2312. 25. J. R. Lee, R. C. Backstrom, J. A. Halbleib, J. P. Quintenz and T. P. Wright, "Gradient B Drift Transport of High Current Electron Beams", J. Appl. Phys. 56(11) (1984) 3175. 26. J. R. Lee, "Pulse Shaping of High-Current Electron Beams with Gradient B Drift Transport", Sandia National Laboratories report SAND86-0920 (1986). 27. J. R. Lee, D. 1. Faucett, J. A. Halbleib, M. A. Hedemann and W. A. Stygar, "Reverse-Field Injection for the VB Drift Transport of a High-Current Electron Beam", to be published in J. Appl. Phys. June (1988). 28. W. Beezhold, 1. J. Lorence, Jr., M. A. Stark, (Sandia National Laboratories); W. Seidler, B. Passenheim and B. Kitterer (JAYCOR); D. L. Shaeffer (Physics International Company); unpublished. 29. D. E. Stilwell, W. D. Davis, R. M. Joyce, F. B. McDonald, J. H. Trainor, W. E. Althouse, A. C. Cummings, T. 1. Garrard, E. C. Stone and R. E. Vogt, "The Voyager Cosmic Ray Experiment", IEEE Trans. Nucl. Sci. NS-26 No.1 (1979) 513. 30. M. A. Hedemann, A. 1. Pregenzer, L. D. Posey, D. C. Evans and P. W. Spence, "Radiation Field Results from the SPEED Triax Diode, A Large Area, High Dose Rate, Short Pulse 1 MV Bremsstrahlung Source", IEEE Trans. Nucl. Sci. NS-32 No. 6 (1985) 4266. 31. A. 1. Pregenzer and J. A. Halbleib, "Accuracy of Coupled Monte-Carlo/Next• Event-Estimator for Bremsstrahlung Dose Predictions", IEEE Trans. Nucl. Sci. 284 J. Halbleib

NS-32 No. 6 (1985) 4405. 32. P. A. Miller, J. A. Halbleib and J. W. Poukey, "Inverse Ion Diode Experiment", J. Appl. Phys. 52 No.2 (1981) 593. 33. M. A. Sweeney and J. N. Olsen, "Monte Carlo Calculations of the Bremsstr• ahlung Radiation Environment Expected in the Future Particle Beam Fusion Accelerator Facility PBFA II", Nucl. Sci. Eng. 89 (1985) 233. 34. M. A. Sweeney, J. A. Halbleib and K. M. Tolk, "Improved Shielding Calculations for the Particle Beam Fusion Accelerator PBFA II", Fusion. Tech. 10 (1986) 656. 35. M. A. Sweeney,J. A. Halbleib, J. B. Mashburn, L. P. Mix and T. N. Simmons, "Comparison of Observed and Predicted Bremsstrahlung Doses in the PBFA• II Facility", Proceedings of the Topical Conference on Theory and Practices in Radiation Protection and Shielding, April 22-24, 1987, Knoxville, Tennessee; Am. Nuc. Soc. Pub. No. ISBN: 0-89448-132-0 (1987) 235. 36. M. A. Sweeney, J. A. Halbleib and B. N. Turman, "Analysis of X-Ray Production in a Multimodule Accelerator", Conference Record - 1983 IEEE International Conference on Plasma Science, May 23-25, 1983, San Diego, California; IEEE Cat. No. 83CH1847-3 (1983) 127. See also M. A. Sweeney, J. A. Halbleib and B. N. Turman, "Diagnosis of Accelerator Load Energy Based on X-Ray Dose" , Bull. Am. Phys. Soc. 27(8) (1982) 1031. 37. C. J. MacCallum, Sandia National Laboratories (unpublished). 38. J. A. Halbleib, R. Hamil and E. L. Patterson, "Energy Deposition Model for the Design of REB-Driven, Large-Volume Gas Lasers", Conference Record - Abstracts, 1981 IEEE International Conference on Plasma Science, May 18-20, 1981, Santa Fe, New Mexico; IEEE Cat. No. 81CH1640-2 NPS (1981) 117. 39. J. R. Lee, "Pulse Shaping of High-Current Electron Beams with Gradient B Drift Transport", Sandia National Laboratories report SAND86-0920 (1986). 40. J. M. Peek and J. A. Halbleib, "Improved Atomic Data for Electron-Transport Predictions by the Codes TIGER and TIGERP: I. Inner-Shell Ionization by Electron Collision", Sandia National Laboratories report SAND82-2887 (1983). 41. J. A. Halbleib and E. L. Patterson, Sandia National Laboratories, (unpublished). 42. M. A. Sweeney, E. L. Patterson and G. E. Samlin, "Numerical Simulation of the Operation on an Electron-Beam Pumped Laser", Conference Record - 1987 IEEE International Conference on Plasma Science, June 1-3, 1987, Arlington, Virginia; IEEE Catalog No. 87CH2451-3 (1987) 26. 43. J. A. Halbleib, "Time-Dependent Monte Carlo Deposition Rate from Flash X• Ray Sources", IEEE Trans. Nucl. Sci. NS-29 No. 6 (1982) 1955. 44. C. N. Vittitoe and J. A. Halbleib, Sr., "EMP Source Terms for an Electron Beam in Bremsstrahlung-Converter Mode", Sandia National Laboratories report SAND82-2955 (1983). 45. J. A. Halbleib, "Coupled Electron/Photon Collisional Transport in Macroscopic Electric Fields", Bull. Am. Phys. Soc. 26(7) (1981) 1062. 46. J. A. Halbleib, Sandia National Laboratories (unpublished). 47. J. A. Halbleib, Sr., M. M. Widner, J. W. Poukeyand J. P. Quintenz, "Enhanced Electron Deposition in Foils", Bull. Am. Phys. Soc. 22 No.9 (1977) 1061. 48. C. J. Crannell, H. Crannell, R. R. Whitney and H. D. Zeman, "Electron-Induced Cascade Showers in Water and Aluminum", Phys. Rev. 184(2) (1969) 426. The EGS4 Code System 12. Structure and Operation of the EGS4 Code System

Walter R. Nelson and David W. O. Rogers*

Radiation Physics Group Stanford Linear Accelerator Center Stanford, California, 94309, U.S.A.

12.1 INTRODUCTION The EGS** system of computer codes is a general purpose package for the Monte Carlo simulation of the coupled transport of electrons and photons in an arbitrary geometry for particles with energies above a few keY up to several TeV. The current version is the EGS4 Code System by Nelson, Hirayama and Rogersl, which is more commonly referred to as EGS4.

With the introduction of EGS4, we created a new manual (SLAC-265) that re• tained much, but not all, of the previous documentation for the EGS3 Code System2 • In particular, we omitted the history prior to EGS3 and, since many of the EGS3 bench• mark comparisons had been published elsewheres- 5 and SLAC-265 was already rather large, we did not duplicate the effort in the EGS4 manual. However, since the EGS3 documentation (SLAC-210) will soon be out of print, it seems appropriate to include some of this information here.

In the sections that follow, the history leading up to the release of EGS4 will be presented, together with a relatively short description of the EGS4 Code System itself. This will then be followed by a series of benchmark comparisons with experiments and with Monte Carlo results of others. Additional comparisons are also presented in Chapter 13.

12.1.1 History Prior to EGS3

Messel and Crawford code - Australia (1958-1970). The first use of an electronic digital computer in simulating high-energy cascades by Monte Carlo methods was reported by Butcher and MesseI6,7, and independently by Varfolomeev and Svetlolobov8• These two groups collaborated in a much publicized work9 that eventually led to an extensive set of tables describing the shower distribu• tion functions-the so-called "shower book" 10.

* Physics Division National Research Council of Canada Ottawa, Canada KIA OR6 ** Electron-Gamma .s.hower

287 288 w. R. Nelson and D. W. O. Rogers

Zerby and Moran code - ORNL (1962-1963). Around the same time period, Zerby and Moran at the Oak Ridge National Laboratory developed a high-energy electromagnetic cascade program based on the Monte Carlo methodll- 13• This ORNL code was motivated by the construction of the Stanford Lin• ear Accelerator Center (SLAC) and by the many physics and engineering problems that were anticipated as a result of high-energy electron beams showering in various devices and structures at that facility. This code has been used by Alsmiller and others14- 18 for a number of studies since its development. Even though the results of Zerby and Moran calculations were used extensively during the initial design of SLAC, the code was not readily available outside ORNL, nor was it maintained, so that today most of the ORNL studies requiring electron-photon transport make use of EGS.

Nagel code - Bonn University (1963-1967). Also during the early 1960's, H. H. Nagel wrote his Ph.D. thesis at Bonn University on electron-photon Monte Carlo19- 21 , and several versions of his code have appeared since then22- 25 , including one that eventually became EGS3. Nagel's original FORTRAN IV program, which we shall simply refer to as SHOWER (not to be confused with SUB• ROUTINE SHOWER of the EGS Code System), represented a very practical tool for the experimental physicist during the middle 1960's-and it was free for the asking!

However, SHOWER was still rather limited in its application. One could initiate radiation transport with energies only up to 1 GeV, and then only for monoenergetic electrons. Except for annihilation, positrons and electrons were treated alike and were followed until they reached a cutoff energy of 1.5 MeV (total energy) and photons were followed down to 0.25 MeV. But this still represented cutoff energies that were, at the time, as low as or lower than those used by ORNL or the Australians. Probably the most limiting constraint of SHOWER, however, was its built-in geometry-one was obliged to use a single cylinder of Pb. To make matters worse, the code was not very modular and a fair amount of effort on the part of the user was involved in order to do even rather simple things, such as a bremsstrahlung spectrum as input. There was nothing wrong with the physics or the Monte Carlo schemes. The simple fact was that SHOWER was both unwieldly and limited in scope.

During the period starting around 1967 up to about 1974, the SHOWER program was modified by Nelson and colleagues at Stanford, who were attempting to make it faster, as well as more useful. However, efforts by R. L. Ford at the High Energy Physics Laboratory (HEPL), where a group led by R. Hofstadter and E. Hughes was continuing their development of large NaI(Tl) Total Absorption Shower Counters (TASC's), soon led to the growing conviction among everyone concerned that a generalized code was really necessary.

12.1.2 The Development of EGS3

The EGS Code System (Version 3) was a joint effort, undertaken over the period 1972- 1978 by R. L. Ford and W. R. Nelson. The sole purpose of the collaboration was to revamp completely the work started by Nagel, but to do it in such a way that further enhancements could easily be made as time progressed-in today's words, to create a program that was versatile, upward-compatible, and very user-friendly. 12. Structure and Operation of the EGS4 Code System 289

When EGS3 was formally introduced in 19782 , it was designed to simulate EM cascades in various geometries and at energies up to a few thousand GeV and down to cutoff kinetic energies of 0.1 MeV (photons) and 1 MeV (electrons/positrons). By means of the PEGS* auxiliary code, radiation transport was made available in any of 100 elements, or any compound or mixture of these elements. In addition to providing more efficient sampling schemes, EGS3 also included some processes that were not part of the original SHOWER program. To lend credibility to our efforts, a fairly extensive set of benchmark comparisons, representing a wide range of applications, were included in the EGS3 manual (SLAC-210). The most important benchmarks, however, have been performed by the multitude of users of the code itself.

Upon reflection, probably the single most important event that made EGS an everyday word in high-energy physics was the discovery of the J / IjJ particle in the Fall of 1974. EGS3 was originally intended to be a tool for high-energy health physicists and accelerator designers, but the "November Revolution", as it is now referred to in the particle physics world, led to a dramatic increase in the use of storage rings and the need for sophisticated EM calorimetry. It is safe to say that EGS3 has played a role in the design of many, if not most, of the electromagnetic shower counters since then.

12.1.3 EGS4 - A Code Greatly Influenced by Medical Physics

Since the introduction of EGS3, there has been a growing need to extend the lower energy limits-i.e., down to 1 and 10 keV for photons and electrons, respectively. Es• sentially, EGS3 has become more and more popular as a general low-energy electron• photon transport code that can be used for a variety of problems in addition to those normally associated with EM cascade showers. While there was a collaborative effort being undertaken by Nelson (SLAC) and Hirayama (KEK) to extend the flexibility of EGS in general, particularly for use around high-energy accelerators, there was also an important low-energy ben.::hmarking effort being done by Rogers, Bielajew, and col• leagues at the NRC in Canada. The efforts of these three laboratories was pooled and the EGS4 Code System became the result1 •

Although EGS is still very heavily used in particle physics, it is interesting to note that of the 260 EGS4 Distribution Tape requests received by the SLAC Radiation Physics Group during 1986, well over half went to hospitals or to organizations involved in medical physics and dosimetry research. The fact that this book is based on a course on electron-photon Monte Carlo is further demonstration of the current strong interest in this field of research.

12.2 GENERAL DESCRIPTION OF EGS4 (AND PEGS4)

EGS is basically an analog Monte Carlo program. That is to say, each and every particle is followed until it reaches its final destiny, usually an energy limit (cutoff) or a discard boundary. Due to the statistical nature of the Monte Carlo method, the accuracy of the results will depend on the number of histories run. Generally, the statistical uncertainties are proportional to the inverse square root of the number of histories26 . Thus, to cut uncertainties in half, it is necessary to r~n four times as many histories. Also, for given cutoff energies, the computer time for a shower history is slightly more than linear in the energy of the incident particle. The point to be made here is that

* Ereprocessor for EGS 290 w. R. Nelson and D. W. O. Rogers analog Monte Carlo calculations can be very time consuming. It is for this reason that the computational task of the EGS4 Code System is divided into two parts. First, a preprocessor c9de (PEGS4) uses theoretical (and sometimes empirical) formulas to compute the various physical quantities needed, and prepares them in a form for fast numerical evaluation. Then another code (EGS4) uses these data, along with user supplied data and routines, to perform the actual simulation.

12.2.1 PEGS4 as a Development Tool

To aid in debugging and to help those interested in studying the various interactions, the EGS4 Code System was expanded beyond the minimum coding necessary to simulate radiation transport. With this in mind, PEGS4 was written in a modular form with over 95 subprograms. These include functions to evaluate physical quantities which are either needed by PEGS4, or are of interest for other reasons. Other routines necessary for operation of EGS4 include the fitting routines and the routine to write the data for a given material onto a data set. Included among the PEGS4 subprograms not needed for the operation of EGS4 itself are routines to plot the functions on the lineprinter or a graphics device, and a routine to compare (on a lineprinter plot) the theoretical final-state density functions with sampled final-state distributions. The latter may be created most easily by means of UCTESTSR*, which is provided on the EGS4 Distribution Tape.

12.2.2 PEGS4 as a Preprocessor for EGS4

As we have stated, the prime use of PEGS4 is to produce material data sets for sub• sequent use by EGS4 itself. The main program of PEGS4 calls some once-only ini• tialization routines and then enters an option loop. After reading in the option that is desired, a NAMELIST read establishes other parameters which may be needed. The action requested is then performed and control returns to the beginning of the loop. This continues until the control input has been exhausted. Options exist for plotting and examining the cross sections themselves, but the most important options are: ELEM, COMP, MIXT, and ENER. The first three tell PEGS that an element, compound, or mix• ture, respectively, is being requested. Additional data are then supplied by the user in order to establish the medium in question.

The ENER option is even simpler-it defines the range of energies, both upper and lower (i.e., cutoffs), which are to be used by PEGS4 when it creates the data for EGS4. The amount of data that the user supplies to PEGS4 is actually quite small (less than about 10 cards), and examples for a variety of material situations are given in SLAC-265 (Appendix 3).

12.2.3 General Implementation of EGS4

The EGS4 code itself consists of two user-callable subroutines, HATCH and SHOWER, which in turn call the other subroutines in the EGS4 code, some of which call two user• written subroutines, HOWFAR and AUSGAB. The latter determine the geometry and output (scoring), respectively. The user communicates with EGS4 by means of various COMMON variables. To use EGS4, the user must write a MAIN program and the subroutines HOWFAR

* !J.ser Qode for Testing .s.ampling Routines 12. Structure and Operation of the EGS4 Code System 291 and AUSGAB. Usually, MAIN will perform any initialization needed for the geometry routine, HOWFAR, and set the values of certain EGS4 COMMON variables which specify such things as names of the media to be used, the desired cutoff energies, and the unit of distance (e.g., centimeters, radiation lengths, etc.). MAIN then calls the HATCH subroutine, which "hatches" EGS by doing once-only initialization and by reading from the data sets prepared by PEGS for the materials requested.

This initialization completed, MAIN may then call SHOWER when desired. Each call to SHOWER results in the generation of one EGS history. The arguments to SHOWER specify the parameters of the incident particle. Therefore, the user has the freedom to use any source distribution desired.

12.2.4 Mortran3 Macros and EGS User Codes The flow of control and data when a user-written program is using the EGS4 code is illustrated in Fig. 12.1.

c (J) /"Tl :0 () o CJ /"Tl

/"Tl c;) (J) o() CJ /"Tl

Block Data (Default)

Figure 12.1. Flow control with user using EGS4.

The detailed information needed to write such user programs, commonly referred to as User Codes, is given in the EGS4 User Manual (SLAC-265, Appendix 2). As an introduction, however, the reader might find it more useful to study first the series of short tutorials provided in Chapter 3 of SLAC-265. 292 w. R. Nelson and D. W. O. Rogers

The entire EGS4 Code System is written in a structured language called Mortran3, a FORTRAN-like language that has been developed at SLAC by Cook27 , and which contains a macro-facility that is very useful. EGS4 contains many macros (i.e., defini• tions), most of which are quite simple once they are understood. Buried within EGS4 are patterns (also called templates), which are no more than strings that get replaced by other strings. Stated explicitly, the Mortran string processor searches throughout EGS4 (and the User Code) for a specified pattern. If a match is found, the template part gets replaced by the replacement part. It is not our intention to discuss Mortran3 in any detail at this time, but it seems appropriate to demonstrate the power of the Mortran macro-facility by means of a simple example.

Let us assume that the user wants to change the way charged particles are trans• ported by EGS. This is really quite easy to do at the User Code level (i. e., above the dashed line in Fig. 12.1). The macro pattern

$CHARGED-TRANSPORTj has been specifically located within SUBROUTINE ELECTR so that during the so-called "Mortran-step", just prior to the usual FORTRAN compilation, a search and replace• ment can be done. Located within a file called EGS4MAC MORTRAN* (provided on the EGS4 Distribution Tape) is a macro

REPLACE {$CHARGED-TRANSPORTj} WITH {X(NP)=X(NP)+U(NP)*VSTEPj Y(NP)=Y(NP)+V(NP)*VSTEPj Z(NP)=Z(NP)+W(NP)*VSTEPj} , which is the default replacement that is used with EGS4-i. e., a simple linear translation along the direction of motion (U, V,W) by the scale factor VSTEP (the step length).

The important point is that the user can override the above macro by placing one of his choice within the User Code. For example,

REPLACE {$CHARGED-TRANSPORTj} WITH {CALL MYTRAN}j would be invoked first and the default macro (in EGS4MAC MORTRAN) would never find a pattern to replace. Of course, the user must now supply SUBROUTINE MYTRAN or an error message will occur during the FORTRAN compilation. Alternatively, the replacement part (CALL MYTRAN) could be the entire subroutine itself. Placing code directly in-line can sometimes help speed up a code rather dramatically.

To summarize, Mortran macros provide the user with an easy and effective way to change the EGS4 code (i.e., below the dashed line in Fig. 12.1) without having to actually edit EGS4 itself. The disadvantage of this approach is that one must take the time to learn something new. The benefits can be significant, however, and the more sophisticated EGS user generally takes advantage of them.

* EGS4MAC.MOR in VAX/VMS notation. 12. Structure and Operation of the EGS4 Code System 293

12.3 SOME BENCHMARK COMPARISONS 12.3.1 Conversion Efficiency of Lead for 30-200 MeV Photons

An experiment to measure the conversion efficiency for 44, 94, and 177 MeV photons incident on lead was performed by Darriulat et a[28 at CERN. By tagging the photons, the mean energy was determined to an accuracy of ±4 MeV. The photon beam, with an area less than 15 X 15 cm2, struck a lead plate of desired thickness (1 to 20 mm) and area (20 x 20 cm2). Immediately following the lead was a large plastic scintillation detector, 28 X 40 cm2 in area and 5-mm thick. An event was counted as a conversion if more than 60 keV was deposited in the scintillator for each incoming photon.

To calculate the conversion efficiency with EGS4, the geometry layout shown in Fig. 12.2 was used, consisting of four regions separated by three semi-infinite planes.

Vacuum

----i----').z Incident Photon

x ( Y nto i paper) [::> : Plane Number o Reg : ion Number Figure 12.2. Geometry layout used in HOWFAR for simulation of the conversion efficiency experiment.

Polystyrene, with a density of 1.032 g/cm3 and consisting of hydrogen and carbon with an atomic ratio H/C=1.lO, was used as the medium for plastic scintillator in region 3. The density of lead was taken to be 11.34 g/cm3 . PEGS4 was used to create the necessary material data with cutoff energies of 0.1 MeV and 1.5 MeV (total energy) for photons and electrons, respectively.

The HOWFAR subprogram portion of the User Code* utilized the macro form of three auxiliary geometry subroutines, PLAN2P, PLANE1 and CHGTR**. The AUSGAB (scoring and/or outputting) subroutine was set up to sum the energy deposition in the plastic (region 3) for each photon. Upon completion of a photon shower initiated by a CALL SHOWER statement in the MAIN driver program, a conversion event was scored, provided that the energy sum in the plastic exceeded 0.060 MeV as dictated by the discrimination level established in the experiment.

* The EGS4 User Code: UCCONEF1 ** These subprograms are described both in SLAC-265 and in Chapter 17. 294 W. R. Nelson and D. W. O. Rogers

The results of the calculation are compared with the experimental data in Fig. 12.3. The agreement is extremely good over the entire lead thickness range for the two energies shown.

In the text describing the experimental results, Darriulat et al point out that the energy distribution in the scintillator showed characteristic peaks corresponding to the production of one, two, or three secondary electrons that are produced in the lead and lose energy as they pass through the scintillator. To check out this observation, the total energy deposition in the scintillator per incident photon was histogrammed, and typical results are shown in Fig. 12.4. Two of the three electron peaks are indeed prominent and are located where one would expect them to be based on a stopping power of ,..., 2 MeV -cm2 / g. This experiment is well-defined and easily simulated. One can conclude that EGS4 can predict photon conversion efficiencies rather well, at least in the energy range 30-200 MeV and for geometries similar to the one described here.

1.0

x a Experlment >- 0.8 u EGS4 z w ti ~ 0.6 l.L.w Z 0 Ui 0.4 0::w >z 0 u 0.2

Figure 12.3 Absolute comparison of EGS4 simulation with a conversion efficiency experiment by Darriulat et al 28 •

15000

E=l77 MeV (/) tpb-lO mm 1-z w tSclnt-5 mm >w 10000 l.L. a 0:: w ro 1: :J Z 5000

o 0 2 3 4 5 ENERGY (MeV) Figure 12.4. Energy distribution in the scintillator (EGS4 calculation). 12. Structure and Operation of the EGS4 Code System 295

12.3.2 Large. Modularized Nal(TI) Detector Experiment

The application oflarge, modularized NaI(Tl) detectors to physics experiments, particu• larly those involving photon spectroscopy around high-energy electron-positron storage rings, has increased considerably during the last decade. An report by Ford et a1 29 describes an experiment that was performed at SLAC to measure, among other things, the energy resolution of a typical detector array consisting of 19 NaI(Tl) hexagonal modules. Although each module itself cannot be expected to provide good energy res• olution at high photon or electron energies-due to the transverse spread of energy in the EM shower (i.e., leakage)-this problem is overcome in a detector made up of an array of such modules.

ENERGY RESOLUTION vs ~ DISPLACEMENT

4 GeV/e e•

OBSERVED .. CALCULATED (STACK OF 7 HEXAGONS) .... Z LU U a: ~ DIRECTION OF LU W--BEAM DISPLACEMENT Q.

z ~5.25" ..'" ....2 :::> ...J o I r '"LU .. ! a: r f ! ..t .. .. I tWA~L I~I __ ~~I __ ~~I __ ~~ __ ~ o 1.0 2.0 3.0 BEAM DISPLACEMENT (in.) Figure 12.5. Comparison between the observed and calculated (EGS3) res• olution at 4 GeV Ic as a function of the displacement of the 0.25 X 0.25 in.2 beam (from Ford et a1 29 ).

Each hexagon was encapsulated in a stainless steel container with a wall thickness of 0.51 mm. The individual crystals were optically coupled at one end to a 0.5-inch thick glass window, through which the crystal volume was viewed by a 3-inch diameter photomultiplier tube. The stainless steel walls cause undesirable effects when the beam trajectory approaches closely or intercepts them, as illustrated in Fig. 12.5. In this figure, the variation of the energy resolution at 4 GeV Ic for an array of 7 modules is shown as a function of the displacement of the trajectory from the axis. No significant loss in the resolution is experienced until the trajectory approaches within about 0.5 inch of the nearest wall. The agreement with the measurements is quite good.

The calculated and observed response of the modular array of 19 hexagons to 0.1 to 4 GeV Ic electrons incident along the axis of the central module is summarized in Fig. 12.6. This figure shows not only the energy resolution obtained when the energies deposited in all 19 crystals are summed, but also those obtained when only the energies 296 W. R. Nelson and D. W. O. Rogers in the central 7 modules or in the central module alone are used. The agreement is observed to be very good. EGS takes into account both the energy leakage fluctuations from the detector volume and fluctuations due to energy absorption in the stainless steel walls surrounding each crystal module. Also shown in Fig. 12.6 is the EGS simulation of 19 crystals without walls-i.e., the best resolution possible with such a system.

RESOLUTION VI ELECTRON ENERGY

10 2: :r ~ l>. ~ z ...Q ::;) -J 0 en "lr-_ w -...l(. T II: - -l _ -J ~9 HEXAGONS I. !7" --x £19 HEXAGONS 0'0" OBSERVED (WITHOUT STN. STL. WALLS) x •• " CALCULATEO 10 ELECTRON ENERGY (GeV) Figure 12.6. Comparison between the observed and calculated (EGS3) reso• lution for detectors consisting of 1, 7, or 19 hexagons (from Ford et a1 29 ).

12.3.3 Longitudinal and Radial Showers in Water and Aluminum at 1 GeV

An experiment was performed by Crannell et a1 30 to measure the three-dimensional distribution of energy deposition for I-GeV showers in water and aluminum. The water target consisted of a steel tank containing 8000 liters of distilled water. The incident beam, less than 1 mm in diameter, entered the water through a 0.13-mm thick aluminum window centered on the square end of the tank (122 X 122 X 460 cm3). The aluminum target, on the other hand, consisted of plates varying in thickness from 0.64 to 2.5 cm, pressed together to form a solid target (61 X 61 X 180 cm3).

Differential, as well as integral, energy deposition data obtained from this experi• ment afford a good benchmark comparison, particularly since i) a reasonably good comparison has been made using the Zerby and Moran codell- 13,16, and ii) Crannell indicates in the paper that the Nagel code (i.e., SHOWER) does not give radial distributions in agreement with the experiment (note: since EGS descends from SHOWER, we are obligated to make this comparison). 12. Structure and Operation of the EGS4 Code System 297

The User Code for this calculation* makes use of $CYLNDR and $PLAN2P, geometry macros contained within the EGS4MAC MORTRAN file on the EGS4 Distribution Tape. Another useful subprogram, ECNSV1, provides a convenient way to keep track of where and how energy is deposited in each cylindrical shell-slab region.

TARGET MATERIAL: ALUMI~ TARGET MATERIAL: WATER - (lI.peflmen (Ctonnel 1969) - Experimenl(CrorneII1969) •• - EGS3 ",,,,,,,. Carlo --- EGS3 Monle Carlo

Ro(hal InlefYCJI o olem Radial '- __ . Inlerval o tolem 1102cm

---I ta 2em 2 to 3cm

2 to 3em 3104cm

4 to oem

510Gem

6107cm

'---- 6 to 8em 710 Scm .,1 0-3~ 8.ol0em , ... 8 to 12em

10 o12cm

o 200 400 o 40 80 120 DEPTH (em) DEPTH (em)

Figure 12.7. Comparison ofEGS with the Crannell et a/ 3D shower experiment in water and aluminum at 1 GeV.

A comparison of the Crannell data with EGS3 is given in Fig. 12.7. The agreement is extremely good everywhere for the water case and reasonably good for the aluminum experiment. The slight discrepancy at large radii in the aluminum comparison is possi• bly attributed to a a mismatch between detector and absorber. CaF2 (Eu) was used as a scintillation detector in the aluminum experiment, whereas anthracene which is a much better match, was used in the water case. Crannell goes into considerable discussion on this in the paper, and the reader is referred to this reference. The calculations have also been repeated using EGS431 , and similar results were obtained.

* UCH20AL 298 W. R. Nelson and D. W. O. Rogers

12.3.4 Track-length Calculations

Track-length calculations are most easily done with EGS by summing the length of the step, TVSTEP, in SUBROUTINE AUSGAB each time a transport (IARG=O) takes place. In the case of photon track lengths, the calculation is simplified because the photon does not lose energy during transport between events. Charged particles, on the other hand, lose energy during the step, and the method of scoring is correspondingly more complicated. The comparisons made below were done using the EGS3 code. However, the same results have also been obtained using EGS4.

Differential photon track length. Alsmiller32 has used the Zerby and Moran codeU - 13 to calculate the differential photon track length for the specific case of 18-GeV electrons incident on a cylindrical copper target having a radius of 11.5 cm and a thickness of 24.5 cm. The results are compared with similar data obtained using EGS3, as shown in Fig. 12.8 where ageement between the codes is quite apparent. Also shown is a solid line corresponding to the track-length formula of Clement33.

10°

~ EO= 18 GeV ~.: I I- <.9 Formula by Clement (1963) Z W ...J :>::: u

10-2 3 5 7 9 II 13 15 17 19 PHOTON ENERGY, k(GeV) Figure 12.8. Differential photon track length. Comparison of EGS with Monte Carlo results using the Zerby and Moran code.

Differential electron track length. In order to score the charged particle track length in SUBROUTINE AUSGAB properly, account must be taken of the continuous energy loss along the track. By determining the energy of the particle at the beginning and the end of the track, the total track length can be fractionated, sorted, and summed in histogram bins corresponding to energy (an example of a track-length scoring algorithm is given in SLAC-2102 ). 12. Structure and Operation of the EGS4 Code System 299

Figure 12.9 compares the charged particle track length, as calculated by EGS3, with that of Zerby and Moranll- 1S for three electron beams (50, 200, and 700 MeV) incident on a 100 cm thick, semi-infinite slab of copper. Cutoff energies of 10 MeV were used in both Monte Carlo simulations, which agree quite well with each other. The same results have also been obtained with EGS4.

10°

5 • 700 MeV 'u • 200 MeV a; '" x MeV , 50 1[; 2 Zerby and Moran :2' Monte. Carlo § 10- 1 -l:L EGS3 Monte Carlo :r: I- l'J z 5 W ..J ><: u« cc 2 I- w d 10"2 f= cc it o 5 w l'Jcc « Q 2

10"3L-__~-L~~~li- __-L~~-LLLLU

10 1 2 5 102 2 5 ENERGY (MeV)

Figure 12.9. Differential electron track length. Comparison of EGS with Monte Carlo results using the Zerby and Moran code.

12.4 SUMMARY OF EGS4 CAPABILITIES AND FEATURES The following is a summary of the main features of the EGS4 Code System, including statements about the physics that has been put into it and what can realistically be simulated.

• The radiation transport of electrons (+ or -) or photons can be simulated in any element, compound, or mixture. That is, the data-preparation package PEGS4 creates data to be used by EGS4, using cross-section tables for elements 1 through 100.

• Both photons and charged particles are transported in random rather than in discrete steps. 300 W. R. Nelson and D. W. O. Rogers

• The dynamic range of charged particle kinetic energies goes from a few tens of keY up to a few thousand GeV. Conceivably, the upper limit can be extended higher, but the validity of the physics remains to be checked.

• The dynamic range of photon energies lies between 1 keVand several thousand GeV (see above statement).

• The following physics processes are taken into account by the EGS4 Code System:

Bremsstrahlung production (excluding the Elwert correction at lowener• gies ).

- Positron annihilation in flight and at rest (the annihilation quanta are fol• lowed to completion).

- Moliere multiple scattering (i.e., Coulomb scattering from nuclei). The re• duced angle is sampled from a continuous (rather than discrete) distribution. This is done for arbitrary step sizes, selected randomly, provided that they are not so large or so small as to invalidate the theory.

- Mlllller (e-e-) and Bhabha (e+e-) scattering. Exact rather than asymptotic formulae are used.

- Continuous energy loss applied to charged particle tracks between discrete interactions.

o Total stopping power consists of soft bremsstrahlung and collision loss terms.

o Collision loss determined by the (restricted) Bethe-Bloch stopping power with Sternheimer treatment of the density effect.

- Pair production.

Compton scattering.

- Coherent (Rayleigh) scattering may be modelled using an independent-atoms approximation (non-default option in EGS4). by means of an option.

- Photoelectric effect.

o Neither fluorescent photons nor Auger electrons are produced or trans• ported in the default version of SUBROUTINE PHOTO.

o Other user-written versions of PHOTO can be created, however, that allow for the production and transport of K- and L-edge photons [see, for example, the discussion of the EGS4 User Code called UCEDGE in Chapter 4 of SLAC-265)].

• PEGS4 is a stand-alone data preprocessing code consisting of 12 subroutines and 85 functions. The output is in a form for direct use by EGS4. 12. Structure and Operation of the EGS4 Code System 301

- PEGS4 constructs piecewise-linear fits over a large number of energy inter• vals of the cross-section and branching-ratio data.

- In general, the user need only use PEGS4 once to obtain the media data files required by EGS4.

- PEGS4 control input uses the NAMELIST read facility of the FORTRAN lan• guage (in Mortran3 form).

- In addition to the options needed to produce data for EGS4, PEGS4 contains options to plot any of the physical quantities used by EGS4, as well as to compare sampled distributions (produced by the UCTESTSR User Code) with theoretical spectra.

• EGS4 is a package of subroutines plus block data with a flexible user interface.

- This allows for greater flexibility without requiring one to be overly familiar with the internal details of the code.

- Together with the macro facility capabilities of the Mortran3 language, this reduces the likelihood that user edits will introduce bugs into the code.

- Flow diagrams for the 13 subroutines of EGS4 are given in Appendix 1 of SLAC-265.

- EGS4 uses material cross-section and branching-ratio data created and fit by the companion code, PEGS4.

• The geometry for any given problem is specified by a user-written subroutine called HOWFAR which, in turn, can make use of auxiliary subprograms.

- Auxiliary geometry routines for planes, cylinders, cones, spheres, etc., are provided with the EGS4 Code System for those who do not wish to write their own.

- Macro versions of these routines are also provided in the set of defining macros (i.e., in the EGS4MAC file) which, if used, generally result in a faster running simulat.ion.

- The MORSE-CG Combinatorial Geometry package can be incorporated into HOWFAR (e.g., see the UCSAMPCG file on the EGS4 Distribution Tape). However, experience indicates that a much slower simulation generally results (of the order of at least a factor of four).

- Transport can take place in a magnetic field by writing a specially designed HOWFAR subprogram (e.g., see Section 4.2 of SLAC-2651). Transport in both electric and magnetic fields can be simulated in a more general manner (e.g., see Chapter 19) by making use of Mortran3 macro templates that have been appropriately placed for that purpose in SUBROUTINE ELECTR. 302 W. R. Nelson and D. W. O. Rogers

• The user scores and outputs information in the user-written subroutine called AUSGAB.

- Auxiliary SUBROUTINE ECNSVl is provided in order to keep track of energy for conservation (or other) purposes.

- Auxiliary SUBROUTINE NTALLY is' provided in order to keep track of the num• ber of times energy has been scored into the ECNSVl arrays (i. e., an event counter).

- Auxiliary SUBROUTINE WATCH is provided in order to allow an event-by-event or step-by-step tracking of the simulation.

• EGS4 allows for the implementation of importance sampling and other variance• reduction techniques-e.g., splitting, path-length biasing, Russian roulette, leading• particle biasing, etc.

• Initiation of the radiation transport:

- An option exists for initiating a shower with two photons from 11"0 decay (i. e., use IQI=2 in the CALL SHOWER statement).

- The user has the choice of initiating the transport by means of a monoener• getic particle, or by sampling from a known distribution (e.g., a synchrotron• radiation spectrum).

- Transport can also be initiated from sources that have spatial and/or angular distributions.

12.5 EGS4 GRAPHICS CAPABILITIES

EGS4 has been coupled* with the SLAC Unified Graphics System (UGS77)S4 to provide a means for displaying particle tracks on UGS77-supported devicess5 • This is done by inserting CALL SHOWPL statements at appropriate places in the EGS4 User Code, attach• ing an auxiliary subprogram package (SHOWGRAF), and creating SUBROUTINE HOWPL to match HOWFAR. SHOWGRAF may be used to create shower displays directly on an interactive IBM-5080 color terminal, supporting three-dimensional rotations, transla• tions, and zoom features, and providing illustration of particle types and energies by color and/or intensity.

Alternatively, SHOWGRAF can produce graphics output data which are sub• sequently operated on by a post-processor system (EGS4PL)S6 for display on two• dimensional devices supported by UGS77. Options exist within EGS4PL that allow for two-dimensional translations and zoom, for creating line structure to indicate par• ticle types and energies, and for turning off particle types altogether. Examples of shower pictures created with the SHOWGRAF package are provided in Figs. 28.2-4, Figs. 28.13-14, and Figs. 28.28.15-16 of Chapter 28.

EGS4PL currently runs under IBM VM/SP and ~!AX VMS operating systems.

* This is a recent addition not found in the EGS4 manual. 12. Structure and Operation of the EGS4 Code System 303

REFERENCES 1. W. R. Nelson, H. Hirayama and D. W. O. Rogers, "The EGS4 Code System", Stanford Linear Accelerator Center report SLAC-265 (1985). 2. R.1. Ford and W. R. Nelson, "The EGS Code System: Computer Programs for the Monte Carlo Simulation of Electromagnetic Cascade Showers (Version 3)", Stanford Linear Accelerator Center report SLAC-210 (1978). 3. W. R. Nelson, "Solution of the Electromagnetic Cascade Shower Problem by Analog Monte Carlo Methods-EGS", in Computer Techniques in Radiation Transport and Dosimetry, edited by W. R. Nelson and T. M. Jenkins, (Plenum Press, New York, 1980). 4. W. R. Nelson, "Application of EGS to Detector Design in High Energy Physics", in Computer Techniques in Radiation Transport and Dosimetry, edited by W. R. Nel• son and T. M. Jenkins, (Plenum Press, New York, 1980). 5. W. R. Nelson, "Application of EGS and ETRAN to Problems in Medical Physics and Dosimetry" , in Computer Techniques in Radiation Transport and Dosimetry, edited by W. R. Nelson and T. M. Jenkins, (Plenum Press, New York, 1980). 6. J. C. Butcher and H. Messel, "Electron Number Distribution in Electron-Photon Showers", Phys. Rev. 112 (1958) 2096. 7. J. C. Butcher and H. Messel, "Electron Number Distribution in Electron-Photon Showers in Air and Aluminum Absorbers", Nucl. Phys 20 (1960) 15. 8. A. A. Varfolomeev and I. A. Svetlolobov, "Monte Carlo Calculations of Electro• magnetic Cascade with Account of the Influence of the Medium on Bremsstrah• lung", SOy. Phys. JETP 36 (1959) 1263. 9. H. Messel, A. D. Smirov, A. A. Varfolomeev, D. F. Crawford and J. C. Butcher, "Radial and Angular Distributions of Electrons in Electron-Photon Showers in Lead and in Emulsion Absorbers", Nucl. Phys. 39 (1962) 1. 10. H. Messel and D. F. Crawford, Electron-Photon Shower Distribution Function, (Pergamon Press, Oxford, 1970). 11. C. D. Zerby and H. S. Moran, "Studies of the Longitudinal Development of High-Energy Electron-Photon Cascade Showers in Copper", Oak Ridge National Laboratory report ORNL-3329 (1962). 12. C. D. Zerby and H. S. Moran, "A Monte Carlo Calculation of the Three-Dimen• sional Development of High-Energy Electron-Photon Cascade Showers", Oak Ridge National Laboratory report ORNL-TM-422 (1962). 13. C. D. Zerby and H. S. Moran, "Studies of the Longitudinal Development of High• Energy Electron-Photon Cascade Showers", J. Appl. Phys. 34 (1963) 2445. 14. R. G. Alsmiller, Jr. and H. S. Moran, "Electron-Photon Cascade Calculations and Neutron Yields from Electrons in Thick Targets", Oak Ridge National Lab• oratory report ORNL-TM-1502 (1966). 15. R. G. Alsmiller, Jr. and H. S. Moran, "The Electron-Photon Cascade Induced in Lead by Photons in the Energy Range 15 to 100 MeV", Oak Ridge National Laboratory report ORNL-4192 (1968). 16. R. G. Alsmiller, Jr. and H. S. Moran, "Calculation of the Energy Deposited in Thick Targets by High-Energy (1 GeV) Electron-Photon Cascades and Compar• ison with Experiment", Nucl. Sci. Eng. 38 (1969) 131. 17. R. G. Alsmiller, Jr. and J. Barish, "Energy Deposition by 45-GeV Photons in H, Be, AI, Cu, and Ta", Oak Ridge National Laboratory report ORNL-4933 (1974). 304 W. R. Nelson and D. W. O. Rogers

18. R. G. AlsmiUer, Jr., J. Barish and S. R Dodge, "Energy Deposition by High• Energy Electrons (50 to 200 MeV) in Water", Nucl. Instr. Meth. 121 (1974) 161. 19. H. H. Nagel and C. Schlier, "Berechnung von Elecktron-Photon-Kaskaden in Blei fur eine Primiirenergie von 200 MeV", Z. Phys. 174 (1963) 464. 20. H. H. Nagel, "Die Berechnung von Elektron-Photon-Kaskaden in Blei mit Hilfe der Monte-Carlo Methode", Inaugural-Dissertation zur Erlangung des Doktor• grades der Hohen Mathematich-Naturwissenschaftlichen Fakultiit der Rhein• ischen Freidrich-Wilhelms-Universitat zu Bonn, 1964. 21. H. H. Nagel, "Elektron-Photon-Kaskaden in Blei: Monte-Carlo Rechnungen fi Primiirelektronenenergien zwischen 100 und 1000 MeV", Z. Phys. 186 (1965) 319; English translation in Stanford Linear Accelerator Center report SLAC• TRANS-28 (1965). 22. U. Volkel, "Elektron-Photon-Kaskaden in Blei fur Primiirteilchen der Energie 6 GeV", DESY report DESY-65/6 (July 1965); English translation in Stanford Linear Accelerator Center report SLAC-TRANS-41 (1966). 23. U. Volkel, "A Monte-Carlo Calculation of Cascade Showers in Copper Due to Primary Photons of 1 GeV, 3 GeV, and 6 GeV, and 6-GeV Bremsstrahlung Spectrum", DESY report DESY-67/16 (1967). 24. D. F. Nicoli, "The Application of Monte Carlo Cascade Shower Generation in Lead", submitted in Partial Fulfillment of the Requirement for the Degree of Bachelor of Science at the Massachusetts Institute of Technology, 1966. 25. H. Burfeindt, "Monte-Carlo Rechnung fUr 3 GeV-Schauer in Blei", DESY report DESY-67/24 (1967). 26. The Monte Carlo Method, edited by Y. A. Shreider, (Pergamon Press, New York, 1966). 27. A. J. Cook, "Mortran3 User's Guide", SLAC Computation Research Group tech• nical memorandum CGTM 209 (1983). 28. P. Darriulat, E. Gygi, M. Holder, K. T. McDonald, H. G. Pugh, F. Schneider and K. Tittel, "Conversion Efficiency of Lead for 30-200 MeV Photons", Nucl. Instr. Meth. 129 (1975) 105. 29. R. L. Ford, B. L. Bero, R. L. Carrington, R. Hofstadter, E. B. Hughes, G. I. Kirk• bridge, L. H. O'Neill and J. W. Simpson, "Performance of Large, Modularized NaI(TI) Detectors", Stanford High Energy Physics Laboratory report HEPL-789 (September 1976); presented at the IEEE 1976 Nuclear Science Symposium and Scintillation and Semiconductor Symposium (New Orleans, LA, October 20-22, 1976). 30. C. J. Crannell, H. Crannell, R. R. Whitney and H. D. Zeman, "Electron-Induced Cascade Showers in Water and Aluminum", Phys. Rev. 184 (1969) 426. 31. Private communication with H. Hirayama (calculations made at SLA C during the early part of 1983). 32. R. G. Alsmiller, Jr., "High-Energy « 18 GeV) Muon Transport Calculations and Comparison with Experiment", Nucl. Instr. Meth. 71 (1969) 121. 33. G. Clement, "Differential Path Length of the Photons Produced by an Electron of Very High Energy in a Thick Target", C. R. Acad. Sci. 257 (1963) 2971; English translation in Stanford Linear Accelerator Center report SLAC-TRANS- 141 (1972). 12. Structure and Operation of the EGS4 Code System 305

34. R. C. Beach, "The Unified Graphics System for FORTRAN 77: Programming Manual", SLAC Computation Research Group technical memorandum CGTM 203 (November 1985 revision). 35. R. Cowan and W. R. Nelson, "Use of 3-D Color Graphics with EGS", Compo Phys. Comm. 45 (1987) 485. 36. R. Cowan and W. R. Nelson, "Producing EGS4 Shower Displays With Unified Graphics", Stanford Linear Accelerator report SLAC-TN-87-3 (1987). 13. Experimental Benchmarks of EGS

David W.O. Rogers and Alex F. Bielajew

Division of Physics National Research Council of Canada Ottawa, Canada KIA OR6

13.1 INTRODUCTION

Monte Carlo codes are very complex and full of approximations. Hence, it is essential to benchmark them against high-quality experimental data in order to establish their accuracy. The purpose of this chapter is to present a variety of experimental bench• marks of the EGS code system1,2 in its low-energy region, i.e., up to 50 MeV. These benchmarks will demonstrate that the code is capable of producing accurate results in a wide variety of simulations. However, this does not mean that this, or any other code, can always be trusted. Whenever a new situation is encountered, it is the user's responsibility to ensure the code works accurately in those circumstances. One of the advantages of using a general purpose code such as EGS or the ITS system3 is that a wide variety of experimental comparisons already have been made, and thus the user's task is lessened. However, one must always ensure that the code is being used prop• erly, and hence each user should attempt to duplicate some appropriate experimental or previously calculated data.

There are a variety of problems in developing good benchmarks. The first is that experimental data of adequate quality often are hard to obtain. To benchmark a code properly, one needs a very well specified and preferably simple experimental setup, and yet this is usually not the case. For example, often it is hard to get a well-specified radiation source (e.g., almost all electron beams have an energy spread, and photon beams include considerable electron contamination in many cases). In a similar vein, radiation detectors often have thresholds or response functions which are not well spec• ified. For example, it is often impossible to measure dose accurately without resorting to Monte Carlo calculations to convert ionization measurements into dose and thereby introduce some circularity into the comparison. Fortunately, more high-quality data are becoming available, and some of these experimental limitations are being overcome. At the same time, more complex simulations are feasible which makes it possible to use even more complex experimental setups for benchmarks.

Another complication in doing meaningful benchmarks is that many calculated quantities are very insensitive to much of the calculational detail. For example, the central axis depth-dose curve of a photon beam is not sensitive to the electron-transport

307 308 D. W. O. Rogers and A. F. Bielajew algorithm used, even in the build-up region. More surprisingly, the central axis depth• dose curve for a broad beam of electrons with energies over 10 MeV is also quite insen• sitive to many of the details of the electron-transport algorithm.

Another difficulty with benchmarks is that various components of the transport algorithm have greater or lesser importance in different materials and/or energy re• gions. Thus, a benchmark demonstrating accuracy at 10 MeV in carbon may be totally irrelevant to a similar calculation in lead because the roles of multiple scattering and energy-loss straggling change so dramatically.

In short, one must always be cautious in applying a Monte Carlo code to a partic• ular problem unless closely related simulations have been carefully benchmarked. The rest of this chapter will review a selection of benchmarks of the EGS code.

13.2 DETECTOR RESPONSE FUNCTIONS 13.2.1 Photon Spectrometers

One of the early major applications of Monte Carlo transport calculations was to deter• mine the absolute efficiency and the response functions of photon spectrometers such as NaI detectors (see, e.g., Zerby·). That is, one calculates the fraction of incident photons which interact in the detector and the fraction of the photon energy deposited in the crystal after it interacts. The EGS code has been benchmarked against a wide variety of experimental data and previous calculations5 . It must be emphasized that these comparisons were primarily a test of the photon-transport algorithms for energies below a few MeV. The overall agreement with experiment is excellent, although one must realize that even very crude algorithms will account well for electron-transport effects in calculating these response functions. Table 13.1 presents a comparison of the calculated results with the very careful experimental measurements by Heath6 of the photofraction for point mono-energetic photon sources placed 10 cm from a 3" x 3" NaI detector (the photofraction is the fraction of the detected particles which deposit all of their energy in the detector). These comparisons are not completely valid since the experimental data include the effects of a 1.18 g·cm-2 beryllium slab to eliminate beta-rays (except at 320 and 835 keV), and this has reduced the experimental data by a few percent, especially at the lower energies. Fig. 13.1 shows that EGS is capable of explaining the effects of the beta absorber which reduces the photofraction somewhat and fills in the valley below the photopeak. In Table 13.1, the agreement between the experimental data and the calculated results is within 1%, which is remarkable given the stated 1% uncertainty in the photon cross sections. The agreement is also good at higher photon energies. However, as the photon energy increases it becomes difficult to extract the photopeak efficiency from the spectrum because the photopeak is not resolved from the escape peaks. Thus in Fig. 13.2, the more meaningful comparison is between the calculated and measured absolute efficiency for counts above 4 MeV generated by 6.13-MeV photons incident on a 5" X 4" NaI detector. The agreement was found to be within the 2% measurement uncertainties. At this energy, the effect of electron transport is to change the photopeak efficiency by a factor of two, and thus this comparison tells us something about the accuracy of the electron-transport algorithm. The disagreement at lower energies is thought to be because of the 511-keV photons generated by the counter's environment.

Similar comparisons to experimental data for germanium detectors of a variety of shapes, and including inert volumes, showed that EGS was able to reproduce a 13. Experimental Benchmarks of EGS 309 germanium counter's relative efficiency curve over a broad range of photon energies5. However, there were problems calculating the absolute efficiencies of these detectors because the physical dimensions of the active volumes of solid state detectors are not nearly as well defined as those of the NaI scintillation detectors.

z II) 10,000

~...: Q BACKSCATTER PEAK II::w 1,000 c.. ...

en • 1.18 gm /cm2 Be I- Z ::> o BARE 0 U 100 661 keY ON 3"x3" NaI

10 400 600 800 E (keV)

Figure 13.1. Measured and calculated response of a 3" X 3" NaI detector with a 1.18 g·cm-2 beta absorber to 3.07 X 107 photons of energy 661 keV from an isotropic source 10 cm away. The histogram is the experimental data, the open circles are the calculations with no absorber. The inclusion of the absorber in the calculations (filled circles) explains the filling in of the valley. There are no free parameters in the comparison (From Ref. 5).

\ ~

~

..., ~ .,"" .\ ~ § 0 <>

2000 4000 6000 8000 energy / keY Figure 13.2. An absolute comparison of the calculated and measured response of a 5" x 4" NaI detector to a known fluence of 6.13-MeV gamma rays (with a 2% contaminant line at 7 MeV). The integrated counts above 4 MeV agree within the 2% experimental uncertainty. From Ref. 5. 310 D. W. O. Rogers and A. F. Bielajew

Table 13.1 Comparison of measured and calculated photofractions for isotropic point sources 10 cm from a 3" X 3" NaI detector with a thin Al jacket. From Ref.5.

E"I Expta EGS (MeV) 0.320 0.825 0.826(3) 0.662 0.536 0.559(2) 0.835 0.484 0.492(2) 1.33 0.357 0.361(5) 2.75 0.225 0.224(2) 3.13 0.207 0.206(4)

a Except for the 320 and 835-keV cases, the experimental data include the effects of a 1.18 g·cm-2 beryllium beta absorber which was not simulated. Experimental data are from Ref. 6.

13.2.2 Electron Detectors

One of the most severe tests of any electron Monte Carlo transport code is the cal• culation of the response functions of small electron spectrometers. These calculations require accurate simulation of bremsstrahlung production, multiple scattering leading to backscatter and, for thin detectors, energy-loss straggling. There is no published direct comparison of EGS-calculated response functions to experimental data. How• ever, Berger et al 7 have presented a detailed set of calculated response functions for silicon detectors subjected to pencil beams of electrons between 0.15 and 5 MeV, and have compared these to extensive experimental results below 1 MeV. The ETRAN cal• culations are generally in very good agreement with experiment except for detectors which are thin with respect to the electron's range (see Figure 8.20 in Chapter 8 by Berger). In this case, the low-energy peak in the calculated response curves tends to be at somewhat too high an energy. Detailed comparisons between ETRAN and EGS cal• culations for electrons on silicon detectors have shown that they are very similar, but there are some minor differences8 • However, for thin detectors the EGS calculations predict the low-energy peak to be in the same place as in the ETRAN calculations, i.e., at a higher energy than the experimental data. These discrepancies deserve further investigation.

Electron detectors are sometimes used to detect positrons. It is worth noting that differences in the transport of electrons and positrons lead to significant differences in the calculated response functions. For example, by ignoring the effects of the annihila• tion photons in order to isolate the differences caused by electron VB positron transport, it has been shown that the full-energy peak efficiency calculated by EGS differs by 13% depending on whether 4-MeV electrons or positrons are incident on a thick ger• manium detector5. EGS uses different stopping powers and inelastic-scattering cross sections for electrons and positrons, although multiple scattering is treated using the same formalism. 13. Experimental Benchmarks of EGS 311

13.3 CALCULATED ION CHAMBER RESPONSE The calculation of ion chamber response to photons is discussed in Chapter 25 by Nahum. These calculations provide a very rigorous benchmark for Monte Carlo codes because the response can be calculated independently using cavity theory. There is some circularity in this approach in the sense that cavity theory makes use of the photon cross sections in the wall and gas materials (via the (p.m/ p) factors) and the wall-to-gas stopping-power ratio (see Chapter 23 by Andreo). The stopping-power ratio is calculated using another Monte Carlo calculation, but its value is close to unity for the carbon-walled chamber we shall consider, and it is not sensitive to the details of the calculation. For example, it is accurate to about 0.1 % to use the ratio of stopping powers at the average energy of the electron spectrum rather than averaging them over the entire spectrum. The fact that the cavity-theory calculation and the direct calculation of the response with a Monte Carlo code use the same input photon cross-section data is actually an advantage in this comparison since it means that it is independent of these data.

In detailed comparisons between the response of an ion chamber calculated using EGS and that calculated using cavity theory, it has been found that for a variety of carbon-walled ion chambers in 6OCO beams, the agreement is within the 1% uncertainty of the Monte Carlo calculations9 • However, using the default step-size algorithm leads to a 40% under-estimate of the chamber's response. In fact, as shown in Chapter 5, calculating the response of ion chambers is a very sensitive test of the electron-transport algorithms in EGS. In developing PRESTA (Chapter 5), several times we found bugs which did not affect other tests, but which caused the calculated ion chamber response to be wrong by a large margin.

On the other hand, it is possible to calculate the correct ion chamber response using much simpler transport models which are not really simulating the electron transport accurately. For example, EGS calculates the correct (within 2%) response for a carbon• walled ion chamber in a 60Co beam, independently of the choice of electron step size i) if all path-length corrections are ignored; ii) if multiple scattering of the electrons is ignored; or iii) if all electron transport is ignored (the (p..n/ p) values for air and carbon happen to be identical at 60Co energies) 10. This is an important example which demonstrates that agreement with experiment is only a necessary, not a sufficient condition to make the assertion that the Monte Carlo transport simulation is adequately modelling the actual physical processes.

13.4 DEPTH-DOSE CURVES 13.4.1 Photon Depth-Dose Curves

Photon depth-dose curves do not represent a particularily stringent benchmark of electron-transport algorithms, but they represent a case of considerable interest in ra• diation dosimetry, and it is important to verify the photon transport part of a code. Mohan (Fig. 21.14, Chapter 21) shows a comparison between the measured and calcu• lated depth-dose curves from a 6OCO therapy unit. In these calculations, the inclusion of scattered photons from the source capsule and collimators is shown to have a minor effect (which in fact makes the agreement with experiment slightly worse). Fig. 13.4 shows a similar comparison for the build-up region of the depth-dose curve from a 6OCO beam. In this case, the experimental data have been corrected to remove the influence of electron contamination of the beam. Once again the scattered photons make only a small contribution to the shape of the build-up curve, although they contribute roughly 312 D. W. O. Rogers and A. F. Bielajew

20% of the dose at dmax. However, in this case they improve the already excellent agree• ment with experiment. Mohan et al ll have shown that EGS can also predict the shape of the depth-dose curve for a 20-MV x-ray beam from an electron accelerator, although electron contamination cannot be removed from the experimental data and since elec• tron contamination was not included in the simulation, agreement is not good in the build-up region. Electron contamination can have substantial effects on the build-up region of depth-dose curves from broad beams.

125

100 (L) fIl 0 <--- expt (Higgins et al) "=' "=' 75 .a(L) ".. 0 fIl histograms - calculations .a 50 ~ dashed - no scatter (L) I:>- ~ solid - with scatter .-~ 25 (L) "..

0 0.0 0.1 0.2 0.3 0.4 0.5 depth / g.cm-2

Figure 13.3. A comparison of the measured (Higgins et a1 12 ) and calculated build-up region of a 60Co depth-dose curve with the effects of electron con• tamination of the beam removed. Taken from Rogers et a1 13•

Fig. 13.4 shows the measured and calculated build-up region for a broad beam of 60CO which has passed through a PMMA filter. The excellent agreement between the calculations and experiment at depths greater than 0.05 cm is typical of a wide variety of comparisons (see Ref. 13), but the calculations clearly underestimate the measured dose at the very surface (1.6 mg·cm-2). Part of the discrepancy can be explained by some approximations introduced into the calculation to speed it up. But even in those cases where the approximations can be removed and the full EGS calculation done, the surface dose is still underestimated, although by only half as much as with the approximations. The reason for the remaining underestimate is not understood, and represents one of the major unresolved discrepancies between measurement and calculations with EGS.

7-Me V photon beam. The above comparisons between calculated and measured photon depth-dose curves were all relative, i. e., they compared the shapes, not the absolute values, of the dose per unit photon fluence. A series of measurements has been reportedl4 which tried to overcome this restriction. A van de Graaff accelerator was used to produce a nearly mono-energetic beam of 7-MeV photons using the 19F(p,D:/,)160 reaction. The photon fluence from this source was known with an accuracy of ±1.1%. The experiment con• sisted of measuring the charge produced in an exposure-calibrated Baldwin-Farmer ion chamber placed at 5-cm depth in a water phantom which was irradiated by a known 13. Experimental Benchmarks of EGS 313

photon fluence. Using the AAPM or NACP protocols for determining the absorbed dose in a high-energy photon beam with an ion chamber, one can deduce the absorbed dose to water per unit incident photon fluence and compare this with the value predicted from a Monte Carlo calculation for the same geometry and the measured spectrum of incident photons. The agreement between the measured value of the dose per unit fluence and the value calculated by EGS was well within the ±1.6% uncertainty in the experimental measurement of the charge collected per unit fluence.

125

:= a 100 "d cd

Q) 75 02 0 "d

0 50

-~ 02 cd * expt (Attn et al) 25 Q) 02 0 "d

0.0 0.1 0.2 0.3 0.4 0.5 0.6 depth / em Figure 13.4. Comparison of calculated (histogram) and experimental depth• dose curves (stars) at an SSD of 72 cm for a 39-cm diameter 60Co beam passing through a 0.7 g·cm-2 filter of PMMA at an SSD of 57 cm. The data are from Attix, et a[15 and the figure is from Ref. 13. The lower histogram shows the dose calculated for the photon beam without electron contamination.

In this study, the finite size of the water phantom (16 cm high by 16 cm wide) reduced the calculated absorbed dose per unit fluence by nearly 3% compared to a semi-infinite slab phantom and hence direct comparison to other published calculations (which were all for ICRU spheres or slab phantoms) was impossible. However, in another study16, it has been shown that EGS predicts the absorbed dose per unit fluence to be 3 to 6% lower than that calculated by several Monte Carlo codes for 6 and 7-MeV photons (see Ref. 16 and the references therein). Since the 7-MeV experimental result was slightly lower than the EGS prediction, it appears to favour fhe EGS calculation.

However, the above interpretation of the measurement ignores the fact that the dosimetry protocols used to convert the ion-chamber reading into an absorbed dose mea• surement are based on Monte Carlo calculations of stopping-power ratios (see Chapter 23), and there are other systematic uncertainties in the protocols. Thus, the experi• mental measurement is best thought of as a constraint on Monte Carlo calculations. Unfortunately, a complete check using stopping-power ratios calculated with EGS has not yet been done.

The same study14 contains another, independent benchmark on the photon trans• port component of EGS. The absorbed dose at 5-cm depth in a water phantom was 314 D. W. O. Rogers and A. F. Bielajew measured as a series of iron plates was placed between the source and the water phan• tom. The attenuation of the 7-MeV photons by the plates was also measured directly. The absorbed dose in the water phantom per unit 7-MeV fluence hitting the phantom increased by 11 % as 1 cm of iron was placed in the path. This was because the iron slabs scattered photons which reached the phantom. EGS predicted this increase within the ± 0.6% uncertainty of the measurements.

13.4.2 Electron Depth-Dose Curves

Electron depth-dose curves in water and other tissue-like materials are of central impor• tance in calculations related to radiotherapy physics. Unfortunately, it is quite hard to get good benchmark data for such a set-up because many experimental parameters can affect the results. Shortt et al 11 have done some very careful measurements in which they measured depth-dose curves in water using silicon diode detectors and parallel• plate ion chambers. The energy of the electron beam prior to leaving the accelerator tube was measured independently to an accuracy of ±100 keY. Figs. 13.5 and 13.6 show a comparison of the measured and calculated depth-dose curves for the nominal 10 and 20-MeV electron beams. It was found essential to do a complete simulation of the experimental set up, including the accelerator exit window (0.013 cm of Ti), beam scattering foil (0.013 cm of Pb) and the air (102 cm), in order to achieve good agreement with the experimental data. The figures also show calculations (shown as diamonds) done using a monoenergetic point source which passed through a vacuum, but with the correct mean electron energy at the phantom surface. These calculations do not predict the data adequately whereas the full simulations are in excellent agreement with the experimental results.

OJ '"Q ." ." OJ of .,Q s~ - experUoent ~ 2 .. boxes - full simulation :;::... diamonds - point source f'" 1

o 2 3 5 6 depth / em

Figure 13.5. Comparison of Shortt et at's11 experimentally measured central axis depth-dose curves in water and those calculated with EGS using PRESTA. The calculations shown by by the diamonds were done for a mono-energetic point source with energy equal to the mean energy of the experimental beam as it hits the phantom surface (10 MeV). The calculations shown by the boxes start with a 10.56-MeV electron beam and explicitly model the titanium exit window, lead scattering foil and 102 cm of air. The curves are arbitrarily normalized at dmax. From Ref. 18 18 . 13. Experimental Benchmarks of EGS 315

3 q.l '"0 "d squares - full simulation "d q.l ,.Q ... 2 0 diamonds - point source .c'"

'"q.l I> stars - experiment :;:l '" ~...

0 0 2 4 6 8 10 12 depth / em Figure 13.6. Same as previous figure except for a 20.84-MeV electron beam which has a mean energy of 20 MeV at the phantom surface. From Ref. 18.

Shortt et at also presented experimental data showing the dose perturbations caused by small cylinders of air or aluminium inserted at various depth in the water phantom. The geometries used corresponded to some standard benchmark geometries with 1-cm diameter cylinders of air (2 cm long) and aluminium (1 cm long) at a depth of 2 mm in a water phantom irradiated by a broad beam of 10 or 20-MeV electrons. These small inhomogeneities cause very dramatic effects, as can be seen from Figs. 13.7 and 13.8. These dose perturbations are well predicted by the EGS4 calculations17, which gives the user considerable confidence in the code's ability to simulate the passage of electrons through the human body.

13.5 BREMSSTRAHLUNG PRODUCTION

A major medical-physics application of Monte Carlo transport techniques concerns the simulation of bremsstrahlung spectra from clinical accelerators. Therefore, it is impor• tant to benchmark Monte Carlo codes against experimental data on bremsstrahlung production. Unfortunately, there is only a small amount of suitable experimental data available. NCRP Report 51 19 presents some data which give, as a function of angle with respect to the initial electron beam, the dose at a depth of 4.5 g·cm-2 in a water phan• tom caused by the bremsstrahlung created in thick targets of tungsten and tantalum. The data are absolute. Figs. 13.10 and 13.11 present a comparison of the experimental data and the results calculated with EGS. The calculations include two approximations which affect the results. Firstly, the calculations were for radiators which were semi• infinite slabs, and hence the dose calculated at 90° to the beam is zero. Secondly, the conversion of photon fluence on the phantom surface to the dose at 4.5 g·cm-2 was done using conversion coefficients for broad beams of photons, and hence in the forward direction, the calculations overestimated the dose for the high-energy beams because the photon beams were actually very narrow. The calculations are in reasonably good agreement with the experimental data except for the very low-energy electron beams (2 and 3 MeV). In these cases, the calculations overestimate the bremsstrahlung yield, 316 D. W. O. Rogers and A. F. Bielajew especially in the forward directions. This presumably reflects the use of inadequate bremsstrahlung cross sections for low-energy electrons in the EGS code. The fact that the code overestimates the yield is surprising because the EGS code ignores the Elwert correction factor to the bremsstrahlung cross section 1, and the effect of this factor would be to increase further the cross section for low-energy electrons*.

"'s 6 .. ",c.!O 5 20 MeV, 2 mm 1 0

"- ~ QI :I QI ::I 3 "-=

"'"..Q '" 1 ...0

Figure 13.7. Depth-dose distributions from a 20-MeV beam of electrons incident on a water phantom with air and Al cylinders at 2-mm depth. The o's repre• sent homogeneous phantom data; the x's, the case with the Al cylinder; the triangles, the case with the air cylinder. The histograms are EGS4 calculated results for a point source of 20-MeV electrons. The arrows show the depths at which radial profiles are presented in the next figure. The experimental data are all normalized to the calculations via one point on the homogeneous curve. From Ref. 17.

As well as the total yield as a function of angle, Monte Carlo codes predict the energy spectrum of high-energy x-rays. The variation in the spectrum as a function of angle and after hardening by flattening filters is of interest in modelling the output from accelerators. Mohan et all! have shown that using EGS, the calculated depth• dose curves on- and off-axis are in good agreement with their experimental results for a 15-MV photon beam (see Fig. 21.5 in Chapter 21). However, it is not clear how sensitive this comparison is to the details of the calculated spectra. Unfortunately, there are only a few measured spectra which can be compared with calculations. Figs. 13.11 and 13.12 present comparisons between EGS3 calculations (which should be the same as for EGS4 in this case), and the absolute experimental measurements of O'Dell et al 20 and the relative measurements of Levy et a1 21 • The figures show that there is reasonable qualitative agreement between the calculations and the experimental data, but that there are some quantitative discrepancies which may reflect problems in the code or in the experiments. Figure 13.11 also presents the results of calculations using ETRAN which is in somewhat better agreement with experiment.

* Thanks to Stephen Seltzer for pointing this out during the lecture! 13. Experimental Benchmarks of EGS 317

13.6 CONCLUSIONS This highly selective set of comparisons of EGS calculations with experimental data demonstrates that the code is capable of simulating a wide variety of situations with considerable accuracy. However, there are some problem areas. The dose very close to the surface of a water phantom in a broad soeo beam is underestimated by calcula• tions. The agreement between the calculated and experimental bremsstrahlung spectra is at best qualitative. In both cases, the discrepancies may reflect the difficulty of the experimental measurements.

6 0.10 em behind 0.60 em behind "'8 '" ~4 1'..• ~... !~. :t"l&~-~ _ i"\ =:t!I ------.... t . : ~ -. ------..-....._. ~ : ...~ ------"""s1 IIJ.£ ~~ ~ '0 - 2 ...... '" '"d :5'" 6 2.10 em behind 3.60 em behind ...... , '"o "" 4

"",Q'" ...o 2 en ,Q 2 cm air cylinder at 2 mm '"

- 1 o - 1 o radial position / em

Figure 13.8. Radial dose profiles behind the air cylinder in a water phantom irradiated by a 20-MeV beam of electrons. The symbols res present the mea• sured data and the histograms are the EGS4 calculations. The slight discrep• ancy in the 3.6-cm case is explained by a minor difference in the experimental and modelled situations. From Ref. 17.

There are a wide variety of situations which have not been discussed here and which one may wish to investigate using EGS. One must be wary of assuming that the good agreement obtained with most of the comparisons presented here will necessarily apply elsewhere. On the other hand, the good results presented should make it clear that there is a good probability that EGS, when used properly, will be able to provide accurate simulations in most applications. 318 D. W. O. Rogers and A. F. Bielajew

CD o~ ...J :J o ...... U '"a: IJJ t• IJJ ~ I > ~ 10~ t• Z IJJ «...J ....> :J Cl IJJ IJJ (/) o o o 25 50 75 toO 125 150 175 Ang le (degrees) Figure 13.9. Absolute comparison of calculated and measured dose at 4.5 g·cm-2 depth in a water phantom which is 1 m from thick (68 and 22.5 g/cm2, semi-infinite targets of tantalum irradiated by 100 and 30-MeV electrons. Ex• perimental data are from Wyckoff et al 22, and the figure is from Ewart and Rogers23.

CD 104 ~ 0 ...J :J 0 U ...... 103 a: '"IJJ t- IJJ :::E I > 102 ~ t- Z IJJ ...J« ....> 10 :J Cl IJJ IJJ (/) 0 0 o 25 50 75 100 125 150 175 Angle (degrees) Figure 13.10. Same as previous figure but for 8, 3 and 2-MeV electron beams incident on a thick tungsten target. Experimental data are from NCRP5119, and the figure is from Ref. 23. 13. Experimental Benchmarks of EGS 319

'... '"

• 10.0 MeV

4 8 12 16 20 PHOTON ENERGY (MeV) Figure 13.11. Absolute comparison of measured and calculated forward-directed bremsstrahlung spectra generated by 10 and 20-MeV electrons incident on a composite target of tungsten (0.49 g·cm-2) followed by gold (0.25 g·cm-2). Experimental data from O'Dell2o have an experimental uncertainty of 10 to 20%. From Ref. 2.

IL EGS Monte Carla results (6 r.f. Pb,Oo-5° cone) T u 10- 1 Qj'" T :;; T > :::.'"

N'"zl~ 10-2 "'~

10-3 ~---'-_-'--_..I...----'_--'--_..L.-..Jl...J o 4 8 12 16 20 24 28 PHOTON ENERGY (MeV) Figure 13.12. Comparison of calculated and measured21 bremsstrahlung spec• tra generated by a 25-MeV beam of electrons incident on a 6 radiation-length thick target of lead. The calculations and experiment are normalized arbitrar• ily in the high-energy portion of the spectrum. Figure from Ref. 2. 320 D. W. O. Rogers and A. F. Bielajew

REFERENCES 1. W. R. Nelson, H. Hirayama and D. W. O. Rogers. "The EGS4 Code System", Stanford Linear Accelerator report SLAC-265 (1985). 2. R. L. Ford and W. R. Nelson, "The EGS Code System (Version 3)", Stanford Linear Accelerator report SLAC-21O (1978). 3. J. A. Halbleib and T. A. Melhorn, "ITS: The Integrated Tiger Series of Coupled Electron/Photon Monte Carlo Transport Codes.", Sandia National Laboratories report SAND84-0073 (1984). 4. C. D. Zerby, "A Monte Carlo Calculation of the Response of Gamma-ray Scin• tillation Counters", in Methods in Computational Physics, Vol. I, (Academic Press, New York, 1963) 89. 5. D. W. O. Rogers, "More Realistic Monte Carlo Calculations of Photon Detector Response Functions", Nucl. Instr. Meth. A199 (1982) 531. 6. R. 1. Heath, "Scintillation Spectrometry, Gamma-Ray Spectrum Catalogue", USAEC report IDO-16880-1(1964). 7. M. J. Berger, S. M. Seltzer, H. A. Eisen and J. Silverman, "Response of Silicon Detectors to Monoenergetic Electrons with Energies Between 0.15 and 5.0 MeV.", Nucl. Instr. Meth. 69 (1969) 181. 8. D. W. O. Rogers, "Low energy electron transport with EGS", Nucl. Instr. Meth. A227 (1984) 535. 9. D. W. O. Rogers, A. F. Bielajew and A. E. Nahum, "Ion chamber Response and AWall Correction Factors in a 60Co Beam by Monte Carlo Simulation" , Phys. Med. Biol. 30 (1985) 429. 10. A. F. Bielajew and D. W. O. Rogers, "Interface Artefacts in Monte Carlo Cal• culations" , Phys. Med. Biol. 31 (1986) 301. 11. R. Mohan, C. Chui and 1. Lidofsky , "Energy and Angular Distributions of Photons from Medical Linear Accelerators", Med. Phys. 12 (1985) 592. 12. P. D. Higgins, C. H. Sibata and B. R. Paliwal, "Determination of Contaminant• Free Build-Up for 60Co", Phys. Med. Biol. 30 (1985) 153. 13. D. W. O. Rogers, G. M. Ewart and A. F. Bielajew, "Calculation of Contamina• tion of the 60Co Beam from an AECL Therapy Source", NRCC report PXNR- 2710, Jan. 1985 (available from the authors). 14. H. Mach and D. W. O. Rogers, "A Measurement of Absorbed Dose to Water Per Unit Incident 7 MeV Photon Fluence", Phys. Med. Biol. 29 (1984) 1555. 15. F. H. Attix, F. Lopez, S. Owolabi and B. R. Paliwal, "Electron Contamination in 60Co Beams", Med. Phys. 10 (1983) 30l. 16. D. W. O. Rogers, "Fluence to Dose Equivalent Conversion Factors Calculated with EGS3 for Electrons from 100 keV to 20 GeV and Photons from 11 keV to 20 GeV", Health Phys. 46 (1984) 89l. 17. K. R. Shortt, C. K. Ross, A. F. Bielajew and D. W. O. Rogers, "Electron Beam Dose Distributions Near Standard Inhomogeneities". Phys. Med. Biol. 31 (1986) 235. 18. D. W. O. Rogers and A. F. Bielajew, "Monte Carlo Techniques of Electron and Photon Transport For Radiation Dosimetry", in The Dosimetry of Ionizing Radiation, edited by K. R. Kase, B. E. Bjarngard, F. H. Attix, Vol III, to be pub. 1988, Academic Press. 13. Experimental Benchmarks of EGS 321

19. NCRP Report 51, "Radiation Protection Design Guidelines for 0.1-100 MeV Particle Accelerator Facilities", National Council on Radiation Protection and Measurements, Washington D.C., 1977. 20. A. A. O'Dell, C. W. Sandifer, R. B. Knowlen and W. D. George, "Measurement of Absolute Thick-Target Bremsstrahlung Spectra", Nucl. Instr. Meth. 61 (1968) 340. 21. 1. B. Levy, R. G. Waggener, W. D. McDavid and W. H. Payne, "Experimental and Calculated Bremsstrahlung Spectra from a 25-MeV Linear Accelerator and a 19-MeV Betatron", Med. Phys. 1 (1974) 62. 22. J. M. Wyckoff, J. S. Pruitt and G. Svensson, "Dose vs Angle and Depth Produced by 20 to 100 MeV Electrons Incident on Thick Targets", Proc. of Int'l Congo on Protection Against Accelerator and Space Radiations, Vol 2, CERN report CERN-71-16 (also reported in NCRP51) 23. G. M. Ewart and D. W. O. Rogers, "Calculated Thick Target Bremsstrahlung Angular Distributions and Shielding Calculations", NRCC report PXNR-2640, (1982). 14. A Comparison of EGS and ETRAN

David W. O. Rogers and Alex F. Bielajew

Division of Physics National Research Council of Canada Ottawa, Canada KIA OR6

14.1 INTRODUCTION Much of the material to be covered in this chapter has been discussed in earlier chapters on ETRAN, EGS and the ITS systems. This chapter explicitly compares and contrasts the codes with the intention of providing some new insights into them. It should be noted that, for the purposes of this chapter, the term ETRAN refers to the original NBS version of ETRANl,2 and to its many descendants, primarily the ITS system from Sandia3 *.

The most obvious difference between the EGS and ETRAN codes concerns the underlying philosophy of the design. ETRAN is a packaged code in which the user need only install the code and then prepare the appropriate input files. The code then pro• duces its standard output which includes many different quantities of interest such as energy and charge deposition within regions, fluence spectra within regions, and energy and angular distributions of particles leaving the geometry of interest. The ITS system consists of a series of codes handling a variety of different, but generalized, geome• try packages (e.g., a multi-material cylindrical package, or the combinatorial geometry package), and hence one has the ability to solve a wide variety of problems. In contrast to this, the EGS system consists of a series of subroutines which handle the radiation transport simulation. It is up to the individual user to write a user code which makes use of a very well specified interface to the transport process. This user code includes a specification of the user's own geometry and the scoring of quantities of interest. This gives the user a great deal more flexibility, but at the cost of needing to know much more about the details of the system. It also means that EGS can more easily be customized to execute efficiently for a particular problem by, for example, taking advantage of special symmetries. Both of these approaches have obvious advantages. In those cases where the problem can be cast within the ETRAN framework, this ap• proach is obviously more efficient for the user. On the other hand, ETRAN is virtually impossible to adapt (except by real experts), and in problems requiring adaptation and flexibility, the EGS approach has many advantages. For medical physics applications,

* Also note that all ETRAN and ITS calculations presented were done with versions which still included the energy-loss straggling error for which Seltzer describes the correction in Chapter 7.

323 324 D. W. O. Rogers and A. F. Bielajew there are now a variety of general purpose user codes available which allow users to tackle many problems without writing their own user codes (see for example the code INHOM distributed with the EGS4 system).

As a final introductory comment, perhaps it is worth noting that EGS and ETRAN do not use the same cross-section data sets as inputs. In general, the ETRAN code uses more up-to-date electron cross-section data (see chapters by Berger and Seltzer). In the rest of this chapter, this aspect of the differences will not be discussed, although there are cases where it can be important (e.g., low-energy bremsstrahlung production).

14.2 CLASS I vs CLASS" ALGORITHMS One of the major differences in the physics of the ETRAN and EGS codes concerns how they handle the creation of secondary electrons. There are generally two classes of electron-transport algorithms for doing this, called class I and class II algorithms by Bergerl . In a class I algorithm such as used in ETRAN, the creation of a secondary electron does not directly affect the primary electron. Instead, the effects of creating secondary electrons are taken into account on a statistical basis by sampling from an energy-loss straggling algorithm during each step and decrementing the primary elec• tron's energy by that amount. The deflections from secondary electron production are then accounted for within the multiple-scattering theory. Thus, in a class I algorithm we have (where, for the time being, we ignore radiative events):

Efinal = Einitial - .6.E(t), (14.1) and Etkporitod = .6.E(t) - Es, (14.2) where Einitial and Efinal are the primary electron's energy at the beginning and end of a condensed-history step of length t, .6.E(t) is a value sampled from an energy-loss straggling formalism, Edeporited is the energy deposited in the medium during the step, and Es is the kinetic energy of any secondary electron created during the step. The energy of the secondary electron Es must be subtracted because it already has been included in .6.E(t) in a statistical sense. The mean value of .6.E(t) is just the total collisional stopping power, but in principle it can range from 0 to !Einitial. Note that when using this technique, it is possible to get negative values of Edepo.ited in individual steps because of the statistical nature of the process. However, its expectation value is positive, as expected.

In a class II algorithm such as used in EGS, the energy of the primary electron is directly correlated to the creation of individual secondary electrons, and the primary electron's direction is also affected by the creation of a knock-on. Thus we have:

E final = Einitial - tL!f - Es, (14.3) and (14.4) where L!f is the appropriately energy-averaged restricted collision stopping power which accounts for energy lost during the step to electrons below a threshold of AE.

So far, the discussion has been in terms of creating secondary electrons, but the same ideas apply to the creation of bremsstrahlung photons. However, for reasons which 14. A Comparison of EGS and ETRAN 325 will become apparent in a moment, ETRAN and EGS both use a class II algorithm to account for the creation of secondary photons.

Fig. 14.1 shows the ratio of the restricted collision stopping power to the total collision stopping power as a function of the threshold kinetic energy for producing a knock-on electron for 2 and 20-MEV electrons stopping in carbon and lead. The important feature to note here is that in all four cases shown, for values of AE = 1 keY (i.e., energy ratios of 5 X 10-5 and 5 X 10-4 for 20 and 2-MeV electrons respectively), the restricted stopping power accounts for well over 50% of the energy loss. This means that even if a class II algorithm explicitly considered the creation of all secondary electrons with energies above 1 ke V, less than 50% of the collisional energy loss would be considered. This is a major reason that the condensed-history technique is an essential part of electron Monte Carlo transport codes.

.. 1.0 •C> "" dashed - carbon ... 0.9 .[ ...""C> ...... solid - lead ..... O.S C.> b..... 0.7 20 ...... '" ..... C.> 2 MeV :s..... 0.6

10-4 10-3 10-2 10-1 maximum knock-on energy / energy of primary Figure 14.1. Ratio of the restricted collison stopping power to total collision stopping power as a function of the maximum kinetic energy of the secondary electrons considered within the restricted stopping power (AE-0.511 MeV in EGS). Data are shown for 2 and 20-MeV electrons stopping in carbon and lead. Data from EGS44 •

Fig. 14.2 is the same as Fig. 14.1 except it is for restricted radiative stopping powers. However, in this case the shape of the curves is very different. For one thing, the energy lost to photons above one-half of the energy of the electron represents a significant portion of the total radiative stopping power (20 - 30%), whereas there are no secondary electrons produced with an energy this high. Secondly, the amount of energy lost via the creation of low-energy photons is a small fraction of the total. For example, for 20-MeV electrons in lead, less than 2% of the radiative energy loss is via photons with energies less than 200 keY. This means that one can use a class II algorithm and explicitly simulate all bremsstrahlung events. There is no need to use a restricted radiative stopping power if a sufficiently small threshold for bremsstrahlung processes is used. For this reason, both EGS and ETRAN use a class II algorithm for the simulation of radiative events. 326 D. W. O. Rogers and A. F. Bielajew

dashed - carbon

solid - lead

10-3 10-2 10-1 maximum photon energy / electron energy

Figure 14.2. Same as in previous figure except for restricted radiative stopping powers.

As discussed above, one difference between EGS and ETRAN is in the simulation of the creation of knock-on electrons. In general, the lack of correlation between the creation of the knock-ons and the transport of the primary electron has little effect. However, one does expect differences. Fig. 14.3 shows the variation in the energy of the primary electrons as a function of angle as a 20-MeV electron beam passes through the titanium exit window of an accelerator. In a class II algorithm, there is a considerable variation in the energy as a function of angle because those few electrons which have been scattered to large angles by a discrete interaction have lost a considerable amount of energy to the secondary, and hence they pull the average energy down (although note that those electrons at large angles due to multiple scattering all have the same energy - except for slight changes due to different path lengths in the titanium). In a class I algorithm, the energy is independent of the angle because there is no correlation between the creation of secondaries and the energy or angle of the primary. Although this difference in algorithms will lead to different results with EGS and ETRAN, it is not expected to have a significant impact in most situations.

The use of class I and class II algorithms in EGS and ETRAN has led to one significant difference. ETRAN uses the Blunck-Leisegang5 modification of Landau's theory6 to account for energy-loss straggling. This formalism is known to have an error in it (see Rogers and Bielajew1, and references therein) which means that the mean energy loss is not given correctly, as is shown in Table 14.1. Errors of similar magnitude have been found for a thin slab of copper. The reason for this error is that the modified Landau energy-loss straggling formalism underestimates the number of large energy-loss events. This was seen in Fig. 6.3 from Chapter 6 in which an explicit calculation was compared to the modified Landau distribution. Seltzer has shown that this problem is the result of a numerical accuracy problem, and has recently implemented a more accurate procedure in ETRAN (see Chapter 7) which removes any discrepancies between EGS and ETRAN in this area. While this error in the energy-loss straggling distribution 14. A Comparison of EGS and ETRAN 327 had only a small effect on the energy spectrum, the rather large error in the mean energy loss lead to sizeable errors in the calculated depth-dose curves, at least in water and other light elements. As we shall see below, the differences between EGS and ETRAN in the calculated depth-dose curves for heavier elements were not so large, which may reflect compensating errors in higher Z elements, or the fact that energy-loss straggling from collisional events becomes less important. Future comparisons with the corrected version of ETRAN should clarify these questions.

20.0 :.~ ~ r:I 0 b ..,Co> ~ 19.5 ~ :ar:I 10-' ..,... -[ ~ III -0 b ~ ....., 19.0 Co> ..,r:I 10-2 ~ .., .....lOG 5 1B.5 10-- 0 10 20 30 40 angle / degrees

Figure 14.3. Variation in primary electron energy as a function of angle after a 20-MeV beam passes through a O.2-mm plate of titanium (histogram) and the number of electrons in each angular bin (dashed curve). The results were calculated with EGS4 which uses a class II algorithm to account for collisional events. A class I algorithm would assign virtually the same average energy to all electrons (straight line).

Table 14.1. Mean energy lost by primary electrons passing through a slab of water with a thickness of ro/40 as calculated7 using the total stopping power or using the ITS (VI) code CYLTRAN. Energy 'lost passing through foil Eo ro/40 CYLTRAN" Stopping powerb Difference (MeV) (em) (keV) (keV) (%) 1 0.0109 18.5 20.6 -10.3 5 0.0638 114.5 128.5 -10.9 10 0.124 243 271 -10.3 20 0.233 520 576 -10.0 50 0.496 1510 1620 -7.0'

" Mean energy lost by primaries as calculated by CYLTRAN. b Energy loss calculated using Berger and Seltzer (1964) stopping powers8. • This value is lower because the radiative energy loss is beginning to playa more important role, and the error does not affect radiative events. 328 D. W. O. Rogers and A. F. Bielajew

14.3 DIFFERENCES IN MULTIPLE SCATTERING

EGS and ETRAN use two different multiple-scattering formalisms, the Moliere4 and the Goudsmit-Saundersonl formalisms respectively. The Moliere theory contains a small an• gle approximation in its derivation, whereas the Goudsmit-Saunderson (G-S) derivation holds for all angles. Therefore, it is often felt that the G-S distribution should be more accurate. However, Bethe9 has shown that by modifying the Moliere formalism, one could obtain a theory which applies for large-angle scattering as well. Fig. 14.4 presents a comparison made by Bergerl of the angular distribution of I-MeV electrons passing through a thin plate of aluminium. The only differences between the two formalisms occur in the very extremes- of the tails .

., o ..... stars - G-S Rutherford triangles - G-S e- Mott diamonds - G-S t Mott

o 25 50 75 100 125 150 175 angle / degrees

Figure 14.4. Comparison of the number of electrons as a function of angle as calculated by various multiple-scattering formalisms for I-MeV electrons passing through a 0.029 g·cm-2 plate of AI. The Goudsmit-Saunderson results are shown for three different scattering cross sections. Based on data from Bergerl .

Fig. 14.5 presents a comparison between angular deflections calculated by the two codes as 500-keV electrons pass through a thin (0.02-cm) slab of water. The EGS cal• culations are broken down into three components, all individually normalized to one electron passing through the slab so that the shapes can be compared. The first compo• nent to examine are those electrons which undergo no discrete interactions while passing through the slab. All their angular deflections are due to the Moliere multiple-scattering formalism; their angular distribution is virtually identical to that predicted by the G-S formalism in CYLTRAN. The second component, which is much less intense (before being normalized), consists of those electrons which had an additional deflection from interactions creating secondary electrons. There are fewer of these events at 0°, and they are more broadly scattered. The third 'component' shown is just the weighted sum of the other components, and is seen to be very similar to the angular distribution of those particles undergoing no discrete interactions. The point being made is that the discrete interactions do have some effect on the angular distributions calculated with EGS because they use a class II electron-transport algorithm. However, this is a rather 14. A Comparison of EGS and ETRAN 329

,... - 500 keY e on 0.02 em H2O - 0 I - 0 w I I ~ 0 QI 1 N 10- 0 .... :=- CQ a e - c... - 0 0 - + NO DISCRETE .s a c:: - 0 .c ... - o WITH DISCI£TE c... a QI c. 0 Ul 10-2 c:: =- a TOTAL - EGS 0 c... - a ..... - U QI '"' .... - -TOTAL - CYLTRAN 0 UJ ... - .. + I I I I I I I I 20 40 60 80 100 ANGLE (degrees)

Figure 14.5. Angular distributions of 500-keV electrons after passing through a 0.02-em slab of water as calculated by CYLTRAN (using the G-S multiple• scattering formalism) and EGS (using the Moliere formalism). The EGS cal• culations are broken into three components, all separately normalized to a total of one electron passing through the slab. They are those electrons which do not create a secondary electron above 10 ke V while passing through the slab, those that do, and all electrons. Only primary electrons are included. Taken from Ref. 10. unusual case. In most situations we have looked at, the angular deflections caused by the creation of secondary particles make virtually no difference - i. e., the approximation used in the ETRAN codes seems to be acceptable.

Fig. 14.5 introduces a minor problem which has been noted in EGSIO. The multiple-scattering formalisms in both EGS and ETRAN replace a Z2 factor by a Z(Z + 1) factor in order to take into account, in an approximate way, multiple scatter• ing from atomic electrons. In the class I algorithm used in ETRAN, this approach is consistent, but in the class II algorithm used in EGS, this leads to some double count• ing because the scattering associated with the creation of secondary electrons above the threshold AE is already explicitly accounted for. This is not expected to be a serious problem because, as noted above, the scattering from electrons rarely plays a significant role. However, the case shown in Fig. 14.5 presents an opportunity to investigate the accuracy of the approximate treatment of multiple scattering from electrons. It is pos• sible with EGS to change the multiple-scattering formalism to include just the Z2 term (by setting $FUDGEMS to 0.0 in PEGS4). Fig. 14.6 presents two histograms showing the calculated angular distributions both with (solid) and without (dashed) the electron component in the multiple-scattering formalism, but only for electrons which did not 330 D. W. O. Rogers and A. F. Bielajew undergo discrete interactions. The angular distributions shown by the stars and boxes show the effect of the angular deflections when secondary electrons greater than 10 keVand 1 keY, respectively, are included in the simulation explicitly, but still with no multiple scattering from electrons. It can be seen that the effect of the deflections from the discrete interactions for all electrons above 1 keY is close to that predicted by the inclusion of multiple scattering from electrons by utilizing the Z(Z + 1) approximation. This is a rather satisfying 'experimental' check of the multiple-scattering formalism for this Z and energy. However, it does not get around the usually unimportant double counting inherent in the EGS code. 0.4,----,----,----,,----,----,----,-----r---.0.OB

r-----j I * I 0.06 : D .9 0.3 : * ----> ,.Q Il ____ _ '-. I i 0.04 fIJ d '------o <---- bQ 0.2 ____ _ Q) -Q) 0.02

O. 1 L-__-L ____ ..I- __---' ____ -L ____ L- __----'- ____..l- __---' 0.00

o ~ ~ 00 00 angle / degrees Figure 14.6. Angular distribution of 500-keV electrons after passing through a 0.02-cm slab of water, as calculated with EGS4 with a variety of differ• ent algorithms for the multiple scattering from atomic electrons. The his• tograms are for electrons which did not create any secondary electrons; for the dashed histogram, the multiple-scattering formalism contained no correc• tion for scattering from electrons (Z2 term only); for the solid histogram, the multiple-scattering formalism included such a correction (Z (Z + 1) term). The symbols represent calculations without the multiple-scattering formalism ac• counting for electrons, but including scattering from all those interactions in which secondary electrons above 10 kev (stars) and 1 kev( boxes) were created.

As another way of comparing the multiple-scattering formalisms in the EGS and ETRAN codes, we have calculated a series of depth-dose curves for broad electron beams incident on slabs of various materials. In these calculations, all forms of energy• loss straggling have been turned off so that the different approaches used by the two codes do not obscure the comparison - i.e., we are doing continuous-slowing-down approximation (csda) calculations with each code, and hence the effect of the energy-loss straggling is eliminated. Furthermore, bremsstrahlung energy deposition is ignored in all the calculations (although radiative stopping powers contribute to the slowing-down process). These calculations depend primarily on the stopping powers used and on the multiple-scattering formalisms .. Figs. 14.7 and 14.8 compare the results calculated for lOO-keV and 30-MeV beams of electrons on water. The results from the two codes are very similar, confirming the earlier observations that the multiple-scattering formalisms are equivalent. 14. A Comparison of EGS and ETRAN 331

CI.> C.) 1::1 :::s1OCI.> c:: ...... CI.> fIl .g histogram - EGS4 -.::l 5 CI.> ~ ~ stars - CYLTRAN JITS fIl ~ C

o~--L---~--~---L---L--~--~--~----~~ 0.0 0.2 0.4 0.6 O.B 1.0 depth / ro

Figure 14.7. Comparison of depth-dose curves from broad parallel beams of 100-keV electrons incident on a slab of water as calculated in the csda approx• imation by EGS and ETRAN/CYLTRAN using the Moliere a.nd Goudsmit• Saunderson multiple-scattering formalisms respectively. The EGS results were obtained using PRESTA, but were identical to an ESTEPE=l % calculation.

Figs. 14.9 and 14.10 present the results of similar calculations, but for electrons incident on a slab of lead. In this case, EGS4 results using several different step-size algorithms are included since, in particular, the lOO-keV calculation is very sensitive to this parameter. In fact, for the 100-keV case the standard ESTEPE algorithm could not be made to reach a small enough step size without turning off the multiple-scattering formalism for a large fraction of the steps (see Chapter 5). This is perhaps not too much of a concern because the codes are not expected to work in this situation since they both assume atomic electrons are unbound, clearly a bad approximation in lead at 100 keY and less. Nonetheless, one observes that, at both energies, the EGS4/PRESTA results are somewhat lower than the CYLTRAN results near the surface of the lead, and they penetrate somewhat further. The exact reason for this needs further study, but the work of Berger and Wang (see Chapter 2) suggests the Moliere distribution may be breaking down for high Z and low energies. Also, at low energies the EGS calculation may still have some step-size dependence because one cannot take small enough steps before the Moliere multiple-scattering theory breaks down.

To summarize, the multiple-scattering formalisms used in ETRAN and EGS are very similar for low-Z materials. There are differences in the test calculations in lead which probably reflect differences in the multiple-scattering formalisms for high-Z mate• rials, but which may reflect other distinctions, such as details of the step-size algorithms which play an important role in these materials. 332 D. W. O. Rogers and A. F. Bielajew

NS 5 (J I ~ ot!l - 4 ·0 -<

stars - EGS4

histogram - CYLTRAN/ITS

0~ __~ __L- __L- __L- __L- __L- __L- __~ __~~

0.0 0.2 0.4 0.6 0.8 1.0 depth / em

Figure 14.8. Same as Fig. 14.7, but for a beam of 3D-MeV electrons. No bremsstrahlung dose is included. From Ref. 7.

NS 2.0 (J + I + ... til histogram - ITS/CYLTRAN ·0 -< 1.5 stars - EGS4/PRESTA ......

~ 0.5

"'s 5 c.> I I>. ...00° 4 10 '-- ~ 3 ~ ::sCI> c: ~ 2 histogram - ITS/CYLTRAN o -0 stars - EGS4/PRESTA -0 ~ 1 diamonds - ESTEPE = 0.5% ]'"' + - ESTEPE = 0.8% CIS oL---~----~----L----L----~--~----~ __~ 0.0 0.2 0.4 O.B O.B depth Ira

Figure 14.10. Same as Fig. 14.9, but for a beam of 20-MeV electrons. In this case, the ESTEPE algorithm is capable of giving accurate reults. No bremsstrahlung dose is included.

14.4 ELECTRON DEPTH-DOSE CURVES

In the previous section, we presented comparisons of csda depth-dose calculations done using EGS and ETRAN. In this section, we present similar comparisons, but for the full calculations which include the effects of energy-loss straggling and the creation and transport of secondary electrons. Fig. 14.11 for 100-keV electrons on water is virtually identical to the csda calculation shown in Fig. 14.7 since energy-loss straggling and secondary electrons playa small role. However, Fig. 14.12 for 20-MeV electrons on water shows a strong difference from the high-energy csda comparison shown in Fig. 14.8. In the calculation with straggle, the error in the energy-loss straggling formalism of ETRAN leads to an underestimate of the dose deposited near the phantom surface, and to too deep a penetration of the electrons before a steeper fall off to nearly the same practical range as EGS (as shown by the tangential lines). This difference in the depth• dose curves in water has been discussed in considerable detail7• In particular, it leads to a change in the specification of the mean energy of an incident electron beam in terms of Rso , the depth at which the dose falls to 50% of its maximum value. This parameter is used in a variety of dosimetry protocols to specify the energy of radiotherapy beams. The EGS values reduce Rso by several percent for a given energy. At the same time, below about 20 MeV the two codes predict very similar values of Rp , the practical range. This is because the error in the energy-loss straggling algorithm concerns just the electrons which create high-energy secondaries, and the electrons which reach and define the practical range have not created any high-energy secondaries. Hence, the practical ranges calculated by the two codes are the same. The figure also shows depth-dose curves as calculated by both ETRAN and CYLTRAN to demonstrate the equivalence of the two versions of what we have been calling ETRAN. By correcting the problem with energy-loss straggling in ETRAN, Seltzer (Chapter 7) has shown that the depth• dose curves for 20-MeV electrons on water are virtually identical when calculated by by EGS or ETRAN. 334 D. W. O. Rogers and A. F. Bielajew

""S20 c I I>, ot.!l 1 015....

Q) c d ~ 10 ;;:: ...... Q) ofIl 't:I 't:I 5 stars -CYLTRAN/ITS Q) ..c ~ diamonds - EGS4/PRESTA ..cfIl as

0~ __L- __L- __L- __L- __L- __L- __L- __L- __~~ 0.0 0.2 0.4 0.6 0.8 1.0 depth / ro

Figure 14.11. Comparison of depth-dose curves for 100-keV electrons incident on a water phantom as calculated by CYLTRAN and EGS (ESTEPE=l%).

Q) c d ~ 2 histogram - EGS4 ;;:: ...... Q) fIl ~ curve/diamonds - ETRAN 't:I 1 Q) ..c ~ stars - CYLTRAN fIl ..c as

o~~--~--L-~--~--L-~---L __L-~~~~

0.0 0.2 0.4 0.6 0.8 1.0 1.2 depth / r 0

Figure 14.12. Comparison of depth-dose curves for a broad parallel beam of 20- MeV electrons incident on a water phantom, as calculated by EGS, ETRAN and CYLTRAN. The tangential lines demonstrate the near equality of the practical range Rp. The differences are ascribed to an error in the energy-loss straggling formalism used in ETRAN. From Ref. 7. (Compare Seltzer's Fig. 7.7 in Chapter 7 which uses the new version of ETRAN). 14. A Comparison of EGS and ETRAN 335

Figs. 14.13 and 14.14 present comparisons of the complete depth-dose calculations by EGS and ETRAN for 100-keV and 20-MeV beams of electrons on lead. With the same caveat as before concerning the accuracy of the code for electrons at 100 keY and below in lead, we note that the 100-keV comparison is very similar to the csda comparison in Fig. 14.9. However, in the complete calculation for the 20-MeV case in Fig. 14.14, the two codes look remarkably similar despite the significant differences in the csda case in Fig. 14.10. The differences caused by the error in the energy-loss straggling formalism app~ar to compensate the use of different multiple-scattering for• malisms. Fig. 14.14 also includes the csda calculation done with CYLTRAN (now including bremsstrahlung energy deposition, unlike in Fig. 14.1O). The full calculation compared to the csda calculation shows the characteristic decrease in dose near the sur• face when the erroneous energy-loss straggling formalism is included in the calculation.

In summary, the two code systems appear to calculate fairly similar depth-dose curves in high-Z materials, but this appears to reflect compensating errors in one or both of the codes.

histogram - ITS/CYLTRAN stars - EGS4/PRESTA

o~~~~~~~~~~~~~ 0.0 0.1 0.2 0.3 0.4 0.5 depth/ro

Figure 14.13. Comparison of calculations by CYLTRAN and EGS4/PRESTA of the depth-dose curve for a broad beam of 100-keV electrons incident on a slab of lead.

14.5 LOW-ENERGY TREATMENT AND TERMINATION OF HISTORIES ETRAN was originally developed to be used for applications at energies of a few MeV and lower, whereas EGS was originally designed for use in high-energy physics appli• cations. Thus, it is not surprising that, in general, ETRAN does a more careful job in the low-energy region than EGS (the default data set circulated with version 3 of EGS considered electron transport only above 1 MeV). However, the basic limitation of both codes in the low-energy region is their use of some cross-section data (in particular, for inelastic electron-electron scattering) which assume atomic electrons are unbound*. For

* Binding is accounted for in electron stopping powers in both codes and in the energy-loss straggling formalism used in ETRAN. 336 D. W. O. Rogers and A. F. Bielajew low-Z materials, the K-shell binding is less than 1 keY, and hence this approximation is quite good down to 10 keVor so. However, for high-Z materials, the K-shell binding is of the order of 100 keY, and hence one must be cautious below a few hundred keY. Both codes also sample the free-electron differential Compton scattering cross section (Klein-Nishina). ETRAN uses the total Compton cross section for bound electrons whereas EGS uses the free-electron cross section here also. The default version of EGS does not generate fluorescent photons or Auger electrons after a photo-electric event, nor does it consider K-shell vacancies after ionization by electrons. However, ETRAN considers all of these processes, at least using an approximate model which assigns all of the cross section to the highest Z element in the material being considered ( a rea• sonable approximation since the cross sections go roughly as ZS or Z·). The EGS4 distribution tape also includes a user written subroutine which allows fluorescent x-rays to be generated following photo-electric events.

"'8 5 t.J I I>. ",t".!:I -< 4 '00

...... - ~ 3 .::I CI) ::1 ;;::: ~2 rn o stars - EGS4/PRESTA "=' , solid histogram - CYLTRAN : "='CI) ..c 1 --I .... 1__ , o rn ..c dashed histogram - CYLTRAN - CSDA '--,___ --, ______as

0~ __L- __~ __~ __~ ___L __ _L __ ~ ___J __ ~~~

0.0 0.2 0.4 0.6 0.8 1.0 depth / rO

Figure 14.14. Comparison of calculations by EGS4/PRESTA and CYLTRAN of the depth-dose curve for a broad beam of 20-MeV electrons incident on a slab of lead. The dashed histogram shows the csda calculation done using CYLTRAN and including the energy deposition by bremsstrahlung.

The SANDYL codel1 (another code based on ETRAN), and the P-codes in the ITS system3 both go one step further and take into account the very complex atomic relaxation events which occur in the K, L, M and N shells each time a vacancy is created in an inner atomic shell. However, according to Halbleib and Melhorn: "in the vast majority of problems, the P-codes give results that are virtually identical to those of the standard codes"3.

The ETRAN system samples the initial angle of the electron which is ejected in a photo-electric event from the Sauter distribution for high energy interactions, and from the Fischer distribution for low-energy interactions (see e.g., Roy and Reed12). The EGS code assumes that these electrons are going in the same direction as the original photon since the distribution is forward peaked except for very low-energy photons, and in this case the multiple scattering of the electrons very quickly will dominate the effects of 14. A Comparison of EGS and ETRAN 337 the initial angular distribution. In an effort to explain the differences between ETRAN and EGS in calculating the dose delivered to a thin slab of LiF sandwiched between two lead slabs which were irradiated by a 6OCO beam (see Fig. 25.12 in Chapter 25 by Nahum), we have implemented a routine to be used with the EGS system to sample the angle for the photoelectron, also using the Sauter distributionl3. Including this effect had only a small impact on the results.

When an electron history is terminated in ETRAN because its energy has fallen below the threshold for transport (ECUT), the remaining energy of the electron is deposited at a point in the current direction of the electron at a distance which is randomly chosen between 0 and the residual range of the electron times DETOUR, a factor which takes into account that the practical range of a particle is much less than its residual range. In contrast, EGS deposits the residual energy on the spot. In cases in which the residual range at ECUT is comparable to the dimensions of some of the regions of interest in a problem, the ETRAN procedure can be a decided improvement.

The ETRAN /CYLTRAN cross-section package includes the Elwert correction factor in the bremsstrahlung cross sections. At energies of a few (~ 3) MeV or less, this fac• tor can lead to a significant increase in the cross section, and thus the cross sections used in EGS are considerably in error because they do not include this factor, How• ever, in cases in which the bremsstrahlung generated by low-energy electrons is being simulated, EGS already seriously overestimates the x-ray yield (see e.g., Fig. 13.11 in Chapter 13). Thus, there appear to be serious problems with the low-energy bremss• trahlung cross-sections in EGS, and for problems in which this is important, one should use the ETRAN system. However, in many applications this shortcoming is of little concern because, compared to higher-energy electrons, low-energy electrons create only a small fraction of the total bremsstrahlung.

The one low-energy area in which the default version of EGS4 is more complete than ETRAN concerns coherent (Rayleigh) scattering, which is not part of ETRAN. In this process, low-energy photons are ocassionally scattered elastically, predominantly in the forward direction. In many cases, this process has virtually no effect on the simulation. However, there are applications in which it can play an important role, for example, in simulating diagnostic x-rays, or in looking at the backscatter of low-energy photons from a high-Z material. EGS4 has an option to include this process in any simulation. It uses the approximation that all the atoms in a material act independently. This is an essential approximation if this effect is to be implemented for an arbitrary material - but it is known to be quite wrong in some important cases, in particular water, where the correlations between atoms at both the molecular and liquid levels have a dramatic effect on the process (see Johns and Yaffe 14 , and references therein). It would not be too difficult to introduce the correct form factors for an arbitrary material, but this is not currently a formal option in EGS4.

14.6 STEP SIZES AND BOUNDARY CROSSING

ETRAN and EGS use quite different approaches to selecting electron step sizes. In ETRAN, step sizes are chosen so that on average an electron's energy decreases by a constant fraction (default of 2-1 = 0.917 in the ITS system). For this step, the energy• loss straggling distribution is sampled. Within this major step, a series of n sub steps of equal length are taken. The value of n increases with Z, the default algorithm in the ITS system leading to values of 4 for aluminium and 16 for tungsten. The smaller steps are taken at higher Z values to ensure that the electron transport continues to 338 D. W. O. Rogers and A. F. Bielajew be accurate, since ETRAN does not employ a pathlength correction (see Chapter 5). Within each of these substeps, the multiple scattering is sampled and the cross sections for the production of knock-on electrons, bremsstrahlung photons and K-shell ionization are all sampled. Only in the case of bremsstrahlung production does the creation of a particle directly affect the primary's energy, the other processes already being included in the energy-loss straggling routines.

When an interface boundary is crossed in ETRAN, the electron's track is interupted at the boundary. Energy-loss straggling and creation of secondary particles are ac• counted for in the normal manner for the partial sub-step to the boundary. However, since the Goudsmit-Saunderson multiple-scattering distributions are pre-tabulated for fixed step-sizes, a Gaussian approximation is used for the multiple scattering in the partial sub-step.

In contrast to the above approach, EGS allows all physical processes and boundaries to affect the choice of step sizes. The step size is chosen as the minimum of the distance in the current direction to the nearest boundary, the randomly chosen distance to the next discrete interaction, and the maximum step size for which the multiple-scattering theory is valid. This procedure requires a multiple-scattering algorithm which can be applied to any step length while the calculations are being done. It is for this reason that the Goudsmit-Saunderson algorithm cannot be implemented within EGS unless further work is done to remove the restriction of using a set of pre-tabulated step lengths. Aside from the differences between class I and class II transport algorithms, the different step-size algorithms used in ETRAN and EGS do not seem to lead to different results.

14.7 SAMPLING PROCEDURES The EGS system prepares as much data as possible ahead of the actual simulation and then, during the simulation, it samples from the appropriate theoretical distributions using the current value of the particle's energy and material. In contrast to this, for many processes the ETRAN system pre-computes cumulative probability distributions (CPD) at a series of grid energies. These are interpolated in energy and/or angle as appropriate. For multiple-scattering, ETRAN samples the same distribution for all the multiple-scattering sub-steps within the larger energy-loss steps (which have a default of 8% energy loss in the ITS system). Often, sampling via a precomputed CPD is faster than sampling from the original distribution, depending on how fast the interpolation schemes are, or how complex the original distribution is.

14.8 TIMING Monte Carlo simulations can consume large amounts of computing time. Therefore, an important question concerns the efficiency of the calculations done by a particular code. The amount of time taken for any particular calculation is very dependent on many details concerning the calculation, and thus it is hard to compare the efficiency of two codes in any sort of general sense. Also, variance-reduction techniques can have very large effects on the efficiency of a particular calculation.

However, we have been running EGS4 (using the standard ESTEPE algorithm distributed with the code), ITS/CYLTRAN and EGS4/PRESTA on the same machine (a VAXll/780) for the same problems in order to prepare various comparisons, and we thought it would be useful to compare the running times. In Table 14.2, we present a summary of some results. 14. A Comparison of EGS and ETRAN 339

Table 14.2. Time per history for identical calculations done with EGS4 (using ESTEPE), EGS4 (using PRESTA) and CYLTRAN (the ITS double precision version for a VAX but with an in-line random number generator which saves about 15%). In all cases broad parallel beams of electrons were incident on slabs of material which were slightly thicker than the practical range of the beam.

CPU time per history, s Material Energy/MeV CYLTRAN EGS4/PRESTA EGS4/ESTEPE water 50.0 0.40 0.26 0.14(def) 20.0 0.29 0.21 0.22(4%) 10.0 0.24 0.18 0.19(4%) 1.0 0.14 0.12 0.35(1%) 0.100 0.13 0.091 0.34(1%) lead 20.0 1.67 0.50 5.0 1.2 0.77 0.500 0.62 0.48 0.100 0.24 0.25 uranium 1.0 0.74 1.35(0.3%) copper 10.0 0.54 0.35 1.0 0.39 0.33 beryllium 10.0 0.21 0.37 1.0 0.12 0.18 0.300 0.20 0.24 0.53(1%) aluminium 1.0 0.34 0.30 0.50(1%)

Rather than specify all the necessary details of each calculation, just let it be said that the parameters were chosen to make these codes run as fast as possible, consistent with getting the "right" answer with all the codes, but without worrying about optimization (so, for example, reasonable values of ECUT were used in all cases). All the cases studied were for broad beam electron depth-dose curves with reasonably good depth• resolution (typically 0.05 to 0.01 times the csda range), and thus lots of boundary crossing was involved (which slows down the PRESTA algorithm). In contrast, the broad beam geometry makes certain details of the calculation unimportant, and thus large ESTEPE values could be used. For calculations requiring more detail, smaller ESTEPE values would be necessary for the EGS/ESTEPE calculations which would thus take longer, whereas the other two calculations would be unchanged.

It appears hard to draw any conclusions from the results, except that CYLTRAN and EGS4/PRESTA run in comparable times for these examples, with EGS4/PRESTA 340 D. W. O. Rogers and A. F. Bielajew becoming more efficient at higher energies. This is because at higher energies, the geometric regions become physically large, and EGS4/PRESTA can use very large step sizes for the lower energy electrons in these problems. In general, the EGS4/PRESTA algorithm should gain in efficiency as boundary crossing plays a smaller role in the simulation, whereas EGS4/ESTEPE and CYLTRAN running times are not so strongly coupled to the size of the geometric regions.

We conclude by emphasizing the futility of trying to make a definitive statement about the relative efficiency of these codes. Firstly, recall that CYLTRAN must be run in double precision on a VAX which slows it down considerably (presumably almost a factor of two). On a machine in which both codes can be run in single precison, there would be a relative gain in efficiency for CYLTRAN. Secondly, the comparisons given are for only one type of calculation. CYLTRAN would definitely be more efficient at calculating bremsstrahlung spectra because the default version of the code has a good variance-reduction option built in, whereas the EGS code can only do an analog simulation of this process until someone designs an efficient way to introduce variance reduction into this part of the code. On the other hand, other forms of variance reduc• tion are more easily added to EGS than to ETRAN. Finally, the calculations were done with identical parameters such as ECUT, although in general, CYLTRAN can afford to be run with a higher value of ECUT than EGS because of its more sophisticated terminal handling of histories.

In summary, the codes have comparable efficiencies, at least in straight-forward geometries.

14.9 MISCELLANEOUS

This section gathers together a variety of other points of comparison between EGS and ETRAN.

In EGS, bremsstrahlung photons start off at a constant angle given (in radians) by the ratio of the electron rest mass to the total energy of the electron. ETRAN samples from an appropriate distribution in which the angle selected is correlated to the photon energy selected. In both cases, the primary electron's direction is not affected.

In a similar fashion, in EGS the angles of both the electron and positron created in pair production are equal and given (again in radians) by the ratio of the rest mass of the electron to the photon's energy. In fact, the mean angle of the electron's angular distribution is given by the ratio of the electron rest mass to the total energy of the electron given off (see, for example, Andersonl5). However, Wong et aIl6 have shown that using the more accurate formulation makes very little difference because, as the EGS4 manual points out, the effects of multiple scattering wash out the initial angular distribution effects, especially at lower energy where both effects are more pronounced. ETRAN samples the angles of the pair from the leading term of the high-energy Bethe• Heitler theory.

In both ETRAN and EGS, the distribution of energies between the pair-produced electron and positron is done by sampling from the appropriate Bethe-Heitler distribu• tion. For efficiency reasons at photon energies below 2.1 MeV, EGS assigns all of the energy to only one of the particles. 14. A Comparison of EGS and ETRAN 341

In EGS, the stopping powers and inelastic-scattering cross sections (M¢ller and Bhabha) for electrons and positrons are distinguished and positron annihilation in flight is taken into account, although the multiple-scattering formalism does not distinguish between the two charge states. No distinction is made in ETRAN between the two charge states except that positrons annihilate when they come to rest. For an initial beam of positrons, it is possible for ETRAN to utilize the appropriate positron cross section throughout the code (with the exception of in-flight annihilation). Both codes (at least as distributed) assume that the bremsstrahlung cross section for positrons is the same as that for electrons, despite the recent work of Berger et a1 11•1s which demonstrates that there are significant differences between the two - in particular the positron cross section is lower for lower energies and high-Z materials.

Acknowledgements: We would like to express our thanks to our colleague Ralph Nel• son for all his work to make the EGS system what it is, and to Stephen Seltzer and John Halbleib who, over the years, have patiently spent many long phone conversations explaining the subtleties (and sometimes the basics) of ETRAN and the ITS system. 342 D. W. O. Rogers and A. F. Bielajew

REFERENCES 1. M. J. Berger, "Monte Carlo Calculation of the Penetration and Diffusion of Fast Charged Particles" in Methods in Computational Physics, Vol. I, (Academic Press, New York, 1963) 135. 2. M. J. Berger and S. M. Seltzer, "ETRAN, Monte Carlo Code System for Electron and Photon Transport Through Extended Media" , ORNL Documentation for RSIC Compute Code Package CCC-107 (1973). 3. J. A. Halbleib and T. A. Melhorn, "ITS: The Integrated Tiger Series of Coupled Electron/Photon Monte Carlo Transport Codes.", Sandia National Laboratories report SAND 84-0073 (1984). 4. W. R. Nelson, H. Hirayama and D. W. O. Rogers. "The EGS4 Code System", Stanford Linear Accelerator report SLAC-265 (1985). 5. O. Blunck and S. Leisegang, "Zum energieverlust schneller Elektronen in dunnen Schichten", Z. Phys. 128 (1950) 500. 6. 1. Landau, "On the Energy Loss of Fast Particles By Ionization", J. Phys. USSR 8 (1944) 201. 7. D. W. O. Rogers and A. F. Bielajew, "Differences in Electron Depth-Dose Curves Calculated with EGS and ETRAN and Improved Energy-Range Relationships", Med. Phys. 13 (1986) 687. 8. M. J. Berger and S. M. Seltzer, "Tables of Energy Losses and Ranges of Electrons and Positrons", NASA report NASA SP-3012 (1964). 9. H. A. Bethe, "Moliere's Theory of Multiple Scattering", Phys. Rev. 89 (1953) 1256. 10. D. W. O. Rogers, "Low energy electron transport with EGS", Nucl. Instr. Meth. A227 (1984) 535. 11. H. M. Colbert, "SANDYL, A Computer Program for Calculating Combined Photon-Electron Transport in Complex Systems", Sandia Laboratories report SLL-74-0012 (1974). 12. R. R. Roy and R. D. Reed, Interactions of Photons and Leptons with Matter, (Academic Press, New York, 1968). 13. A. F. Bielajew and D. W. O. Rogers, "Photoelectron Angle Selection in the EGS4 Code System", NRC report PIRS-0058 (1986). 14. P. C. Johns and M. J. Yaffe, "Coherent Scatter in Diagnostic Radiology", Med. Phys. 10 (1983) 40. 15. D. W. Anderson, Absorption of Ionizing Radiation, (University Park Press, Bal• timore, 1984). ' 16. J. W. Wong, W. S. Ge, S. Monthofer and S. S. Hancock, "Spatial Distribution of Charged Particles Emitted by Pair Production", Med. Phys. 14 (1987) 474 (abstract ). 17. M. J. Berger and S. M. Seltzer, "Stopping Powers and Ranges of Electrons and Positrons", NBS report NBSIR-82-2550-A (1982). 18. 1. Kim, R. H. Pratt, S. M. Seltzer and M. J. Berger, "Ratio of Positron to Electron Bremsstrahlung Energy Loss: An Approximate Scaling Law", Phys. Rev. A33 (1986) 3002. Low-Energy Monte Carlo 15. Low-Energy Monte Carlo and W-Values

B. Grosswendt

Physikalisch-Technische Bundesanstalt D-3300 Braunschweig Federal Republic of Germany

15.1 INTRODUCTION Electrons in the low-energy range of about 1 keY or less play an important role in many fields of radiation research for two reasons: firstly, they are created in large numbers during the passage of all kinds of ionizing radiation through matter, and secondly, they have a linear energy transfer comparable to that of low-energy protons and a-particles, and accordingly they are responsible for the greater part of radiation damage observable in any material. A detailed understanding of the action of low-energy electrons in matter therefore is required in many contexts. In the fields of dosimetry, for example, the determination of the absorbed dose in water or the air kerma is of great practical importance, but in most experiments only the amount of ionization produced by secondary electrons within the sensitive volume of a dosimeter can be measured. The results of ionization measurements therefore must be converted to quantities based on energy absorption or energy transfer, either by calibration or numerically using an appropriate conversion factor. The most frequently used conversion factor is the so• called W-value, which is the mean energy required to produce an ion pair upon complete slowing down of a charged particle. Its relation to the primary particle kinetic energy T, and to the mean number Ni of ionizations produced (ionization yield), is given by

T W(T) = Ni(T) . (15.1)

Due to the fact that the ionization yield Ni(T) is directly proportional to the particle energy T for higher values of T, as will be seen later, the W-value is energy independent in the high-energy range. This is also the fact in the case of the differential value w(T) of the mean energy to produce an ion pair. It is defined as the quotient dT j dN of the mean energy dT lost by a charged particle of energy T in traversing a thin absorbing medium, and the mean number dN of ionizations produced when dT is completely dissipated in the medium. By simple mathematical manipulation, it can be shown using Eqn. 15.1 that w(T) = W(T)j[l- (TjW(T))(dWjdT)]. This implies that w(T) = W(T) in the high-energy range where the W-value is energy independent. Both quantities, W(T) as well as w(T), are therefore particularly well suited for converting quantities based on the measurement of ionization produced e.g., by secondary electrons of unknown energy distribution, into those based on absorbed energy. In the following,

345 346 B. Grosswendt the W-value, or the mean ionization yield, is discussed in relation to other more general transport properties of electrons such as the spatial distribution of energy transfer in matter. To do this, let us consider primary electrons of kinetic energy T emitted at the origin of a cylindrical coordinate system in the direction of its z-axis, and totally slowed down in a gaseous medium of volume V and density p. As a consequence of elastic and inelastic scattering events, a spatial energy transfer distribution d2 E(z, r) and an ionization number distribution d2 N(z, r) are created. If dE is the energy transferred to a mass element d(pV) = 27rrdrd(zp), the energy transfer distribution can be described by the energy 27r[dE/d(pV)]rdrd(zp) transferred at a depth z of a mass per area between zp and zp+d(zp) in cylindrical shells around the z-axis at a distance r of a mass per area between rp and rp + d(rp). Accordingly, the spatial distribution of ionization events can be described by 27r[dN/d(pV)]rdrd(zp).

After normalization of these spatial distributions to the number of source electrons, the integration with respect to rp yields the average spatial energy transfer or ionization number distributions, D.E(zp) and D.N(zp), per incident electron.

D.E(zp) = Lt~)] D.(zp) = 27rD.(zp) 100 [dfp~)] rdr , (15.2)

D.N(zp) = [d1~)]D.(zp) = 27rD.(zp) 100 [dt:v)J rdr . (15.3)

These distributions show the typical shape of depth-dose curves, with a maximum for small values of zp as compared with the electron mass range followed by a straight line segment of negative slope and a long tail for large values of zp.

The integration of the axial distributions of Eqn. 15.2 and Eqn. 15.3 within the limits -00 < z < 00 now leads to the primary electron energy T and the mean number Ni of ionizations produced during total electron slowing down:

(15.4)

(15.5)

Therefore, Eqn. 15.4 or Eqn. 15.5, combined with Eqn. 15.1, relates the W-value with the more general spatial energy transfer and ionization number distributions caused during particle slowing down. Because of the stochastic nature of radiation interactions however, the quantity Ni does not represent the actual ionization yield of a single electron, but only the expectation value of the probability distribution concerning the number of ionizations. 15. Low-Energy Monte Carlo and W-Values 347

Let P(T, j) be the probability that exactly j ionizations are produced upon the complete slowing down of the particle, with Ei P(T,j) = 1. If the summation runs over all possible values of j, then Ni follows from

00 Ni = LjP(T,j) . (15.6) ;=0

Eqn. 15.6 is a special case of the definition of the moments M" of P(T, j) for v = 1:

00 M" = Lj"P(T,j). (15.7) i=O

The variance (72 of the mean number of ionizations Ni is given by

00 (72 = LU - Ni )2 P(T,j) , (15.8) i=O which is a special case of the central moments m" of P(T,j) for v = 2 as defined by

00 m" = LU - NitP(T,j). (15.9) i=O

In the same sense, the spatial energy transfer and ionization number distributions of Eqn. 15.2 and Eqn. 15.3 can be interpreted as the expectation values of the spatial distribution of energy deposition, or ionization number production, with respect to the stochastically distributed spatial energy transfer or ionization points, often called the particle track or microdosimetric inchoate distribution. The complete description of a real particle track is, in principle, much more complicated and includes the spatial coordinates of each interaction point, the properties of the primary particle after the interaction, and also those of all secondary particles set in motion by the interaction.

For a more detailed discussion of this subject, let us again assume primary electrons of kinetic energy T entering a medium. The primary electrons collide with the molecules of the stopping medium, lose part of their kinetic energy, and at the same time produce excited or ionized molecules. The secondary electrons emitted by impact ionization also join the collision process and lead to a succession of spatially distributed energy transfer points in the form of an electron cascade.

The understanding of such a track structure, and of quantities such as the mean number Ni of ionization or the W-value, is part of the objective of electron degradation theory, and leads to a deeper insight into collision physics. This is especially true if not only expectation values are studied in detail but also statistical distributions as, for example, that of the number of ionizations. One approach to obtaining information on these subjects is the Monte Carlo simulation of the history of a great number of 348 B. Grosswendt primary electrons in different materials. In the following sections, typical features of a low-energy electron Monte Carlo transport model, and sets of cross sections necessary for performing the calculations, are presented. Some results concerning W values and the statistical fluctuation of the ionization yield are discussed afterwards.

15.2 LOW-ENERGY ELECTRON MONTE CARLO TRANSPORT MODEL For electrons in the energy range of less than a few keY, only a very limited number of inelastic interactions is necessary for complete particle slowing down. Hence, instead of using the continuous-slowing-down approximation (csda) as necessary for fast electrons, the history of each electron can be followed directly from one interaction point to another. The transport calculations therefore are usually performed on the assumption that the whole path length of an electron during its slowing down can be subdivided into a number of sublengths connecting successive points of interaction within the stopping medium. If there is no external electromagnetic field, these sublengths are assumed to be straight lines; otherwise, they are curved. At each point of interaction, the electron is scattered elastically or inelastically, and changes its direction or loses energy if an ionization or excitation event takes place. The term "excitation event" here implies all non-ionizing inelastic events, including dissociation of molecules. In the case of impact ionization, a secondary electron is set in motion which can also be scattered within the stopping medium, and therefore must be handled in the same way as the primary. In general, only single ionization is considered and the possibility of multiple ionization in a single electron collision is neglected. Photons emitted after impact excitation and electrons with kinetic energies below the minimum ionization threshold of the stopping material usually are assumed to be absorbed at their point of origin. Moreover, it is assumed that an excitation event does not change the direction of the primary electron. Because elastic collisions of electrons contribute negligibly to their slowing down, elastic scattering and the change of direction in impact excitation and ionization, of course, are of importance only if spatial energy transfer and ionization number distributions are to be studied. They can be neglected if only W-values and ionization probability distributions are of interest.

Bearing this in mind, the following cross section data are needed for performing W-value calculations: firstly, the cross sections a~~l(T) for the excitation of state n with energy loss kn by an electron of energy T; secondly, the cross section a1~(T) for the impact ionization with threshold 1m; and thirdly, the differential ionization cross section [da1~(T, E)/dEjdE describing the ionization impact with threshold 1m of an electron of energy T and energy loss between E and E + dE resulting in two electrons, one having the energy T - E and the other the energy E - 1m.

The effect of these cross sections on the number of ionizations produced upon total electron slowing down can be demonstrated in a simple manner by the following integral equation of the Fowler type for Ni(T) 1.

Ni(T) = Lpi~~(T)Ni(T - kn ) + LP~::!(T){ 1+ n m (15.10) 15. Low-Energy Monte Carlo and W-Values 349 where p~~l(T) = u~:l(T)lutot(T) is the probability of impact excitation to the state n, utot(T) is the total cross section for inelastic collisions, i.e., the sum of every ion• ization and excitation cross section for an electron of kinetic energy T, p~::!(T) = u!::! (T) I utot(T) is the probability of impact ionization with ionization threshold 1m, and [dq(m)(T,E)ldE] = [du!::!(T,E)ldE]/u!::!(T) is the differential probability of en• ergy loss between E and E + dE with respect to the ionization state with threshold energy 1m.

The right-hand side of Eqn. 15.10 enumerates the contributions to Ni(T), classi• fied according to the next inelastic collision that the electron of energy T undergoes. With the probability p!~l(T), this interaction is an impact excitation to the state n with energy loss k... Hence, the scattered electron will have the energy T - k .. , and the mean number of ionizations produced subsequently by this electron; including its secondaries, is Ni(T - k .. ). Therefore, the contribution of the excitation event to Ni(T) is p!~l(T)Ni(T - k .. ). With the probability p~::!(T), the next inelastic event is an ion• ization with threshold 1m, and the two resulting electrons of energy T - E and E - 1m supply the contribution to Ni(T) given by the integral term of Eqn. 15.10 multiplied by p~::!(T). The contribution of the ionization event itself is p~::!(T).

Any Monte Carlo procedure to determine Ni or W can be seen as a direct and very simple method to solve Eqn. 15.10. Its kernel is the sampling of the type of interaction for each electron of energy T and the determination of the corresponding energy loss using appropriate differential probability distributions. Fig. 15.1 shows a schematic flow diagram suitable for performing the calculation. After assignment of initial values for the primary electron energy T = To and the number of ionizations j = 0, the type of interaction is sampled from the probabilities p!~)(T), n= 1,2, ... for excitation, and p~::!(T), m = 1,2,... for ionization. In the case of an excitation event with energy loss k .. , the electron energy is diminished and the procedure is repeated with the new energy T - k ... If an ionization takes place, the number of ionizations is enlarged by 1; then the energy loss E is determined from the differential distribution, and the primary energy T is set equal to T - E. Now two possibilities with respect to E must be tested: (1) if E < 1m + I mi .. , which means that the energy of the secondary electron E - 1m is less than the smallest ionization threshold I mi .. of the stopping medium and cannot cause further ionizations, the history of the primary electron with energy T is continued and the whole procedure is repeated; (2) if the kinetic energy of the secondary exceeds Imi .. , it must be handled in the same way as the primary. Because of the fact that the secondary particle energy is less than that of the primary particle by definition, it comes to rest after a smaller number of inelastic interactions. The calculations with respect to the primary therefore are interrupted at this point after storing its energy, and the history ofthe secondary with energy T = E - 1m is followed as described above. If the energy is less than I mi .. , the history of the primary particle, or that of another secondary electron, is continued. This procedure is repeated until the primary particle energy as well as that of all secondaries is less than the smallest ionization threshold, Imi ... In this case, the number of ionizations is stored to allow peT, j) to be calculated. The whole procedure is repeated for Nmaz primary particle histories. To get a smooth probability distribution, Nmaz must be on the order of 5 X 105 • The use of such large values for Nmaz is necessary if central moments of higher order with respect to peT, j) are to be determined. This implies a practical upper electron energy of the order of 10 ke V for performing the simulations because of the necessary computer time. 350 B. Grosswendt

number of ioniza tions : j = 0 number of secondaries:; = 0 primary energy: T= To

storeP(To ,j)

normalization and output

Type of interaction nlnl (T) p.lml (T) ~exc Ion

calculate !----i_--ll energy loss ~~~ E

store T·(i)=T-E

Figure 15.1. Flow diagram for the Monte Carlo simulation of ionization yields.

15.3 INPUT CROSS SECTIONS

As should be clear from the last section, the task of calculating W-values and ionization number probability distributions is a very elementary one if all essential electron impact cross sections for the desired stoppirig medium are known with satisfactory accuracy (in this context, essential means all cross sections that contribute by more than about 98% to the stopping power at all electron energies). Unfortunately, such comprehensive data sets are seldom available and must be constructed from different sources, which, however, are often fragmentary in the range of the different variables involved, such as electron energy, and show systematic discrepancies with each other. A powerful procedure for testing the consistency of the cross-section data available, and to bridge possible gaps in the data, has been developed by Platzman2, and summarized by Fan03 and Inokuti et a1 4 • It is based on the fact that electron collision cross sections are closely related among one another and to other properties, and are subject to a number of general constraints. One constraint, which follows from the Bethe theory extensively discussed by Inokuti5, and Inokuti et a1 6 , is the asymptotic behaviour of cross sections 15. Low-Energy Monte Carlo and W-Values 351 with respect to excitation and ionization, as well as of the differential ionization cross sections at fixed energy loss E at higher values of T given by

abc u(T) = TInT + T + T2 + ... , (15.11 ) where u(T) stands for u •.,c(T), uion(T), and dUion(T, E)/dE, a, b, and c are constants. From Eqn. 15.11, it is obvious that a plot of Tu(T) as a function of In T should approach a straight line for higher values of T, and moreover should smoothly and monotonically approach zero at the threshold energy. This type of plot, the so-called Fano plot, therefore can be used to interpolate and extrapolate data more easily than the direct plot of u(T) versus T.

Another constraint is the E dependence of the differential ionization cross section dUion(T, E)/dE, which should be similar to that of the Bethe cross section in the case of low-energy transfer, the so-called glancing collisions5, and should approach that of the Mott cross section in the case of large energy transfer. The differential cross section therefore can be written in the following semi-empirical form:

dUion(T,E) =const. [R dJ 1 (4TRb(E)) dE T EdE n E2 + (15.12) 1 1 1)] ,

Comprehensive sets of electron cross sections constructed along the lines described above have been published for H2 by Gerhartl3 , for N e by Soongl4, for Ar by Eggarter15, and for H20 vapor by Paretzke and Bergerl6 . For liquid water, cross section sets have been derived, for example, by Hamm et al l7 and by Ritchie et al l8 . These data are com• pleted by cross-section sets published by Jain and Kharel9 for CO2, CO, H20, CH4, andNH3 , and by Green and coworkers for planetary gases He20- 22 , H22l-23, N22l,22,24-26, 0 22,25,26, 0 221,22,25-27, C022,28, C0222,29,30, and H20 vapor3l, as well as liquid water32. Ionization cross sections and secondary electron distributions for some additional gases, such as Ne, Ar, Kr, Ne, CH4, C2H2, and NO, have been published by Green and Sawada33. All these latter cross-section data are given by simple mathematical functions fitted to experimental and theoretical cross sections. Unfortunately, the distributions used by 352 B. Grosswendt

Green and Sawada have the wrong shape for slow secondary electrons, as pointed out by Kim9, and moreover exhibit a slightly different high-energy behaviour in the Fano plot than expected from the Bethe theory. Despite these short-comings, it is felt that the semi-empirical cross sections of Green and co-workers are of great importance for many low-energy Monte Carlo applications, first because the semi-empirical functions have been determined for the molecules of most importance for dosimetry and atmospheric radiation physics, and secondly, because the functions are very simple and therefore useful even on small computers. As an example, the differential ionization cross section of Green and Sawada is given by

2 dUion(T, E) = A(T) r (15.13) dE (E - Eo)2 + f2 ' where A(T), Eo, and r are parameters depending only on the primary electron energy T. The energy of the secondary electron E now follows from Eqn. 15.13 and a random number 1/ uniformly distributed between 0 and 1, using the conventional Monte Carlo sampling techniques:

(15.14) with Tm = (T - 1)/2; I is the ionization threshold.

15.4 RESULTS CONCERNING IONIZATION YIELDS Because of its importance in various fields of radiation physics, the literature on the mean number of ionizations or the W-value, determined both experimentally and the• oretically, is extensive, though not at all satisfactory from the point of view of en• ergy range covered or accuracy to which W is known. For a detailed survey, see the data collection of the International Commission on Radiation Units and Measurements (ICRU)34. Only part of the theoretically determined ionization yields has been derived by the Monte Carlo approach because it takes a great amount of computer time, and its numerical results cannot easily be translated into basic physical ideas; in contrast, analytical methods such as the use of the Fowler equation or the Spencer-Fano method, provide a deeper insight into, and a fuller understanding of, the general physics, but are not applicable to the solution of more complicated problems. Because of the scope of the present work however, only results obtained by the Monte Carlo procedure are discussed in the following. Numerical results concerning W-values or mean ionization yields for low-energy electrons in H2 have been published by Terrissol and Patau35 , Garvey and Green36, and Grosswendt37; for electrons in He by Terrissol et aZS8 , and in He and Ne by Grosswendt39; in Ar by Unnikrishnan and Prasad40 , Parikh4!, and Grosswendt39; in Kr and Xe by Dayashankar et al 42•43• Ionization yields in water vapor were calculated by Paretzke and Berger16, and by Terrissol et al 38, in liquid water by Turner et al 44 and by Paretzke et al 45 , in N 2 and air by Terrissol et al 38 and by Grosswendt and Waibel46. Moreover, some results are given for O2 and CO2 by Terrissol et al 38; for CH4 and methane based TE gas by Waibel and Grosswendt47 and Grosswendt48 .

To give an impression of some of these results, the energy dependence of the W• value for low-energy electrons in H2 is shown in Fig. 15.2. It can be seen from the figure that the energy dependence of the Monte Carlo results agrees quite well with measurements of Combecher49, whereas the absolute values are slightly different. The 15. Low-Energy Monte Carlo and W-Values 353 data of Garvey and Green36 are systematically higher than the experimental data in the whole energy region, in contrast to those of Grosswendt37 which are rather too low for energies T > 22 eV. The same holds for the results of Terrissol and Patau35• The structure of the energy dependence of the experimental W-value near 30 eV, however, is very similar to that of the Monte Carlo results of Grosswendt37, and might be seen as one example of the fact that the W-value in any case does not increase monotonically with decreasing electron energy. 150

eV

100

tw

50

ee -14* x 6i + + 0 0

OL-~~~~~LU~~-W~~ 101 102 103 eV 104 T---')_-- Figure 15.2. Energy dependence of the mean energy W-expanded per ion pair formed for electrons in H2: - experimental results of Combecher49, 0 Monte Carlo results of Garvey and Green36, + Monte Carlo data of Grosswendt37, 0 Monte Carlo results of Terrissol and Patau35 j !:::. theoretical data of Eggarter50 derived from the Fowler equation.

To demonstrate the quality of results determined by solving the Fowler equation in Fig. 15.2, the very recent results of Eggarter50 are also presented. They are systematically larger than the experimental data for energies < 60 eV, and smaller for energies> 450 eV, but show nearly the correct energy dependence. Despite its importance for a gen• eral understanding of radiation interaction (see, for example, the fundamental studies of Inokuti and co-workers51,52), only little information concerning the probability distri• butions P(T,j) for ionization can be found in the literature. Some results for electrons in Hz and the noble gases have been calculated by Grosswendt37,39, Unnikrishnan and 354 B. Grosswendt

Prasad40 , as well as by Dayashankar et a1 42•43 ; probability distributions in TE gas and water have been published by Grosswendt48 and Paretzke et a1 45 , respectively. To demonstrate the typical change of the curve shape of ionization number distributions with increasing electron energy, the probability distribution peT, j) is given in Fig. 15.3 in a three-dimensional plot for electrons in He39• The lower part of the figure showing the results for T < 600 eV clearly indicates the change of the curve shape from a Poisson-like distribution in the very low-energy region to that of a Gaussian• like distribution for higher energies. Moreover, a marked decrease of the maxima of the distributions accompanied by a simultaneous increase of their half-widths with in• creasing energy can immediately be seen in the figure. Similar results can be found for other stopping media. In Fig. 15.4, some probability distributions for electrons in liquid water and water vapor determined by Paretzke et a1 45 are compared with results for TE gas (volume fraction 64.4%CH4 , 32.4%C02 , 3.2%N2 )48. It can be seen that the results for TE gas agree very well with those for water vapor despite its very different atomic composition, whereas the mean values of the distributions are larger for liquid water than for water vapor, with the half widths at the same time being smaller. If the probability distributions peT, j) are known, the mean number Ni of ionizations according to Eqn. 15.6 and its variance, given by Eqn. 15.8 or higher moments M and central moments m according to Eqn. 15.7 and Eqn. 15.9, can be calculated.

0.4

I0.2

20 40 100 120 j ---1--

P(T,j)

j - Figure 15.3. Three-dimensional plot of the probability P(T,j) that exactly j ionizations are produced from the complete slowing down of electrons of energy T in He39• 15. Low-Energy Monte Carlo and W-Values 355

T=20eV 50eV 0 ~ 100eV t .5 ? 200eV P(T.j) 0 ! ? A~ li\ \ t \ ~5> : f: !~ 0 ~A! 111111 LLu llJiM1111~ 0 5 10 0 5 10 0 5 10 0 5 10 j Figure 15.4. Probability distribution P(T,j)- for electrons of energy T = 20 eV, 50 eV, 100 eV, and 200 eV totally slowed down in liquid water 0-----• and water vapor ~ ------45, as well as in methane-based TE gas (+ )48.

Mv(T) (T/evt 1

Figure 15.5. Energy dependence of the ratio M..,(T)/T'" of the moments M..,(T) and the primary energy T for electrons slowed down in TE gas48 j 0 derived from W measurements of Combecher49. 356 B. Grosswendt

In Fig. 15.5, the energy dependence of the ratio Mv(T)/Tv is shown for v = 1,2, ... ,5 for electrons totally slowed down in TE gas48 . The results for v = 1 are compared with the experimental data of Combecher. The agreement is very satisfac• tory. If one looks at the energy dependence, it follows from the figure that the quantity Mv(T)/Tv is first a function increasing with energy, apart from some oscillations in the low-energy region which might be due to wrong cross sections, and secondly, remains almost constant for higher energies. This means that Mv(T) is approximately propor• tional to T , an important result which was pointed out by Inokuti and co-workers&l·52 from basic considerations, and which could also be shown for other gases37.39 . For v = 1, this constant is equal to the reciprocal of the high-energy W-value, i.e., the W-value for energies above about 1 keY.

In Fig. 15.6, the energy dependence of the central moments mv(T), v = 2,3, ... ,5 is also shown for TE gas. For energies T > 100 eV, this energy dependence can be described by a power law mv(T) = const. TP, with powers p R:J v /2 if v is an even number, and p R:J (v -1)/2 if v is an odd number. Note that for electrons slowed down in TE gas, the central moments with v= 3 or 5 are negative; hence, the values of m3(T) and ms(T) are shown in the figure instead of the moments themselves. For the special case v = 2, the central moment m2(T) is the variance of Ni(T) which, according to the early work of FanoS1, should be proportional to the mean number of ionizations 0-2 = F Ni , where F, the so-called Fano factor, is a constant of proportionality of the order of magnitude of unity and approaches a constant value for higher electron energies. The constant with respect to the power law of the central moment for v = 2 therefore is equal to the high-energy value of F/W. Monte Carlo results on the Fano factor have been published e.g .. for electrons in H2, the noble gases, H20, TE gas, CO2, CH4, and an Ar - CH4 gas mixture by different authors37.39-43.4s.48.53. As an example, in Fig. 15.7 the energy dependence of the Fano factor for electrons in TE gas48 is compared with that of liquid water and water vapor of Paretzke et a1 45 • Whereas the Fano factors of TE gas and water vapor agree very satisfactorily in the whole energy range studied, the Fano factor is systematically smaller in the liquid, except at the lowest energies.

15.5 CONCLUSION

It can be stated that most of the numerical data concerning statistical-yield fluctuations and Fano factors known at present, in particular if one considers the generalized Fano factors introduced by Inokuti et al52 , have been obtained by the Monte Carlo method. If the necessary cross sections are known with satisfactory accuracy, the Monte Carlo method, because of its simplicity, can be seen to be a very powerful tool for deriving information on W-values, or what is the same, on ionization yields and their statistical fluctuations. 15. Low-Energy Monte Carlo and W-Values 357

Figure 15.6. Energy dependence of central the moments mv(T) according to Eqn. 15.9 for electrons totally stopped in TE gas48 •

,, \, ~, \

o

F ~ water vapor ~L YEgas -tr'rr-e--Jo--o--+ liquid water

Figure 15.7. Energy dependence of the Fano factor F for electrons slowed down in liquid water - , and water vapor + - - + ,45, as well as for methane-based TE gas 0 48• 358 B. Grosswendt

REFERENCES 1. M. Inokuti, "Ionization Yields in Gases under Electron Irradiation", Radiat. Res. 64 (1976) 6. 2. R. L. Platzman, "Energy Spectrum of Primary Activations in the Action of Ionizing Radiation", in Radiation Research, edited by G. Silini (North-Holland Publishing Comp., Amsterdam, 1966) 20. 3. U. Fano, "Platzman's Analysis of the Delivery of Radiation Energy to Molecules", Radiat. Res. 64 (1975) 217. 4. M. Inokuti, D. A. Douthat, A. R. P. Rau, "Degradation Spectra and Ionization Yields of Electrons in Gases," Proc. 5th Symp. on Microdosim. (1975) 977. 5. M. Inokuti, "Inelastic Collisions of Fast Charged Particles with Atoms and Molecules - The Bethe Theory Revisited", Rev. Mod. Phys. 43 (1971) 297. 6. M. Inokuti, Y. Itikawa, J. E. Turner, "Addenda: Inelastic Collisions of Fast Charged Particles with Atoms and Molecules - The Bethe Theory Revisited", Rev. Med. Phys. 50 (1978) 23. 7. Y.-K. Kim, "Angular Distributions of Secondary Electrons in the Dipole Ap• proximation", Phys. Rev. A6 (1972) 666. 8. Y.-K. Kim, "Energy Distribution of Secondary Electrons. I. Consistency of Ex• perimental Data", Radiat. Res. 61 (1975) 21. 9. Y.-K. Kim, "Energy Distribution of Secondary Electrons. II. Normalization and Extrapolation of Experimental Data", Radiat. Res. 64 (1975) 205. 10. Y.-K. Kim, "Energy Distribution of Secondary Electrons", Radiat. Res. 64 (1975) 96. 11. Y.-K. Kim, T. Noguchi, "Secondary Electrons Ejected by Protons and Elec• trons", Int. J. Radiat. Phys. Chern. 7 (1975) 77. 12. H. C. Tuckwell, Y.-K. Kim, "Effects of Partial Cross Sections on the Energy Distribution of Slow Secondary Electrons" , J. Chern. Phys. 64 (1976) 333. 13. D. E. Gerhart, "Comprehensive Optical and Collision Data for Radiation Action. I. H2", J. Chern. Phys. 62 (1975) 821. 14. S. C. Soong, "Inner-Shell Contributions to Electron Degradation Spectra", Ra• diat. Res. 67 (1976) 187. 15. E. Eggarter, "Comprehensive Optical and Collision Data for Radiation Action. II. Ar", J. Chern. Phys. 62 (1975) 833. 16. H. G. Paretzke, M. J. Berger, "Stopping Power and Energy Degradation for Electrons in Water Vapor", Proc. 6th Symp. on Microdosim. (1978) 749. 17. R. N. Hamm, H. A. Wright, R. H. Ritchie, J. E. Turner, J. P. Turner, "Monte Carlo Calculation of Transport of Electrons through Liquid Water", Proc. 5th Symp. on Microdosim. (1975) 1037. 18. R. H. Ritchie, R. N. Hamm, J. E. Turner, H. A. Wright, "The Interaction of Swift Electrons with Liquid Water", Proc. 6th Symp. on Microdosim. (1978) 345.

19. D. K. Jain, S. P. Khare, "Ionizing Collisions of Electrons with CO2 , CO, H20, CH" and NH", J. Phys. B: Atom. Molec. Phys. 9 (1976) 1429. 20. A. T. Jusick, C. E. Watson, L. R. Peterson, A. E. S. Green, "Electron Impact Cross Sections for Atmospheric Species. 1. Helium", J. Geophys. Res. 72 (1967) 3943. 15. Low-Energy Monte Carlo and W-Values 359

21. 1. R. Peterson, A. E. S. Green, "The Relation between Ionization Yields, Cross Sections and Loss Functions", J. Phys. B (Proc. Phys. Soc.) Ser. 2,1 (1968) 1131. 22. C. H. Jackman, R. H. Garvey, A. E. S. Green, "Electron Impact on Atmospheric Gases 1. Updated Cross Sections", J. Geophys. Res. 82 (1977) 5081. 23. W. T. Miles, R. Thompson, A. E. S. Green, "Electron Impact Cross Sections and Energy Deposition in Molecular Hydrogen", J. Appl. Phys. 43 (1972) 678. 24. R. S. Stolarski, V. A. Dulock, C. E. Watson, A. E. S. Green, "Electron Impact Cross Sections for Atmospheric Species. 2. Molecular Nitrogen", J. Geophys. Res. 72 (1967) 3953. 25. L. R. Peterson, S. S. Prasad, A. E. S. Green, "Semi-empirical Electron Impact Cross Sections for Atmospheric Gases", Can. J. Chern. 47 (1969) 1774. 26. A. E. S. Green, R. S. Stolarski, "Analytic Models of Electron Impact Excitation Cross Sections", J. Atmosph. Terr. Phys. 34 (1972) 1703. 27. C. E. Watson, V. A. Dulock, R. S. Stolarski, A. E. S. Green, "Electron Impact Cross Sections for Atmospheric Species. 3. Molecular Oxygen", J. Geophys. Res. 72 (1967) 3961. 28. T. Sawada, D. L. Sellin, A. E. S. Green, "Electron Impact Excitation Cross Sections and Energy Degradation in CO", J. Geophys. Res. 77 (1972) 4819.

29. D. J. Strickland, A. E. S. Green, "Electron Impact Cross Sections for CO 2'', J. Geophys. Res. 74 (1969) 6415. 30. T. Sawada, D. J. Strickland, A. E. S. Green, "Electron Energy Deposition in CO2", J. Geophys. Res. 77 (1972) 4812. 31. J. J. Olivero, R. W. Stagat, A. E. S. Green, "Electron Deposition in Water Vapor, with Atmospheric Applications", J. Geophys. Res. 77 (1972) 4797. 32. G. J. Kutscher, A. E. S. Green, "A Model for Energy Deposition in Liquid Water" , Radiat. Res. 67 (1976) 408. 33. A. E. S. Green, T. Sawada, "Ionization Cross Sections and Secondary Electron Distributions", J. Atmos. Terr. Phys. 34 (1972) 1719. 34. ICRU, Report 31, "Average Energy Required to Produce an Ion Pair", Inter• national Commission on Radiation Units and Measurements, Washington D.C., (1979). 35. M. Terrissol, J. P. Patau, "Simulation du transport d'electrons d'energie inferieure a un keY par une methode de Monte-Carlo", Proc. 4th Symp. on Microdosim. (1973) 717. 36. R. H. Garvey, A. E. S. Green, "Energy-Apportionment Techniques Based upon Detailed Atomic Cross Sections", Phys. Rev. A14 (1976) 946. 37. B. Grosswendt, "Determination of Statistical Fluctuations in the Ionization Yield of Low Energetic Electrons in Hydrogen", Nucl. Instr. Meth. 198 (1982) 403. 38. M. Terrissol, J. Fourmenty, J. P. Patau, "Determination theorique des fonctions microdosimetriqes pour des electrons de basse energie dans les gaz", Proc. 5th Symp. on Microdos. (1975) 393. 39. B. Grosswendt, "Statistical Fluctuations of the Ionization Yield of Low-energy Electrons in He, Ne and Ar", J. Phys. B: Atom. Mol. Phys. 17 (1984) 139l. 40. K. Unnikrishnan, M. A. Prasad, "Energy Deposition by Electrons in Argon", Radiat. Res. 80 (1979) 225. 360 B. Grosswendt

41. M. Parikh, "Energetic Electron Degradation Spectra and Initial Yields in Argon", J. Chem. Phys. 73 (1980) 93. 42. Dayashankar, M. A. Prasad, K. Unnikrishnan, "Energy Degradation of Electrons in Krypton", Phys. Lett. 90A (1982) 402. 43. Dayashankar, K. Unnikrishnan, "Ionization-Yield Fluctuations in Xenon Due to Energy Degradation of Electrons", Phys. Lett. 99A (1983) 81. 44. J. E. Turner, H. G. Paretzke, R. N. Hamm, H. A. Wright, R. H. Ritchie, "Com• parative Study of Electron Energy Deposition and Yields in Water in the Liquid and Vapor Phases", Radiat. Res. 92 (1982) 47. 45. H. G. Paretzke, J. E. Turner, R. N. Hamm, H. A. Wright, R. H. Ritchie, "Cal• culated Yields and Fluctuations for Electron Degradation in Liquid Water and Water Vapor", J. Chem. Phys. 84 (1986) 3182. 46. B. Grosswendt, E. Waibel, "Transport of Low Energy Electrons in Nitrogen and Air", Nucl. Instr. Meth. 155 (1978) 145. 47. E. Waibel, B. Grosswendt, "Spatial Energy Dissipation Profiles, W values, Back• scatter Coefficients, and Ranges for Low-Energy Electrons in Methane", Nucl. Instr. Meth. 211 (1983) 487. 48. B. Grosswendt, "Degradation Spectra and Statistical Ionization Yield Fluctua• tions for Low-Energy Electrons in TE Gas", Proc. 8th Symp. on Microdosim. (1982) 165. 49. D. Combecher, "Measurement of W values of Low-Energy Electrons in Several Gases", Radiat. Res. 84 (1980) 189. 50. E. Eggarter, "Theory of Initial Yields of Ions Generated by Electrons in Binary Mixtures", J. Chem. Phys. 84 (1986) 6123. 51. A. R. P. Rau, M. Inokuti, D. A. Douthat, "Variational Treatment of Electron Degradation and Yields of Initial Molecular Species", Phys. Rev. A18 (1978) 971. 52. M. Inokuti, D. A. Douthat, A. R. P. Rau, "Statistical Fluctuations in the Ion• ization Yield and Their Relation to the Electron Degradation Spectrum", Phys. Rev. A22 (1980) 445. 53. B. Grosswendt, E. Waibel, "Statistical Ionization Yield Fluctuations and Deter• mination of Spatial Ionization and Energy Absorption for Low Energy Electrons" , Radiat. Prot. Dosim. 13 (1985) 95. 16. Electron Track Simulation For Microdosimetry

Akira Ito

Cyclotron Laboratory The Institute of Medical Science The University of Tokyo 4-6-1 Shirokanedai, Minato-ku, Tokyo, 108 Japan

16.1 INTRODUCTION Microdosimetry is the study of the primary physical processes of spatial and temporal distribution of energy deposition in biological targets, and the correlation of this with radiobiological effects. Theoretical and experimental approaches have been developed by many pioneers. Recently, emphasis has been placed on the basic understanding of the physical tracks1- •• In particular, the importance of the low-energy electron or delta ray to a biological target has been recognized. As the range of the low-energy electron is short, the size of the biological target, and the end effect in question, should clearly be stated. The dosimetric and micro dosimetric concepts and quantities applicable for describing the biological responses should be chosen carefully.

For example, to study the radiobiological effects of tritium beta rays which are important for the risk assessment of future fusion reactors, there are at least three approaches. First, the macro- or average-absorbed dose (in Gy) may be used when the concentration of tritium in the specimen is known (in Bq/l). The absorbed dose is the first approximation quantity for radiation effects. It does not, however, always predict the biological responses well, especially in terms of Relative Biological Effectiveness (RBE). Second, as the cellular distribution of tritium is important, the average dose concept is no longer applicable for describing the short-range effects of tritium beta rays (f'V 1 Jlm). The distribution of tritium in a cell, and the microscopic dose distribution, should be studied, taking the range distribution of tritium beta rays into account. The evaluation of RBE has been attempted, based on measurements of the energy-deposition spectra in the micrometer sphere and following the theory of dual radiation actions. Although the RBE values of tritium beta rays to reference radiations are calculated, they do not always agree with experimental results; also, the model does not explain the radiobiological mechanism behind the RBE values. And third, to understand the basic mechanism of radiation action on the DNA and other macromolecules, it is essential to know the spatial distribution of individual ionization and excitation events along the electron track at the nanometer level, to consider the physicochemical processes of production, diffusion and decay of the reactive chemical species, and also to take the structure and sensitivity of the molecular target into consideration.

361 362 A. Ito

The electron track simulation method is most appropriate for the third approach. In this chapter, the outline of the simulation task is drawn. The electron collision cross-section data set is described, as this is essential information in simulation work. Then, my own electron track simulation Monte Carlo code (ETRACK) is described. The results of the electron track simulation are discussed in two ways: 1) deduction of basic physical quantities, and 2) assessment of radiobiological effects.

16.2 OUTLINE OF THE ELECTRON TRACK SIMULATION The spatial distribution of the ionization and excitation events along a radiation track, called the track structure, is the basic quantity in characterizing the physical aspect of radiation effects. To simulate the track structure on the computer, basic information on the electron cross section, an electron transport model, and the initial electron en• ergy spectrum in water are needed. Fig. 16.1 illustrates the flow and relationship of such information. Among others, the evaluation of the electron collision cross section in water has basic importance. We need an accurate, comprehensive set of cross-section data for track simulation to be used for microdosimetry. The electron transport model could be very precise if all the known atomic-collision types are included. However, the main interaction types pertinent to track simulation for microdosimetry are exci• tation, ionization and subsequent secondary electron production. Again, an accurate and manageable transport model should be constructed. The primary electron energy spectrum must be provided for the radiation in question. For monoenergetic electrons and beta-ray spectra, it is readily available. Photons, such as 6OCO gamma rays and 200-300 kVp x-rays are used as the reference radiation. Their initial secondary electron energy spectra in water should be calculated prior to the electron track simulation. Photon transport can be performed accurately since the basic interaction types are photoelectric and incoherent (Compton) scattering, and their cross sections are rela• tively well-known. Track simulation for charged particles such as protons, heavy ions, or negative pi-mesons, is more complicated. Fast neutrons involve the additional step of neutron-charged particle interactions.

Given the cross-section data for water, electron-transport model and the primary electron energy spectrum, the Monte Carlo program can simulate the electron track on the computer using a random number generator as an essential tool. Thus, it also is very important to use a well-tested random number generator in order to simulate the truly stochastic processes of energy deposition. The resulting track-structure information can be stored on a magnetic medium, called the track-structure file, preferably for repeated use by the application programs.

16.3 EVALUATION OF THE ELECTRON CROSS SECTION The Monte Carlo program requires a comprehensive set of electron interaction cross• section data and related information in the energy region between 10 eVand 1 MeV. We use the inelastic cross-section data compiled by Paretzke and Berger6- S• They cross-compared inelastic cross-section data of their own evaluation in the energy region between 10 eVand 10 keY. This data was for water vapor rather than for liqiud water on which the data are very limited. The general agreement was fairly good; hence, they concluded that many interesting electron energy degradation quantities can be calculated, to an acceptable accuracy, using these cross-section data (See Refs. 6-8 for further details.). The elastic scattering cross section was calculated from the equation given in the textbook by Mott and Massay9. The energy spectrum of secondary electrons (delta rays) was calculated from the equation by Mfilller9 , after adjustment of the total inelastic cross section to be consistent with Paretzke's data. 16. Electron Track Simulation for Microdosimetry 363

',....------/k"'~ Evaluation of Cross Section

Plot of Cross Section

Check Results Track Structurer--___-' Fi Ie

Output Electron Basic Application Track Analysis Programs

Figure 16.1. General flow of the electron track simulation by the Monte Carlo method. The evaluated electron cross-section data, a correct electron• transport model, the electron source spectrum, and a good random number generator are the key requirements. The resulting track-structure data are stored on a file for subsequent analysis.

Fig. 16.2 shows evaluated electron cross-section data between 10 eV and 100 keY. For the convenience of the track simulation, the ionization cross section was divided into three parts, namely, soft ionization, hard ionization with delta-ray energy above 12.6 eV, and the oxygen inner shell (Ols) ionization that yields the Auger electron (523 eV). With this classification, the slowing-down electrons will be calculated down to the cutoff energy of12.6 e V (first ionization potential of H20). The right-side ordinate of Fig. 16.2 is the mean free path (in nm) of the electron, which is used directly to decide the interaction distance. In the Monte Carlo program, the type of interaction is chosen from the fractional cross-section table at the given energy shown in Fig. 16.3. In the case where the elastic scattering of the electron with the atom is not important, it is possible to ignore this portion; a total inelastic cross section can instead be used. The fractional cross section is then re-plotted to exclude elastic scattering. Energy is transferred at the point of interaction; this energy consists of atomic potential energy and, in the case of ionization, kinetic energy of the secondary electron. 364 A. Ito

105~~~~~~~~~~~~~~~..= ....=... .~. .. ~... ,~ .... ~....~ .,.~.... ~.. 1~2

r-.-=.. -=.. " ... -' ..;.c •.:. •.•"" ., f.... -!..• c- ..±. -!., -!..•-! .•+ .• I 1~ I ::::::::::?::::'!::::?::;:!::;:!:!

n ~~~~~~~~~~~~~~~rH~ __ +--+-+~f+~-::-:: -:::-::~:; -:::-::~:: -;::~::"'::~f-:~f-.::f-~:~~::~lO 0 \... Q) .::::::: ::~:::::r ::~::~ :~:r ~:; .... •.•.•.•• "1"" .~ •••• ~ •• ~ .~ •• ~. :': Q) n \... 0 Q) '"\... 0 Q) ..... :::1lHljjj -0::1 o ...... -= 10 2 ~~~~~==~~~~r.=±~~~'li~~~~~~ 101 C o U c c U o ...... s::::...... o Q) n.- (f) " 1::::t.:::±:::±±::i:±±=±:::-:±:::i:±-±::±:I=.:::±::::±~~!:iP.~ri::-::~~~ 10 2 ~ ~ 10 1 :::. :::::i:::::r::; :n:::n: ::::::::::::::T:i::n:i:;T ;:::::::::::::::rru:::;:: .. L o ,···· ..... ·;.···;··}·..:._I_:·:· ...... •. : ...... :. ... : •• .:. •..:.. :.:.: ••••••.•• ••;; •.••••••••••• ~ ..... :.I.I LL L ...... :...... :.. ... ' .~ ..:..l.:.: ...... i...... :. ... =.. ~.~.i.~.i ...... i ..... ;.... i •. ..:. . .:..:.:.: U -.- ······i·····.;··· .. ; .~.i.i.i ...... i·····~···i··~·+·;· ;·~- ······-··-;-·-·-~····i··+·.;-i-i-;

j 10 0 t::::::t:::·:::.:::±:i::::::::±:-!:t: ·~1±1+i!1~1f::- :::::::::::::::::±:i:::::::::±:-r:±I::i:I ±!±:!!±!~-··::::···::::··:~:I:::::::::±:t :±1::i:l::i:1~!j~1 .17::::-=::::i:::±::-i~d 10 3 ::: : :::::!:::::~::: ::~ :~:!. ;:!: :::: ::::::~:::::~:::~::~: ~:lt~' ...... :: ::~: :::: ~ ::::~:: ~:~:~: ~ :; ::::: ::: ::;:::: :::::::!::!:r.:t:::: ::: ::::::i:::::t::: ::~:rnT :::::':::T::::r::i::t:Y:iIr ::::::::::i::::: ~ :::: i ::!:!: ~ :;: ~ :::::::::t:::t:::t::t:t:t:U ... -" j. , •• ,.(,., •••••, ·.)·j·;·i· ...... j ••••• .Q. ••• j ··~·.Q.·i .;. i ...... j .•.. -c···· i·. ~.i.j -i.j -...... ~ .. ,- ,!, ... ~ -, i'~' '~'H ··ITllll!:iJl:!ll11-1-tj:1HlI::Ef:!ttl~ 1~1~~~~~-=~~±±++-=±-~~~~~~~~10ij :: :::::::~::: ::±:: .:::::!::: ::: ::::::::::~:::::t::~:: tt; .::~: :::: ::::::~ ::::: ~:: :l::t ±:l:~: 1 ., ••••••• j ••••• .(,. •• j···j·O·j·(·j· ···· · ···· i ····-;··· i ·.~·: ·(·i ··.·i ..... {.... i .. ';.(..i.i.i .. ·.. ····~·····t·· ~···~·t·i·~·i· ..... ·.... i..... ! .. ·:··t.·~ :'j'i' ··········:·····i····:··t·t·~·~· i .. ·······;·····r· ~··1·TT~T ····· .. ··T····;-·T·r 'rn ·········r .. ··r··T·~·T"~·~·j ...... ~ ..... ! .. :··~·t·~·:':· ····-·· .. ·:·····t···~··t·· .~.~.~. · '·'·'··' · ;·"··r···~··t·t·l·l·l : ! ~ : : : : : ; ;: .. . : : : : : : ·······!·····t .. i···~·t·~tl· ··········~·· · .. f ·.. ~··t· ...... ~... +·t'i'~·! lcr2~ __ ~i ~.i ~j~i ~' ~' wq~~~i __ ~' ~i~i ~~~-J' __ ~~i ~i ~i ~ii~~ __ ~~~~ 10 1 102 lOY El ect ron energy (eVl Figure 16.2. Evaluated cross section of the electron in water vapor. Most of the data is taken from Paretzke and Berger6 • The interaction type is classified into elastic (+), total excitation (*), total ionization (x) of which delta-ray (above 12.6 eV) creation (.6.) and Auger electron creation (0) are separately evaluated. The right-side of the ordinate is the reciprocal of the cross section, or the mean free path of the electron. 16. Electron Track Simulation for Microdosimetry 365

Fig. 16.4 illustrates the partition of the mean inelastic energy transfer between the various modes, i.e., the excitation event, the atomic potential, and the kinetic energy of the secondary electrons. Part of the atomic potential energy given to the 1s state (K shell) of oxygen is eventually released as an Auger electron (with Auger yield = 0.994). Thus, partition to the Auger electron is plotted separately. The mean kinetic energy of the secondary electron is partitioned into soft ionization and hard ionization. Soft ionization refers to secondary electrons which have kinetic energy less than a given cutoff energy (12.6 eV, first ionization potential energy), and thus cannot generate

IO Or-~~~~~~~~~~~;~; ~. ~, ~r-~, ~~.~.~..~ ..

90 ::·::::: :.F.:i.~Fm j... ~~Y .. :: ~~~~ ~ ·:. ..~ t ~~.. Y .· ·r. '~.·:!.·i.· ~~r... ? j~~ t:ltl' i j ~ j ~ ~ ! j ...... ~ ... ··t···i···~ -t-~.~.~ ...... ,: .. .,.~., .~ .. 7'7 -;' ;.:, ,.. ,... ,.. :.... ,j'" ,~ .. t· t';'j '1 ,... ,... ,.~ -.. "i""~" t,;, '~'1' 1 :::! 80 ,,, , ,,,,,~,,,,,~,,, ~ ,,,~,~,;,~,:,,,, ····-!-····+··-!--f·t·j·j·j --··-·····f·····~-·,·!··+·}-l,i·~ ··········~·····i,,,·}·,f·~,,~·l·; ...... L... i ... L.LLUJ ...... i." .. ~, .. i .. ~.L,Li, ..... "... ~ ...,.~",.L.l.L,U ,.. ,.... "L".;., .. ~"l .t.u.~

~ 70 '::::::'I:::rrH:,I, ::::::J ~rr::il, :fI. · ~:: .:r .. r.:,.Ir:...:I.:::.::T:.::l..T.r:I:.II! J: 60 ...... + .... ;.. i···~·H+·. ··· .. ·· .. ·j· .... t···H·t·H+ ·· .. ····j·····i .. ·j·H·H-j ...... :.H.++.l.~.F~ ~ ."+rH'iTi ...... i··t .. :··tTl'Ii· ...... L.... i.... ,.. ;.;.U., ...... : : : : : : : ' oSOr-~~~~+-~-+~~~~~~~~--~~~~ G ..~ .. :... :.:.:.:...... !. .. L.i..Lli.ii, , : f :k 11 ott o~ : ~;; i ~ l qo ··::::t::r:.t:tttJ:'. ·:"::"::::::l:::i:·n:iJr :::::.::::'JJ.fH" ::::·::··i:··IlIr{.:l :s 30 .. H.. H-+t++HH ...... i ..... ~ ... i.. ;.; '. :.'...... ····:····L. .. !··t·~·i.i.! ...... ! .. LH·HH ., l- LL 20

10

O~~~~~~~~~~·· ~· ~~·~· --~~~~~~uw 10 1 102 103 El ec j ron. ener gy Figure 16.3. Plot of the fractional electron cross section in water vapor derived from the absolute cross-section data in Fig. 16.2. The electron simulation Monte Carlo program (ETRACK) uses this table directly to determine the interaction type.

another delta ray. Vvith hard ionization, the secondary electron has gained more than 12.6 eV, and thus may create another delta ray. To separate the portion of the kinetic energy for soft ionization and hard ionization, the secondary electron energy is evaluated from the M¢ller equation in such a way that the mean energy transfer is consistent with Paretzke's data6 . When hard ionization takes place, the random delta-ray energy is taken from the pre-calculated integral energy spectrum for the incident energy. The angular distribution of the primary electron and delta rays is not well known, so it is simply calculated from their momentum ratio, assuming that the collision is with a free electron. 366 A. Ito

:;: Q)

L• Q) ~~Or-~~~~r-~~~~r-~~~~+-~~~~ ~30~::: _···:_j~::: ~J_::I~j~,I.:I::~!I~:·:_::::_::: ~!: ~1_::: ~II~:I~i ~fi ~::'~':: _::':~! ~:::~::::~I: ::~!:!~I~i '~'· _::::_··::~i ~J~:·::~I ~!:f~j~11 c " , '"'' ; ; ; ; ;;;; ...... ~ :::::::::f::rFITlll·· ......

i_ 20 f3]']']'1' .~..~~~~~~~~ ..... ~. ~: , ······· ~'···Ru~r"5··(~"··~r-·I·fH+h:~·-t·Hffi::: ---_ .. --.. : .....:. __.; ..~ .l.:_:.i ...... ~ .....~ ... ~_.~ . ~ ..~_~_~ " ...... ,i..... ~",i .. ~,;.i.i, i ···,······.· .... t .... i··.;.·'O'·j·i ·i ..... · ··}-····,····}··~·C · ·}·H i ~!!!! it : ~ ! ! l i l i ~;; ;;; p : :::':: : : :;:: ::: ·.. ,····+····f .. ·i .. ·; :,H+ ...... " ; ::::::, ' "'--":"--':'''';''1'; ':';'; ; 1:: l::: ::::::::.:::::·:f' ::;.. l.r:::::: ,.:::: Skf:j! l~i~t i :~~ ::: : i~k n 1:~ ~ :: ~yjL::Ltt::tn ...... -.. ; ···t·· ··j··1·t,j,H- ········,·!·····t···I··t' t·l · ~'j ··········I·····;····)··t·t·'·I·I·········1····,;.·+·t· 1,1,1"[ 102 103 10~ lOS EI ec1 ron energy (eV)

Figure 16.4. Mean inelastic energy transfer at a collision point as a function of the electron kinetic energy. This is further partitioned into excitation potential energy, Auger electron, soft collision and hard collision. (See text for details.)

16.4 DESCRIPTION OF AN ELECTRON TRACK SIMULATION Monte Carlo PROGRAM (ETRACK)

The general flow of the electron track simulation Monte Carlo program (ETRACK) is illustrated in Fig. 16.5. After reading in the basic physical data as described previously, an electron is generated and its track is followed. First, the interaction distance is determined by the exponential random number whose average value is equal to the mean free path of the electron at a given energy. Second, the type of interaction is chosen by comparing a uniform random number with the fractional cross-section table shown in Fig. 16.3. Each interaction type is processed in a separate subroutine; ELSCAT for elastic scatter, EXCITE for excitation, IONIZE for soft ionization with no delta ray, DELTA for hard ionization with an energetic delta ray, and AUGER for Auger electron generation. For each interaction type chosen, the position, direction, and energy transfer is recorded in the track structure file on magnetic tape or disk in an event-by-event manner. The new direction and energy of the scattered electron is calculated, and the electron track is followed until it loses all its kinetic energy below the cutoff energy, which is usually set at 12.6 eV (the first ionization potential of water). 16. Electron Track Simulation for Microdosimetry 367

( START)

1 ----' ~Basic ,J ,I Input Data I--~ Initialize~: Data I ~ --1----- i I SO~CE I Generation of electron

follow history start of Delta/Auger It!.. ---

COLIDE ( Interaction Distance and ;;;;-)-1 ITYPE=ll 2 31 4 5'

'------NO~Cutoff I Save p~rimary- I . ? c.}' I Yes t , __1- [ I I OUTPUT I ~ TRh.'END I Restore Primary, I I 1

End 0 Track?·/'"',..!N""oc.....,,;::C~o~n~t Previousi~n...,ue~1 Track // Yes

'------~NO~All istor ? Q .. --i sJ~;s ~ yI ( STOP)

Figure 16.5. General flowchart of the electron simulation Monte Carlo program (ETRACK). (See text for explanation.)

When a delta ray or an Auger electron is generated, the primary electron track information is transferred to the 'STACK'; the newly generated electron then is followed. After the delta ray or Auger electron history has been terminated, the previous electron is retrieved from the 'STACK'. Up to eight generations of delta rays can be traced in this manner in the ETRACK program.

All the interaction events are recorded on a track structure file. For example, 4000 complete electron track structures of tritium beta rays (average energy 5.7 ke V) can be stored on one volume of magnetic tape (2400 ft, 1600 BPI, about 30 MB).

This electron simulation Monte Carlo program (ETRACK) was run on both PDP- 11/34 (RSX-11M) and VAX-11/750 (VMS) computers. The physical random-number generator MIKY was used instead of a conventional mathematical pseudorandom number generator. See Section 16.11 for a discussion of physical random-number generators. 368 A. Ito

16.5 RESULTS OF ELECTRON TRACK SIMULATION

In the following section, results of, and outcomes from, the electron track simulation are discussed. First, two-dimensional plots of the track structures are demonstrated for some example radiations. Second, basic physical quantities derived from the track structure, such as slowing-down spectra, electron range and point kernel (radial dose distribution from a point source) are briefly discussed. Third, the applications of the electron track structure in microdosimetry are discussed. Realistic track structure pat• terns in a cell nucleus are shown, and used for the subsequent analysis of radiobiological responses. Lineal-energy spectrum, which measures the energy deposition by a single track in a given microscopic sphere, can be simulated very well at the micron level. The proximity function, which measures the distance distribution among all energy de• position (or ionization) events in a track, can be calculated from the track structure (nanometer level).

Fig. 16.6 illustrates sample plots of the electron tracks in question. Figure 16.6( a) is a two-dimensional plot of a secondary electron from 60Co gamma rays. The ionization hits are plotted as dots. The range is as long as 700 fLmj it creates several energetic secondary electrons and also lower energy delta rays. Figure 16.6(b), with superimposi• tion, shows ten secondary electron tracks from 280-kVp x-rays. The range in this case varies from a few to 20 fLm. At the end of each track, densely ionized delta rays can be observed. Initial electron spectra for both gamma rays and x-rays were calculated by the photon Monte Carlo program (PTRACK). Figure 16.6(c) shows the three-dimensional plot of five randomly selected beta-ray tracks from tritium. The ionization events (0) and excitation events (x) are plotted. The range is around 1 fLm. Figure 16.6(d) is the plot of a 523-eV oxygen Auger electron. In this case, ionization (0), excitation (x), and elastic scattering events (.) are plotted. The range is as short as '" 30 nm, which is comparable to the size of the DNA molecule shown as a reference.

Fig. 16.7 shows a sample listing of an electron track structure file for a 1000-eV electron in water. Every interaction event is recorded with its interaction type (elastic, excitation or ionization), kinetic energy (E.), energy transfer (E108.), interaction distance (dR ), and position (x,y,z) (not shown in Fig. 16.7).

16.6 BASIC PHYSICAL QUANTITIES DERIVED FROM ELECTRON TRACK STRUCTURE

Various physical quantities can be derived from the electron track structure. Some examples of slowing-down spectra, range-energy curves, and point kernels (i.e., radial dose distribution from a unidirectional point source) for tritium beta rays are shown, and comparisoIls with other data are discussed.

The slowing-down, or degradation, spectrum refers to the energy spectrum of the electron during complete slowing down through many multiple scatterings in the media (water). It is expressed as the sum of the track length at kinetic energies between E and E + dE. In other words, it gives the probability of any matter (chemical species or biological targets) in the media to interact with the electron at a given energy. It can be calculated by integrating the interaction distances at electron energies between. E and E + dE from the electron track structure. Fig. 16.8 shows an example of the slowing• down spectrum for tritium beta rays in water, in units of cm/eV. Contributions of primary beta rays, delta rays, and Auger electrons, are shown separately. The fraction of Auger electrons at 450 eV amounts to about 15%. ~ \ ?' / ,~ :/ " m / '., • / .•..•. _.:::.~-.'I ~

,.' (... . . ',~../ :''"...... ~- \ \ ~ jJ o ,/ -' ::::J

l~ ..L..,:;.:::~.~:«(>/ :;i I" n \ I ,~ '"7<:" 'r/' V'> j 3' \ c OJ """\\ r+ MM f:Co60 (a) MM 1: 280kVP ·. (b) o· z-y Plane z-x Plane ::::J I I I I I I I I I 0- o 200 400 600 800 0 4 8 12 16 20 .... (J..Lm) (J..Lm) s:: n'

oQ..

~ Y DNA Molecu le 3' 500 , (5882 r"I r+~ ""\(~ Nucleotides) ,'" //./ -< (c) DNA Molecule (117 Nucleotides) (d) o ~-+$~~ • '1-- -"-" t \ '. L- - 500 'b "-'. '-, ... _.• ' . fr ~ .;i

Electron '\ Track \ f 523 eV ELS Trilium /3-ray \ -1 0 z-x Plane In Water

-500 o 8 16 24 32 40 ( nm) Figure 16.6. Two-dimensional plots of the electron tracks in water for a) 60Co

gamma ray, b) 280-kVp x-ray, c) tritium beta ray and d) 523-eV oxygen Auger W 0'< electron. <0 370 A. Ito

COMMENT ; 1000 EV MONO-ENERGY ELECTRON IN WATER, 31-AUG-81

START OF PRIMARY TRACK FROM 1000.0 (EV) EVENT. ELASTIC EXCITE IONIZE EE(EV) ELOSS(EV) DR(NK) ••• 1000.0 13.4 1.60 2 ATOMIC POTENTIAL ••• 986.6 16.4 2.97 ( TO BE CONTINUED FROM 942.3 EV ) START OF DELTA-RAY (FIRST LEVEL) FROM 27.9 (EV) EVENT. ELASTIC EXCITE IONIZE EE(EV) ELOSS(EV) DR(NK) 1 ••• 27.9 11.1 0.01 2 ••• 16.7 9.7 0.01 3 ABSORPTION ••••••••••• 7.1 7.0 0.00 ( CUT OFF HERE ! ) CONTINUATION OF PRIMARY TRACK FROM 942.3 (EV) EVENT. ELASTIC EXCITE IONIZE EE(EV) ELOSS(EV) DR(NK) 3 ATOMIC POTENTIAL ••• 942.3 16.4 1.29 ( TO BE CONTINUED FROM 895.0 EV) START OF DELTA-RAY (FIRST LEVEL) FROM 31.0 (EV) EVENT. ELASTIC EXCITE IONIZE EE(EV) ELOSS(EV) DR(NK) ••• 31.0 18.8 0.24 2 ABSORPTION ••••••••••• 12.1 12.1 0.00 ( CUT OFF HERE ! ) 12.1 CONTINUATION OF PRIMARY TRACK FROM 895.0 (EV) EVENT. ELASTIC EXCITE IONIZE EE(EV) ELOSS(EV) DR(NK) 4 ••• 895.0 13.5 3.65 5 ••• 881.5 24.8 6.35 6 ••• 856.7 13.5 0.85 7 ••• 843.2 24.7 1. 76 8 ••• 818.5 13.5 3.55 9 ••• 805.0 13.5 2.37 10 ••• 791.5 13.5 2.12 11 ATOMIC POTENTIAL ••• 778.0 16.4 5.89 ( TO BE CONTINUED FROM 736.6 EV )

Figure 16.7. Sample listing of an electron track structure file for a 1000-eV electron in water. Every interaction along the track is recorded for subsequent use.

The electron range is also an important quantity. Generally, the csda range is given in tableslO• However, each electron has a different integral pathlength (sum of all collision distances) due to energy-loss straggling. Fig. 16.9 is a plot of individual pathlengths as a function of initial kinetic energy. There are 4000 electron tracks from the tritium beta-ray spectrum. The solid line is the mean csda range. The actual fluctuation of the individual electron pathlengths can be seen. The mean value of the pathlength is very close to the mean range. Also, we have comparedll the results of a different kind of range calculation with those by Chmelevski et a[12; (1) pathlength, (2) straight range (A distance between initial position and the last cutoff position at E < 12.6 eV), and (3) projected range (Z coordinate of the last transfer point when the electron starts at Z = 0 and in the direction of the Z axis). Agreement is good except in the low-energy region where small discrepancies were observed. This perhaps is due to differences in the (generally poorly-known) cross sections of low-energy electrons.

Finally, the radial dose distribution from a unidirectional point source of tritium beta rays was calculated. Fig. 16.10 compares the results with other data1S,14. The three data sets agree well at distances up to 3 /lm. However, the present Monte Carlo result gives a higher dose beyond a radius of 3 /lm. This is attributed to range straggling which the Monte Carlo method includes. 16. Electron Track Simulation for Microdosimetry 371

... .,o enCL ~106H-----~-F~~~~r-~~~~+---~.~. ~.. o 0> C J: en~1~7~-- ~--t---~~rl---~~-+-+~----1

Figure 16.8. The slowing-down spectrum of tritium beta rays in water calculated from the electron track structure. Contributions of the primary beta rc;ty, delta ray and Auger electrons are shown separately.

. 0 2 4 6 8 10 12 14 16 18 20 Electron Energy (keVl Figure 16.9. Plot of the pathlength (integral of collision distances until E < 12.6 eV) of individual electron tracks from 4000 tritium beta rays. Solid line is the csda mean range. 372 A. Ito

g lO-I b=~~~~~~~~~+ .....L o RObertaon and Hough U

"~""...... ,1 "'--' ...... •••••• - •...... ~ ...... ': . ·· .. ····1 10-6 1-::.. :::: . :-:-,-. -. +-. -: "i.!,:,."-".. :.+- ..-i -:=+.:=::::-:-'.:,r.::::::":.. +. ... - ='::Zt:-.;: -:::.-+. ~rr-t--+--HIt:fI-r:::-I ••••••• , p •• , ·······i··· .• ,., . :.·::~t:::: ::: ;. lO - 7 L::~~··:·~.1"_~:_::··L···~·t _····_· L·~~··~···~·"!~·· .._ .. ~_;__ . ~~~ __~~~~ 0. 0 l.O 2. 0 3. 0 4.0 S. O 6. 0 7. 0 8.0 Rodi 01 Oi st once ()J 11. )

Figure 16.10. Radial dose distribution of a unidirectional point source of tritium beta rays in water. The Monte Carlo result shown in the histogram agrees well with analytical calculations by Robertson and Houghes13, as well as by Shiragai14 , up to 3 p.m. However, the Monte Carlo result gives a higher dose beyond 3 p.m because it considers range straggling.

The cross-checks of the basic physical quantities derived from the electron track structure by the ETRACK program generally agree well with other independent data in the energy region above 500 e V. Some discrepancies at lower energies are mainly attributed to the uncertainty of the differential and double-differential cross sections.

16.7 PATTERNS AND THE PROXIMITY FUNCTION IN CELL NUCLEUS

When the mammalian cell nucleus is uniformly exposed to a radiation field, a character• istic track pattern is observed. When the electron range is longer than the diameter of a cell nucleus (5 p.m), which is the case with secondary electrons from gamma rays and x-rays, most of the tracks cross the target ('Crosser'). When the range is shorter than a cell nucleus, which is the case for tritium beta rays and delta rays, most of the tracks are generated in the cell nucleus and terminate within the same cell nucleus ('Insider'). Intermediate cases are the 'starter', where the track is created in the target and then leaves the target, and also the 'stopper', where the track is created outside the target and terminates in the target. 16. Electron Track Simulation for Microdosimetry 373

a) Co-60 Gamma-ray (d=5pm. D=lGy) b) 280 kvp X-ray (d=5~m. D=lGy)

. . ol'·'···.

- ' ~.

. '.;. " , :'-~ .

. : .

', ' ::. : .. :::~:. : ' \ . ":" ' ~~~ '" .

' . . ' .' .' '. ": :...... , .

c) H-3 Beta-ray (d=5pm. D=lGy) d) 523 eV Delta-ray (d=5pm. D=lGy) Figure 16.11. Plots of electron tracks in a cell nucleus of 5 tim diameter for a) 60Co gamma rays, b) 2S0-kvp x-rays, c) tritium beta rays and d) 523- eV oxygen Auger electrons, randomly generated from the electron tracks in Fig. 16.6.

Fig. 16.11 shows the track patterns for a) 60Co gamma ray, b) 2S0-kV x-ray, c) tritium beta ray and d) 523-eV oxygen Auger electron, in the cell nucleus of 5 tim diameter. The average dose is 1 Gy, and the number of ionization events, which are shown as dots in Fig. 16.11, is about 1.36 X 104 for each case. These track patterns are generated from the primary electron tracks, shown in Fig. 16.6, which deposited their energy within the sphere of 5 tim diameter, and with each track segment then randomly rearranged in the cell nucleus. By this randomization process, the track pattern which a cell nuclues might encounter has been simulated. The differences among the track patterns are evident. The 6OCo gamma ray gives both numerous sparse single hits and densely ionizing delta rays. The x-ray is similar to the gamma ray, but more short tracks are included. The tritium beta-ray tracks consist of separated short tracks. 523- eV delta rays give scattered, very short tracks all over the sphere. The average number of ionization events on a single track is 60, SO, 120 and 16, respectively. 374 A. Ito

The distribution of the ionization events in a sphere gives the energy-deposition spectrum f(y) which is measurable with experimental microdosimetry using a wall-less spherical proportional counter. In the case of tritiumbeta rays, y spectra were simulated from the electron track structure randomly distributed in the spherical target volume shown in Fig. 16.11(c). This simulates a "wall-less water-vapor" proportional chamber.

Fig. 16.12 shows the average lineal energy, both in frequency (YF) and in dose (fJD), as a function of sphere diameter between 0.5 and 5 pm. The results are compared with the experimental data of Ellett and Brady15. The experimental y values give higher results. Part of the difference is attributed to the difference in the stopping power between water vapor and tissue-equivalent gas (propane base). The ratio, however, between the Monte Carlo calculation and experimental results, was constant for all spherical sizes.

10.------,

Average y for 3 H 13- ray

-- Experimental (8raby 1972)

Theoretical (Ito 1981)

-:c ~ " '"~ y~'6 > 0(

Sphere Diameter (JIm) Figure 16.12. Average lineal energy of the tritium beta ray. Theoretical calculations are derived from the Monte Carlo track-structure analysis. Ex• perimental data was measured with a wall-less proportional counter. The difference is attributed to stopping power between water vapor and tissue• equivalent gas.

The proximity function t(x), the probability distribution of the distance among ionization events, was calculated for the track patterns shown in Fig. 16.11. It is important to calculate the proximity function with the random track segments in the target because it provides the information, not only on the intratrack effect, but also on the intertrack effect. Fig. 16.13 shows the integral form of the proximity function T( x) for each electron track. The integrated number of neighboring ionization events around any given hit within the distance x(nm) is plotted. The intratrack effect has a shorter interaction distance. The distance to find any next neighboring ion hit (next hit distance) ranges from 0.9 nm (523 eV) to 1.6 nm (60Co). In the distance region between 0.1 and 10 nm, lower energy electrons have more probability of being found in close proximity. The probability of finding the neighboring ion hits within 5 nm is 2.9 (60 Co), 3.1 (x-ray), 3.6 (tritium) and 6.0 (523 eV). On the other hand, the interaction 16. Electron Track Simulation for Microdosimetry 375 distance for intertrack effects is much greater at this dose level (1 Gy). The next hit distance ranges from 90 nm (60CO) to 120 nm (523 eV).

Chemical species having a long diffusion distance could interact with each other. However, when the absorbed dose increases, the next hit distance for the intertrack effect becomes shorter, while the proximity function for the intratrack effect is unchanged. This will partly explain the nonlinear term for radiobiological effects at high doses.

, ,;. .... Inter-track

. i .j I··";;' C 31 01~~~~~~~~~#*~~~~~~~

Figure 16.13. The probability of neighboring ionization events existing around an ionization hit within a distance x(nm) for 60Co gamma rays, 280 kVp x• rays, tritium beta rays and 523 eV oxygen Auger electrons in the cell nucleus of 5 11m diameter.

16.8 CALCULATION OF THE DSB PROBABILITY OF DNA

The most important biological target for radiation effects is the DNA molecule16. The vital information for life is coded as the base sequences of DNA. The information on DNA is copied to RNA and translated into the protein to maintain its biological activi• ties. If the DNA molecule is fatally damaged, self-reproduction is inhibited and the cell no longer can divide. A double strand break (dsb) on the double helix DNA constitutes such fatal damage. On the other hand, radiation damage on the DNA base, or single strand break (ssb) of DNA, can be repaired very efficiently by various enzymic repair mechanisms.

The mammalian cell nucleus (3-10 11m diameter) contains as much DNA as 5.2x 109 nucleotide pairs (3.4 x 1012 dalton, 5.6 pg). The number of ionization events and subse• quent reactive radical species created in the cell nucleus is proportional to the absorbed dose. For example, let us consider a cell nucleus of 5 11m diameter (64 pg), as shown in Fig. 16.11, which receives an absorbed dose of 1 Gy. There are'" 1.36 x 10" ionization hits [assuming G(Ion) value=3.3] and", 8.87 x 103 OH· radicals (G(OH·) value = 2.95) 376 A. Ito after initial recombination within the spur in the cell nucleus. The ssb is assumed to take place either by ionization or OH· radical hits on the molecules composing the backbone of the DNA strand (-O-P-O-C5-C4-CS-). As a first approximation, the ssb probability is assumed to be proportional to the mass of backbone of DNA (1.7 pg). Under such simplified conditions, 360 ssbs (fCell/Gy) by ionization hits, and 235 ssbs (fCell/Gy) by OH· radical hits, are created randomly in the cell nucleus. This figure is consistent with the experimental results of ssb frequency (several hundreds ssb/Cell/Gy).

The dsb of DNA takes place around a ssb, when the second ssb occurs in close proximity. Fig. 16.14 illustrates the models for dsb of DNA. They are categorized into the intratrack effect, or single-track effect, and the intertrack effect. In each case, strand break mechanisms are classified into direct action and indirect action.

Intra - track effect Inter - track effect (low Dose) (High Dose)

Direct ~ lft 100 + 100 ~' I/ t 'xl ... ,;" .~ ...... ------l----- S Ixl " ·r

tt ~ ! .p \ 100 + Oli- ,g.-A~ 9H'A' l~~ /\". , t IXI '(j . f' ' " , \, ,-- ~,s;'. '-00;"- ",' \~ P"" ".X' , .-r --I''. SI., '! , Indirect ~!~?/ OH· · OH· ~ -----~ " , ~ Ixl t )'H~ . -"01<' POt«)M(',X) .. ~~ ~~hr-', S (.) , Qa~-J

Figure 16.14. A model of the dsb breaks of DNA with electron tracks. The direct action with ionization hits and the indirect action with the OH· radicals are considered. Also, the intratrack effect and the intertrack effect are treated separately.

At low-dose level, the electron tracks are created sparsely, both in time and in space, so that DNA lesions are caused by the single-track effect. Direct action takes place when an electron creates two neighboring ssbs on the pair strand of DNA within close proximity « 5 nm). Therefore, the distance distribution of the neighboring ionization events along the track (proximity function) is important. Also, the probability function of the DNA molecule that finds the pair strand at a given distance (target function) 16. Electron Track Simulation for Microdosimetry 377 should be calculated. On the other hand, indirect action is mainly caused by 0 H . radicals, as the contribution of H· radicals, or hydrated electrons (e-aq), are relatively small17• The diffused OH· radicals will cause a dsb when they attack DNA strands in close proximity. The diffusion and decay of OH· radicals in the cell environment is very important in this case. At high-dose levels, where ionization and OlI·-radical density are high, the intertrack effect plays an important role. In particular, the OH· radicals, which have a longer diffusion distance, dominate the dsb of DNA.

16.9 THE DSB PROBABILITY AND RBE The dsb probabilities of an intratrack effect were calculated for both direct and indirect actions by ItolS for the electron tracks in Fig. 16.12. The results are tabulated in Table 16.1 from Ref. 18.

Table 16.1. Probability or DNA dsb and the RBE relative to 6OCo.

Radiation Direct Indirect Total RBE relative to Ion+Ion Ion+Orr· OH· + OH· dsb 6OCo 6OCo gamma 0.74 0.80 0.02 1.56 1.0 280 kvp X 0.78 0.89 0.02 1.69 1.08 Tritium Beta 0.88 1.06 0.02 1.76 1.26 523 eV Auger 1.50 1.59 0.03 3.12 1.99 DNA dsb/ssb (in %) (where DNA ssb = 580 /Gy/Cell)

The direct action (Ion+Ion) dsb probability is calculated to vary between 0.74 (60CO) and 1.50 (523 eV) in % of total ssb. As the DNA dsb target function s(x) to find the pair strand has maximum probability at a distance between 1 and 2 nm, the dsb probability is higher for lower energy electrons whose hit distance is short. The direct action dsb depends on the track structure, and cannot be influenced by chemical modifications.

The probability of dsb by indirect action is calculated to vary from 0.80 to 1.59 (Ionization + OH·) and from 0.02 to 0.03 (OH· + OH·) in % of total ssb. Indirect dsb occurs more frequently than direct dsb. As indirect action depends on the diffusion of OH· radicals, it has a longer interaction distance, and the yield is strongly influenced by chemical modifiers such as radical scavengers or sensitizers around a DNA molecule. As indirect dsb takes place more efficiently when the initial distance of hits is short, it also depends on the track structure. Thus, the lower energy electron also has a higher yield of indirect dsb. Indirect dsb by OH· + OH· is an order of magnitude smaller than the case of ds];> by Ion + OH·. The yield is linear with absorbed dose at low-dose levels.

The total dsb probability was calculated to be 1.56, 1.69, 1.96 and 3.12% of the total ssb, for the four radiation qualities respectively. The ratio for x-rays is comparable to the experimental value (irreparable strand breaks) of about 1% in mouse V79 cells19•

Finally, the RBE of dsb compared to 60CO gamma rays was calculated. RBE values of 1.1,1.3 and 2.0 were derived for x-rays, tritium beta rays and 523-eV electrons. These 378 A. Ito figures are reasonable for many of the radiobiological data on cell killing or chromosome aberration for x-rays and tritium beta rays. However, the present dsb calculation does not consider any chemical modifiers, nor any repair processes. Only the initial damage by DNA dsb is calculated.

16.10 CONCLUDING REMARKS

The simulation of the electron track structure by the Monte Carlo method is extremely useful in analyzing the two classes of problems: Class J) physical energy deposition by the electron itself, and Class II) subsequent physicochemical and biological effects. At present, various problems can be solved using the electron track structure Monte Carlo method with acceptable accuracy. However, for better understanding and more accurate analysis of radiobiological responses in light of the first physical principle, correct, absolute, and comprehensive cress-section data are needed in liquid water in the energy region below 300 ke V .

16.11 THE USE OF THE PHYSICAL RANDOM NUMBER GENERATOR. MIKY

The Random Number Generator (RNG) is the key component in any Monte Carlo computation. Mathematical, or pseudo-RNGs supplied by a computer manufactuer, are commonly used. None of these pseudo-RNGs, however, are truly random. Hence, their use involves inherent risks of various kinds. The most commonly used multiplicative congruential RNGs are known to "fall in the planes" 20. Some RNGs are known to have very short "pseudo-periods"21. Tests of a RNG are done for frequency, serial, gap, poker, moments, and so on. Some RNGs give substantially erroneous results on simulated phenomena22 . Careful choice and tests may reduce the pitfalls due to pseudo-RNGs23. MIKY is a set of physically generated random numbers, developed by Miyatake, Inoue, Kumahora, and Yoshizawa2" that has an excellent randomness compared to the conventional arithmetic RNGs. Thus, MIKY was implemented on a PDP-ll computer, running on a RSX-llM operating system. The MIKY distribution tape, having 3 x 107 single-digit random numbers, was copied onto a disk file, after conversion into 5 X 106 six-digit random numbers (real type, 0 ::::; R < 1) for later use. A set of MACRO subroutines, callable from FORTRAN programs, was prepared to get the random numbers efficiently from the MIKY disk file.

The speed of getting one random number was compared with conventional arith• metic RNGs. MIKY gave one random number in 35 J-ts, whereas the DEC standard RNG (f = RAN(il,i2), for F4P V3.0) took as much as 685 J-tS on a PDP-ll/34A with a Floating Point Processor (FP-llA) and a CASHE memory (KK-llA). Thus, MIKY was found to be 20 times faster. Also, the randomness was tested for frequency, run, Poker, serial, etc. The MIKY RNG passed all the randomness tests.

Fig 16.15 shows a result of the frequency test for the MIKY RNG. The Chi-square distribution from 10,000 independent samplings is shown. The measured distribution is very close to the theoretical one. A pseudo-RNG (DEC Old) was examined for the same frequency test. Fig. 16.16 shows the chi-square distribution after 10,000 trials. The resulting distribution is different from the theoretical one. This pseudo-RNG (DEC Old) was replaced by a new RNG, at the expense of generating time (the old RNG took 88 J-ts and the new RNG takes 685 J-tSi IBM-360 RANDU takes 197 J-ts on PDP-ll/34). 16. Electron Track Simulation for Microdosimetry 379

0.9

0.8

0.7 >• r- ~ 0.6 .-J gE O. 5 H----'--+-'-··-'---'---'---"d- co ~O.Lj CL 0.3

0.2

0.1 O. 0 U::------~L---'----'-----=----'---:I;:::::=:~~~.....;..3:-l?0 0.0 0.5 .

Figure 16.15. Chi-square distribution for a frequency test with physically• generated random numbers (MIKY). The result of 10,000 trials agreed well with the theoretical expectation.

1.0

0.9 . fDECQLDJ

0.8 fREfDOME.9

0.7 .SAMPLE=Y99 >- r- ~0.6 .-J gE 0.5 co ~O.Lj CL 0.3

0.2

0.1

0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 REDUCED CHI-SQURRE Figure 16.16. Chi-square distribution for frequency test with a pseudorandom number generator (DEC OLD). The result of 10,000 trials shows discrepancies from the theoretical expectation. This RNG was replaced by a new one. 380 A. Ito

The MIKY physical RNG is superior to arithmetic RNGs in most respects, except for the length of the random numbers, which is limited by the size of the disk file, i.e., 20 MByte for 5 X 106 numbers* .

Acknowledgement: The author thanks Prof. Shigefumi Okada of Kyoto University for his encouraging and inspiring discussions on this subject, and Ms. Sayuri Noguchi for preparation of the original manuscript. This work was supported in part by grants from the Japanese Ministry of Education.

* The MIKY RNG data tape is available from Dr. Inoue of Hiroshima University upon request. 16. Electron Track Simulation for Microdosimetry 381

REFERENCES 1. H. G. Paretzke, G. Lentheld, G. Burger and W. Jacobi, "Approaches to Physical Track Structure Calculations" , in Proc. of the Fourth Symp. on Microdosimetry, EUR 5122 d-e-f, (1974) 123. 2. H. G. Paretzke, "An Appraisal of the Relative Importance for Radiobiological Effects of Slow Electrons", in Pmc. of the Fifth Symp. on Microdosimetry, EUR 5452 d-e-f," (1976) 41. 3. R. N. Hamm et al., "Spatial Correlation of Energy Deposition Events in Irradi• ated Liquid Water", in Pmc. of the Sixth Symp. on Microdosimetry, (Harwood Academic Publishers, 1978) 179. 4. M. J. Berger, "On the Spatial Correlation of Ionization Events in Water", in Pmc. of the Seventh Symp. on Microdosimetry, (Harwood Academic Publishers, 1980) 521. 5. V. P. Bond, "RBE of 3H Beta Rays and Other Low-LET Radiations at Low Doses and/or Dose Rates", in Pmc. of the Sixth Symp. on Microdosimetry, (Harwood Academic Publishers, 1978) 69. 6. H. G. Paretzke and M. J. Berger, "Stopping Power and Energy Degradation for Electrons in Water Vapor", in Proc. of the Sixth Symp. on Microdosimetry, (Harwood Academic Publishers, 1978) 749. 7. M. J. Berger, "Discussion of Electron Cross Section for Transport Calculations", Argonne National Laboratory report ANL-84-28, (1984) 1. 8. H. G. Paretzke, "Cross Sections Needed for Investigations into Track Phenomena and Monte-Carlo Calculations", Argonne National Laboratory report ANL-84- 28, (1984) 9. 9. N. F. Mott and H. S. W. Massey, The Theory of Atomic Collisions, Third Edition, (Oxford Univ. Press, 1965). 10. M. J. Berger and S. M. Seltzer, "Stopping Powers and Ranges of Electrons and Positrons", National Bureau of Standards report NBSIR 82-2550-A (1982). 11. M. Nenoi, T. Kanai and A. Ito, "Estimation of Interaction Function l'(x) with Sparsely Ionizing Radiation", Radiat. Res. 122 (1987) 1. 12. D. Chmelevski, A. M. Kellerer, M. Terrissol and J. P. Patau, "Proximity Func• tions for Electrons up to 10 keV", Radiat. Res. 84, (1980) 219. 13. J. S. Robertson and W. V. Houghes, "Intranuclear Irradiation with Tritium• labeled Thimidine", in Proc. First Nat!. Biophysics Conf., edited by H. Quastler, (Yale Univ. Press, 1959) 278. 14. A. Shiragai, "Comments on Estimation Methods of Absorbed Dose Due to Tri• tium", J. Radiat. Res. 13, (1972) 208. 15. W. H. Ellett and L. A. Braby, "The Microdosimetry of 250 kVp and 65 kVp X• Rays, 60Co Gamma Rays, and Tritium Beta Particles", Radiat. Res. 51, (1972) 229. 16. K. H. Chadwick and H. P. Leenhouts, The Molecular Theory of Radiation Biol• ogy, (Springer-Verlag, Berlin, 1981). 17. R. Roots and S. Okada, "Estimation of Life Times and Diffusion Distances of Radicals Involved in X-Ray-Induced DNA Strand Breaks or Killing of Mam• malian Cells", Radiat. Res. 64, (1975) 306. m A.~

18. A. Ito, "Calculation of Double Strand Break Probability of DNA for Low LET Radiations Based on Track Structure Analysis" , in Nuclear and Atomic Data for Radiotherapy and Related Radiobiology, edited by K. Okamoto, (IAEA publica• tion, TECDOC series, Vienna, 1987) 413. 19. K. Sakai and S. Okada, "Radiation-Induced DNA Damage and Cellular Lethality in Cultured Mammalian Cells", Radiat. Res. 98 (1984) 79. 20. G. Marsaglia, "Random Numbers Fall Mainly in the Planes", Proc. Nat. Acad. Sci. 61 (1968) 25. 21. P. R. Nicholson, J. M. Thomas and C. R. Watson, "Characterization of PDP-11 Pseudo-Random Number Generators", Proc. of Digital Equipment Corporation Computer Users' Society, San Francisco (1978) 853. 22. R. L. Morin, D. E. Raeside and J. E. Goin, "Monte Carlo Advice", Med. Phys. 6, (1979) 305. 23. J. R. Ehrman, "The Care and Feeding of Random Numbers", in Stanford Linear Accelerator Center SLAC VM Notebook, SLAC (1981). 24. H. Inoue et al., "Random Numbers Generated by a Physical Device", J. Royal Statistics Soc., Series C Applied Statistics 32, (1983) 115. General Techniques 17. Geometry Methods and Packages

Walter R. Nelson and Theodore M. Jenkins

Radiation Physics Group Stanford Linear Accelerator Center Stanford, California 94309, U.S.A.

17.1 MATHEMATICAL CONSIDERATIONS The trajectory of a particle in a Monte Carlo calculation can be described by position and direction vectors x = xi+yj+zk (17.1 ) and v = ui + vj + wk , (17.2) respectively, where (x,y,z) are the coordinates of the particle at point P(x,y,z) (e.g., see figures below), and (u, v, w) are its direction cosines (the symbol' denotes a unit vector). These quantities, together with such things as particle type, energy, weight, time, etc., define the state function of the particle.

In general, one is interested in determining the point of intersection P'(x', y', z') of the vector tV with any given surface that constitutes the geometry for the problem at hand. To be more specific, the distance t = IP- P'I is generally needed in order to compare with the actual transport distance that is about to be used in the simulation. In the following sections, we will develop the basic mathematical expressions for the intersection of the particle trajectories with plane and conic surfaces (i.e., cylinders, cones, and spheres). We will show how they are used in the EGS4 Code System!, which should be typical of the way it is done elsewhere. The remainder of this chapter will then be devoted to a general survey of some of the more prominent geometry packages currently being used in electron-photon Monte Carlo.

17.1.1 Intersection of a vector with a plane surface.

A plane surface can be described by the vector to a point P" on the surface

(17.3) and a unit vector normal to the surface

(17.4)

385 386 W. R. Nelson and T. M. Jenkins

p a ~===---

x Figure 17.1. Vector diagram of particle intersecting a plane surface. as shown in Fig. 17.1.

The condition for the intersection point pI to lie on the plane is obtained from the vector dot product (17.5) From the diagram we see that

(17.6) which leads to the solution (6 - X)-l~ t = A A (17.7) U·N Equation 17.7 is indeterminant when U . N = 0, corresponding to the physical situation in which the particle travels parallel to the plane. The particle travels towards the plane when t > 0 and away from it when t < O. Expanded into its components, Eqn. 17.7 becomes t = ~(C...::.l_-_x-,):..-n...::.l_+_(~C.::...2_---'y~)_n.::...2 _+....;(,-cs=--_z,"-)n--=.s (17.8) unl + vn2 + wns

17.1.2 The PLANE1 algorithm available in EGS4.

SUBROUTINE PLANE1 (see Fig. 17.2) of the EGS4 Code System provides an example of an algorithm for determining the intersection of a particle trajectory with a plane surface using the equations developed above. The components of X and U are available through COMMON/STACK/. NP is the stack pointer-i.e., the particle currently being followed-of which there can be as many as 40 in the default version of EGS4. The vectors that define the plane, 6 and N, are described by arrays PCOORD and PNORM, respectively, and are passed in COMMON/PLADTA/ (for a maximum of 100 planes in this example).

Except for parameter INPT, which allows for more efficient determination of t > 0 ("hit") versus t < 0 ("miss"), the algorithm in PLANE1 is based precisely on the equations developed in the previous section.

17.1.3 Intersection of a vector with a cylindrical surface.

The example we will use is a right circular cylinder of radius R whose axis of rotation lies along the z-axis as shown in Fig. 17.3 . The intersection point pI is located by 17. Geometry Methods and Packages 387

SUBROUTINE PLANE1(IPLN,INPT,IHIT,TPLN); "______------, " Input: IPLN Plane identification number INPT - 1 Surface normal points AWAY fro. current region --1 Surface normal points TOWARDS current region " Output: IHIT - 0 Particle travels AWAY fro. surface (a miss) - 1 Particle travels TOWARDS surface (a hit) - 2 Particle travels PARALLEL TO surface (a miss) TPLN Distance to surface (when IHIT-l) 11 ______------______11 COKKON/STACK/E(40),X(40),Y(40),Z(40),U(40),V(40),W(40), DNEAR(40) , WT(40) ,IQ(40) ,IR(40) ,NP; DOUBLE PRECISION E; COKKON/PLADTA/PCOORD(3,100),PNORK(3,100);

UDOTN-PNORK(l,IPLN)*U(NP) + PNORK(2,IPLN)*V(NP) + PNORK(3,IPLN)*W(NP); UDOTNP-UDOTN*INPT;

IF (UDOTNP.EQ.O.O) [ IHIT-2; "Parallel to z-axis (indeterminant)" ] ELSEIF (UDOTNP.LT.O.O) [ IHIT-O; "Traveling away froll surface"] ELSE [ "Traveling towards surface---deterlline distance (TPLN)" IHIT=l; TPLN=PNORK(l,IPLN)*(PCOORD(l,IPLN)-X(NP» + PNORK(2,IPLN)*(PCOORD(2,IPLN)-Y(NP» + PNORK(3,IPLN)*(PCOORD(3,IPLN)-Z(NP»; TPLN=TPLN/UDOTN; ]

RETURN; END;

Figure 17.2. Listing of SUBROUTINE PLANE1. vector X', whose k-component is defined by

v = (X'.k)k = (z + tW)k , (17.9)

SInce X' = X + tiT = (x + tu)i + (y + tv)j + (z + tW)k . (17.10) The radial vector ii = X' - V = (x + tu)i + (y + tv)j (17.11) can then be squared (i.e., ii· ii) to give the quadratic equation

(17.12)

The solution. of this equation can be written

-f3 ± Jf32 - (ry t= """:""-"":""';---'- (17.13) where a = u2 + v 2 , f3 = xu + yv , (17.14) ,= x2 + y2 - R2 388 w. R. Nelson and T. M. Jenkins z I

Figure 17.3. Vector diagram of particle intersecting a cylindrical surface.

A few special cases are worth discussing. First, since

(17.15) a = 0 implies that w = ±1, so that the particle travels parallel to the cylinder surface and does not intersect it (except for the trivial case where ,= 0 - i.e., on the surface itself). Note that 0 :::; a :::; 1 also follows from Eqn. 17.15.

Second, when I < 0, the particle travels inside the cylinder, in which case the solution of interest is the positive one, as depicted by the solid line portion of (a) in Fig. 17.4.

8----~- - ---Co)

- ---Cb)

-8- --- Ce)

8------Cd)

Figure 17.4. Possible trajectories intersecting a cylinder (starting and inter• section points indicatd by squares and crosses, respectively).

Finally, when I > 0, the particle travels outside the cylinder and several situations present themselves. If j32 < ai, there are no real solutions, which represents a particle completely missing the cylinder (e.g., see (d) in Fig. 17.4). The "grazing" solution, corresponding to j32 = ai, is also not of particular interest to us. However, when j32 > ai, two real solutions exist-either both negative (j3 > 0) or both positive (j3 < 0), as depicted by (c) and (b), respectively, in Fig. 17.4. Of course, we are only interested in the smaller positive solution.

One can easily find solutions for particle trajectories intersecting other conic sur• faces (e.g., spheres and cones), including those that have been translated and rotated relative to the defining coordinate system. In the following section, we will show how the above equations are put to use in an auxiliary subprogram that comes with the EGS4 Code System. 17. Geometry Methods and Packages 389

17.1.4 The CYLNDR algorithm available in EGS4.

SUBROUTINE CYLNDR (see Fig. 17.5) of the EGS4 Code System provides an example of an algorithm for determining the intersection of a particle trajectory with a cylinderical surface using the equations developed above. As in the case of SUBROUTINE PLANE1, the components of X and iT are available from EGS4 by means of COMMON/STACK/. The cylinder radius is passed in COMMON/CYLDTA/ (for a maximum of 75 cylinders in this example). To make the algorithm more efficient, the user is expected to supply pre-knowledge about whether the current position of the particle is inside (INCY=1) or outside (INCY=O) the cylinder.

A difficulty can arise for a particle traveling very close to the cylinder surface, particulary at a glancing angle. For a given machine precision, the quadratic solutions, together with the way EGS4 uses them, can result in a particle being "stepped" sideways to the surface. Albeit a very small amount, this can cause enough error in the true position of a particle to confuse the user's boundary tracking program (i.e., SUBROUTINE HOWFAR), generally with the result that the program gets caught in an infinite loop. An attempt to resolve such difficulties has been addressed by Stevenson2, and those portions of the algorithm involving the parameter DELCYL are a direct result of this study. Aside from that, the algorithm in CYLNDR is based precisely on Eqn. 17.13 and Eqn. 17.14.

As a practical matter, the use of a DELCYL-value of 1. OE-4 cm has been found to work satisfactorily in most situations. For very small cylinders, the user may need to reduce the value to 1.0E-5 (or even 1.0E-6). However, for very large cylinders (e.g., R ~ 10 cm or larger), a choice of 1. OE-3 may be required in order to avoid the infinite-loop syndrome. The user can experiment with DELCYL and change its value in a dynamic way, as dictated by the particular problem at hand. Alternatively, one can select a small number for DELCYL, such as 1. OE-6, and perform the entire calculation with higher precision. The AUTODBL option associated with FORTRAN compilers on IBM computers provides an easy way to accomplish this feat without having to re-code EGS4.

17.2 GEOMETRY CONSIDERATIONS IN THE EGS4 CODE SYSTEM 17.2.1 The EGS4 User Code concept.

The EGS4 code itself consists of two user-callable subroutines, HATCH and SHOWER, which in turn call the other subroutines in the EGS4 code, some of which call two user• written subroutines, AUSGAB and HOWFAR. The latter determine the output (scoring) and geometry, respectively. The uSer communicates with EGS4 by means of various COMMON variables, and is required to write a MAIN driver program which, together with AUSGAB and HOWFAR (and any auxiliary subprograms), consititute what is commonly referred to as the EGS4 User Code (e.g., see Chapter 12).

In general, any geometry initialization needed by HOWFAR is performed in MAIN. Although the user most typically "hard codes" the various geometry parameters within MAIN itself, there really is no restriction on how this is done. To this end, "card" -input systems similar to those used in MORSE-CG3 or in FLUKA 4,5 can be implemented, even though this is not a general feature of EGS4. An example of how to adapt the MORSE• CG system to EGS4 is provided by the User Code called UCSAMPCG MORTRAN (file #58 on the EGS4 Distribution Tape). 390 w. R. Nelson and T. M. Jenkins

SUBROUTINE CYLNDR(ICYL,INCY,IHIT,TCYL); tt ______It " Input: ICYL Cylinder identification number INCY - 1 Current particle position is INSIDE cylinder - 0 Current particle position is OUTSIDE cylinder " Output: IHIT - 1 Particle trajectory HITS surface - 0 Particle trajectory MISSES surface TCYL Distance (shortest) to surface (vhen IHIT=l) ,, ______------______11 COMMON/STACK/E(40) ,X(40) ,Y(40) ,Z(40) ,U(40) ,V(40) ,V(40) , DNEAR(40), VT(40) ,IQ(40) ,IR(40) ,NP; DOUBLE PRECISION E; COMMON/CYLDTA/CYRAD2(75); DATA DELCYL/1.0E-4/; "Close approach parameter for easy getavay"

ALPHA=U(NP)*U(NP) + V(NP)*V(NP); IF (ALPHA.EQ.O.O) [ IHIT-O; "Parallel to z-axis (indeterminant)" ] ELSE [ BETA=X(NP)*U(NP) + Y(NP)*V(NP); GAMMA=X(NP)*X(NP) + Y(NP)*Y(NP) - CYRAD2(ICYL);

"Define sOlie local variables next" ACYL=SQRT(ALPHA); BCYL=BETA/ACYL; CCYL-GAMMA; ARGCY-BCYL*BCYL - CCYL;

IF (ARGCY.LT.O.O) [ IHIT-O; "Illaginary solutions" ] ELSE [ "Real solutions (treating lIachine precision difficulties)"

IF (ABS(CCYL).LT.DELCYL.AND.INCY.EQ.O.AND.BCYL.GE.O.O)[ IHIT-O; ELSEIF (ABS(CCYL).LT.DELCYL.AND.INCY.EQ.l.AND.BCYL.LT.O.O)[ IHIT-l; TCYL--2.0*BCYL/ACYL; ] ELSE [ IF (INCY.EQ.l.AND.CCYL.GE.O.O) [ IHIT-l; TCYL-DELCYL; ] ELSEIF (INCY.EQ.O.AND.CCYL.LE.O.O) [ IHIT-l; TCYL=DELCYL; ] ELSE [ "Normal hit-or-lliss solutions" IF (CCYL.LT.O.O) [ "Inside cylinder" IHIT-l; TCYL-(-BCYL + SQRT(ARGCY»/ACYL; ] ELSEIF (BCYL. LT. 0.0) [ "Outside (and 1I0ving tovards) cylinder" IHIT=l; TCYL=(-BCYL - SQRT(ARGCY»/ACYL; ] I ELSE [ IHIT=O; "Outside (but 1I0ving avay froll cylinder)" ] I ] I ] RETURN; IEND;

Figure 17.5. Listing of SUBROUTINE CYLNDR.

17.2.2 Specifications for (and an example of) HOWFAR.

On entry to the geometry subprogram HOWFAR, EGS4 has determined that it would like to transport the top particle on its stack (identified by NP) by a straight-line distance USTEP. The state-function parameters of the particle are available to the user via COM• MON/STACK/ as described previously. The user controls the transport by setting the variables USTEP, IDISC, IRNEW and, in some instances, DNEAR(NP). 17. Geometry Methods and Packages 391

Except for the last variable, which is in COMMON/STACK/, they are available to the user via COMMON/EPCONT/. The various ways in which they may be changed, and the way EGS4 will interpret these changes, is discussed in great detail in the EGS4 User ManuaP, and we will not duplicate that effort here. Instead, we will give a simple HOWFAR example using the PLANE1 and CYLNDR routines discussed above, which should provide the reader with a reasonable idea of how geometries are generally handled in the EGS4 Code System.

Consider a cylindrical target struck by an incident electron beam as shown in Fig. 17.6. The cylinder of rotation about the z-axis is identified by Box 1. . There are four regions of interest-the target (region 2) and three vacuum regions upstream, downstream, and surrounding the target. The extent of the target along the z-direction is determined by two end planes, identified by Triangles 1 and 2 that point in the direction of the defining unit normal vectors:

PNORM(l, 1) = 0.0 PNORM(2, 1) = 0.0 (17.16) PNORM(3, 1) = 1.0 , and PNORM(l, 2) = 0.0 PNORM(2, 2) = 0.0 (17.17) PNORM(3, 2) = 1.0 . The target length, T, is

T = PCOORD(3, 2) - PCOORD(3, 1) (cm) (17.18)

(all the other PCOORD-values are zero), and the radius, R is defined by

(17.19) where all the quantities are (generally) defined in MAIN and passed to HOWFAR in COMMON/PLADTA/ and COMMON/CYLDTA/, as we have indicated earlier.

Y Ix lnto p!lperl

4 v!lcuum 2

1 Rlcml 3 v!lcuum vacuum 2 t!lrqet e z be!lm Tlcml

Figure 17.6. Cylinder of rotation about the z-axis bounded by two planes. 392 w. R. Nelson and T. M. Jenkins

With EGS4, the actual transport of electrons and photons is performed in sub• routines ELECTR and PHOTON, respectively. In the default version (i.e., no electric or magnetic fields), a simple translation is done; namely,

(17.20) where Xnew is the new vector position of the particle after being translated the distance e (i.e., variable USTEP in the code). However, before the transport takes place, HOWFAR is called by one of these subprograms in order to allow the user the opportunity to "interact" with the process-e.g., shorten e, stop a process entirely, etc.

For purpose of illustration, we will assume that the incident electron initially starts out in region 2, and that all particles leaving the target are to be "discarded". Referring to the listing given in Fig. 17.7, the way this is accomplished is to change IDISC=O (default) to IDISC=l (i.e., from no-discard to discard) whenever a particle is somewhere other than in region 2, and to return to the calling program. EGS4 will handle the rest, including calling AUSGAB in order to allow the user to score quantities of interest-e.g., backscattered electrons, forward bremsstrahlung, lateral energy escape, etc.

SUBROUTINE HOWFAR; 11 ______11 " Cylinder of rotation about the z-axis bounded by two planes. " _____ ------______11 COMIN/CYLDTA,EPCONT,PLADTA,STACK/; "(See text for explanation)"

IF (IR(NP) .NE.2) [ IDISC=l; "Discard particles outside the target" ]

ELSE [ "Track particles within the target"

CALL CYLIfDR(1,l, IHIT, TCYL); "Check the cylinder surface" IF (IHIT.EQ.l) [ "Surface is hit---make changes if necessary" IF (TCYL.LE.USTEP) [USTEP-TCYL; IRNEW=4;] ]

CALL PLANE1(2,l,IHIT, TPLIf); "Check the downstream plane" IF (IHIT.EQ.l) [ "Surface is hit---make changes if necessary" IF (TPLIf.LE.USTEP) [USTEP-TPLIf; IRNEW=3;] ] ELSE IF (IHIT.EQ.O) [ "Heading backwards" CALL PLANE1(1,-l,IHIT,TPLIf); "To get TPLIf-value (IHIT~l, a must)" IF (TPLIf.LE.USTEP) [USTEP-TPLIf; IRNEW=l;] "Make necessary changes" ]

RETURIf; EIfD;

Figure 17.7. A simple example of how to write SUBROUTINE HOWF AR.

In this example, all of the particle transport takes place in region 2 where it is HOWFAR's job to decide whether or not the current size of e =USTEP is such that Xnew remains within the target boundaries. Specifically, one must determine the distance from the particle's current position to a point of intersection with a boundary, denoted by t in the previous sections involving subroutines CYLNDR and PLANE1. If e < t, the translation is allowed and one can return to the calling program without further ado. 17. Geometry Methods and Packages 393

On the other hand, both USTEP and IRNEW (the new region in which the particle will eventually end up), will have to be changed if'- 2: t.

Since one does not know a priori which surface is intersected "first", all surfaces must be checked in order to obtain the smallest t. A little reflection should convince the reader that it does not matter in which order the calls are made, but all surfaces must be checked. The small exception to this statement, of course, involves two parallel planes; namely, if the first plane is "hit", one does not have to bother with the second plane. Furthermore, if it is known beforehand that the radiation prefers (statistically) to strike one plane rather than the other, the user can take advantage of the fact and force a call to the preferred plane first (e.g., the "downbeam" plane).

Together with earlier discussions about PLANE1 and CYLNDR, the listing in Fig. 17.6 demonstrates how the user should construct SUBROUTINE HOWFAR, at least for the ex• ample presented. However, the statement COMIN/CYLDTA,EPCONT,PLADTA,STACK/;, still requires some clarification. Without going into detail at this point, suffice it to say that a COMIN statement is a macro that is part of the EGS4 Code System, providing the user with a compact way of writing COMMON statements. The EGS4 Code System is written entirely in the Mortran3 language6 which has a macro facility for accomplishing such tasks (additional information on Mortran3 macros is provided in the EGS4 Code System documentation1 ).

17.2.3 Auxiliary geometry subprograms available with EGS4.

A variety of subprograms, designed to aid the user in creating relatively sophisOticated User Codes, are available with the EGS4 Code System*. In the previous sections, we have taken a close look at CYLNDR and PLANE1, including the mathematics that forms the basis of them. We will not go into further mathematical detail in this section since the basic idea and methods are the same. Instead, we will simply itemize which subroutines are available, and provide a terse description of their main features and use. Each routine is self-documented by means of commentary contained within the coding.

PLANE1 - Determines if particle trajectory strikes a planar surface. Returns trajec• tory distance (TPLN) (see Section 17.1.2). CYLNDR - Determines if the particle trajectory strikes a cylindrical surface. Returns trajectory distance (TCYL) (see Section 17.1.4). CONE - Determines if the particle trajectory strikes a conical surface. Returns trajectory distance (TCON). SPHERE - Determines if the particle trajectory strikes a spherical surface. Returns trajectory distance (TSPH). CHGTR - Changes USTEP and IRNEW whenever USTEP is larger than the trajectory distance (TPLN, TCYL, TCON, TSPH). FINVAL - Determines the coordinates of the particle trajectory at the point of an intersection with a given surface.

* e.g., see file GEOMAUX MORTRAN (#26) on the EGS4 Distribution Tape. 394 w. R. Nelson and T. M. Jenkins

PLAN2P - Determines the intersection point for two parallel planes by calling PLANE1 twice (when necessary), and CHGTR if a plane is hit. PLAN2X - Determines the intersection point for two crossing planes by calling PLANE1 twice (always), and CHGTR if a plane is hit (PLAN2X is slightly less efficient than PLAN2P). CYL2 - Similar to PLAN2P, but for concentric cylinders. CON2 - Similar to PLAN2P, but for concentric cones. SPH2 - Similar to PLAN2P, but for concentric spheres.

As an example that demonstrates how the writing of SUBROUTINE HOWF AR can be simplified with the aid of the routines listed above, consider the example given in the previous section (see Fig. 17.7). Using SUBROUTINE PLAN2P, eight lines of code involving two calls to PLANE1 can be replaced by a single call as shown in Fig. 17.8 .

SUBROUTINE HOIiFAR; IIt ______11 I ::_~::~~~~_~~_~~:~:~~~_~b_"..~:_t~_e_~:~:~_~~~~~~_~:_~~~_::~~~: ______,," ICOI'IIN/CYLDTA ,EPCONT .PLADTA ,STACK/; "(See text for explanation)" IF (IR(NP) .NE.2) [ IDISC=l; "Discard particles outside the target" ]

IELSE [ "Track particles vithin the target"

CALL CYLNDR(1,l,IHIT, TCYL); "Check the cylinder surface" I IF (IHIT .EQ.1) [ CALL CHGTR(TCYL.4);] "Change if necessary"

I CALL PLAN2P(2.3.1.1.1.-1); "Check the dovnstream plane first and" I " then the upstream one if necessary"

RETURN; I END; Figure 17.8. Simplification of the previous SUBROUTINE HOWF AR listing.

We have also made use of CHGTR, which admittedly is somewhat trivial in the present example and, in fact, may even slow things down due to "overhead" costs involved in calling subprograms. However, we do gain in modularity and code readability and, as we shall discuss shortly, one can completely eliminate this overhead with the use of macro equivalents in place of the conventional CALL statements. Before explaining this, however, we will give another very practical example.

Consider an electromagnetic cascade shower counter made up of alternating slabs of material along the z-axis (e.g., Pb-scintillator layers) and bounded in the x- and y-directions by two pairs of planes to form a right parallelepiped-i.e., a box of slabs. Since we have already pointed out that (almost) all surfaces must be checked to see if they are hit, it becomes apparent that SUBROUTINE HOWFAR consists primarily of three successive PLAN2P calls: Two x-planes - CALL PLAN2P(IPLNX1, IRGNX1, 1, IPLNX2, IRGNX2, 1) ; Two y-planes - CALL PLAN2P(IPLNY1, IRGNYl, 1, IPLNY2, IRGNY2, 1) ; 17. Geometry Methods and Packages 395

Many z-planes - CALL PLAN2P(IPLNZ1 > IRGNZ1 > 1 > IPLNZ2, IRGNZ2, -1) ;.

The reader should consult the code listing for PLAN2P for an explanation of the calling parameters. The important point in this example is the significant reduction of coding effort as a result of using PLAN2P. With more complicated geometries, this becomes even more apparent, as demonstrated in a recent paper by Nelson and Jenkins7.

17.2.4 Mortran3 and macro forms of the geometry routines.

The EGS4 Code System relies heavily on extensions to the Mortran3 language in the form of a set of macros that reside in the file called EGS4MAC MORTRAN (#20 on the EGS4 Distribution Tape)* . One of the new features in EGS4 is the addition of a set of geometry macros that perform the same task as the subroutines listed in the previous section, but in a more efficient way since the sole purpose of each macro is to place subroutine code directly in-line. In other words, the overhead of performing an external call is obviated.

We will not go into any detail on geometry macros other than to give a terse example to show how things work. Consider in Fig. 17.8 the statement

IF (IHIT.EQ.1) [ CALL CHGTR(TCYL,4); ] , which performs the same task as the few lines of code in Fig. 17.7. The equivalent macro template would look like

IF (IHIT.EQ.1) [ $CHGTR(TCYL,4);] , which would get operated on by the following replacement macro:

REPLACE {$CHGTR(#,#);} WITH {IF({P1}.LE.USTEP) [USTEP={P1}; IRNEW={P2};]}

The first # assigns TCYL to {P1}, the second # assigns 4 to {P2}. The replacement code is then inserted directly at the location of $CHGTR, such that the resultant code looks like:

IF (IHIT.EQ.1) [ IF (TVAL.LE.USTEP) [USTEP=TCYL; IRNEW=4;] ]

In other words, we get the same in-line code that we originally started with (see Fig. 17.7)-i.e., we get modularity, readability, and speed.

With this digression behind us, suffice it to say that the geometry macros are only slightly more complicated than the example above. They are used in the same way as

* See Chapter 12 for a discussion of Mortran3 macros. 396 w. R. Nelson and T. M. Jenkins

CALL statements are used. Thus, the example that we have been following could have been written as shown in Fig. 17.9 .

17.2.5 Other EGS4-related geometry packages.

A general purpose EGS4 User Code to do Cartesian coordinate dose deposition studies has been designed by Rogers8. This User Code, called XYZWRN.MOR, makes use of $PLAN2P and associated macros to construct SUBROUTINE HOWF AR. Rectangular parallel beams of photons or electrons are incident on the x-y surface at an arbitrary angle relative to the z-direction. Every voxel (volume element) can have different materials and/or varying densities (e.g., for use with CT data input). Voxel dimensions are completely variable in all three directions.

SUBROUTINE HOWF AR; ,,------______" ' ::_~~:~~~:_~:_:~:~::~~_~b_~~::~_e_::.:'::~_~~~~~~_~:_:~~_~:~~~:------':' ICOKIIf/CYLDTA,EPCOIfT,PLADTA,STACK/; "(See text for explanation)"

'IF (IR(IfP).NE.2) [IDISC-1; "Discard particles outside the target" ] IELSE [ "Track particles within the target" $CYLIfDR(l,l, IHIT, TCYL); "Check the cylinder surface" I IF (IHIT.EQ.1) [ $CHGTR(TCYL,4); "Change if necessary"]

$PLAIf2P(2 3 1 1 1 -1)· "Check the downstream plane first and" I ' , , ", " then the upstream one if necessary" , RETURIf; EIfD;

Figure 17.9. SUBROUTINE HOWFAR listing using macros instead of CALL state• ments.

A similar Cartesian geometry package has been designed by Stevenson9 • Although it is not presented with any particular application in mind, it does demonstrate how one uses the $PLANE1 and $PLAN2P (also called $PLANE2) macros to create a rectilinear system of volume elements, using a region numbering system similar to that used in FLUKA·,6.

General purpose geometry routines are convenient for many applications, although the ease in not having to write the code can cost CPU time. When the geometry is very regular, as is so often the situation, substantial savings (e.g., 20% to 40%) can be obtained using special purpose coding8 . 17. Geometry Methods and Packages 397

17.3 COMBINATORIAL GEOMETRY Geometry routines were devised to make defining complex geometries relatively simple for the user. An early, and relatively good, example of such a routine is Combinatorial Geometry, developed by MAGpo and subsequently adapted for use in the MORSE3 Monte Carlo neutron and gamma-ray transport program. It has since been added to FLUKA4,5, ACCEPTll, and to the EGS4 Code System in the form of a specifically designed User Code called UCSAMPCG (file #58 on the Distribution Tape).

The combinatorial method of specifying input zones in solid bodies is usually more intuitive and simpler than specifications in terms of boundary surfaces. Combinato• rial Geometry describes complex three-dimensional configurations by the use of unions, differences and intersections (OR and AND boolean algebraic equations) of simple ge• ometric bodies. The geometric description subdivides the problem space into zones which are the result of combining one or more of the following simple bodies.

RPP - Right Parallelepiped with sides that are parallel to x, y, and z, the minimum and maximum values of which are specified. BOX - RPP arbitrarily oriented in space. Three mutually perpendicular vectors are specified. SPH - Sphere. The vertex and a scaler denoting the radius, R, are specified. RCC - Right Circular Cylinder. The vertex of the center of the base, the height vector, and a scaler denoting the radius, R, are specified. REC - Right Elliptical Cylinder. The coordinates of the center of the base ellipse, the height vector, and two vectors in the plane of the base defining the major and minor axes are specified. TRC - Truncated Right Angle Cone. The vertex of the center of the base, the height vector, and two scalers denoting the radii of the upper and lower bases are specified. ELL - Ellipsoid. Two vertices denoting the foci, and a scaler, R, denoting the length of the major axis are specified. . WED - Right Angle Wedge*. Three mutually perpendicular vectors are specified. ARB - Arbitrary Convex Polyhedron (4, 5 or 6 sides). An index to each vertex is assigned, and the x, y and z coordinates for each are given. Each of the six faces are then described by a four-digit number giving the indices of the 4 vertex points in that face. (If there are less than six faces, zeros are entered for the non-existent vertices.) For each face, these indices must be entered.in either clockwise or counterclockwise order.

All body types except the right parallelepiped may be oriented arbitrarily with respect to the x, y and z coordinate axes describing the space, but the RPP body type must have sides which are parallel to x, y and z.

A special operator notation involving the symbols (+), (-) and (OR) is used to describe the intersections and unions of these simple bodies. Whenever a body appears in a zone description with a (+) operator, it means that the zone being described is wholly contained within the body, whereas if the body appears in a zone description with a ( - ) operator, it means that the zone being described is wholly outside the body. If the

* Also denoted RAW. 398 W. R. Nelson and T. M. Jenkins body appears with an (OR) operator, it means that the zone being described includes all points in the body. Sometimes, a zone may be described in terms of subzones lumped together by (OR) statements. Subzones are formed as intersects, and then the zone is formed by the union of these intersects.

When (OR) operators are used, there are always two or more of them, and they refer to all body numbers which follow.· That is, all body numbers between (OR)'s, or until the end ofthe zone cards for that zone, are intersected together before the (OR)'s are performed. The body types (except RPP) are shown in Fig. 17.10, and the right parallelepiped (RPP) is shown in Fig. 17.11.

Table 17.1 gives the input required for these geometrical bodies as required by the Combinatorial Geometry package used either in MORSE or EGS.

Table 17.1. Input Required for Various Bodies in Combinatorial Geometry.

ITYPE IALP Real data defining Particular Body Number of 3-5 7-10 11-20 21-30 21-40 41-50 51-60 61-70 cards needed BOX assigned Vx Vy Vz Hlx Hly Hlz lof2 by user H2x H2y H2z H3x H3y H3z 2 of 2 RPP or by Xmin Xmax Ymin Ymax Zmin Zmax 1 SPH code if Vx Vy Vz R 1 RCC left Vx Vy Vz Hx Hy Hz 1 of 2 blank. R 2 of 2 REC Vx Vy Vz Hx Hy Hz lof2 Rlx Rly Rlz R2x R2y R2z 2 of 2 ELL Vlx Vly VIz V2x V2y V2z lof2 R 2 of 2 TRC Vx Vy Vz Hx Hy Hz lof2 Rl R2 2 of 2 WED or Vx Vy Vz Hlx Hly Hiz lof2 RAW H2x H2y H2z H3x H3y H3z 2 of 2 ARB Vix VIy Vlz V2x V2y V2z 1 of 5 V3x V3y V3z V4x V4y V4z 2 of 5 V5x V5y V5z V6x V6y V6z 3 of 5 V7x V7y V7z V8x V8y V8z 4 of 5 Face descriptions 5 of 5

Note: ii = (HIx, HIy,Hlz), etc. 17. Geometry Methods and Packages 399

BOX SPH RCC

R

REC TRC ELL

WED ARB Figure 17.10. Body types and required input dimensions (Combinatorial Ge• ometry). 400 W. R. Nelson and T. M. Jenkins

Z

Ymln ymzuc / ;' / ;' / ;' / ------Zmox

/-----+---r----+------~y

7---+-----,,1- --- Zm,n x

Xmox ---- ...I<::______-Y

Figure 17.11. Right parallelepiped (RPP) with required dimensions (Combi• natorial Geometry).

17.3.1 Constructing bodies using Combinatorial Geometry.

A problem geometry is constructed by 1. Defining the location and orientation of each body required for specifying the input zone. 2. Specifying the input zones as combinations of these bodies. 3. Specifying the volumes of the input zones (if necessary). 4. Specifying the material in each input zone.

As an example, taI:'~ the intersection of a sphere and a cylinder as shown in Fig. 17.12. The location and orientation of each body (SPH and RCC) would be spec• ified with cards similar to those of Table 17.1. The various zones, described by the input cards, are shown under the different combinations. If the zones can be consid• ered as common (e.g., the same material), as in Fig. 17.12( c), the description would be OR +10R +2. That is, the zone is composed of either the sphere (Body 1) or the cylin• der (Body 2). If there are to be two zones as shown in Fig. 17.12(b), or Fig. 17.12( d), the zone descriptions would be; (b) Zone 1: +1; Zone 2: +2 -1; and (d) Zone 1: +1 -2; Zone 2: +2. Where three zones are defined as in Fig. 17.12(e), the zones would be described by; Zone 1: +1 -2; Zone 2: +2 -1; Zone 3: +2 +1. Note that in all these examples, each zone is described only once. That is, a particle can be in only one zone. In Fig. 17.12(c), the geometry defines only one zone. But in the others, there are multiple zones, and each zone is defined in only one place. It is important that every spatial point in the geometry be located in one, and only one, zone since otherwise the program will fail (with appropriate error messages).

When constructing a geometry, one should also try to avoid defining bodies which share a common boundary surface since the program might have difficulties determining in which zone a particle is located. A small overlap is preferable. At the same time, the user must make certain that the overlap is contained in only one of the zones. For example, if Fig. 17.13 were composed of two cylinders which butted against each other, they shouldn't share a common surface as in figure17.13(a), but should overlap as in Fig. 17.13(b). The size of the overlap isn't important since it subsequently disappears in the zone description given below Fig. 17.13(b). 17. Geometry Methods and Packages 401

(al n/BOOYl (b) (e) \gJ_____ BODY 2 INPUT ~ ZONE G ~ 1 OR -lOR -2· 2

(d) (e) ...... (8) o\gJ ·1 -2 2 ·2 -1 3 -2 ·1·

• Indlc~tes sh~ded zone Figure 17.12. Zone descriptions for a sphere (SPH) and a cylinder (RCC) input.

(a) (b) (e)

V BODY 3 Goon 0 Q .. ••• • •••• I I ••••• I I·····••••• ~OOY 2 I·····• •••• I ••••• I I..... I ..... 1- •••• l .....: .....: ':...J l .....: .....: :..! l .....: .....: :...! INPUT ZONE 1 +1 +1 2 +2- +2 -I· 3 +3 -2 -1

• Indlcates shaded zone

Figure 17.13. Zone description for two cylinders (RCC) which share a common boundary.

The universe for Combinatorial Geometry is defined (limited) by the outermost body which must enclose all other bodies within it. In order to turn tracking off (i.e., to have a discard region), this last body is defined as a null region. Fig. 17.13(c) illustrates this last point with the addition of Body 3 surrounding the two cylinders. In 402 W. R. Nelson and T. M. Jenkins the zone description, Zone 3 would be a null zone, and all particles entering it would immediately be discarded.

17.3.2 An example of a complex MORSE·CG geometry.

As an example of a reasonably complex geometry constructed with Combinatorial Ge• ometry, a typical medical accelerator room (with roof removed for better viewing) is shown in Fig. 17.14 . This geometry was used for transmission as well as room scattering studies for both neutrons and gamma rays. The input code that created the geometry is shown in Fig. 17.15.

Medlc~l Acceler~tor Room

Figure 17.14. A typical hospital medical accelerator room.

A more recent trend in geometry packages has been to define relatively simple shapes using, for example, Combinatorial Geometry (with additional shapes, such as trapeziods, tubes, segments of cylinders and cones, etc), and then to replicate these shapes anywhere in space (i. e., with any orientation and any size) to form complex geometries where repetitive shapes are found, as is the case in many of the high-energy particle accelerator detectors. However, one should realize that there is a trade-off between ease of use of a geometry package and computer time. The geometries that allow the user to construct intricate geometries most easily may not have the best tracking algorithms.

17.4 GEOMETRY PACKAGES IN ETRAN, ITS AND FLUKA 17.4.1 ETRAN

The ETRAN code12 *, as it is distributed by the Radiation Shielding Information Center (RSIC) at the Oak Ridge National Laboratory, is capable of treating: 1. A set of homogeneous, semi-infinite slabs (I-D). 2. A set of homogeneous, concentric, right circular cylinders of finite length (3-D).

* Also see Chapters 7 and 8. 17. Geometry Methods and Packages 403

COMMENT - A MEDICAL ACCELERATOR ROOK INPUT GEOMETRY RPP 1 -396.5 518.5 -976.0 411.8 -244.0 290.0 RPP 2 -366.0 366.0 -945.5 381.3 -132.0 234.0 ARB 3 -152.5 -738.1 -131.99 -122.0 -738.1 -131.99 -122.0 -738.1 233.99 -152.5 -738.1 233.99 -152.5 -655.8 -131.99 0.0 -610.0 -131.99 0.0 -610.0 233.99 -152.5 -655.8 233.99 1234. 4158. 8567. 7326. 6512. 3784. ARB 4 -121.99 -738.1 -131.99 54.9 -738.1 -131.99 54.9 -738.1 233.99 -121.99 -738.1 233.99 365.99 -228.8 -131.99 365.99 -427.0 -131.99 365.99 -427.0 233.99 365.99 -228.8 233.99 1234. 4158. 8567. 7326. 6512. 3784. \/ED 5 -365.99 -404.0 -131.99 0.0 -69.0 0.0 289.7 0.0 0.0 0.0 0.0 365.98 ARB 6 -106.8 -403.99 -131.99 -76.3 -403.99 -131.99 -76.3 -403.99 233.99 -106.8 -403.99 233.99 -183.0 -320.3 -131. 99 -76.3 -320.3 -131.99 -76.3 -320.3 233.99 -183.0 -320.3 233.99 1234. 4158. 8567. 7326. 6512. 3784. \/ED 7 -365.99 -403.99 -131.99 0.0 266.7 0.0 259.18 0.0 0.0 0.0 0.0 365.98 \/ED 8 365.99 381.29 -131. 99 0.0 -228.8 0.0 -228.7 0.0 0.0 0.0 0.0 365.98 ARB 9 518.49 -975.99 -131.99 518.49 -274.5 -131.99 518.49 -274.5 233.99 518.49 -975.99 233.99 366.01 -975.99 -131.99 366.01 -427.0 -131.99 366.01 -427.0 233.99 366.01 -975.99 233.99 1234. 4158. 8567. 7326. 6512. 3784. RPP 10 30.5 365.99 -975.99 -945.51 -131.99 233.99 RPP 11 -400.0 520.0 -980.0 415.0 -250.0 300.0 END CON 10000R +1 -2 -9 -100R +50R +60R +70R +80R +4 OR +3 VAC OR +2 -3 -4 -5 -6 -7 -80R +90R +10 NUL +11 -1 END 1 1 1000 0

Figure 17.15. Combinatorial Geometry input used by MORSE to produce Fig. 17.14.

Many cylinders (or slabs) may be specified in a single run, but they are treated independently. That is to say, events that occur within an inner cyli~der are useful in obtaining outer cylinder results, but outer cylinder interactions have no effect' on the results obtained for the inner cylinders. The same is true for the slab geometry-i.e., an individual Monte Carlo run may involve 10 or 20 thicknesses of the same material, with the thin slab interactions used directly in obtaining the results for the thicker slabs (but not the other way around).

A version of I-D ETRAN at the National Bureau of Standards13 also has been developed in order to handle multilayer slabs of different materials. The version that is currently distributed by RSIC, however, is severely limited in its application to a large number of real physical problems, primarily because it lacks a general, versatile, and easy to use geometry package. Overcoming this limitation was the original motivation for the development of the TIGER series, which we shall briefly discuss next. 404 W. R. Nelson and T. M. jenkins

17.4.2 ITS: The Integrated TIGER Series

TIGERl", CYLTRAN15, and ACCEPTll are the base codes that constitute what is called the Integrated TIGER Series (ITS)*. They are all based on ETRAN, and differ from one another primarily in their dimensionality and geometric modeling.

The geometry of the TIGER codes is the simplest of the ITS members. It is strictly a one-dimensional code that is essentially the same as the multilayer NBS version of ETRAN: 1. A particle trajectory is described in terms of the z-coordinate of position and the z-direction cosine. 2. Layers are stacked along the positive z-axis beginning at o.

The CYLTRAN codes employ a fully three-dimensional description of particle tra• jectories with the material geometry consisting of a right circular cylinder of finite length, the axis of which coincides with the z-axis. CYLTRAN is particularly useful for radiation fields that exhibit cylindrical symmetry, such as electron or photon beams.

The ACCEPT codes provide a method for electron-photon transport through three• dimensional multimaterial geometries described by the Combinatorial Geometry scheme developed by MAGIlo (see earlier section).

17.4.3 The FLUKA hadronic cascade code.

FLUKA4.5 is a modular program for computing hadronic and electromagnetic cascades in matter. Designed primarily for use by the high-energy particle physics community, FLUKA may be of interest to a more general audience because: 1. It has been coupled with the EGS4 code in such a manner that electron-photon transport can be done completely within the FLUKA environment, thereby ob• viating the need to create User Codes (note: the hadron interaction option can be turned off). 2. Several geometry packages are available for general use, and user-defined pack• ages can be implemented in a relatively simple way. 3. Some of the FLUKA geometry packages stand on their own, and provide methods and algorithms that might be of use in other electron-photon codes. FLUKA provides its own cylindrical, Cartesian, and spherical-conical geometry pack• age. In addition, FLUKA provides access to a modified version of the Combinatorial Geometry package that has been described earlier. All the geometries provide multi• region, multi-medium environments. FLUKA87 is written in FORTRAN 77, and the user input consists of option cards which are sometimes followed by data cards specific to the option card given. The documentation for FLUKA is rather extensive, representing many man-years of effort by the scientists at CERN-Leipzig-Helsinki who developed the system. In particular, those sections of the User's Guide4•5 concerned with geometries are well-written, and the card input system is nicely conceived.

* Also see Chapter 10. 17. Geometry Methods and Packages 405

REFERENCES 1. W. R. Nelson, H. Hirayama and D. W. O. Rogers, "The EGS4 Code System", SLAC-265 (1985). 2. G. R. Stevenson, "A Cylindrical Geometry Package for HOWFAR in the EGS Elec• tron Gamma Shower Program", CERN internal report HS-RP /TM/80-78 (1980); also of related interest: G. R. Stevenson and T. Lund, "A Spherical-Conical Ge• ometry Package for HOWFAR in the EGS Electron Gamma Shower Program", CERN internal report HS-RP/TM/81-30 (1981). 3. E. A. Straker, P. N. Stevens, D. C. Irving and V. R. Cain, "MORSE-CG, General Purpose Monte-Carlo Multigroup Neutron and Gamma-Ray Transport Code With Combinatorial Geometry", Radiation Shielding Information Center (ORNL) report CCC-203 (1976). 4. P. A. Aarnio, A. Fasso, H. J. Moehring, J. Ranft and G. R. Stevenson, "FLUKA86 User's Guide", CERN Divisional report TIS-RP /168 (1986). 5. P. A. Aarnio, J. Lindgren, J. Ranft, A. Fasso and G. R. Stevenson, "Enhance• ments to the FLUKA86 Program (FLUKA87)", CERN Divisional report TIS• RP /190 (1987). 6. A. J. Cook, "Mortran3 User's Guide", SLAC Computation Research Group tech• nical memorandum CGTM 209 (1983). 7. W. R. Nelson and T. M. Jenkins, "Writing SUBROUTINE HOWFAR for EGS4", Stanford Linear Accelerator report SLAC-TN-87-4 (1987). 8. Private communication with D. W. O. Rogers, National Research Council of Canada (1987). 9. G. R. Stevenson, "A 3-Dimensional Cartesian Geometry Package for HOWFAR in the EGS Electron Gamma Shower Program", CERN Internal report HS• RP /TM/80-60 (1980). 10. W. Guber, J. Nagel, R. Goldstein, P. S. Mettelman and M. H. Kalos, "A Geomet• ric Description Technique Suitable for Computer Analysis of Both the Nuclear and Conventional Vulnerability of Armored Military Vehicles", Mathematical Applications Group, Inc. report MAGI-6701 (1967). 11. J. A. Halbleib, "ACCEPT: A Three-Dimensional Electron/Photon Monte Carlo Transport Code Using Combinatorial Geometry", Sandia National Laboratories report SAND 79-0415 (1979); Nucl. Sci. Eng. 75 (1980) 200. 12. M. J. Berger and S. M. Seltzer, "ETRAN Monte Carlo Code System for Electron and Photon Transport Through Extended Media", Oak Ridge National Labora• tory (RSIC) report CCC-107 (1968). 13. Private communication with S. M. Seltzer (August 1987). 14. J. A. Halbleib and W. H. Vandevender, "TIGER, A One-Dimensional Multilayer Electron/Photon Monte Carlo Transport Code", Nucl. Sci. Eng. 57 (1975) 94. 15. J. A. Halbleib and W. H. Vandevender, "CYLTRAN: A Cylindrical- Geometry Multimaterial Electron/Photon Monte Carlo Transport Code", Nucl. Sci. Eng. 61 (1976) 288. 18. Variance-Reduction Techniques

Alex F. Bielajew and David W. O. Rogers

Division of Physics National Research Council of Canada Ottawa, Canada KIA OR6

18.1 INTRODUCTION

In this chapter, we discuss various techniques which may be used to make calculations more efficient. In some cases, these techniques require that no further approximations be made to the transport physics. In other cases, the gains in computing speed come at the cost of computing results which may be less accurate since approximations are in• troduced. The techniques may be divided into 3 categories: those that concern electron transport only, those that concern photon transport only, and other more general meth• ods. The set of techniques we discuss does not represent an exhaustive list. There is much reference material available, and we only cite a few of theml - 5 • An especially rich source of references is McGrath and Crawford's reportS which contains an annotated bibliography. Instead, we shall concentrate on techniques that have been of consider• able use to the authors and their close colleagues. However, it is appropriate to discuss briefly what we are trying to accomplish by employing variance-reduction techniques.

18.1.1 Variance Reduction or Efficiency Increase?

What we really mean to do when we employ variance-reduction techniques is to reduce the time it takes to calculate a result with a given statistical uncertainty or variance. Analogue Monte Carlo calculations attempt to simulate the full stochastic development of the electromagnetic cascade. Hence, with the calculated result is associated a variance s2, and a statistical uncertainty related to s. The method by which s2 is estimated will not be discussed here. Let us assume that it is calculated by some consistent method. If the variance is too large for our purposes, we run more histories until our criterion is satisfied. How do we estimate how many more histories are needed? Assuming we can do this, what do we do if it is too expensive to simulate the requisite number of histories? We may need a more subtle approach than reducing variance by "grinding out" more histories.

Let us say we devise some "tricks" that allow us to reduce the variance by, say, a factor of 10 using the same number of histories. Let us also imagine that this new subtle approach we have devised takes, say, 20 times longer on average to complete a particle history. (For example, our variance-reduction technique may involve some detailed,

407 408 A. F. Bielajew and D. W. O. Rogers expensive calculation executed for every particle step.) Although we have reduced the variance by a factor of 10, we take 20 times longer to calculate each particle history. We have actually reduced the efficiency by a factor of two! To add to the insult, we have wasted our own time implementing a technique which reduces efficiency!

We require a reliable figure of merit which we may use to estimate gains in efficiency of a given "variance-reduction" technique. It is common to use the efficiency, f, defined by 1 (18.1) f = s2T' where T is a measure of the computing time used (e.g., CPU seconds). The motivation for this choice comes from the following: We assume that mean values of quantities calculated by Monte Carlo methods are distributed normally. It then follows that for calculations performed using identical methods, the quantities s2 Nand s2T, where N is the number of histories, are constant, on average. That both are constant is so because N should be directly proportional to T. By considering the efficiency to be constant, Eqn. 18.1 may be used to estimate the total computing time required to reach a given statistical accuracy, if a preliminary result has already been obtained. For example, if one wishes to reduce the uncertainty, s, by a factor of 2, one needs 4 times as many his• tories. More importantly, Eq. 18.1 allows us to make a quantitative estimate ofthe gain (or loss!) in efficiency resulting from the use of a given "variance-reduction" technique, since it accounts not only for the reduction in variance but also for the increased com• puting time it may take to incorporate the technique. In the aforementioned example using Eq. 18.1, we would obtain f(with subtletY)/f(brute force) = 0.5, a reduction of 1/2. In the following sections, we attempt to present more successful variance-reduction techniques!

18.2 ELECTRON-SPECIFIC METHODS 18.2.1 Geometry Interrogation Reduction

This section might also have been named "Code optimization" or "Don't calculate what you don't really need", or something equivalent. We note that there is a fundamen• tal difference between the transport of photons and electrons in a condensed-history transport code. Photons travel relatively long distances before interacting, and their transport steps are often interrupted by boundary crossings (e.g., entering a new scor• ing region or element of the geometry). The transport of electrons is different, however. In addition to having its step interrupted by boundary crossings or the sampling of discrete interactions, the electron has other constraints on step-size. These constraints may have to do with ensuring that the underlying multiple-scattering theories are not being violated in any way (Chapter 5), or the transport may have to be interrupted so that the equations of transport in an external electromagnetic field may be integrated (Chapter 19). Therefore, it is often unnecessary to make repeated and expensive checks with the geometry routines of the transport code because the electron is being trans• ported in an effectively infinite medium for most of the transport steps. The EGS4 code6 , has an option that allows the user to avoid these redundant geometry subrou• tine calls. With this option switched on, whenever the geometry must be checked for whatever reason, the closest distance to any boundary is calculated and stored. This variable is then decremented by the length of each transport step. If this variable is greater than zero, the electron cannot be close enougb to a boundary to cross it, and the geomety subroutines are not interrogated. If this variable drops to zero or less, the geometry subroutines are called because a boundary crossing may occur. 18. Variance-Reduction Techniques 409

There is no additional approximation involved in this technique. The gain in trans• port efficiency is slightly offset by the extra calculation time that is spent calculating the distance to the closest boundary. (This parameter is not always needed for other aspects of the particle transport.) As an example, consider the case of a pencil beam of I-MeV electrons incident normally on a 0.3-cm slab of carbon divided into twelve 0.025-cm slabs. For this set of simulations, transport and secondary-particle creation thresholds were set at 10 keY kinetic energy, and we used EGS4, setting the energy loss per electron step at 1% for accurate electron transport7 at low energies. The case that interrogates the geometry routines on every step is called the "base case". We invoke the trick of interrogating the geometry routines only when needed, and call this the "RIG" (reduced interrogation of geometry) case. The efficiency ratio, f(RIG)/f(base), was found to be 1.34, a significant improvement. (For EGS-perts only: This was done by calculating DNEAR in the HOWFAR routine of a planar geometry code. A discussion of DNEAR is given on pages 256-258 of the EGS4 manua16 .)

Strictly speaking, this technique may be used for photons as well. For most practi• cal problems, however, the mean free path for the photons in the problem is of the order of, or greater than, the distance between boundaries. For deep-penetration or similar problems, this may not be true. However, this technique is usually more effective at speeding up the electron-transport part of the simulation.

The extra time required to calculate the distance to the closest boundary may be considerable, especially for simulations involving curved surfaces. If this is so, then the efficiency gain may be much less, or efficiency may be lost. It is advisable to test this technique before employing it in "production" runs.

18.2.2 Discard Within a Zone In the previous example, we may be interested just in the energy deposited in the planar zones of the carbon slab. We may therefore deposit the energy of an electron entirely within a zone if that electron's range is less than the distance to any bounding surface of the zone in which it is being transported. We note that we make an approximation in doing this-we neglect the creation and transport of any bremsstrahlung photons that otherwise may be created. For the worst possible case in this particular example, we will be discarding electrons that have a range which is half of the zone thickness, i.e., having a kinetic energy of about 110 keY. The radiative yield of these electrons is only about 0.07%. Therefore, unless we are directly interested in the radiative component of the electron's slowing-down process in this problem, the approximation is an excellent one. For the above example, we realize a gain in the efficiency ratio, f(zonal discard + RIG)/f(base), of about 2.3. In this case, the transport cutoff, below which no electron was tranported, was 10 keY. If we had used a higher cutoff, the efficiency gain would have been less.

Before adopting this technique, the user should carefully analyze the consequences of the approximation-the neglect of bremsstrahlung from the low-energy electron com• ponent.

18.2.3 PRESTA!

In Chapter 5, we discussed a new electron transport algorithm, PRESTA. This algo• rithm, by making improvements to the physical modelling of electron transport, allows 410 A. F. Bielajew and D. W. O. Rogers the use of large electron steps when one is far away from boundaries. This algorithm may therefore be considered to be a variance-reduction technique since it saves comput• ing time by employing small steps only where needed, i.e., in the vicinity of boundaries and interfaces. Continuing with the present example, we calculate the gain in efficiency ratio, f(PRESTA)/f(base), to be 6.1. RIG is always switched on with PRESTA, so actually it is fairer to calculate the efficiency ratio, f(PRESTA)/f(RIG), which was found to be 4.6. If we allow zonal discard as well, we calculate the efficiency ratio, f(zonal discard+PRESTA)/f(zonal discard + RIG), to be 3.1. There is a briefdiscus• sion in Chapter 5 on when PRESTA is expected to run quickly. Basically, the fewer the boundaries and the higher the transport cutoffs, the faster PRESTA runs. A detailed discussion is given in the PRESTA documentation8 •

18.2.4 Range Rejection

As a final example of electron variance reduction, we consider the technique called "range rejection". This is similar to the "discard within a zone" except for a few differ• ences. Instead of discarding the electron (e.g., stopping the transport and depositing the energy "on the spot") because it cannot reach the boundaries of the geometrical el• ement it is in, the electron is discarded because it can not reach some region of interest. For example, a particle detector may contain a sensitive volume where one wishes to calculate energy deposition, or some other quantity. Surrounding this sensitive volume may be shields, converters, walls etc, where one wishes accurate particle transport to be accomplished, but where one does not wish to score quantities directly. Electrons that cannot reach the sensitive volume may be discarded "on the spot", provided that the neglect of the bremsstrahlung photons causes no great inaccuracy.

As an example of range rejection, we consider the case of an ion chamber9. In this case, a cylindrical air cavity, 2 mm in depth and 1.0 cm in radius, is surrounded by 0.5 g/cm2 carbon walls. A flat circular end is irradiated by 1.25-MeV ,-rays incident normally. This approximates the irradiation from a distant source of 60Co. This is a "thick-walled" ion chamber, so-called because its thickness exceeds the range of the maximum-energy electron that can be set in motion by the incident photons. This sets up a condition of "near charged-particle equilibrium" in the vicinity of the cavity. The potential for significant saving in computer time is evident, for many electrons could never reach the cavity. We are interested in calculating the energy deposited to the air in the cavity, and we are not concerned with scoring any quantities in the walls. The range-rejection technique involved calculating the closest distance to the surface of the cavity on every transport step. If this distance exceeded the csda range of the electron, it was discarded. The omission of residual bremsstrahlung-photon creation and transport was negligible in this problem. The EGS code was used. The secondary• particle creation thresholds, as well as the transport cutoff energies, were set at 10-keV kinetic energy. (For EGS-perts only: ECUT=AE=0.521 MeV, PCUT=AP=O.OI MeV, and ESTEPE=O.OI for accurate low-energy simulation.) A factor of 4 increase in efficiency was realized in this case.

Range rejection is a relatively crude but effective method. The version described above neglects residual bremsstrahlung, and is applicable when the discard occurs in one medium. The bremsstrahlung problem could be solved by forcing at least some of the electrons to produce bremsstrahlung. The amount of energy eventually deposited from these photons would have to be weighted accordingly to keep the sampling game "fair". Alternatively, one could fully transport a fraction, say j, of the electrons, and weight 18. Variance-Reduction Techniques 411 any resultant bremsstrahlung photons by 1/f. The other problem, the one of multi• media discard, is difficult to treat in complete generality. The difficulty is primarily a geometrical one. The shortest distance to the scoring region is the shortest geometrical path only when the transport can occur in one medium. The shortest distance we need to calculate for range rejection is the path along which the energy loss is a minimum. It is not difficult to imagine that finding the "shortest" path for transport in more than one medium may be very difficult. For special cases, this may be done or approximations may be made. The "payoff" is worth it as large gains in efficiency may be realized, as seen in the above example.

18.3 PHOTON-SPECIFIC METHODS 18.3.1 Interaction Forcing

In problems where the interaction of photons is of interest, efficiency may be lost because photons leave the geometry of the simulation without interacting. The probability distribution for a photon interaction is:

(18.2) where 0 5 ,x < 00 and ,x is the distance measured in mean free paths. It can easily be shown that sampling ,x from this distribution can be accomplished by the following formula*: ,x = -In(1 - e), (18.3) where e is a random number uniform on the range, 0 5 e < 1. Since,x extends to infinity and the number of photon mean free paths across the geometry in any practical problem is finite, there is a non-zero and often large probability that photons leave the geometry of interest without interacting. If they don't interact, we waste computing time tracking these photons through the geometry.

Fortunately, this waste may be prevented. We can force these photons to interact. The method by which this can be achieved is remarkably simple. We construct the probability distribution, (,x)d,x _ e-l.d,x (18.4) P - foA e-l. I dN ' where A is the total number of mean free paths along the direction of motion of the photon to the end of the geometry. This ,x is restricted to the range, 0 5 ,x < A, and ,x is selected from the equation,

(18.5)

We see from Eq. 18.5 that we recover Eq. 18.3 in the limit A --+ 00. Since we have forced the photon to interact within the geometry of the simulation, we must weight the quantites scored resulting from this interaction. This weighting takes the form,

(18.6)

* It is conventional to use the expression, ,x = -In(e), since both 1 - eand eare distributed uniformly on (0,1), but the former expression executes more slowly. However, it has a closer connection to the following mathematical development. 412 A. F. Bielajew and D. W. O. Rogers where w' is the new "weighting" factor and w is the old weighting factor. When in• teraction forcing is used, the weighting factor, 1 - e-A , simply multiplies the old one. This factor is the probability that the photon would have interacted before leaving the geometry of the simulation. This variance-reduction technique may be used repeatedly to force the interaction of succeeding generations of scattered photons. It may also be used in conjunction with other variance-reduction techniques. Interaction forcing also may be used in electron problems to force the interaction of bremsstrahlung photons.

On first inspection, one might be tempted to think that the calculation of A may be difficult, in general. Indeed, this calculation is quite difficult and involves summing the contributions to A along the photon's direction through all the geometrical elements and materials along the way. Fortunately, most of this calculation is present in any Monte Carlo code because it must possess the capability of transporting the photons through this geometry! This interaction-forcing capability can be included in the EGS code in a completely general, geometry-independent fashion with only about 30 lines of code1o .

The increase in efficiency can be dramatic if one forces the photons to interact. For example, for ion-chamber calculations similar to those described in Section IS.2.4 and discussed in detail elsewhere9 , the efficiency improved by the factor 2.3. In this calculation, only about 6% of the photons would have interacted in the chamber. In calculating the dose to skin from contaminant electrons arising from the interaction of 60CO (e.g., 1.25-MeV ,'s) in 100 cm of air10, the calculation executed 7 times more efficiently after forcing the photons to interact. In calculating the dose from 60Co directly in a O.OOI-cm slice of tissue where normally only 6 x 10-5 of the photons interact, the efficiency improved by a factor of 260010,11.

18.3.2 Exponential Transform. Russian Roulette. and Particle Splitting

The exponential transform is a variance-reduction technique designed to enhance ef• ficiency for either deep-penetration problems (e.g., shielding calculations) or surface problems (e.g., build-up in photon beams). It is often used in neutron Monte Carlo work, and is directly applicable to photons as well.

Consider the simple problem where we are interested in the surface or deep pene• tration in a simple slab geometry with the planes of the geometry normal to the z-axis. We then scale the interaction probability making use of the following formula:

~ = A(l - Cp), (lS.7) where A is the distance measured in the number of mean free path's, ~ is the scaled distance, p is the cosine of the angle the photon makes with the z-axis, and C is a parameter that adjusts the magnitude of the scaling. The interaction-probability distribution is (IS.8) where the overall multiplier 1 - C p is introduced to ensure that the probability is correctly normalized, i.e., 1000 p(A)dA = 1. For C = 0, we have the unbiased probability distribution e->'dA. One sees that for 0 < C < 1, the average distance to an interaction 18. Variance-Reduction Techniques 413 in the forward direction is stretched* . For G < 0, the average distance to the next interaction is shortened in the forward direction. Examples of a stretched and shortened distribution are given in Fig. 18.1. In order to play the game fairly, we must obtain the appropriate weighting function to apply to all subsequent scoring functions. This is obtained by demanding that the overall probability be unchanged. That is, we require

w'p(>.)d>. = wp(>.)d>., (18.9) where w' is the new weighting factor and w is the old weighting factor. Solving Eqn. 18.9 for w' yields, (18.10)

Finally, we require a technique to sample the stretched or shortened number of mean free paths to the next interaction point from a random number. It is easily shown that >. is selected using the formula:

>. = -lnW/(1 - GJl), (18.11) where eis a random number chosen uniformly over the range, 0 < e::; 1.

1.5",\,\".-,,-.-..-,,-.-.,-,,-.-.,-,,-.-.,-, ·· ·· .c 1.0 .c o'" ~ Co s:: ....o C=O +' U \~<'" 0.5 ~ C1J'" +' s:: -----~~-Dt------:~~,,~:: ______

sh 0 rt~~~-~-g------______----...... -- _.. -- 0.0~-L~~-L-L~~-L~~-L~~-L-L~~~-- 0.0 0.5 1.0 1.5 2.0 2.5 z (mfp 's)

Figure 18.1. Examples of a stretched (G = 1/2) and shortened (G = -1) distribution compared to an unbiased one (G = 0). In all three cases, Jl = 1. For all three curves Ioco p(>.)d>. is unity. The horizontal axis is in units of the number of mean free path's (mfp's).

* Note that the average number of mean free paths to an interaction, (>.), is given by (>.) = It >.p(>.)d>. = 1-~'" 414 A. F. Bielajew and D. W. O. Rogers

In order to use the exponential transform for photons going in any direction, one must obey the restriction, 101 < 1. "Path-length stretching" means that 0 < a < 1, i.e., forward-directed photons are made to penetrate deeper. "Path-length shortening" means that -1 < a < 0, i.e., forward-directed photons are made to interact closer to the surface. For studies of surface regions, one may use a stronger biasing, e.g., a ::; -1. If one used a ::; -1 indisciminantly, then nonsense would result for particles going in the backward direction, i.e., I" < O. Sampled distances and weighting factors become negative. It is possible to use a ::; -1 for special, but important cases. If one restricts the biasing to the incident photons which are directed along the axis of interest (i.e., I" > 0), then a ::; -1 may be used. If one uses this severe biasing, then as seen in Eqn. 18.10, weighting factors for the occasional photon that penetrates very deeply can get very large. If this photon backscatters and interacts in the surface region where one is interested in gaining efficiency, the calculated variance can be undesirably increased. It is advisable to use a "splitting" techniquel, dividing these large-weight particles into N smaller ones each with a new weight w' = w / N, if they threaten to enter the region of interest. Thresholds for activating this splitting technique and splitting fractions are difficult to specify, and choosing them is largely a matter of experience with a given type of application. The same comment applies when particle weights become vary small. If this happens and the photon is headed away from the region of interest, it is advisable to play "russian roulette"l. This technique works as follows: Select a random number. If this random number lies above a threshold, say a, the photon is discarded without scoring any quantity of interest. If the random number turns out to be below a, the photon is allowed to "survive" but with a new weight, w' = w / a, insuring the fairness of the Monte Carlo "game". This technique of "weight windowing" is recommended for use with the exponential transform12 to save computing time and to avoid the unwanted increase in variance associated with large weight particles.

Russian roulette and splitting* can be used in conjunction with the exponential transform, but they enjoy much use by themselves in applications where the region of interest of a given application comprises only a fraction of the geometry of the simulation. Photons are "split" as they approach a region of interest, and made to play "russian roulette" as they recede. The three techniques, exponential transform, russian roulette and particle splitting, are part of the "black art" of Monte Carlo. It is difficult to specify more than the most general guidelines on when they would be expected to work well. One should test them before employing them in large scale production runs.

Finally, we conclude this section 'Nith an example of severe exponential-transform biasing with the aim of improving surface dose in the calculation of a photon depth-dose curvelO• In this case, 7-MeV "'!'s were normally incident on a 30-cm slab of water. The results are summarized in Table 18.1.

In each case, the computing time was the same. Therefore, the relative efficiency reflects the relative values of 1/82 • As a decreases, the calculational efficiency for scoring dose at the surface increases while, in general, it decreases for the deepest bin. The relative efficiency was defined to be unity for a = 0 for each depth bin. For the deepest bin, there is an increase initially because the mean free path is 39 cm. At first, the number of interactions in the 10-30 cm bin increases! Note that as a is decreased, the number of histories per unit computing time decreases. This is because more electrons

* According to Kahnl , both the ideas and terminology for russian roulette and split• ting are attributable to J. von Neumann and S. Ulam. 18. Variance-Reduction Techniques 415 are being set in motion, primarily at the surface. These electrons have smaller weights, however, to make it a "fair game".

Table 1B.1. This series of calculations examines a case where a gain in the com• putational efficiency at the surface is desired. Each calculation took the same amount of computing time. In general, efficiency at the surface increases with decreased C while efficiency worsens at depth.

C Histories Relative efficiency on calculated dose lOs 0-0.25 em 6.0-7.0 em lO-30 em

0 100 =1 =1 =1 -1 70 1.0 3.5 -3 55 1.4 1.2 0.6 -6 50 2.7 2.8 0.1

18.4 OTHER TRICKS 18.4.1 Sectioned Problems, Use of Pre-Computed Results

One approach to saving computer time is to split the problem into separate, manage• able parts using the results of previous Monte Carlo simulations as part of another simulation. These applications tend to be very specialized, although unique problems sometimes demand unique approches. For illustration, we shall present two related examples.

Fluence-to-dose conversion factors for monoenergetic, infinitely broad electron and photon beams incident normally on semi-infinite slabs of tissue and water have been calculated previouslyll,lS. These factors, called J(E(Z), vary with depth, z, and with the energy of the beam, E, at the water surface. Dose due to an arbitrary incident spectrum as a function of depth D(z) is calculated from the following relation:

(18.12) where

Another example is the study of the effects of scatter in a 60Co therapy unit14• The calculation was divided into three parts. Firstly, the source capsule itself was modelled, and the phase-space parameters (energy, direction, position) of those parti• cles leaving the source capsule and entering the collimator system were stored. About 2 X 106 particles were stored in this fashion, taking about 24 hrs of VAX 11 /780 CPU time for executing the simulation. These data were then used repeatedly in modelling 416 A. F. Bielajew and D. W. O. Rogers the transport of particles through the collimators and filters of the therapy head. The approximation inherent at this stage of the calculation is that the interaction between the source capsule and the rest of the therapy head can be ignored. However, since the capsule is small with respect to the therapy head, and we are interested in calculating the effects of the radiation somewhat downstream from the therapy head, the approx• imation is an excellent one. Another aspect of this calculation was that the effect of the contaminant electrons downstream from the therapy head was studied. Again, this part of the calculation was "split off" and done by the method described previously. That is, Eq. 18.15 was used to calculate the depth-dose profiles in tissue.

By splitting the problem into 3 parts, the total amount of CPU time used to simulate the 60Co therapy head14 was 5-16 hours of VAX 11/780 CPU time for each geometry. If we had attempted to simulate the problem entirely without "dividing and conquering", the amount of CPU time required would have been 10 to 100 times longer.

18.4.2 Geometry-Equivalence Theorem

A special but important subset of Monte Carlo calculations is normal beam incidence on semi-infinite geometries, with or without infinite planar inhomogeneities. The use of a simple theorem, called the "geometry-equivalence" or "reciprocity" theorem, provides an elegant technique for calculating some results more quickly. First we prove the theorem.

Imagine that we have a zero radius beam coincident with the z-axis impinging on the geometry described above. We "measure" a response that must have the form f(z, Ip!), where P is the cylindrical radius. This functional form holds true since there is no preferred azimuthal direction in the problem. If the beam is now shifted off the axis by an amount p', then the new functional form of the response must have the form, f(z, Ip - pI!), by translational symmetry. Finally, consider that we have a finite circular beam of radius Ph, and we wish to integrate the response over a finite-size detection region with circular radius Pd. This integrated response has the form,

IP'I~P~ JIPI~Pd F(z, Ph, Pd) = J dp' dp f(z, Ip - p'!), (18.13)

where Jlpl~Pd pdp is shorthand for J02'" drjl Jci d dp. If we exchange integration indicies in Eq. 18.16, then we obtain the reciprocity relationship,

{18.14)

What Eqn. 18.14 means is the following: If we have a circular beam of radius Ph and a circular detection region of radius Pd, then we calculate the same response if we had used a circular beam of radius Pd and a circular detection region of radius Ph! The gain in efficiency comes when we wish to calculate the response of a small detector in a large-area beam. If one does the calculation directly, then much computer time is squandered tracking particles that may never reach the detector. By using the reciprocity theorem, one calculates the same quantity faster. 18. Variance-Reduction Techniques 417

In an extreme form, the reciprocity theorem takes the form15,

limF(z, Pb, f) = limF(z, f, Pb), (18.15) f~O (-+0 which allows one to calculate the "central-axis" depth dose for a finite-radius beam by scoring the dose in a finite region from a zero-area beam. The gain in efficiency in this case is infinite! The radius, Pb, can even be infinite to simulate a broad beam.

A few remarks about the reciprocity theorem and it's derivation should be made. If the response function, f(z, IpD, has a finite lateral extent, then the restriction that the geometry should be semi-infinite may be relaxed as long as the geometry, including the inhomogeneous slabs, is big enough to contain all of the incident beam once the detec• tion region radius and the beam radius are exchanged. Electron-photon beams produce infinitely wide response functions owing to radiation scatter and bremsstrahlung-photon creation. In practice, however, the lateral tails often contribute so little that simulation (and experiments!) in finite geometries is useful. Also, in the above development it was assumed that the detection region was infinitely thin. This is not a necessary approxi• mation, but this detail was omitted for clarity. The interested reader is encouraged to repeat the derivation with a detection region of finite extent. The derivation proceeds in the same manner, but with more cumbersome equations.

18.4.3 Use of Geometry Symmetry

In the previous section, we saw that the use of some of the inherent symmetry of the geometry realized considerable increase in efficiency. Some uses of symmetry are more obvious, for example, the use of cylindrical-planar or spherical-conical simulation geometries if both the source and target configurations contain these symmetries. Other uses of symmetry are less obvious, but still important. These applications involve the use of reflecting planes to mimic some of the inherent symmetry.

For example, consider the geometry depicted in Fig. 18.2. In this case, an infinite square lattice of cylinders is irradiated uniformly from the top. The cylinders are all uniform and aligned. Clearly, one cannot model an infinite array of cylinders. If one tried, one would have to pick a finite set, and decide somehow that it was big enough. Instead, it is much more efficient to exploit the symmetry of the problem. It turns out that in this instance, one needs to transport particles in only 1/Sth of a cylinder! To see this, we find the symmetries in this problem. In Fig. 1S.2, we have drawn three planes of symmetry in the problem, planes a, b, and c. There is reflection symmetry for each of these planes. Therefore, to mimic the infinite lattice, any particles that strike these reflecting planes should be reflected. One needs to transport particles only in the region bounded by the reflecting planes. Because of the highly symmetric nature of the problem, we only need to perform the simulation in a portion of the cylinder, and the "response" functions for the rest of the lattice are found by reflection.

The rule for particle reflection about a plane of arbitrary orientation is easy to derive. Let U be the unit-direction vector of a particle, and ii be the unit-direction normal of the reflecting plane. Divide the particle's direction vector into two portions, ull parallel to ii, and U.L perpendicular to ii. The parallel part gets reflected, uil = -ulI, 418 A. F. Bielajew and D. W. O. Rogers and the perpendicular part remains unchanged, u~ = U.L. That is, the new direction vector is U' = -ull + U.L. Another way of writing this is,

u-, = u- - 2(-u·n-)• n. (18.16)

Applying Eqn. 18.16 to the problem in Fig. 18.2, we have: For reflection at plane a,

(u~,u~,u~) = (-UZ,u1"UZ ). For reflection at plane b, (u~,u~,u~) = (U.,,-ulI,UZ ). For reflection at plane c, (u~, u~, u~) = (-ulI , -U." u,,). The use of this reflection technique can result in great gains in efficiency. Most practical problems will not enjoy such a great amount of symmetry, but one is encouraged to make use of any available symmetry. The saving in computing time is well worth the extra care and coding.

: y-axis

Figure 18.2. Top end view of an infinite square lattice of cylinders. Three planes of symmetry are drawn, a, b, and c. A complete simulation of the entire lattice may be performed by restricting the transport to the interior of the three planes. When a particle strikes a plane, it is reflected back in, thereby mimicking the symmetry associated with this plane. 18. Variance-Reduction Techniques 419

REFERENCES 1. H. Kahn(1956), in Symposium on Monte Carlo Methods, edited by H. A. Meyer, (John Wiley and Sons, New York, 1956) 146. 2. J. M. Hennersley and D. C. Handscomb, Monte Carlo Methods, (John Wiley and Sons, New York, 1964). 3. E. J. McGrath and D. F. Crawford, "Techniques for Efficient Monte Carlo Simu• lation, Vols. I, II, and III", Radiation Shielding Information Center, Oak Ridge National Laboratory report ORNL-RSIC-38. (1975). 4. G. A. Carlsson, "Effective Use of Monte Carlo Methods", Department of Radi• ology, Linkoping University, report ULI-RAD-R-049. (1981). 5. T. Lund, "An Introduction to the Monte Carlo Method", CERN report HS• RP /067 (1981). 6. W. R. Nelson, H. Hirayama and D. W. O. Rogers "The EGS4 Code System", Stanford Linear Accelerator report SLAC-265 (1986). 7. D. W. O. Rogers, "Low Energy Electron Transport with EGS", Nuci. Instr. Meth. in Phys. Res. 227 (1984) 535. 8. A. F. Bielajew and D. W. O. Rogers, "PRESTA - The "Parameter Reduced Electron-Step Transport Algorithm" for Electron Monte Carlo Transport", Na• tional Research Council of Canada Report PIRS-0042, and Nuci. Instr. Meth. B18 (1987) 165. 9. A. F. Bielajew, D. W. O. Rogers and A. E. Nahum A E, "The Monte Carlo Sim• ulation of Ion Chamber Response to 6OCo - Resolution of Anomalies Associated with Interfaces", Phys. Med. BioI. 30 (1985) 419. 10. D. W. o. Rogers and A. F. Bielajew, "The Use of EGS for Monte Carlo Calcu• lations in Medical Physics", RCC report PXNR-2692 (1984). 11. A. F. Bielajew and D. W. O. Rogers, "Calculated Buildup Curves for Photons With Energies up to 60CO", Med. Phys. 12 (1985) 738. 12. J. S. Hendricks and T. E. Booth, in Monte-Carlo Methods and Applications in Neutronics, Photonics and Statistical Physics, edited by R. Alcouffe, R. Dautray, A. Forster, G. Ledanois and B. Mercier, (Springer-Verlag, Berlin, 1985) 83. 13. D. W. o. Rogers, "Fluence to Dose Conversion Factors Calculated with EGS3 for Electrons from 100 ke V to 20 Ge V and Photons from 11 ke V to 20 Ge V" , Health Phys. 46 (1984) 891. 14. D. W. O. Rogers, G. M. Ewart, A. F. Bielajew and G. vanDyk, "Calculation of Electron Contamination in a 60CO Therapy Beam", paper IAEA-SM-298/48 in Dosimetry in Radiotherapy, Vol. 1, (IAEA, Vienna, 1989). 15. ICRU, "Radiation Dosimetry: Electron Beams with Energies Between 1 and 50 MeV", ICRU Report 35, ICRU, Washington D.C. (1984). 19. Electron Transport in E and B Fields

Alex F. Bielajew

Division of Physics National Research Council of Canada Ottawa, Canada KIA OR6

19.1 INTRODUCTION

In this chapter, we discuss the fundamentals of electron transport in static external electric and magnetic fields in vacuum and dense media. By "static" and "external" is meant that macroscopic E and/or B fields are set up in the region where the electron transport is taking place. For example, a high-energy particle detector may be placed in a constant magnetic field so that the momentum of charged particles may be analyzed. The external fields are considered to be static in the sense that they do not change with time during the course of the simulations. This is not a fundamental constraint, but is imposed for simplicity. The bulk of the discussion concerns the theoretical viabil• ity of performing electron transport in dense media in the presence of external fields. The trajectories of particles in this case can be quite complicated. The particles can be subjected to a myriad of forces - de-accelerations due to inelastic processes with orbital electrons and nuclei, elastic deflections due to attraction or repulsion in the nu• clear electric field, accelerations or de-accelerations by the external electric field, and deflections by the external electric and magnetic fields.

In comparison to the effects of the internal processes of multiple scattering and inelastic collisions, the effect of the external fields can be quite dramatic. Electric field strengths can be as high as 2 MV /(g/cm2). The rate of a charged particle's change in energy due to this field can be equal in magnitude to the rate of energy loss of high• energy electrons in matter. We wish to establish a method, even if it is a "brute force" one, that will allow us to do charged-particle transport under these circumstances. We do not wish to treat the effects of the external fields as perturbations on the field-free transport in media. Yet, we don't wish to discard all the theoretical work that has been achieved in field-free transport. Rather, we shall retain what we know about inelastic energy-loss mechanisms and multiple scattering, and attempt to include the effect of the external fields, albeit in a simple-minded fashion.

We commence the chapter with a "review" discussion of charged-particle transport in external fields. We set up the equations and then solve them in vacuum. The vacuum solutions will playa role in the benchmarking of the differential equations as modelled in the Monte Carlo code. We then prove formally under which circumstances the vacuum transport equations can be "tacked on" to the field-free transport with little error. In

421 422 A. F. Bielajew general, it will be shown that the vacuum external-field transport can be superimposed upon the field-free transport as long as the charged-particle steps are short enough.

In the last part of the chapter, we discuss the practical application of the external• field transport equations to a Monte Carlo code. As a specific example, we shall use the EGS codel with which the author is most familiar. The method of application discussed will be very general, and may be applied equally well to any condensed• history electron transport code. The ITS codes2 have also been adapted for transport in external fields3- 5 .

19.2 EQUATIONS OF MOTION IN A VACUUM

In this section, we derive the differential equations of motion for charged particles moving in arbitrary electric and magnetic fields, and we consider transport in a vacuum.

The change in momentum with time due to electric E and magnetic B fields is given by the Lorentz force equation6 ,

dp ...... dt = e(E + v x B), (19.1 ) where p is the momentum, t is the time, e is the charge of the particle (negative for negatrons and positive for positrons), and v is the velocity. This equation may be cast in the following form: (19.2) where c is the speed of light, iJ is the velocity in units of the speed of light, iJ = vic, , is the familiar relativistic factor (1 - iJ· iJ)-l/2, ds is the differential path length that can be related to the time, t, through the relation, ds = (ds/dt)dt = (cf3)dt, and mo is the rest mass of the charged particle.

If on~ ~pli!:s the differential operator to the left-hand side of Eqn. 19.2, express~s d, as ,3 f3(f3 . d(3), and takes the inner product of both sides of the equation with f3, one obtains ... d{3 e -. -+ -t -t -+ .... -d = 2f3 (E - f3(E·f3) + cf3xB). (19.3) s moc, One can perform similar manipulations to derive a particularly useful form of this equation, dii e ...... -d 2f32 (E - u(E· u) cf3u x B), (19.4) s = moc , + where ii is the unit direction vector of the charged particle*. We note, from Eqn. 19.4, that ii . (dii/ ds) = O. This means that the change in the direction vector is transverse to its direction. This also follows directly from the fact that the magnitude is, by fiat, unity, i.e., ii· ii = 1, from which it follows that d(ii· ii) = 2ii·dii = O. We shall make use of Eqn. 19.4 when applying external-field transport to a Monte Carlo code.

* This equation, with B = 0, was stated erroneously in a previous referencel3. The error, ii, mistakenly set to Pin Eq. 19.4, was inconsequential since f3 ~ 1 in the calculations associated with that report. 19. Electron Transport in Eand B Fields 423

19.2.1 Special Cases: E =Constant, B = 0; B =Constant, E = 0

In this section, we present the solutions for the special cases of uniform electric and magnetic fields in vacuum. These solutions may be used by Monte Carlo codes for transport in vacuum, and they may be used for "benchmarking" the transport equations. Rather than repeat derivations which demand much careful arithmetic manipulation, we simply present the solutions, and encourage the reader to verify them at his or her leisure.

It can be shown that the general solution for the equation of motion in the constant electric-field case takes the form,

moc2,o ( (eEX1.) . ( eEx1. )) xII = --E- cosh 2 (3 -1 (3l1o smh 2 (3 , (19.5) e moc,o 1.0 + moc,o 1.0 where XOI,1.) is the distance from the origin in the direction (parallel, perpendicular) to E, ,0 is the value of, at xII = °and Xl. = 0, E = lEI, and (3(11,1.)0 is the initial component of ~ in the direction (parallel, perpendicular) to E.

This equation contains two interesting limiting cases. In the "weak-field" limit (WFL), e~.,) --+ 0, Eqn. 19.5 becomes moc "10 .Lo

1. (3110X1. eExl 1m XII ---t --+ . (19.6) W F L (31.0 2moc2,o(3io

The non-relativistic limit (NRL) may be obtained from Eqn. 19.6 by taking the limit, ,0 --+ 1, with the result, eEt2 lim xII ---t vllot + -2-' (19.7) NRL mo where VOI,1.)O = c(3(II,1.)O, and t, the time, is expressed as t = X1.o/V1.o. The NRL solution, a parabola, should be familiar from elementary mechanics.

Note that the previous three equations, Eqs. 19.5 - 19.7, are valid for positrons (e = leI) and negatrons (e = -leI). As well, if the particle starts out in a direction opposing the electric force, then XII can be a double-valued function of Xl. or t. Addition• ally, as a consequence of the decomposition into parallel and perpendicular components, the motion always takes place in the plane defined by (3110 and (31.0.

The equation of motion in a constant magnetic field should be familiar from el• ementary mechanics for either relativistic or non-relativistic charged particles. The development of the solution is considerably simplified owing to the fact that the mag• netic field does not alter the particle's energy. This results directly from the force and the particle's velocity being always perpendicular. The equations of motion are,

Xl.1 = P12 cos (eB;II)) + P11 sin (eB;II) , (19.8) eB ((1- PII eB PII 424 A. F. Bielajew and

Xl.2 -_ --Pil (1 - cos (eBXII))--0- + -Pi2.sm (eBXII)--0- . (19.9) eB PII eB PII

In these equations, B = rEiI, and the direction of motion has been resolved into 3 components, (Xl.l, Xl.2, XII), forming an orthogonal triad with XII aligned with the B-field. The momentum is also resolved into the three components, (plot, Pl.2, PI!)' with initial values, (pil' Pi2' pfI) when (Xil' xi" xfI) = (0, 0, 0). The component of momentum parallel to B is a constant of the motion, i.e., PII = pfI, as is the energy and the velocity in the direction of the B-field.

As seen from Eqs. 19.8 and 19.9, the motion in the Xl.l-Xl.2 plane is a circle centered at (pi2/eB, -Pil/eB) with radius Ip1/(eB)1 where Ip11 = J(P1l)2 + (P12)2. For relativistic particles, one should substitute pO = mo'Yoc(3, and pO = movo may be used for non-relativistic particles. In other words, the particle "spirals" along the B• axis with a constant speed and radius. If one "sights" along the B-axis, positrons move in the counter-clockwise direction while negatrons move in the clockwise direction.

19.3 TRANSPORT IN A MEDIUM The transport of charged particles in media with external electric and magnetic fields can be quite complicated and difficult to handle theoretically. However, we still can ac• complish the simulation of charged-particle transport in media with certain approxima• tions. In this section, we state these approximations using a very general development.

Consider the motion of a charged particle in a medium with external electric and magnetic fields. Besides the coupling to the external fields, the particle is acted upon by elastic and inelastic forces. Assuming the medium is isoptropic and homogeneous, the equation of motion takes the general form, dp - - - _ _ di = Fret(E(t)) + F m.(E(t)) + F.m(x(t), E(t), u(t)), (19.10) where p is the momentum, t is the time, Fret is the force due to inelastic (retarding) forces, FmI is the force due to elastic (multiple §.cattering) forces, and F.m is the force due to external (~lectric and magnetic) forces. We may integrate Eqn. 19.1O implicitly to obtain,

v = Vo + mo~(E) it dt' {Fret(E(t')) + Fm.(E(t')) + F.m(:x(t'),E(t), u(t'))}. (19.11)

x = Xo + vot + it dt"v(t"). {19.12)

These are very complex equations of motion with much interplay among all the constituents. Fret accounts for inelastic processes having to do mostly with electron• electron interactions and bremsstrahlung photon creation in the nuclear field. This force affects mostly E, the energy, and consequently v, the magnitude of the velocity, 19. Electron Transport in Eand B Fields 425

V. There is some deflection as well, but angular deflection is dominated by multiple scattering. Fret couples to Fm. and Fem because they all depend on E. Fm. accounts for elastic processes having to do mostly with deflections caused by the nuclei of the medium. It changes the direction of the velocity, although the energy lost to the nuclear recoil usually can be ignored. Consequently, Fm. couples to Fem since the latter depends on U, the direction of motion of the particle. (By definition, U is a unit vector.) Fem accounts for the interaction with the external electric and magnetic fields. It depends on E, u and also X, if the external fields are spatially dependent. Fem can alter both the magnitude and direction of V, thereby coupling directly to both Fm. and Fret. Moreover, outside the integral in Eqn. 19.11 is an overall factor of l/(mo,(E)), owing to the fact that the mass changes when the energy changes.

To complicate matters even further, we do not know the exact nature of Fret and Fm.' For "microscopic" Monte Carlo methods where we model every electron interaction, we can only say something about the momentum before and after the interaction. For complete rigour, one would have to solve the quantum-mechanical equations of motion incorporating external fields. However, unless the external fields are very strong, they may be treated in a perturbation formalism for microscopic Monte Carlo methods.

In this chapter, we restrict ourselves to "condensed-history" Monte Carlo, for which the arguments are even more subtle. For complete rigour, one should incorporate the external fields into the Boltzmann transport equation and solve directly. To our knowl• edge, this has not been attempted. Instead, we attempt to superimpose the transport in the external fields upon "field-free" charged particle transport, and discuss what approximations need to be made.

A major difficulty arises from the fact that during a condensed-history transport step, the trajectory of the particle is not known, nor are the exact forms of Fret and Fm. known. Instead, we wish to make use of the already existing statistical treatments of Fret and FmOl for example, Bethe-Bloch slowing-down theory7-9 and Moliere10•ll multiple-scattering theory. If the external fields are different for the different possible particle trajectories, we face an unresolvable ambiguity. Therefore, we must demand that the transport steps be small enough so that the fields do not change very much over the course of the step. With this approximation, Eqn. 19.11 becomes,

v = Vo + mo~(E) 1t dt' {{Fret(E(t'))) + (F m.(E(t'))) + Fem(xo, E(t'), u(t'))}, (19.13) where Xo is the position at the beginning of the particle's step, and the (F) 's denote that we are now employing the statistical formulations of the physical effects produced by {F}.

We make the further approximation that the energy does not change very much during the course of a particle step. With the approximation of small energy losses, Eqn. 19.13 becomes

where Eo is the energy evaluated at the beginning of the particle's step. 426 A. F. Bielajew

Finally, we make the approximation that the direction angle 11 does not change much over the course of the step. While this can be accomplished by reducing the size of the step for most of the charged-particle steps, occasionally large-angle scatterings associated with single nucleus-electron interactions will occur. Therefore, this approxi• mation must break down at least some of the time. Fortunately, multiple scattering is dominated by small-angle events with only relatively few large-angle ones12. With this approximation,

t - ...... v = Vo + (E ) {{Fret(Eo») + (F",.(E o») + F.",(xo, Eo, llo)}. (19.15) mo, 0 where 110 is the direction vector evaluated at the beginning of the particle's step. At this point, by virtue of the approximations we have made, the F's are decoupled. Note that nothing has been said about the sizes of the F's relative to one another, except that they do not perturb the "force-free" trajectory too much.

To make a closer connection to applying the equations to Monte Carlo simulations, we recast the equations so that they are dependent upon the total path length of the step, s, rather than the time. To this end we write,

t = f' ds, (19.16) 10 v which, to 1st-order may be written,

_ s t- - (1 + ~V(Eo») , (19.17) Vo Vo where ~v(Eo) = Vo - v(Eo) accounts for energy losses in the relationship expressed by Eqn. 19.16. From Eqn. 19.15, we find that to 1st-order in the F's,

s ...... ~v(Eo) = - (E) 110 • {(Fret(Eo») + (F",,(Eo») + F.",(xo, Eo, llo)}, (19.18) mo, 0 Vo where 110 is the direction vector evaluated at the start of the electron step. Using Eqs. 19.12, 19.15 - 19.18, we find that the new direction vector, 11 = vflYl takes the form, 11 = 110 + ~ll, (19.19) where,

(19.20)

The F 1.'S are the components of the F's in the direction perpendicular to 110 . In similar fashion we find that, ...... S ~ .... x = xo + uos + 2" u. (19.21 ) 19. Electron Transport in Eand B Fields 427

We may write Eqn. 19.21 slightly differently as,

- _ - - ;.(l) ;.(l) X - Xo + llOS + (.L,roHm. + (.L,om , (19.22) where the f{l)'s are the 1st-order perturbations of the trajectory due to inelastic slowing down plus multiple scattering, and the deflection and energy change in the external electric and magnetic fields. Since the f{l)'S are decoupled, we may calculate ~!roHm. as if there were no external fields, and ~!om as if the transport had occured in a vacuum. We note that, to 1st-order, the deflections are perpendicular to the initial trajectory. If we had carried out the analysis to 2nd-order in the if's, we would obtain an equation of the form, - _ - - ( (2») ;.(l) ;.(l) ;.(2) X - Xo + lloS 1 - (II + (.L,roHm. + (.L,om + (.L , (19.23) where 'if) is the 2nd-order perpendicular deflection, and 1 - (~2) can be identified as the path-length correction, a quantity that accounts for the curvature of the charged• particle step. Rather than transporting the charged particle the full path length S in the initial direction, this distance must be shortened owing to the deflection of the particle during the step. This is seen to be a 2nd-order effect. We have seen in Chapter 5 that these corrections can be quite large if one attemps to use large step-sizes in field• free simulations. Clearly, if the 1st-order effects of multiple scattering, slowing down or speeding up, and deflection due to the external field are to be considered small in a perturbation sense, then one should ensure the the 2nd-order affects should be commensurately smaller. A reliable method for calculating path-length corrections in field-free transport is discussed in Chapter 5. The terms 'if) and (fI2) contain <'mixing terms" between the field-free and external-field components. To our knowledge, no theory has yet been developed to encompass all of these effects. Until the time that such a theory is developed, we restrict ourselves to charged-particle steps that are small enough so that the 1st-order equations discussed herein are valid. However, if we make this restriction, we may transport particles in any medium with any external-field configuration, no matter how strong.

19.4 APPLICATION TO MONTE CARLO - BENCHMARKS

At the start of a transport step in a medium with external fields present, we know xo, the position, iio, the unit direction vector, Eo, the energy (which one may easily use to find vo, the magnitude of the velocity, or f30 = vole), Do, the macroscopic electric field strength at xo, and Ho, the macroscopic magnetic field strength at xo. In the condensed-history approach to electron Monte Carlo transport, aside from those discrete interactions which we consider explicitly, the interactions are grouped together and treated by theories which consider the medium in which the transport takes place to be a homogeneous, bulk medium. Therefore, using the macroscopic fields, 13 and H, rather than the microscopic fields, :E and ii, is consistent with the condensed-history approach. After the transport step, we wish to know xf, ii" and E" the final position, unit direction vector, and energy.

If we make the identification,

if.L,om(XO, Eo, iio) = e(Do - iio(iio . Do) + Yo X Ho), (19.24) 428 A. F. Bielajew as the transverse force on the particle due to external fields, we may use the results of the previous section and write directly the equation for ii"

(19.25) where the deflection due to multiple scattering and inelastic collisions is:

s - - ~iim.,ret = (E ) 2 {(F .L,rot(Eo)) + (F .L,m.(Eo))}, (19.26) mOl 0 Vo and the deflection due to the external electromagnetic field is:

es - - - ~ii.m = (E) 2 (Do - iio( iio . Do) + vo X H o). (19.27) mOl 0 Vo

Using the results of the previous section, we may also write directly the equation for XI> --- S(A_ A-) X, = Xo + UoS + 2 UUm.,ret + uU..... (19.28) We may also have written these equations from the results of the vacuum transport section, Section 19.2, by superimposing the inelastic- and multiple-scattering forces on the transport in a vacuum but with external electric and magnetic fields.

We wish to ensure that the new direction vector remains properly normalized, i.e., lii,1 = 1. Therefore, after the transport has taken place according to Eqs. 19.24 - 19.28, we normalize ii, using,

(19.29)

Note that there is no inherent contradiction introduced by Eq. 19.29 because the nor• malization uses terms that are higher than 1st-order in the ~ii's. We remark that ii, is already normalized to 1st-order in the ~ii's, consistent with our development. For repeated transport steps, the ii, we calculate becomes the iio of the next step. We have assumed that llO is properly normalized in our development. Unless we obey this normalization condition to all orders, the 2nd and higher-order terms will eventually accumulate and cause substantial error.

Finally, we calculate the final energy from the equation,

E, = Eo - ~Eret + eDo . (x, - xo), (19.30) where ~Eret is the energy loss due to inelastic collisions. It takes the form,

~Eret = 1· ds'ldE/ds'l, (19.31 ) where IdE/ds'l is the stopping power. 19. Electron Transport in Eand B Fields 429

At this point we should re-state the constraints we must impose so that our method is valid. The condition that the fields not change very much over the transport step takes the form, (19.32) and, rH(x,) - ii(xo) I _ c 1 ~ = UH ~ • (19.33) IH(xo)1 The constraint that the energy does not change very much over the transport step takes the form, (19.34) and, (19.35)

Finally, the constraint that the direction does not change very much over the transport step takes the form, (19.36) and, (19.37) Recall that all of the constraints expressed by Eqs. 19.32 - 19.37 can be satisfied by using a small electron step-size except IL\.iim.,r.tl ~ 1, which must be violated some of the time if the multiple-scattering model includes large-angle, single-event Rutherford scattering. We have argued that this occurs infrequently enough so that the error introduced is negligible. The error, if any, would show up in high-Z media subjected to strong magnetic fields, since high-Z media produce more large-angle events, and the magnetic force is direction dependent.

An algorithm for electron transport in media subjected to external electric and magnetic fields would be of the following form: • Choose a total path length for the step, s, consistent with the constraints ex• pressed in Eqs. 19.32 - 19.37. • Calculate ii, using Eqs. 19.25 - 19.27. The inelastic- and elastic-scattering por• tions, (F .l,r.t(Eo)) and (F .l,m.(Eo)), can be calculated using any slowing-down and multiple-scattering theory. • Transport the particle according to Eq. 19.28*. • Normalize the direction vector, ii" using Eq. 19.29**.

* The EGS code simplifies the procedure even morel3 and ignores the lateral trans• port caused by (F .l,rd), (F .l,m.), and F.l,.m at this stage. If the steps are short enough, accurate electron transport will be accomplished through the deflection of the direction vector, ii. (For a discussion of the importance of lateral transport during an electron step caused by multiple scattering, see Chapter 13.) ** Although this guarantees that the direction vector remains normalized, a different procedure has been adopted for the EGS code. This normalization was applied for the electric-field deflection only. For deflection by the magnetic field, only the components transverse to the magnetic field are normalized. Within the struc- 430 A. F. Bielajew

• Calculate the new energy according to Eqs. 19.30 and 19.31. • Repeat.

To proceed further would entail detailed knowledge of a specific Monte Carlo code. This would take us beyond the scope of this chapter. Rather, we assert that this has been done both for the EGS code13 and the ITS codes3- 5 • We simply give examples of some benchmark calculations.

First, we present a test of transport in a vacuum in a constant electric field. In Fig. 19.1, we compare electron trajectories calculated using the EGS Monte Carlo code and the exact analytical solution, Eq. 19.5, discussed in Section 19.2.1. In this example, the electric field is aligned with the ordinate with a field strength of 511 kV fcm. Three positron trajectories are depicted with energies 0.1, 1.0, and 10 MeV initially directed at a 45° angle against the electric field. We also show two negatron (e-) trajectories with energies 2.0 and 20 MeV initially directed perpendicular to the electric field. The Monte Carlo calculated trajectories are depicted by the solid lines, while the exact analytical solutions are shown with dotted lines. The electron step-size limits were obtained by letting the change in kinetic energy be, at most, two percent (the constraint expressed by Eq. 19.35), and the deflection be, at most, 0.02 (the constraint expressed by Eq. 19.37). We see only minor evidence that the neglect of lateral transport during the course of a step causes error. This error can be further reduced by shortening the step-sizes even more.

10

.- S 5 <:.) -...... (1) ...... 0 cd 1-0 cd ~ :><1 -5

10 15 20 25 30 35 x_perpendicular (cm) Figure 19.1. Electron trajectories influenced by a constant electric field in vacuum calculated using the EGS4 Monte Carlo code compared with an exact analytical solution.

Next we present a test of transport in a vacuum in a constant magnetic field. In Fig. 19.2, we compare O.l-MeV negatron and 1.0-MeV positron trajectories calculated using the EGS Monte Carlo code and the exact analytical solution, Eqs. 19.8 and 19.9, also discussed in Section 19.2.1. In this case, the magnetic field is aligned with the abscissa with a field strengh of 0.17 Tesla. The initial velocities are directed 45° away from the

ture of the EGS code, this guarantees that the parallel component of ii does not change thereby,imposing conservation of momentum in the direction parallel to the magnetic field. 19. Electron Transport in Eand BFields 431

B-axis. The analytic solutions are represented by the solid lines. The agreement is excellent in both cases.

The "vacuum" tests were designed to verify that the differential equations of elec• tron transport in external fields are correctly being integrated during the course of the electron simulation. We have verified that the relatively crude "first-order" method presented in Section 19.3 works acceptably well in vacuum. If more accuracy is desired, then one merely has to shorten the electron step-size. We may have been much more sophisticated if we had been interested in rapid and accurate transport in vacuum. For constant, or nearly constant fields, we may have employed the exact analytic solutions expressed in Section 19.2 into the Monte Carlo code directly. If the simulations take place in finite geometries, then one would have to find solutions of the intersection points of the electron trajectories with the surfaces enclosing the geometry in which the simulation takes place. This is a straightforward but tedious problem for most classes of bounding surfaces, for example, cones or cylinders.

1 MeV I .. cd ~ .~ 0 '0 ~ ~ -1

..Q) Po -2 :><'

10 15 20 25 30 35 x_parallel {em}

Figure 19.2. Electron trajectories influenced by a constant magnetic field in vacuum calculated using the EGS4 Monte Carlo code compared with exact analytical solutions.

For electron transport in spatially varying fields or simulations in tenuous media where the multiple scattering and retardation forces may be considered to be perturba• tions upon the deflections and energy losses due to the external fields, one may use more sophisticated integration techniques to obtain electron trajectories more quickly. For example, the ITS series of codes uses a fifth-order Runge-Kutta technique to integrate the external-field transport equationss- 5•

When one is interested in electron transport in dense media, unless more theoretical work is done to elucidate the complex interplay of multiple scattering, retardation and external field forces, one must resort to a method similar to that presented at the start of this section. We have verified that this method works well, and now wish to perform benchmarks in dense media. Unfortunately, the author is not aware of clear-cut benchmark experiments of this sort. Instead, we present three examples which verify at most semi-quantitatively, that external-field transport in dense media is possible.

The first example is that of a 10-MeV negatron through 10 m of air with a magnetic field with strength 0.17 T aligned with the electron's initial direction of motion. An "end-on" view is shown in Fig. 19.3. Because the initial direction and the magnetic field 432 A. F. Bielajew are aligned, there is no deflection by the magnetic field until the particle is deflected by the medium. The initial deflection, a creation of a low-energy 6-ray in the left-hand side of the figure, caused the initial deflection in this example. Afterwards, the negatron spirals in the counter-clockwise direction. The electron-transport cut-off energy in this example is 10 keY, at which point the simulation of 6-ray transport is seen to come to an abrupt end. An interesting feature of this example is that a relatively high-energy 6-ray was produced spiralling many times before losing all its energy. The 6-ray actually drifted about 20 cm in the direction of the magnetic field.

10 MeV e

Figure 19.3. An "end-on" view of a lO-MeV negatron being transported through air. The particle's initial direction and the magnetic-field direction are normal to and out of the plane of view. The vertical axis is lO-cm long.

A second example is a representation of electric-field focussing of electrons shown in Fig. 19.4. In this case, a uniformly charged cylinder with radius 0.5 cm, is placed 2.5-cm deep in a water target. The target is irradiated by a broad, parallel beam of 10-MeV negatrons. The electric potential is <.fI(p)[MV] = -In(2p), which is zero on the surface of the cylinder, and is sufficent to "bind" any electron which strikes the surface of the water. The variable p is the distance away from the cylinder's axis measured in centimeters. The electric field outside the water is zero as well as inside the cylinder. We note the strong focussing of the electrons, although in a few cases, the multiple scattering turns particles away from the cylinder.

Finally, we present in Fig. 19.5, an example that provides some quantitative sup• port for the viability of electron transport in dense media in the presence of strong electric fields. It is somewhat related to the previous example, having to do with fo• cussing of electrons into a cylinder held at high potentiaP3.

When electrons bombard insulating materials and come to rest as a result of slow• ing down in the medium, they become trapped and can set up large electric fields in the medium. Subsequent bombardments may be greatly perturbed by the presence of these fields, and this may result in a deleterious effect on the accuracy of dosime• try measurments in these insulating materials13 . It was common practice to place a radiation-detection device in a cylindrical hole drilled in a plastic phantom, and subject it to electron bombardment. The air in the cylinder becomes ionized and conducting during bombardment, and large amounts of positive charge may be induced to the cylinder to bring its surface to constant potential. Therefore, electrons are focussed to the cylinder causing one to measure an artificially high response in the detector. These 19. Electron Transport in Eand B Fields 433 effects were calculated by Monte Carlo methods13, but the benchmark of the Monte Carlo code is not "clear-cut". The electric-field strength had to be estimated, and it was further assumed that the electric-field strength at the surface of the cylinder was proportional to the total amount of charge having bombarded the insulating material.

Given these approximations, a comparison of the calculated and measured enhance• ment of detector response is presented in Fig. 19.5. In this comparison, the incident electron energy was 5.7 Me V, and the cylinder in which the detector was placed was 0.7 cm in diameter placed at a depth of 1.5 cm in a polymethylmethacrylate (PMMA) medium. In this case, the maximum field strength exceeded 1 MV / cm! We note that the experimental and calculated results agree to within the 1% accuracy of the simulations.

10 MeV e In water

Figure 19.4. 10-MeV electrons incident on water containing a uniformly charged cylinder. The charge on the cylinder strongly focusses the electrons. The horizontal axis is 5-cm long.

JOr;:l 0.20 CZl Z 0 I:l.o CZl JOr;:l ~ 0.15 ~ JOr;:l CZl -

REFERENCES 1. W. R. Nelson, H. Hirayama and D. W. O. Rogers "The EGS4 Code System", Stanford Linear Accelerator report SLAC-265 (1985). 2. J. A. Halbleib Sr., and T. A. Melhorn "ITS: The Integrated TIGER Series of Electron/Photon Monte Carlo Transport Codes", Sandia National Laboratories report SAND84-0073 (1984). 3. J. A. Halbleib Sr., "Theoretical Model for an Advanced Bremsstrahlung Con• verter", J. Appl. Phys. 45 (1974) 4103. 4. J. A. Halbleib Sr., and W. H. Vandevender, "Coupled Electron/Photon Transport in Static External Magnetic Fields" , IEEE Trans. Nuc. Soc. NS-22 (1975) 2356. 5. J. A. Halbleib Sr., and W. H. Vandevender, "Coupled Electron Photon Collisional Transport in Externally Applied Electromagnetic Fields", J. Appl. Phys. 48 (1977) 4103. 6. J. D. Jackson, Classical Electrodynamics, (John Wiley and Sons, Inc., New York, 1975). 7. H. A. Bethe, "Theory of Passage of Swift Corpuscular Rays Through Matter", Ann. Physik 5 (1930) 325. 8. H. A. Bethe, "Scattering of Electrons", Z. Phys. 76 (1932) 293. 9. F. Bloch, "Stopping Power of Atoms with Several Electrons", Z. Phys. 81 (1933) 363. 10. G. Z. Moliere, "Theorie der Streuung Schneller Geladener Teilchen. I. Einzel• streuung am Abgeschirmten Coulom-Feld", Z. Naturforsch. A2 (1947) 133. 11. G. Z. Moliere, "Theorie der Streuung schneller geladener Teilchen II Mehrfach• und Vielfachstreuung", Z. Naturforsch. A3 (1948) 78. 12. M. J. Berger, Methods in Computational Physics, Vol. I, (Academic Press, New York, 1963) 135. 13. J. A. Rawlinson, A. F. Bielajew, D. M. Galbraith, and P. Munro, "Theoretical and Experimental Investigation of Dose Enhancement Due to Charge Storage in Electron-Irradiated Phantoms", Med. Phys. 11 (1984) 814. Applications 20. Electron Pencil-Beam Calculations

Pedro Andreo*

Radiation Physics Department Karolinska Institute 10401 Stockholm, Sweden

20.1 INTRODUCTION Electron pencil beams have become an important research tool in different applications of electron dosimetry. The simplified configuration needed to perform Monte Carlo calculations of electron pencil beams contrasts with the increasing degree of sophistica• tion of special-purpose simulations that follow as closely as possible certain geometrical configurations. Most of the existing calculations on pencil beams are based on a de• tailed simulation of electron transport, but they are usually obtained in an homogeneous medium (water in most of the cases) and very little computational effort is required to scored the desired distributions.

Despite this simplicity, the range and importance of the applications of electron pencil beams has increased considerably in the last few years, mainly in the field of radiotherapy treatment planning, but also in parametric studies, and in providing data on dosimetry and information on elementary electron phenomena. The basic steps to perform electron pencil-beam calculations will be considered here together with some details on the most common applications.

20.2 POINT-MONODIRECTIONAL BEAMS Electron pencil beams are also known as point-monodirectional beams, and can be considered as the most elementary type of electron beam. Although there are no re• strictions regarding the energy spectrum of the electrons, all the existing calculations consider the beam to be composed of electrons having the same energy. The term, point-monodirectional-monoenergetic beam, is a general characterization of an electron source that in most of the available Monte Carlo codes will be described by an initial state (E,r,O) with energy Eo, position (0,0,0) and direction cosines (0,0,1).

When such beams impinge on a homogeneous medium and the distribution of absorbed dose is determined, most of the important properties of the penetration of electron beams into matter can be observed. Fig. 20.1 shows isodose contours of a narrow 10-MeV electron beam incident on water as experimentally determined using a

* Formerly at Seccion de Fisica, Hospital Clinico Universitario, Zaragoza, Spain

437 438 P. Andreo small semiconductor diode detectorl. The broadening of the isodoses with increasing depth due to multiple scattering can clearly be seen, as can the very rapid decrease of absorbed dose on the central axis since, in such narrow beams, the loss of electrons scattered out from the axis is not compensated by in-scattering as occurs in broad beams. We note the absence of primary electrons beyond the electron range, where the absorbed dose decreases very rapidly in all directions. Bremsstrahlung photons produced in water are responsible for the departure from the typical "onion-shape" of the electron distribution, their predominance in the forward direction showing that they are produced mainly at the beginning of the electron trajectories, before the initial direction is modified by scattering.

10

E <..>

5

o ~ __ ~~~ __ ~~~~~~~~~~~~~~4-~~~~ 10 5 o 5 10 em Figure 20.1. Experimental isodose contours of a 10-MeV electron pencil beam in water. Isodose are normalized to 100 at the depth of the practical range, 4.9 cm. (From Brahme1).

20.3 COMPUTATIONAL DETAILS.

The computation of pencil-beam distributions using the Monte Carlo method requires a simple scoring geometry, usually cylindrical, as shown in Fig. 20.2. Electron tracks do not need to be divided into further sub-segments when they cross different regions since the medium is homogeneous. Once the step sizes are selected, the energy loss along the step can be scored in each region as a proportion of the path within that region. Obviously, it is the responsibility of the user of any Monte Carlo code to ensure that adequate electron step-sizes are chosen (see Chapter 5). 20. Electron Pencil-Beam Calculations 439

One of the main properties of point-monodirectional absorbed-dose distributions Dpm(z, r) is that they can be used to generate central-axis distributions of plane-parallel beams of any size Dpp(z, 0), according to the so-called reciprocity relationship2.3

Dpp(z, R) = laR Dpm(z, r) 27rrdr. (20.1)

The same property can be applied in experimental electron dosimetry as depth doses for broad beams can be measured using narrow beams and detectors with diam• eters larger than the electron range (see ICRU 19724 and 19843 for details).

INCIDENT PENCIL BEAM INCIDENT PENCil BEAM DEPTHS

RAOII- ~±-~~~=*~~~r-+-~O•

o (z,r) pm ds d.

Figure 20.2. The geometry most commonly used in Monte Carlo electron pencil-beam calculations.

Although this aspect will be considered in greater detail later on, it is important to stress here the advantages of this type of calculation from a computational point of view. Monte Carlo calculated, plane-parallel broad-beam dose-distributions usually are computed following a scheme similar to the one described for point-monodirectional beams except for the neglect of the radial dependence when energy deposition is scored. The computational effort· required to add such radial dependence is small compared to the amount of information available from the elementary beam distributions, and should be considered as a further step when writing a code. On the other hand, the number of histories required to produce a distribution with little statistical noise will be increased, and so will the computation time. 440 P. Andreo

20.4 MONTE CARLO CODES FOR PENCIL-BEAM CALCULATIONS.

The Monte Carlo codes generally used to simulate electron pencil beams have already been described in previous lectures, most of the calculations having been done by Berger and Seltzer5 using ETRAN. In this section, the Monte Carlo code developed by us will be described briefly.

Our code, named MCEF, couples the transport of photons and electrons as a whole electromagnetic cascade, where all interactions of primary particles and succesive generations are accounted for. The cascade can be initiated by monoenergetic photons or electrons, a given bremsstrahlung- or electron-straggling spectrum, or a mixture of the two. It was originally developed to study in detail the physics of the penetration of electron beams in water6- 8, and has been applied further to treatment planning with Gaussian electron beams9,lo and to the calculation of physical quantities in electron• and photon-beam dosimetryll-15. A limitation of the code is that it can be used only for a semi-infinite wate medium in the energy range 100 keV-50 MeV. Furthermore, it has not yet been prepared for general distribution, and therefore is not properly documented.

The code MCEF uses a direct simulation of the photon interactions considering individual photoelectric absorption, Compton scattering and pair-production interac• tions. Electron transport is simulated dividing the tracks into many small different segments within which angular deflections and energy losses are small, and thus can be grouped and described by multiple-collision models. Large deflections and energy losses are considered individually by direct simulation at the end of each step.

The transport of photons is followed until their energy falls below the Monte Carlo photon cutoff Tout,ph, where the energy is deposited locally, or the photon interacts outside the phantom, in which case the energy is scored separately for energy-balance purposes. Normal values of Tout,ph are between 10 keV and 50 keV for primary photon energies in the range 100 keV to 50 MeV. Energy, position and directional parameters of the electrons and positrons generated are scored separately and followed after the history of the primary and scattered photons has been completed.

The grouping of the electron interactions mentioned above corresponds to a Class II scheme, or mixed procedure (see Berger16). It has been modified by Andreo and Brahme 17 to account for the influence of the parameters governing the classification of interactions in a consistent way. The classification into large and small events is done with two cutoff parameters, .0. and 8, that limit the maximum energy loss and deviation that are described by a multiple-collision theory, csda for energy losses and multiple scattering for angular deflections. Variations greater than .0. and 8 are considered individually, in a manner similar to that of photon interactions, by sampling from the corresponding cross section for delta-ray production18 and nuclear scattering19,20. Bremsstrahlung interactions are considered using the Koch and Motz21 cross-section package.

The two parameters, .0. and 8, require special attention, as theories accounting for restrictions in energy losses and angular deviations are needed. The well-known restricted stopping-power formula accounts for energy losses less than a given cutoff, .0.. Energy straggling is already performed in mixed procedures by directly sampling energy 20. Electron Pencil-Beam Calculations 441 losses between ~ and half the maximuIll energy of the incident electron from the l'''hJIIer cross-section. Straggling bclween a and ~ can be considered with the use of a restricted straggling-theory17 since, depending on the magnitude of ~, the lack of straggling ill energy losses less than ~ would be equivalent to the use of a simple csda model. As already pointed out in previous works (see Andreo17,22), this treatment of energy-loss straggling produces better agreement with experimentally determined electron depth• dose distributions than that using the Landau-Blunck-Leisegang theory which predicts less average energy loss than the stopping-power theory. Multiple-scattering theories restricted in the maximum deviation angle are needed in order to describe scattering through angles less than 8 in a consistent way. An expression for the restricted mean angle (i. e., a restricted scattering power) has been given by Andreo and Brahme17. The use of two groups to classify angular deviations produces a better treatment of large scattering angles that are not accounted for properly with the use of a single Gaussian or a multiple-scattering theory such as that from Moliere23,24.

The transport of electrons and positrons is followed until their kinetic energy is smaller than the Monte Carlo electron cutoff Tcut,., where the energy is deposited lo• cally, and two annihilation photons are created in the case of positrons. The cutoff value is governed by the dimensions of the voxel used to compute energy-deposition distributions, the corresponding csda range of an electron having an energy of Tcut,. being equal to half of the smallest dimension of the voxel.

20.5 APPLICATIONS.

As mentioned before, most of the applications of electron pencil-beam calculations arc included in the field of electron dosimetry. Some of the most common applications will be discussed in this section.

20.5.1 Absorbed-dose distributions.

A Monte Carlo generated radial energy-deposition pattern of a la-MeV electron pencil beam is shown in Fig. 20.3a for five different depths in water. Each histogram rep• resents the amount of energy deposited in the indicated ring around the central axis of the beam, according to the geometry described in Fig. 20.2. The shapes of these histograms show very clearly the influence of multiple scattering, the initictlly narrow beam being broadened as it penetrates into the medium. Absorbed-dose distributions are obtained by dividing the energy deposition by the mass of each ring which produces the radial profiles shown in Fig. 20.3b.

The shape of the dose profiles provides a severe test for any Monte Carlo code regarding energy losses and scattering when compared to experimental data. Fig. 20.1 shows such comparison for a narrow 20-MeV electron beam in water, as measured by Lax et al lO• The histograms correspond to Monte Carlo data from Berger and Seltzer25, (dashed) and from our code17, (solid). The agreement between calculated and measured profiles is very good considering that the comparison includes several orders of magnitude in absorbed dose. Similar agreement can be observed in Fig. 20.5 for the complete isodose distribution of a lO-MeV electron pencil beam. Again, the Monte Carlo calculated distribution is compared with the experimental one including 6 orders of magnitude in isodose contours. 442 P. Andreo

'00 c e 10 MeV u 41 Ii 1.5 >- (; E 050 'h. 0.20

0.25

0.2 0.3 O.1.r/ro 0.00 < z/ r. < 0.05 z o

ii:0: ()5 0.50 o III (D « 0.40-< x/ r.-<0.45 0.80(z/r.<0.85 0.25 w~ z UJ

0.1 0.2 0.3 rl ro

Figure 20.3a. Radial profiles at different depths (expressed as fractions of the csda range, To) of the energy deposition in water from an electron pencil beam of 10 MeV. Histograms represent the energy deposited in a ring around the central axis of the beam (From AndreoS ).

,6'

l1J III 0 C C l1J ,62 III 0:: 0 III «III

l1J >

3 ~...J ,6 l1J a:

104~ ____~~~ ______~ ______~ __~ ____~ ______~ ______~~ o rlro Figure 20.3b. Absorbed-dose profiles, obtained by dividing the energy deposition in Fig. 20.3a by the mass of each ring (From Andreo6 ). 20. Electron Pencil-Beam Calculations 443

Rad ius (e m ) Figure 20.4. Comparison between experimental and Monte Carlo calculated radial absorbed-dose profiles for a 20-MeV electron pencil beam at several depths in water. Solid line, experimental data from Lax et aiIO. Solid his• togram, MC data from our code. Dashed histogram, MC data from Berger and Seltzer25• (From Andreo and Brahme17.)

II t;),;:. /.<> .~(. .... '" . \' \ 1 ~. i . ,

Figure 20.5. Comparison between experimental (right) and Monte Carlo cal• culated (left) isodose contours of a lO-MeV electron pencil beam in water. MC data obtained with our code. 444 P. Andreo

As already mentioned, plane-parallel broad-beam distributions can be generated from pencil-beam data. Central-axis depth-dose curves for beams of different radii can be calculated from Eqn. 20.1 when the absorbed-dose distribution of a point• monodirectional beam is known. Fig. 20.6 shows such calculations for lO-MeV beams where the transition from narrow beam to broad beam can be observed in terms of multiple scattering as described in relation with Fig. 20.1. Electrons scattered out from the central axis are responsible for the very rapid decrease of the depth-dose curve for narrow beams. When the size of the field is increased, this loss is compensated by electrons scattered into the axis, the result being a larger build-up and a less rapid decrease of the steep part of the distribution.

A superposition method has been described by Berger and Seltzers to convolve point-monodirectional isodose distributions with a beam-geometry function that char• acterizes the incident electron beam by the intensity distribution of elementary beams incident on the phantom surface. Isodose distributions derived in this manner for 10 and 20-MeV beams with diameters of 5 and 10 cm are shown in Fig. 20.7. They have been calculated for the ideal case where the intensity of a circular beam has a constant value inside a given field and vanishes elsewhere (a step function), which explains the narrow penumbra of the distributions. The method could easily be extended to model more complicated and realistic situations.

20.5.2 Energy distributions.

The energy spectrum of electron beams at different depths in a medium has important implications in the dosimetry of clinical electron beams. Some of the parameters used to describe the electron spectrum, as the mean and most-probable energy, have received considerable attention, and the depth variation of the mean energy has been considered as an important parameter for selecting dosimetric quantities such as stopping-power ratios or perturbation factors. Although the calculation of electron spectra will be considered in more detail in a lecture on stopping-power ratios, calculations related to the mean energy of primary electrons in the context of pencil beams will be described here.

According to ICRU26, the particle fluence is defined as the differential quotient of the number of particles dN that cross a sphere of cross-sectional area da:

~ = dNjda. (20.2)

This definition is not practical from a calculational point of view as it is based on the geometrical properties of a sphere. However, by multiplying both dN and da by the mean cord length27 i = 4VjA , a more practical definition can be obtained. As idN is equal to the mean path length ds in the sphere, and ida is the volume dV of the sphere, Eqn. 20.2 becomes ~ = dsjdV, (20.3) which is independent of the shape of the volume, and can be interpreted as the track length per unit volume. 20. Electron Pencil-Beam Calculations 445

2.4

'0' 2.0 '"E u ~ ::2: 1.6

0.8

0.4

0.4 0.8 1.2 Zlro

Figure 20.6. Central-axis depth-dose curves from 10-MeV electron circular beams with different radius, R, obtained from a pencil-beam absorbed-dose distribution using Eqn. 20.1 (from Andre06 ).

Using the geometry described in Fig. 20.2, the spectrum of primary electrons calculated for a 20-MeV point monodirectional beam incident on a semi-infinite water phantom is shown in Fig. 20.8 at a depth of 4.4 cm. The spectra pertain to circular rings at different distances from the central axis and in addition, the spectrum for a broad beam (a) at that depth also has been included. It can be seen that there is a substantial change in the shape of the spectrum which will produce a variation of the mean energy of electrons across the beam. Such variation is shown in Fig. 20.9, where the mean energy has been computed according to

(20.4)

(see Andreo and Brahmell). For comparison, a simple analytical model based on a Gaussian function of the form

- - _(l!:)2 Epm(z, r) = Epm(z,O)e '0 (20.5) has been included, where ro is the csda range. 446 P. Andreo

~~Q!..AMl'f!.l~_1 0 ~ E 10 MeV U 2 N :J: I- 0.. w 4 0 10 6 0

2 E u ~ 4 N :J: I- 0.. 6 W 0 8

10

4 2 0 2 4

I· ·BlAMOIAMfIiR 10.m----J 0 10 E u ~ 2 N I

0 80

2 7O~V E u 4 N :J: I- 0.. w 6 0 8

10

8 6 4 2 0 2 4 6 8 RADIAL DISTANCE P (em)

Figure 20.7. Isodose distributions in a water phantom irradiated with 10 and 20-MeV electron beams with circular fields sizes obtained with the superposi• tion method from electron pencil-beam data. (From Berger and Seltzer5). (from ref. 6). 20. Electron Pencil-Beam Calculations 447

1.0

20 MeV depth: 0.45 - 0.50 '0 ,adloua

a I 0.0 - 1.0 '0 0.75 b : 0.0 - 0.1 '0

C I 0.1 - 0.2 '0 d • 0.2 - 0.3 '0

0.50

0.25

d

o ~c~ __~ ______~ ______~ ____~~~ __~ __~

o 5 E (MeV) 10 15

Figure 20.8. Spectra of primary electrons in water for a 20-MeV electron pencil beam for different radial positions at a depth of 4.4 em (z/ro between 0.45-0.50). (From Andreo and Brahmell). 448 P. Andreo

16

, 20 MeV 14 ,, ,, Depth , a: 0.20 - 0.25 \ ro \ \ b: 0.50 - 0.55 ro \ 12 \ \ C : 0.70 - 0.75 ro \ \ \ d: 0.95 - 1.00 ro \ \ ~ \ 10 \ >OIl \ \ ~ \ \ \ ... \ ... \ -~ 8 ...... \ \ N ... \ , , \ I ILl , \ , \ \ \ \ 6 \ \ \ \ \ \ \ \ \ 4 \

2

o o 0.2 0.4 0.6 0.8 1.0

r / ro Figure 20.9. Variation of the mean energy of primary electrons with lateral position from the central axis for a 20-MeV electron pencil beam at different depths in water. Dashed curves correspond to Eq. 20.5. (From Andreo and Brahmell). 20. Electron Pencil-Beam Calculations 449

20

15

~ ~

0. N 10 IW

5

0.25 0.50 0.75 lOO 1.25 1.50

z / ro Figure 20.10. Mean energy of primary electrons for 20-MeV circular electron beams as a function of depth (z/ro) for beam radii of 0.1,0.2,0.3,004,0.5 and

1.0 times the csda range, 1'0' (From Andreo and Brahmell.)

The mean energy at the central axis of plane-parallel broad beams with different radii can be obtained in a manner similar to depth-dose distributions using point• mono directional beam data (d. Eq. 20.1) according to

-( ) foR Epm(z, r)

Results for 20-MeV circular beams are shown in Fig. 20.10 where, as in Fig. 20.6, the transition from narrow-beam to broad-beam can be observed. The mean energy at the central axis decreases with increasing beam radius until a saturation value is reached for beams of a diameter of about 1'0. As before, the decrease can be explained in terms of the compensation of electrons scattered out from the central axis which will have lower energy than those remaining, almost without being scattered. 450 P. Andreo

20.5.3 Pencil beams as "benchmarks" for treatment-planning algorithms.

During recent years, there has been an increased interest in the use of pencil-beam algorithms for electron-treatment planning. Most of them are based on the use of a Gaussian solution to some small-angle or full-diffusion transport equationZ8 , and those using the so-called Fermi-Eyges theoryz9,3o have now reached a satisfactory degree of development. Common aspects of the algorithms are that they use primary electron fluence (i.e., the above Gaussian solution) to describe absorbed dose, and use some type of correction and/or fitting procedure to overcome the intrinsic limitations of the ap• proximations in the theory regarding scattering through small angles, range straggling, etc. Excellent reviews already have been published31- 33, and here we will not consider further details of the algorithms.

The Monte Carlo method has proven to be a very efficient tool for testing pencil• beam algorithms, the shape of predicted beam profiles in water being a particularly severe test, as has been shown by Lax et apo using our code and ETRAN, or by Mohan et aIM and Bielajew et al 36 using EGS. Similar tests have been performed for central-axis depth-dose calculations using Monte Carlo data, such as those shown in Fig. 20.6. Such comparisons have demonstrated the weakness of a pure Gaussian function to describe the electron beam, and the need to include certain parameters to modify the Gaussian distribution. 20. Electron Pencil-Beam Calculations 451

REFERENCES 1. A. Brahme, " Physics of Electron Beam Penetration: Fluence and Absorbed Dose", in Proc. of the Symp. on Electron Dosimetry and Arc Therapy, edited by B. J. Paliwal, (American Institute of Physics, New York, 1982) 45. 2. R. M. Sternheimer, "Multiple Scattering Correction for Counter Experiments", Rev. Sci. Instr. 25 (1954) 1070. 3. ICRU (International Commission on Radiation Units and Measurements), "Ra• diation Quantities and Units", ICRU Report 33 (1984) 4. ICRU (International Commission on Radiation Units and Measurements), "Ra• diation Dosimetry: Electrons with Initial Energies Between 1 and 50 MeV", ICRU Report 21 (1972.) 5. M. J. Berger and S. M. Seltzer, "Theoretical Aspects of Electron Dosimetry", in Proc. of the Symp. on Electron Dosimetry and Arc Therapy, edited by B.J. Pali• wal, (American Institute of Physics, New York, 1982) 1. 6. P. Andreo, "Monte Carlo Simulation of Electron Transport in Water. Ab• sorbed Dose and Fluence Distributions", Dept. of Nuclear Physics, University of Zaragoza, Spain, report FANZ/80/3 (1980). 7. P. Andreo, "Aplicacion del Metodo de Monte Carlo a la Penetracion y Dosimetria de Haces de Electrones", Thesis, University of Zaragoza, Spain (1981). 8. P. Andreo and A. Brahme, "Fluence, Energy Fluence and Absorbed Dose in High Energy Electron Beams", Department of Radiation Physics, Karolinska Institute, Stockholm, report RI 1982-05 (1982); see also Acta Radiol. Suppl. 364 (1983) 25. 9. A. Brahme, 1. Lax and P. Andreo, "Electron Beam Dose Planning Using Discrete Gaussian Beams: Mathematical Background", Acta Radiol. Oncol. 20 (1981) 147. 10. 1. Lax, A. Brahme and P. Andreo, "Electron Beam Dose Planning Using Gaussian Beams. Improved Radial Dose Profiles", Acta Radiol. Suppl. 364 (1983) 49. 11. P. Andreo and A. Brahme, "Mean Energy in Electron Beams", Med. Phys. 8 (1981) 682. 12. P. Andreo and A. Brahme, "Stopping Power Data for High Energy Photon Beams", Phys. Med. BioI. 31 (1986) 839. 13. A. Brahme and P. Andreo, "Dosimetry and Quality Specification of High Energy Photon Beams", Acta Radiol. Oncol. 25 (1986) 213. 14. P. Andreo and A. E. Nahum, "Stopping-Power Ratio for a Photon Spectrum as a Weighted Sum of the Values for Monoenergetic Photon Beams", Phys. Med. BioI. 30 (1985) 1055. 15. P. Andreo and A. E. Nahum, "Influence of Initial Energy Spread in Electron Beams on the Depth-Dose Distribution and Stopping-Power Ratios", Proceedings of the XIV ICMBE and VII ICMP, Espoo, Finland (1985) 608. 16. M. J. Berger, "Monte Carlo Calculation of the Penetration and Diffusion of Fast Charged Particles", In Methods in Computational Physics, Vol.i, Statistical Physics, edited by B. Alder, S. Fernbach and M. Rotenberg, (Academic Press, New York, 1963) 135. 17. P. Andreo and A. Brahme, "Restricted Energy Loss Straggling and Multiple Scattering of Electrons in Mixed Monte Carlo Procedures", Radiat. Res. 100 (1984) 16. 18. C. M!1Iller, "Zur Theorie des Durchgangs Schneller Elektronen Durch Materie", 452 P. Andreo

Ann. Physik. 14 (1932) 531. 19. W. A. McKinley and H. F. Feshback, "The Coulomb Scattering of Relativistic Electrons by Nuclei", Phys. Rev. 74 (1948) 1759. 20. 1. V. Spencer, "Theory of Electron Penetration", Phys. Rev. 98 (1955) 1597. 21. H. W. Koch and J. W. Motz, "Bremsstrahlung Cross-section Formulas and Re• lated Data", Rev. Mod. Phys. 31 (1959) 920. 22. P. Andreo, "Monte Carlo Simulation of Electron Transport. Principles and Some General Results", in The Computation of Dose Distributions in Electron Beam Radiotherapy, edited by A. E. Nahum, (University of Umea, Sweden, 1985) 80. 23. G. Moliere, "Theorie der Streuung Schneller Geladener Teilchen. I. Einzelstreu• ung am Abgeschirmten Coulom-Feld", Z. Naturforsch. A2 (1947) 133. 24. G. Moliere, "Theorie der Streuung Schneller Geladener Teilchen. II. Mehrfach• und Vielfachstreuung", Z. Naturforsch. A3 (1947) 78. 25. M. J. Berger and S. M. Seltzer, "Tables of Energy Deposition Distributions in Water Phantoms Irradiated by Point-monodirectional Electron Beams with Energies from 1 to 60 Me V and Applications to Broad Beams" , National Bureau of Standards report NBSIR 82-2451 (1982). 26. ICRU (International Commission on Radiation Units and Measurements), "Radiation Dosimetry: Electron Beams with Energies between 1 and 50 MeV", ICRU Report 35 (1980). 27. Kellerer, "Considerations in the Random Traversal of Convex Bodies and Solu• tions for General Cylinders", Radiat. Res. 47 (1971) 359. 28. A. Brahme, "Elements of Electron Transport Theory", in The Computation of Dose Distributions in Electron Beam Radiotherapy, edited by A. E. Nahum, (University of Umea, Sweden, 1985) 72. 29. E. Fermi, cited by Rossi and Greissen in Cosmic Ray Theory, Rev. Mod. Phys. 13 265 (1941). 30. 1. Eyges, "Multiple Scattering with Energy Loss", Phys. Rev. 74 (1948) 1534. 31. F. Niisslin, "Computerized Treatment Planning in Therapy with Fast Electrons: A Review of Procedures for Calculating Dose Distributions", Medicamundi 24 (1979) 112. 32. K. R. Hogstrom, "Electron Beam Modeling and Dose Calculation Algorithms in Treatment Planning Computers", in Continuing Education Course at the 1982 AAPM Annual Meeting, Med. Phys. 9 (1982) 645. 33. A. Brahme, "Brief Review of Current Algorithms for Electron Beam Dose Plan• ning" , in The Computation of Dose Distributions in Electron Beam Radiotherapy, edited by A.E. Nahum, (University of Umea, Sweden, 1985) 271. 34. R. Mohan, R. Riley and J. S. Laughlin, "Electron Beam Treatment Planning: A Review of Dose Computation Methods", in Computed Tomography in Radiation Therapy, edited by C. C. Ling, C. C. Rogers and R. J. Morton, (Raven Press, New York, 1983) 229. 35. A. F. Bielajew, D. W. O. Rogers, J. Cygler and J. J. Battista, "A Comparison of Electron Pencil Beam and Monte-Carlo Calculational Methods", in The Use of Computers in Radiation Therapy, edited by I. A. D. Bruinvis, et aI, (Elseviere Science Publishers B.V., Holland, 1987) 65. 21. Monte Carlo Simulation of Radiation Treatment Machine Heads

Radhe Mohan

Memorial Sloan-Kettering Cancer Center 1275 York Avenue New York, New York 10021, USA

21.1 INTRODUCTION Monte Carlo simulations of radiation treatment machine heads provide practical means for obtaining energy spectra and angular distributions of photons and electrons. So far, most of the work published in the literature has been limited to photons and the contam• inant electrons knocked out by photons. This chapter will be confined to megavolt age photon beams produced by medical linear accelerators and 6OCo teletherapy units. The knowledge of energy spectra and angular distributions of photons and contaminant elec• trons emerging from such machines is important for a variety of applications in radiation dosimetry. Examples include: 1. Energy spectra enable us to make accurate prediction of dose using new methods of calculations of the type described in Chapter 26, entitled "Dose Calculations for Radiation Treatment Planning" . 2. They aid us in improving the state of the art of treatment planning by intelli• gently altering beam characteristics based on available spectral information, and by providing an explanation of underlying principles of a number of radiation dosimetry issues including: Electron contamination and its effect on surface dose, and development of tech• niques to reduce contamination. Variation of output with field size. Effect of blocks on output. Effects of spectral variations across the beam on the beam flatness ("horns"). Effect of incident energy on the magnitude of horns. It has been shown that, for the same flattening filter, the magnitude of horns is a more sensitive function of energy of incident electrons than the central-axis depth dose. Effect of angular spread of incident primary photons on the shape of the beam boundary. The boundary of the photon fluence profile is very sharp just beneath the collimator or a block, but becomes progressively more diffuse as a function of distance due to photon angular spread. Spatial spread resulting from the angular distribution of photons may be convolved with fluence for calculating dose near the beam boundaries.

453 454 R. Mohan

3. The knowledge of energy spectra is valuable for designing treatment machine heads and flattening filters. 4. Energy spectra provide useful information for the use of linear accelerators as CT scanners employing megavolt age photon beams for imagingl . 5. Energy spectra and angular distributions may be used in the determination of factors for converting ionization chamber measurements to dose (e.g., stopping• power ratios, mass energy absorption coefficients, etc.).

For many of the applications, it is also necessary to know the variation of the energy spectrum as a function of radial distance from the central ray. The bremsstrahlung energy spectrum in the central part of a photon beam is somewhat harder than in the region near the edge of the beam. The flattening filter not only hardens the beam as a whole, but further enhances the relative hardness near the center. Consequently, the relative depth doses in tissue vary as a function of distance from the central ray of the beam. The variation of the spectrum across the beam is also important when calculating transmission through blocks, compensating filters, or other beam modifiers.

Photon energy spectra may be obtained by measurementsl - S. However, the ex• tremely high intensity of therapeutic photon beams makes direct measurement of energy spectra virtually impossible. Some of the methods described in the literature employ reconstruction techniques to obtain the energy spectra from measured narrow-beam transmission data. Others deduce energy spectra from the spectroscopy of Compton• scattered photons. The results of these measurements are not entirely reliable. Fur• thermore, there are no data available in the literature for angular distributions.

Alternatively, Monte Carlo techniques may be employed to simulate relevant com• ponents of treatment machine geometry, and to generate the energy spectra and angular distributions of photon beams and contaminant electrons produced by linear accelera• tors and 60CO teletherapy devices. An advantage of the Monte Carlo method is that it can be used to obtain angular distributions, quantities which cannot be measured experimentally. Secondly, the energy spectrum can be generated, not only in the cen• tral part of the beam, but simultaneously in regions away from it. Another advantage is the possible savings in manpower at the expense of computer time. The Monte Carlo computer programs can be set up in such a way that they can be applied to most treatment machines with similar geometries. The dimensions and materials used in various components in the machine head (e.g., primary collimator, flattening filter, etc.) are specified as input to the program. Therefore, a different accelerator can easily be described by modifying the input.

Monte Carlo methods provide, at least in theory, an accurate means of obtaining the energy spectra. The geometry of the treatment head, to a good approximation, can be modelled. The accuracy of the technique is limited primarily by the approximations of the theoretical basis of the Monte Carlo program, the accuracy of the cross-section data, and the amount of computer time available to reduce the statistical uncertainties. To confirm the validity of the energy spectra and angular distributions generated by the Monte Carlo programs, one may calculate dose distributions using these data9 , and compare the results of calculations with measured depth-dose data. Another check of the accuracy of energy spectra may be the comparison of calculated half-value layers in water with measurementslO• 21. Monte Carlo Simulation of Radiation Treatment Machine Heads 455

Several investigators have modelled the linear accelerator and 6OCO teletherapy• machine heads in order to obtain energy spectra and other information relevant to radiation therapyll-1.. The results of some of these investigations are summarized in the following sections. The Monte Carlo code employed in all the investigations covered in this chapter was EGS, versions 316 and 416 .

21.2 MONTE CARLO SIMULATIONS OF LINEAR ACCELERATOR HEADS

The geometry of a typical treatment machine head is illustrated schematically in Fig. 21.1. Various components of the head relevant in Monte Carlo calculation of spectra are numbered sequentially. Component 1 is the target button. This is where most of the bremsstrahlung photons are produced. It is usually made of tungsten or gold. Com• ponent 2 is the target backing, and is made of copper to provide fast heat dissipation. Component 3 is the primary collimator. It is made of heavy material such as tungsten or uranium. The opening of the primary collimator is a circular cone with the apex located at the electron-beam bombardment point. Component 4 is the flattening filter. It is thicker in the middle and thinner near the edge in order to produce a relatively flat radiation field at depth. The bremmstrahlung spectra are harder at the center and softer away from the center of a field. This is further accentuated by the shape of the flattening filter. Component 5 is the monitor chamber. Component 6 is the secondary collimator. It follows the divergent shape of the primary collimator. Component 7 is the collimating jaws.

Electron Beam

~SeCOndary Collimator

Figure 21.1. A typical treatment machine head.

Other than the initial electron beam energy, the most dominant factors which in• fluence the energy spectrum and angular distribution are the target and the flattening filter. For Monte Carlo simulations, the target button and backing can easily be mod• elled as a pair of slabs. The shape of the flattening filter is quite complicated however, and is more difficult to model. Mohan and Chui10 developed a special geometry package to model the exact shape of the flattening filter, assuming it to have a flat base and a profile that can be described by a series of points. The flattening filter then may be 456 R. Mohan considered to be composed of a sequence of sections of right circular cones. The primary and secondary collimators also may be modelled with the aid of circular cone sections.

The jaws may be assumed to form a rectangular cone diverging from the apex located at the electron-beam bombardment point. The effect of collimating jaws on energy spectra and angular distributions is small. Therefore, both the upper and the lower jaws may be assumed to be at the same level.

While Mohan and Chui developed their own software to model the acclerator head in the manner described above, one may also employ one of the available geometry packages. For example, Petti et al 12 used a cylindrical geometry package to simulate the head geometry of a Varian Clinac-35 treatment machine, as illustrated in Fig. 21.2a. Fig. 21.2b illustrates the manner in which the flattening filter, lower primary collimator, and adjustable photon jaws are simulated by cylindrical sections. The air column between the collimators was simulated in the same manner, but with fewer cylinders.

(0) 2S AI. V • (b) Bremsstrahlung beam Irom copper target -Target Flatlenlng Iliter copp.r Air st•• ' • run",r.n

Beam monitor 1 '0 chamber .. 5 u:>< ~,

- ~ D.. -.. photon..... ja,,"ws I ;; g ':'0 lO ... ,., '0 .I: ." "'u. , 1 tung,r.n • copper · a;-·, . , z

~ ror.tlon.' .~/N observer/on point

Figure 21.2. (a)Schematic drawing of the Clinac-35 treatment head. (b) The embedded cylinders used to simulate the flattening filter, lower primary collimators, and photon jaws.

Typically, the energy spectrum is scored in a plane perpendicular to the central ray at a specified distance from the x-ray target (see Fig. 21.3). Photons emerging from the target pass through the flattening filter and other components of the collimating system on their way to this plane. The plane is divided into annular regions around the central ray as shown. For each annular region, the number of photons within each energy interval crossing the plane of interest is recorded. The ith annulus corresponds to the space between radii ri-l and rio Similarly, the jth energy bin corresponds to the interval between Ej - 1 and Ej • 21. Monte Carlo Simulation of Radiation Treatment Machine Heads 457

Figure 21.3. Scoring of energy spectrum and angular distribution.

Angular distributions may be recorded over the entire plane as a function of the angle between two vectors ii and V, where vector v joins the source S with the point of observation, and vector ii defines the direction of the photon. For each observation point P, the direction of vector v is fixed, and the direction of ii describes the angular distribution of photons relative to v. These data yield the angular spread at the point of observation relative to the initial direction of incidence, and are of value in calculating spreading of the fluence profile boundaries as a function of the distance from the jaws.

As an example, results of calculations for a Varian Clinac-20 (15 MV) are included here from Mohan and Chui's worklO• Clinac-20 15-MV photons are produced by bom• barding a tungsten target with a 15-Mev electron beam. The energy spectra are plotted in Fig. 21.4. The solid line is the spectrum at the center of the field (in a circular region of 2-cm radius), the dashed line is the spectrum in the annular region between 10 and 15-cm radii. As one would expect, the spectrum at the center of the field is harder than the spectrum away from the center of the field. The mean energy is 4.11 MeV in the central part of the beam (between 0 and 2 cm from the central ray) and is 3.28 MeV in the region between 10 and 15 cm from the central ray. In the absence of both. the flattening filter and the collimating system, i.e., with only the target and the target backing present, the mean energies in the same regions are 2.79 and 2.45 MeV, respec• tively. The flattening filter hardens the beam more in the central part of the beam than in the peripheral region.

Mohan and Chui verified the validity of computed energy spectra by comparing measured values of TMRs in water with those calculated utilizing the energy spectra on the central axis and at a distance of 12 cm from the central axis. The calculated and the measured data for a 10-cm X 10-cm field are plotted in Fig. 21.5. The measured data were obtained in polystyrene for 10-cm X 10-cm field sizes defined at the isocenter, and 12 cm laterally from the isocenter. The agreement beyond the region of electronic build• up is good. Within the build-up region, however, there is discrepancy due to electron contamination of the photon beam which is not taken into account by the theoretical dose-calculation model used. 458 R. Mohan

15 MV Photons Energy Spectra at 100 em from Source

Figure 21.4. Photon Energy Spectra for 15-MV photon beams from a Clinac- 20.

1.0 15 MV Photons Field size 10 em x 10 cm ' ...... 0.8

a: Calculated •• -'.-.• ---,•.•. , ______•• , :::::E 0.6 -'.-~-~- ~ Measured

x Central Ray 0.4 (Mean Energy = 4.11 MeV)

0 12 em from the Central Ray 0.2 (Mean Energy = 3.28 MeV)

o 10 20 30 Depth (cm)

Figure 21..5. Comparison of calculated (using Monte Carlo energy spectra) and measured TMR's for 15-MV photons from a Clinac-20. 21. Monte Carlo Simulation of Radiation Treatment Machine Heads 459

The angular distribution of Clinac-20 15-MV photons is plotted III Fig. 21.6. Each photon travelling from the target to the scoring plane is "tagged" appropriately, depending upon the component of the treatment head from which it scattered. Mohan and Chui found that 93.5% of the Clinac-20 photons arriving at the scoring plane had suffered no collisions, 2.8% scattered from the primary collimator, 3.5% from the flattening filter, and 0.2% from secondary collimator and the jaws. Even the scattered photons travel mostly in the forward direction. The net angular spread is small. For most dose-computation applications, one may neglect the angular spread, and assume that all photons originate from the target. For calculation of dose near the boundaries defined by blocks or collimators however, the angular spread of incident photons is important. The primary fluence at points near the boundaries of blocks and collimator jaws may be computed by convolving spatial spread resulting from angular spread with fluence distribution of an idealized point source. Mohan and Chui estimated that for the Clinac-20 15-MV photons, the contribution of photons scattered by the treatment• head components increases the width of the penumbra by approximately 2 mm at 100 cm from the source.

10 15 MV Photons Angular Distribution

Angle (Degrees)

Figure 21.6. Angular distribution of 15-MV photons from a Clinac-20.

21.2.1 Electron Contamination An important consideration in external-beam therapy is the amount of surface dose delivered to the patient. The entrance dose varies as a function of field size. In addi• tion, the" depth of maximum dose depends on the field size. It has been demonstrated experimentally, as well as with Monte Carlo calculations, that the electrons produced in the accelerator collimation system and in the air between the treatment head and the patient are responsible.

Petti et a/ ll•12, employing cylindrical geometry to approximate various components of the ~ccelerator, carried out Monte Carlo simulation of the treatment head of a Varian Clinac-35 accelerator. The electron distribution is shown in Fig. 21.7. The number of electrons reaching the patient's surface is much smaller than the number of photons. The statistical fluctuations in the electron distributions tend to be much larger than 460 R. Mohan those for the photon distributions, and make the results relatively unreliable. A number of variance-reduction techniques may be employed, as pointed out by Rogers14 and mentioned in the next section. The depth-dose curves calculated using both electrons and photons as incident particles, and with photons only, are shown in Fig. 21.8. The former are plotted along with measured depth dose in Fig. 21.9.

o • Calculated Data 35 tJ. • Smoothed Dot a

Q) '0 u 30

.0'" ~ 0 (j')z 0 a: I-u W ...J W U. 0 a: w III :::E =>z 5

4 6 8 10 12 14 16 18 20 E (MeV) Figure 21.7. Electron energy distribution for a Clinac-35 at the phantom surface.

o • Don Due to Electron, ond Photons 120 /I • Don Oue to Photon, only o 110 o /I /I A 0 100 /I • g tl! 90 II 8 II 8 ~ 80 ILl U II a: 0 ~ TOt- /I II o /I 60f-

50 /I

2 3 4 5 6 1 8 9 10 " 12 13 14 15 16 DEPTH IN POLYSTYRENE (gm/cm2)

Figure 21.8. Comparison of depth dose calculated with and without electron contaminants for a Clinac-35. 21. Monte Carlo Simulation of Radiation Treatment Machine Heads 461

UJ o 8 o o ~ 80 o ~ o UJ o Q. 10f-o

60 o ; Calculated Dose Field Size' 28128 em' at 80em SSD 50 1:>. Measured Dose (Biggs & Ling, 19191 Field Size" 28128 em' at aOem SSD 1 2 3 4 5 6 1 a 9 10 11 12 13 14 15 16 DEPTH IN POLYSTYRENE (gm/cm2) Figure 21.9. Comparison of measurements and calculations including electron contaminants for a Clinac-35.

Petti et al12 further studied the origin of contaminating electrons. They examined three sources: (a) the flattening filter and monitor chamber, (b) the collimating system components downstream from the monitor chamber, including the jaws, and (c) the air column between the patient and the source. They found that for small source-to• surface distances (80-100 cm), the electrons originating from the flattening filter and the monitor chamber accounted for 70% of the contamination, 13% originated from the collimating system, and 17% were produced in intervening air column. On the other hand, at 400-cm distance from the source, 61% were produced in air, 34% originated from the flattening filter and monitor chamber combination, and only 5% resulted from interactions in the collimation system.

21.3 SIMULATION OF 6OCO TELETHERAPY HEADS

The following differences between the 60Co machine and linear accelerators must be considered when modelling the former: 1. The photons are emitted isotropically, and are essentially monoenergetic (1.17 and 1.33 MeV). Scattered photons form a much larger fraction of photons reach• ing the patient. 2. The source is about 1.5 to 2 cm in diameter. In contrast, the linear accelerator source size is of the order of 2 mm in diameter. The large source size and a greater proportion of scattered photons in the beam result in much greater angular spread which has a significant effect on the shape of boundaries defined by collimators and blocks. 3. There is no flattening filter, and therefore no significant change in the shape of beam profiles with depth due to the changing energy spectrum as a function of radial distance from the central ray of the beam. 4. The output of 60Co teletherapy units vary with field size. This variation can be explained as resulting from a change in the fraction of scattered photons reaching 462 R. Mohan

the point of observation. Variation of output with field size is also observed for linear accelerators, but is smaller in magnitude. Furthermore, it is difficult for the case of accelerators to separate the variation of response of the monitor chamber to back-scattered photons and electrons from actual change in output. For both the accelerators and the 6OCo machines, it has been observed that field blocking does not significantly affect the output so long as the collimator opening is kept fixed.

( a )

SOURCE DRJ'WE:R

Cutaway dlllWlng 01 Theralron 780 SolllCllhelld

SOURCE CAPSULE T SOURCE SOURCE HOUSING T (LEAD) 6J 3SI,c:tn PRIMARY DEFINER ·uol (TUNGSTEN) I \ .... I \ r"

LOWER COLLIMATOR 11.t3 .... (LEAD)

J

(b)

Figure 21.10. Cross-sectional views of the Theratron 780 showing the place• ment of the source capsule, source housing, primary definer, and lower collimator. a) Diagram of the actual machine (courtesy of AECL). b) Diagram of the source and collimator regions used in Monte Carlo simulation. 21. Monte Carlo Simulation of Radiation Treatment Machine Heads 463 60CO units have been studied by a number of investigators employing Monte Carlo simulationsls,14. Han et al 14 approximated the complex geometry of a Theratron-780 6OCo unit (manufactured by Atomic Energy of Canada Limited) by a source capsule, the source housing, and the collimator assembly. Fig. 21.10a is a cutaway diagram of the Theratron 780 source head. It shows the source capsule, source housing, and the two principal sections of the collimator assembly: the upper primary definer, and the lower collimator of interleaved, adjustable jaws. Fig. 21.10b shows the corresponding geometry used in computer simulation.

Fig. 21.11a shows a detailed drawing of the actual teletherapy source capsule. Fig. 21.11 b shows the corresponding computer simulation. To generate energy spectra, Han et ailS followed histories of 2,000,000 photons from the decay of 6OCO at random points inside the source volume. The initial energies of the photons produced are divided equally between 1.17 and 1.33 Me V. As a consequence, approximately 750,000 particles (99.5% photons and 0.5% electrons) reached a scoring plane located just above the collimator region. The photon energy spectrum is shown in Fig. 21.12a. Unscattered primary photons make up 71.6% of the total number of photons, almost evenly divided between 1.17 and 1.33 MeV. The rest of the spectra consist almost entirely of Compton• scattered photons mixed in with a small number of unscattered bremsstrahlung and positron annihilation photons. This is shown in Fig. 21.12b. The sharp peak at about 500 keV in the spectrum in Fig. 21.12b is due to the annihilation of positrons with electrons.

Fig. 21.13a shows the energy spectrum of photons crossing a plane at 80 cm dis• tance from the source for the largest field size (35 x 35 cm). Fig. 21.13b shows the energy spectrum of all photons except the unscattered primary photons. Superim• posed on this spectrum are the individual contributions of Compton photons which last scattered from the source region, and of Compton photons which last scattered from the collimator region. The tissue-air ratios (TAR) in water, calculated employing the DPB model9 , using the spectrum shown in Fig. 21.13, and assuming a monochromatic photon beam of energy of 1.25 MeV, are plotted along with the measured data in Fig. 21.14. It is apparent that even though the scattered photons comprise nearly one-third of the total number of photons, their effect on the central-axis dose is not significant.

The photon energy spectrum changes as a function of field size. As the field size gets larger, the number of higher energy scattered photons increases. Fig. 21.15a shows the number of primary photons vs field size and the number of all other photons except primary vs. field size. The number of primary photons remains relatively constant as the field size increases, while the scattered photons increase in number with increasing field size. As is apparent from Fig. 21.15b, the number of Compton-scattered photons from the source region remains relatively constant for different field sizes, while the contributions from the tungsten definer and a.djustable collimator increase with larger field sizes. The contributions from the tungsten definer increase somewhat more rapidly than from the adjustable collimator. It would then seem that the observed increase in the output with increasing field size is due almost entirely to scattered photons from the primary tungsten definer and from the adjustable collimator. In fact, it can be demonstrated that the scattered photons from t.he primary definer and the upper portions of the adjustable collimator contribute most to the observed increase in output with field size. It should be apparent from purely geometrical considerations that the blocks shield a point on the isocenter from the primary definer and the upper portions of the adjustable collimator for only very small square block apertures. This explains the experimentalyobserved variation (or lack thereof) of output for blocked field sizes. 464 R. Mohan

( 0) SHIELDING DISCS, 0.050 IN. (0.1 2CM) TUNGSTEN (OPTIONAL)

ACTIVE FUS ION WELD

CAP SCREW CAP WASHER -SPRING STEEL (AS REOUIRED) x « ::E_ tt~~~:2:t2fllf:::::::=-SPACERS MAGNETIC ST. STL TYPE 416 z::E (IF REOUIRED) - u ~~ ___ ACTIVE I.lATERIAL ""'"""""o· .:-: ~ NICKEL PLATED COBALT 60 PELLETS 0·0394 IN (j·Omm)DIA X 0·0394 11\ II {)Omm) .LENGTH SLOTTED SPA CER-No416 ST .STL. ~[ (IF REOU IR ED) WALL THICKNESS-INNER- 0·035 IN · (O·oaCM) ! ~1~1~. I.~-,,~A: DIA~ . NOM--l~I OUTER -0·025IN. (O·OW,! ) o 0 . B DIA.MAX .---! 0-

1..- (b) TUNGSTEN 0.12 em

STEEL BACK PLATE 2.44 em ...I ...I ...I ...I ~ ~ ...I ...I W W W W t; t;

COBALT 60 (6 gm/em3) 1.0 em 1 STEEL FRONT PLATE 0.11 em

I r------1.5 em -I IT t--1.83 cm----j

Figure 21.11. Cross-sectional views of the source capsule. a) Diagram of the actual source capsule (courtesy of AECL). b) Diagram of the source capsule geometry used in our Monte Carlo ·simulation. 21. Monte Carlo Simulation of Radiation Treatment Machine Heads 465

ENERGY SPECTRUM OF PHOTONS FROM SOURCE REGION - PHC7ION8 _ SOURCE REGION 300000 15000 - ALL SOURCE REGION PHOTONS EXCEPT PRIMARY (0) (b)

250000 z iI z 5 200000 : 10000 I ~ i! S 150000 ~ I .,..~:z: lDOOOO 5000

50000

0 0 250 500 750 1000 1250 1500 250 500 750 1000 1250 1500 _ .. KEY ENERGY IN KEY

Figure 21.12. Energy spectrum of photons from the source region. a) All photons from the source region. b) All photons except the unscattered primary photons.

ENERGY SPECTRUM OF PHOTONS 10000 ( a ) AT 10 em SSD 800 (b) ALL PHOTONS EXCEPT PRIMARY 35 em X 31 em F.'., r. OlD 17.1_ -:I$ ... X3S ... F.8. ,.Otot7.5cm COMPTON PHOTONS SCATTERED _SOURCE_ 7500 600 z FAOM COWMATOA AEGION Ii >w "S .,~400 z ~ ..:z: 2500 200

O+-~~-~--r--'--~~~ o 250 500 750 1000 1250 1500 250 500 750 1000 1250 1500 ENERGY IN KEY ENEAQY IN KEV Figure 21.13. Energy spectrum of photons which reach 80 cm SSD after having passed through the collimators. a) Spectrum of all photons. The field size is 35 cm x 35 cm, and the photons are collected in a radial bin extending from 0.0 to 17.5 cm. b) Spectrum of all photons except the primary unscattered photons. Superimposed on it are the contributions from Compton photons last scattered from the source region, and Compton photons last scattered from the collimator region. 466 R. Mohan

-- MEASURED DATA ----- 1.25 MoV PHOTONS 1.0 ...... 1.17 AND 1.33 _ PHOTONS ALONG WITH SCATTEIIED PHOTONS

i.. 0.8 ~ Ii: 0.6 ..I I OA !.. I!! 0.2

O~-----~------~------L------~ o 10 20 30 40 DEPTH(cm) Figure 21.14. Comparison of tissue-air ratios (TAR) measured or calculated using the spectra shown in Fig. 21.13, or assuming a monoenergetic 1.25 MeV photon beam.

_OF NUMBER OF PRIIIAR\' AND SCATTER PIfOIONS ... Flao SIZE COMPTON SCATTERED PHOTONS ... FIELD SIZE

• ~ AND 1ICAnEII-1tADIAL _ ... 1_. • ALL COIIPIOII PHOrONI - RADIAL .. 0 ID 5 CIII. + PHOIONS SCAnERED FROM SOURCE REGION • I'HImIIIS SCATTERED __ 1UNGStEJI DEANER .---+-- 9DO • I'HImIIIS sc:ATTEAED __ LOWEll COLIMATER 2500 (ol (bl

+ + + + + + + + + + + +

• • • • • • • • 1200 • • 100 •

O~~--~~--~--r--r--~~ O~-T--~--r-~--'-~r-~--' o 5 m U • H • " • o 5 m U • H • " • wocmt OF OIIE SIDE OF IIQIIIIIE .... LENGTH OF OIIE SIDE OF SQUIUIE fIELD Figure 21.15. Number of photons versus field size. a) Number of primary and scattered photons versus field size and b) Number of Compton-scattered photons versus field size. 21. Monte Carlo Simulation of Radiation Treatment Machine Heads 467

21.3.1 Electron Contamination

For 60Co teletherapy units, the electron contamination is a more serious problem than for linear accelerators. For large fields, the dose in the build-up region can .exceed the dose at nominal depth of maximum by as much as 15 percent. Rogers et a[l" by Monte Carlo simulations, have investigated the sources of electron contamination, and examined the efficacy of contamination-reduction techniques. As noted above, the number of electrons exiting from the source region in the direction of the patient is extremely small compared to the number of photons. Rogers et al employed a variety of variance-reduction techniques to improve computing efficiency by up to a factor of 100, and obtain a statistical uncertainty of a few percent for the electron component of depth dose. They estimated that at 80 cm from the source, the dose at the surface due to contaminant electrons is 45% of the peak photon dose, roughly half of it originating in the source capsule and the air colUinn, and the other half being generated by the interaction of particles initially travelling in the direction of the collimators. They also found that the electron dose is a strong function of field size. The electron dose for a 5-cm X 5-cm field is an order of magnitude less than for a broad beam.

Electron dose can be reduced by the use of filters. Copper makes an excellent filtering material. Rogers et al found that a thin sheet of copper, which absorbs all the electrons originating in the source, generates 40% fewer electrons of its own. This is because the electrons are scattered much more in the copper, and hence have a shorter effective range. Furthermore, electrons escaping the filter have a larger mean square angular spread, and therefore fewer of them reach the patient.

21.4 SUMMARY It has been demonstrated in many recent investigations that the Monte Carlo method is an excellent tool for modeling radiation treatment machine heads, and gen• erating the energy spectra and angular distributions of photon beams produced by linear accelerators and 6OCO teletherapy machines, and for studying other characteristics of photon beams. In most instances, information provided by the Monte Carlo techniques cannot easily and accurately be obtained by other means. Current experimental tech• niques to measure the energy spectra of intense photon beams are crude and unreliable. Monte Carlo results can, however, be verified indirectly by comparing quantities such as depth dose, the transmission values of various materials, etc., calculated using energy spectra and angular distributions with the corresponding measured data. Knowledge of both energy spectra and angular distributions is essential for accurate dose calcula• tions and for a variety of other applications in radiation therapy. Application of Monte Carlo techniques in this and other areas of radiation dosimetry is leading to a significant advancement in the state of the art of radiation therapy. 468 R. Mohan REFERENCES 1. W. Swindell, "A 4-MVCT Scanner for Radiation Therapy; Spectral Properties of the Theraphy Beam", Med. Phys. 10 (1983) 347. 2. R. E. Bentley, J. C. Jones, S. C. Lillicrap, "X-Ray Spectra from Accelerators in the Range 2 to 6 MeV", Phys. Med. Bio1.12 (1967) 301. 3. K. A. Jessen, ACTA Radio. Ther. Phys. BioI. 12 (1973) 561. 4. 1. B. Levy, R. G. Waggener, W. D. McDavid, W. H. Payne, "Experimental and Calculated Bremsstrahlung Spectra from a 25-MeV Linear Accelerator and a 19-MeV Betatron", Med. Phys. 1, (1974) 62. 5. 1. B. Levy, R. G. Waggener, A. E. Wright, "Measurement of Primary Brems• strahlung Spectrum from an 8-MeV Linear Accelerator", Med. Phys. 3 (1976) 173. 6. R. Nath, R. J. Schulz, "Determination of High-Energy X-ray Spectra by Pho• toactivation", Med. Phys. 3, (1976) 133. 7. P. H. Huang, K. R. Kase, B. E. Bjarngard, "Simulation Studies of 4-MV X-ray Spectral Reconstruction by Numerical Analysis of Transmission Data", Med. Phys. 9 (1982) 695. 8. P. H. Huang, K. R. Kase, B. E. Bjarngard, "Reconstruction of 4-MV Brems• strahlung Spectra from Measured Transmission Data", Med. Phys. 10 (1983) 778. 9. R. Mohan, C. S. Chui, 1. Lidofsky, "Differential Pencil Beam Dose Computation Model for Photons", Med. Phys. 13 36 (1986). 10. R. Mohan, C. Chui, 1. Lidofsky, "Energy and Angular Distribution of Photons from Medical Linear Accelerators", Med. Phys. 12 (1985) 592. 11. P. L. Petti, M. S. Goodman, T. A. Gabriel, R. Mohan, "Investigation of Buildup Dose from Electron Contamination of Clinical Photon Beams", Med. Phys. 10 (1983) 18. 12. P. L. Petti, M. S. Goodman, J. M. Sisterson, B. J. Biggs, T. A. Gabriel, R. Mo• han, "Sources of Electron Contamination for the Clinac-35 25-MV Photon Beam", Med. Phys. 10 (1983) 856. 13. K. Han, D. Ballon, C. Chui, R. Mohan, "Monte Carlo Simulation of a Cobalt-60 Beam", Med. Phys. 14 (1987) 414. 14. D. W. O. Rogers, G. M. Ewart, A. F. Bielajew, G. Van Dyk, "Calculation of Electron Contamination in a Co-60 Therapy Beam" , in Proceedings of the IAEA International Symposium on Dosimetry in Radiotherapy, Vol. 1, IAEA, Vienna, Austria (1988) 303. 15. R. 1. Ford and W. R. Nelson, "The EGS Code System (Version 3)", Stanford Linear Accelerator report SLAC-210 (1978). 16. W. R. Nelson, H. Hirayama and D. W. O. Rogers. "The EGS4 Code System", Stanford Linear Accelerator report SLAC-265 (1985). 22. Positron Emission Tomography Applications of EGS

A. Del Guerra* and Walter R. Nelson t

Department of Physics, University of Pisa, Piazza Torricelli 2, 1-56100, Pisa, Italy

22.1 PRINCIPLES OF POSITRON EMISSION TOMOGRAPHY The use of positron emitters in medical imaging was first suggested by Wrenn and co• workers1, and Sweet2, and the first prototype Positron Emission Tomography (PET) scanner was built by Brownell and Sweet3 as shown in Fig. 22.1. A radiopharmaceutical labeled with a positron emitter is distributed within a biological target. The emitted /3+ annihilates with an electron of the surrounding tissue to produce two photons of 511 keY back-to-back (in the CM system). By detecting these two gammas in coincidence, one constrains the annihilation event to the volume spanned by the two detectors. The original distribution of radioisotope is reconstructed from a series of "projections" , obtained by moving and rotating the scanner around the patient. The "electronic collimation" avoids the use of passive collimators (typical transmission efficiency 10-4 - 10-3) which are necessary in Single Photon Emission Computed Tomography (SPECT) with ")'-emitting radionuclidesj this implies not only a reduction of the radiation dose delivered to the patient, but also a higher spatial resolution and a better quantization of the imaging results.

22.2 PHYSICAL PROCESSES IN PET Many physical processes are involved in the detection in coincidence of the two annihi• lation quanta.

22.2.1 Positron Emitters The commonly used positron emitters are listed in Table 22.1, together with the lifetime and the mean energy of the (3 spectrum4• Because carbon, nitrogen and oxygen are the most important constituent elements of the human body, the so-called physiological /3+ isotopes ( llC, 13N, 150 ) are particularly attractive for the monitoring of any

* Present address: Department of Physics University of Napoli Napoli, Italy t Radiation Physics Group Stanford Linear Accelerator Center Stanford, California 94309, U.S.A.

469 470 A. Del Guerra and W. R. Nelson metabolic pathway. However, due to their very short lifetime, a dedicated cyclotron ("in loco") is required for their production. 18F is also commonly used, especially to label deoxiglucose (FDG). 68Ga and 82Rb, although non-"physiological" isotopes, are readily available from commercial generators, and their use is increasing rapidly.

DETECTOR #2

~ ______~Electronic Coincidence

Figure 22.1. Schematic drawing of the first PET scanner5 •

Table 22.1. Positron Emitters Most Used in PET. Radioisotope TI Mean energy of 13+ spectrum (minutes) Tmean (MeV) lle 20.4 0.385 13N 10.0 0.491 15 0 2.0 0.735 18F 109.8 0.242 (82S r) => 82Rb 1.3 1.410 (68Ge) => 68Ga 68.1 0.740

22.2.2 Positron Range

The 13+ is emitted with a continuous energy spectrum. Before annihilating "at rest" (the in~flight annihilation probability is at most a few percent), the positron travels a finite range, which depends upon its energy (see Table 22.1). The range in tissue varies from a fraction of a mm for 18F (which has the lowest energy) to several mm for 82Rb (e.g., see Derenz06 ). 22. Positron Emission Tomography Applications of EGS 471

22.2.3 Positron Annihilation The annihilation of the positron at rest with an electron at rest would imply the emission of two '"('s in opposite directions. However, due to the Fermi motion, the distribution is almost Gaussian around 180°. In water (tissue), the FWHM is""' 0.5 degrees7 •

22.2.4 Scatter in Tissue The mean free path of a 511-keV photon in water is ""' 10 cm. The human head or chest are roughly two mean free paths thick. This results in a heavy loss of coincident photons; only 20-30% of the two '"('s will reach the detector unaffected. Furthermore, a fraction of the scattered photons will still produce coincidence events, contributing a distributed background which gives rise to "projections" from "non-existing source" positions, thus smearing the resolution and decreasing the contrast of the object.

22.2.5 Interaction Within the Detector The detection of the two photons depends upon the attenuation coefficient and the stopping power of the detector, and its "quantum efficiency", i.e., the efficiency in converting the energy loss into useful digital information. The type of detector chosen, (scintillator crystal, gaseous detector, etc.) is a major part of the design of a PET camera, as will be discussed in the next section.

22.3 THE PET CAMERA The design of PET cameras varies greatly. From the original two detector scanner moving and rotating around the patient (e.g., Fig. 22.1), more complex solutions have been implemented, such as two opposite and rotating Anger cameras, a planar annular ring of scintillators, large-area planar geometries, and multi-ring configurations (see Fig. 22.2).

For the design of a PET camera, the following requirements are important:

- a high detection efficiency for 511-keV '"(-rays, - a short temporal resolution for the coincidence, - a high spatial resolution, - a large solid angle coverage.

Of course, not all the parameters can be optimized at the same time. A solution which best fulfills one requirement may be rather loose on another; hence, different designs cannot be compared against one or two parameters, but only in terms of their global performance-i.e., the quality of the image obtained in a given time with a certain amount of activity concentration in tissue.

22.3.1 Scintillator Multicrystal Detector

The first generation PET camera was made as a single ring of NaI(Tl) scintillatc;>r crys• tals( e.g., Fig. 22.2b). In order to increase the sensitivity, other scintillators are now used (see Table 22.2). Bismuth Germanate (Bi4Ges012, simply called BGO) is the highest 472 A. Del Guerra and W. R. Nelson density, highest Z scintillator material available. Although the photopeak-to-Compton ratio is not as good as for NaI, (thus producing a slightly worse energy resolution at 511 keY), BGO is the best scintillator for totally absorbing 511-keV photons in small crystals; this makes it possible to reduce the crystal size, and thus increase the spatial resolution without adding cross-talk and multiple-hit problems. Furthermore, to in• crease the axial solid-angle coverage, multi-ring solutions (e.g., Fig. 22.2d) usually are adopted.

a) b)

[ ) d)

Figure 22.2. Examples of PET camera configurations: a) two opposite Anger cameras; b) single ring; c) large-area hexagon planar camera; d) multi-rings.

An alternative solution is the use of faster scintillators to decrease the time coinci• dence window, and thus to reduce the accidental contamination so as to have a better signal-to-noise ratio. Additionally, by measuring the difference in time between the arrivals of the two photons, one could identify directly the source position (or at least reduce the position uncertainty) along the line-of-flight. This technique is called Time Of Flight PET (TOFPET). With scintillators such as CsF and BaF2 which give a time resolution of 400 ps, it is possible to determine the source position to within'" 6 cm along the photon's flight path. 22. Positron Emission Tomography Applications of EGS 473

Table 22.2. Main Properties of Various Scintillators Used in PET Cameras.

Crystal NaI(Tf) BGO CsF BaF2 Density (g/cm3 ) 3.67 7.13 4.64 4.89 Atomic Number 11,53 83,32,8 55, 9 56, 9 Linear Attenuation 0.34 0.92 0.44 0.47 Coefficient at 511 keV (cm-1) Scintillation 250 300 5 0.8/620 Decay Time (ns) Emission 410 480 390 225/310 Wavelength (nm) Energy Resolution > 7% > 10% 23% 13% at 511 keV (FWHM) Index of refraction 1.85 2.15 1.48 1.59 Hygroscopic Yes No Very No

A spatial resolution of 3-4 mm is considered to be the practical limit for PET imaging, if one takes into account the contributions due to the non-collinearity of the two photons, the positron range, and the limited statistics. Thus, it is not worthwhile building detectors with an intrinsic resolution much below that value. To improve the spatial resolution in a discrete system such as in a many-scintillator PET camera, one can simply use smaller crystals, for instance 4 mm instead of the standard 8-10 mm size. This, of course, implies the use of appropriate phototubes and electronics, increasing the cost and the complexity of the tomograph.

22.3.2 Gas Detector Another interesting approach is the use of large-area, position-sensitive gaseous detec• tors, such as the MultiWire Proportional Chamber (MWPC). This type of detector8 was originally designed for high-energy physics experiments. It basically consists of a gas-filled chamber with three wire planes, the central one kept at positive voltage. If an ionization is produced in the gas region, the electrons are drifted towards the anode plane and produce an avalanche (around an anode wire) which can be detected directly from the anode signal or from the signals induced on the cathode planes.

The use of MWPC's for PET is particularly attractive because of their very good spatial resolution: some hundreds of /lm are easily achieved along the anode wire direc• tion, whereas in the other direction, the spatial resolution is determined by the anode wire pitch (typically 2 mm). Furthermore, large areas are easy to cover. However, an obvious limitation in the use of MWPC's for ,-imaging arises from the difficulty of stopping the photon in the gas. For instance, the photopeak efficiency at 511 keV in 1 cm of Xe at 10 atmospheres is still much below 1 %. It is then necessary to use high-Z, high-density converters.

22.4 USE OF MONTE CARLO CODES IN TOMOGRAPH DESIGN

The construction of a tomograph calls for a careful planning of the detector type, the detectors arrangements, the diameter of the gantry, etc. Although experimental work on a smaller prototype is mandatory, a complete simulation of the tomograph by Monte Carlo technique is very useful to evaluate the performance and optimize the design. 474 A. Del Guerra and W. R. Nelson

Monte Carlo radiation transport codes have been succesfully used by various authors to cover individual aspects of a PET design.

Derenz06 has studied the annihilation point-spread function of various positron emitters in water in order to deconvolve the range contribution from the reconstructed image and obtain a better spatial resolution. The same physical process has been investigated by !ida et a19 , who studied the reduction of the annihilation point-spread function by applying a magnetic field, expecially for high-energy emitters such as 82Rb. DerenzolOand Derenzo and Rilesll have studied the efficiency of various scintillator• photomultiplier systems to tailor the dimension of the crystal and the thickness of the absorption septum between two crystals, to minimize cross-talk, and to maximize the spatial resolution.

The idea ~f using Monte Carlo codes to optimize the geometry of a single-slice tomograph for a given field of view has been pursued by Lupton and Keller12. They have evaluated the accidental/true coincidence ratio and the scattered-coincidences profile for an annular ring geometry and various object/gantry diameter ratios. In this case, a Monte Carlo code could give unique information because it can "tag" the scattered and unscattered events, which are indistinguishable in the experimental situation.

In all of these examples and in many others, Monte Carlo codes are used to simulate specific sections, or selected performances of a tomograph. In the next section, we will present an example of a very ambitious program we have been carrying out to simulate completely a large-area 3-D tomograph by means of the EGS4 code1S .

22.5 AN APPLICATION: USE OF EGS4 FOR THE HISPET DESIGN

A fully 3-D large-area positron camera (HIgh Spatial resolution Positron Emission Tomograph) has been proposed14 which consists of six modules arranged so as to form the lateral surface of a hexagonal prism (see Fig. 22.3). The type of detector chosen is a gaseous detector with an appropriate high-density converter. We have developed, as a converter, a matrix of resistive lead-glass tubing15 (also see Section 28.2.4).

An electric field is applied along the converter so that the photoelectrons (produced by the I interaction within the converter walls) are drifted out of the converter to the avalanche region of the MWPC. A schematic drawing of a MWPC equipped with a lead-glass tube converter is shown in Fig. 22.4.

Each module of HISPET will have two MWPC's, each with two I-cm thick con• verter planes of lead-glass tubing (0.5 mm and 0.6 mm, inner and outer diameter, re• spectively). In order to study the efficiency of this type of converter and to evaluate the performance of the tomograph, two different EGS4 User Codes (UCCELL and UCPET) were used, as described in the following sections. 22. Positron Emission Tomography Applications of EGS 475

Figure 22.3. The HISPET project. For simplicity only three modules are shown1•.

22.5.1 The Converter Efficiency Code (UCCEll)

The converter is made of glass capillaries of high lead content, fused to form honeycomb matrices. In order to calculate the efficiency versus photon energy, an EGS4 User Code (UCCELL) with a relatively simple geometry, based on the unit cell concept, was created16-i.e., two concentric cylinders inside a box (see Fig. 22.5).

Photons randomly irradiate the top face of the cell at 90° (±3.5°), the same geom• etry as used in the experimental measurement17 • All particles are transported inside the cell until they reach energy cutoffs (10 keY and 1 keY for e- and " respectively), exit the top or bottom, or exit the sides. In the latter case, in order to characterize fully the converter with a multitude of contiguous holes, the particles are re-transported into the unit cell by making the appropriate coordinate translation while maintaining the direction of motion. If the electron produced by the photon interaction enters the inner region of the cell, it is considered to be detected, irrespective of its energy at that point. 476 A. Del Guerra and W. R. Nelson

ANODE PLANE

Figure 22.4. Schematic drawing of a MWPC equipped with a lead-glass tube converter plane for PET imaging14.

Figure 22.5. The unit cell geometry used in UCCELL.

The probability that an electron will reach a hole depends on its energy, where it is created, and its direction of motion. In UCCELL, the energy released by photon interactions was scored into two separate histograms-detected events (i. e., when an electron enters a hole), and total events. The ratio of the distributions provided a measure of the detection probability (averaged over position and direction) as a function of the electron kinetic energy. The probability table thus created by UCCELL was then 22. Positron Emission Tomography Applications of EGS 477 used in the second EGS4 User Code (UCPET) for the study of the general performance of HISPET (see next section)* .

Various combinations of inner and outer diameters were chosen with the length of the cell fixed at I cm. Various types of lead glass with different percentages of PbO and different densities were also simulated. Figure 22.6 shows the calculated efficiency of the converter as a function of the photon energy for three ID 10D tube converters, compared with our experimental data17 obtained with 80% PbO lead glass (density 6.2 g/cm3 ). Figure 22.7 shows the calculated efficiency for 51l-keV photons versus the diameter of the tube at a fixed OD lID ratio of 1.2 for the various lead glass types. Figure 22.8 shows the calculated efficiency versus wall thickness for a given inner diameter of the tube for various percentages of lead content. These results have made it possible to optimize the lattice of the converter.

~5~------~ "-__- O.48/0.60mm

5.0

~ O.911L10mm ~ w 1.33/1.59mm 2.5

Ey (keV)

Figure 22.6. Calculated efficiency of a I-cm thick converter as a function of the photon energy for different combinations of inner and outer diameters (solid lines) j o-experimen tal data18.

22.5.2 Evaluation of the HISPET Performance (UCPET) A full three-dimensional simulation of the HISPET was performed using the EGS4 User Code, UCPET19 , according to the following scheme:

1. Generation of the positron coordinates, direction, and energy. The latter was sampled according to the energy spectrum of the selected radioisotope; theo• retical beta spectra20 (corrected for screening) were introduced into EGS4 in the form of look-up tables. The following isotopes were considered: llC, 13N, 16 0, 18F, 19Ne, 38K, 68Ga, and 82Rb.

* Note: Limiting the energy loss along a charged particle track by means of ESTEPE (e.g., see Chapter 5) had not been introduced at the time of these calculations (1982). 478 A. Del Guerra and W. R. Nelson

10.0 E('I.)

7.5 0.0'_12 1.0. - .

5.0

2.5

0.5 1.0 1.5 2.0 Tube outer diameter (mm) Figure 22.7. Calculated efficiency of a l-cm thick converter as a function of the tube outer diameter for an outer/inner diameter ratio of 1.2. Note: Var• ious PbO proportions correspond to commercial glasses used experimentally; efficiency curve for pure lead drawn for comparison18•

2. Transport and annihilation of the positron in the phantom. In principle, due to the structure of EGS4 itself, it is possible to simulate any type of phantom, although only cylindrical and spherical geometries were implemented in the sim• ulation. The positron was followed in the phantom until it reached the lower energy cutoff of 10 keV, when it was forced to annihilate as if at rest. In ad• dition to Bhabha scattering and continuous energy loss, EGS4 also considers annihilation in flight as a discrete Monte Carlo process. Once the annihilation takes place, the angular distribution is properly taken into account for two pho• tons, both for annihilation at rest and in flight, the latter probability being at most a few per cent for the highest energy radioisotope. Figure 22.9 shows the differential probability of annihilation per unit distance from the source in water for various radioisotopes.

3. Transport of annihilation quanta from within the phantom to the detector. Dur• ing this step of the program, all charged particles that are generated are immedi• ately discarded. If the photon emerges from the phantom with an energy greater than the cutoff energy (1 keV), it is further transported to the detector. 22. Positron Emission Tomography Applications of EGS 479

6 ~71%PbO

4 ~51%PbO ~34%PbO

2

50 100 Wall thickness (f'm) Figure 22.8. Efficiency of l-cm thick converter of 0.5-mm inner diameter versus the wall thickness for various Pb compositions18.

4. Simulation of the three-dimensional geometry of the detector. The implementa• tion of any particular geometry is left up to the discretion of the user of EGS4. Accordingly, the hexagonal prism geometry of HISPET was simulated by means of the geometry macro package that is distributed with the EGS4 Code System. The geometry was fairly precise, and included the four planes of converters and the two MWPC sensitive regions for each module.

5. Interaction of the photon within the detector. To simulate the interaction of the photon within the detector, the actual lead-glass honeycomb geometry was ap• proximated by a solid converter, the density of which was reduced by the packing fraction (i.e., the ratio of the covered to total area).

6. Scoring of the events. Those events where only one or both photons detected are accounted for, and the single and coincidence rates are tabulated. Finally, to study the spatial resolution of HISPET, simple histograms are produced using coincidence events both in opposite and non-opposing modules.

The SLAC Unified Graphics System21 was also used in UCPET in order to provide a means of visualizing the various interaction sequences that lead to both good and bad event scenarios. An example is presented in Fig. 22.10 showing two orthogonal views of the tomograph and two photon tracks (i.e., solid and dotted lines). 480 A. Del Guerra and W. R. Nelson

1.0

...... -' , /. . I 1'- •., i' .• ~ 0.8 ... L' !J • III .• ','.' .. .. I • C .. "• .... ::I • " , >0.6 ,', . Rb-82 ~ ••. , . ..~ • ...... :a •.., 0-15 C 0.4 . • " "' ';', ' .. • ..... ~ I F-18 ' ... b,,'.:J, .. ' 0.2 \;.1 •'+ :,

0.8 1

Figure 22.9. Differential probability of annihilation per unit distance from the source in water for several (3+ radioisotopes (each curve is normalized to a maximum value of 1.0 for purposes of comparison19).

Information related to the event history for each photon is printed in the two corre• sponding boxes, according to the following key:

S - Indicates the sector where the interaction took place (1-6). The value of 0 is assumed if the interaction is in the phantom itself. D - Indicates which converter ("detector") is involved (first, second, third or fourth). T - Identifies the type of interaction (C for Compton, P for photoelectric). GEl - Photon energy before the interaction. GE2 - Photon energy after the interaction. EKE - Electron kinetic energy. TD - Distance of the interaction point from the end of the tube, which is pro• portional to the drift time to the MWPC.

In the example presented in Fig. 22.10, the first photon (solid line and solid box) interacts twice in sector 1, both times in converter 3. The first results in a Compton electron of 277 keY at a distance of 0.501 cm from the end of the tube; the second in a photoelectron of 146 keVat a distance of 0.581 cm from the end of the tube. The Compton or photoelectron is assumed to be detected with a probability as given by the appropriate probability table calculated from the unit cell simulation (i. e., UCCELL). 22. Positron Emission Tomography Applications of EGS 481

CAS I': 37

Ii S D T GF: I F.Kf. Gf.2 TD(Ot) Ct: \ EKE Ct:2 TD(CM) I I 3 C ;; \1 277 23·\ 0:,01 :,11423 0:1:15 2 I 3 P 234 14(; 0 ;,!l1

S # 1

y

,,------.-:.

Figure 22.10. Typical display of a "good" 2-, event19(see text for an expla• nation).

The second photon (dotted line and dotted box) makes only one interaction, re• sulting in a 423-keV photoelectric electron in the third converter of sector 4 (0.335 cm from the end of the tube).

If one photon produces more than one detected electron in the same module, only the earliest electron (that nearest to the anode plane of the MWPC) is retained. The real coordinate along the thickness of the converter is substituted by half the thickness so as to account for the parallax error. The x and y finite resolution of the MWPC are also simulated directly within EGS4/UCPET by sampling from appropriate Gaussian distributions, and the final position is checked against spatial cutoffs.

22.5.3 Image Reconstruction From EGS4-Simulated Data Output

Output from EGS4 (i.e., "events") are stored onto permanent memory for subsequent analysis, as if it were real. In particular, the coincidence data are analyzed by the 3-D filtered back-projection algorithm22 , which in fact has been tested and optimized on the simulated data, and is now used on the first real data from a HISPET prototype23•

Figure 22.11 shows the spatial resolution which is obtained taking a profile through the central plane of the reconstructed image for a point-like 18F source embedded in a head water phantom at the center of the tomograph. 482 A. Del Guerra and W. R. Nelson

1.00 Fi Itered ...III C ::J ...>• ...E 0.75 ~... 4( FWHM 0.50 -- 4mm

0.25

o 10 20 50 60 Pixel number

Figure 22.11. Spatial resolution of a 18F point-like source embedded in a 10 cm radius water phantom (£phere), at the center of the HISPET tomograph. Data were simulated by Monte Carlo and reconstructed by the filtered back• projection 3-D algorithm19 .

It is also possible to produce a sample of the accidental coincidences by randomly regrouping two-by-two the interactions points. Again, the scattered events in the phan• tom keep their signature in the simulation, and their effects on the spatial resolution can be singled out and quantified.

22.6 SUMMARY The simulation of HISPET by means of the EGS4 Code System has proved to be very useful, especially as an aid in selecting the design parameters of the project. In particular, the user has the capability of using software to turn on and off various processes and quantities of interest. This feature of EGS4 has been used to: -independently study the geometric and total efficiency of the modules and of the entire tomograph; -investigate the different contributions to the spatial resolution-i.e., due to source localization, positron range, two-gamma non-collinearity, parallax er• ror, intrinsic detector resolution, Compton scattering in the phantom, etc.

Because the simulation is purely analog in nature, it is inherently inefficient. About 200 positrons/second can be generated, and the subsequent progeny tracked, on an IBM-3081 mainframe computer. One can, however, take advantage of the modularity of EGS4/UCPET in order to produce a large sample of data, and then to apply the various software cuts, which simulate the configuration being studied. 22. Positron Emission Tomography Applications of EGS 483

REFERENCES 1. E. R. Wrenn, M. L. Good and P. Handler, "The Use of Positron Emitting Ra• dioisotopes for the Localization of Brain Tumours", Sci. 113 (1951) 525. 2. W. H. Sweet, "Uses of Nuclear Disintegration in the Diagnosis and Treatment of Brain Tumours", New Eng!. J. Med. 245 (1951) 875. 3. G. 1. Brownell and W. H. Sweet, "Localization of Brain Tumours With Positron Emitters", Nucleonics 11 (1953) 40. 4. E. Browne and R. B. Firestone, Table of Radioactive Isotopes, edited by V. S. Shirley, (John Wiley & Sons, New York, 1986). 5. A. Del Guerra, "Positron Emission Tomography", Physica Scripta T19 (1987) 481. 6. S. E. Derenzo, "Precision Measurements of Annihilation Point Spread Function for Medically Important Positron Emitters", in Positron Annihilation, edited by R. R. Hasiguti and K. Fujiwara, (Japan Institute of Metals, Sendai, Japan, 1979); 819. 7. P. Colombino, B. Fiscella and 1. Trossi, "Study of Positronium in Water and Ice from 22 to -144°C by Annihilation Quanta Measurements", Nuovo Cimento 38 (1965) 707. 8. F. Sauli, "Principles of Operation of MultiWire Proportional and Drift Cham• bers", CERN report 77-06 (1977). 9. H. Iida, I. Kanno, S. Miura, M. Murakami, K. Takahashi and K. Uemura, "A Simulation Study of a Method to Reduce Positron Annihilation Spread Distribu• tions Using a Strong Magnetic Field in PET", IEEE Trans. Nucl. Sci. NS-33 (1986) 597. 10. S. E. Derenzo, "Monte Carlo Calculations of the Detection Efficiency of Arrays of NaI(T£), BGO, CsF, Ge, and Plastic Detectors for 511 keY Photons", IEEE Trans. Nuc!. Sci. NS-28 (1981) 131. 11. S. E. Derenzo and J. Riles, "Monte Carlo Calculations of the Optical Coupling Between Bismuth Germanate Crystals and Photomultiplier Tubes" , IEEE Trans. Nucl. Sci. NS-29 (1982) 191. 12. 1. R. Lupton and N. A. Keller, "Performance Study of Single-Slice Positron Emission Tomography Scanners by Monte Carlo Techniques" , IEEE Trans. Med. Imag. MI-2 (1983) 154. 13. W. R. Nelson, H. Hirayama and D. W. O. Rogers, "The EGS4 Code System", Stanford Linear Accelerator Center report SLAC-265 (1985). 14. A. Del Guerra, G. K. Lum, V. Perez-Mendez and G. Schwartz, "The HIS PET Project: State of the Art", in Positron Annihilation, edited by P. C. Jain, R. M. Singru and K. P. Gopinathan, (World Scientific Publishing Co., Singa• pore, 1985); 810. 15. M. Conti, A. Del Guerra, R. Habel, T. Mulera, V. Perez-Mendez and G. Schwartz, "Use of a High Lead Glass Tubing Projection Chamber in Positron Emission Tomography and in High Energy Physics", Nuc!. Instr. Meth. A255 (1987) 207. 16. A. Del Guerra, V. Perez-Mendez, G. Schwartz, and W. R. Nelson, "Design Con• siderations for a High Spatial Resolution Positron Camera with Dense Drift Space MWPCs", IEEE Trans. Nuc!. Sci. NS-30 (1983) 646. 484 A. Del Guerra and W. R. Nelson

17. R. Bellazzini, A. Del Guerra, M. M. Massai, W. R. Nelson, V. Perez-Mendez and G. Schwartz, "Some Aspects of the Construction of HISPET: HIgh Spatial Resolution Positron Emission Tomograph", IEEE Trans. NucI. Sci. NS-31 (1984) 645. 18. A. Del Guerra, A. Bandettini, M. Conti, G. De Pascalis, P. Maiano, C. Rizzo, and V. Perez-Mendez, "3-D PET with MWPCs: Preliminary Tests with the HIS PET Prototype", NucI. Instr. Meth. A269 (1988) 425. 19. A. Del Guerra, M. Conti, W. R .Nelson, R .Porinelli, and C. Rizzo, "3-D Imaging with a 3-D PET: A Complete Simulation of the HISPET Tomograph", in Intern. Workshop on Physics and Engin. of Computerized Multidimensional Imaging and Processing, edited by O. Nalcioglu, Z. H. Cho, and T. F. Budinger, (SPIE 671, 1986) 34. 20. E. J. Konopinski and M. E. Rose, "The Theory of Nuclear Beta-Decay", in Alpha-, Beta-, Gamma-Ray Spectroscopy, K. Siegbahn (Ed.) (North-Holland Publishing Co., 1965); p. 1327; U. Fano, "Tables for the Analysis of the Beta Spectra", National Bureau of Standards, Applied Mathematics Series, Volume 13 (1952). 21. R. C. Beach, "The Unified Graphics System for FORTRAN 77: Programming Manual", SLAC Computation Research Group technical memorandum CGTM 203 (November 1985 revision). 22. C. Rizzo, M. Conti and A. Del Guerra, "Evaluation of the Imaging Capabilities of HIS PET" , Physica Medica 1 (1987) 19. 23. A. Del Guerra, A. Bandettini, M. Bucciolini, M. Conti, G. De Pascalis, P. Ma• iano, V. Perez-Mendez and C. Rizzo, "First Experimental Results from a High Spatial Resolution PET Prototype" , Proc. Ninth Annual Conf. IEEE Engin. in Med. and BioI. Society, Boston, MA (November 13-16, 1987); 1010. 23. Stopping-Power Ratios for Dosimetry

Pedro Andreo*

Radiation Physics Department Karolinska Institute 10401 Stockholm, Sweden

23.1 INTRODUCTION The determination of the absorbed dose at a specified location in a medium irradiated with an electron or photon beam normally consists of two steps: (1) the determination of the mean absorbed dose to a detector by using a calibration factor or performing an absolute measurement, (2) the determination of the absorbed dose to the medium at the point of interest by calculations based on the knowledge of the absorbed dose to the detector and the different stopping and scattering properties of the medium and the detector material. When the influence of the detector is so small that the electron fluence in the medium is not modified, the ratio of the mass collision stopping power of the two materials accounts for the differences in energy deposition, and provides a conversion factor to relate the absorbed dose in both materials. Today, all national and international dosimetry protocols and codes of practice are based on such procedures, and the user easily can carry out these steps using tabulated data to convert a measured quantity to absorbed dose in the irradiated medium at the location of interest. Effects due to the spatial extension of the detector are taken into account using perturbation correction factors (see Chapter 25).

The Monte Carlo method has become the most common and powerful calcula• tional technique for determining the electron fluence (energy spectra) under different irradiation conditions. Cavity theory is then used to calculate stopping-power ratios. In this chapter, we will consider the different steps needed to evaluate s-ratios, em• phasizing the different types of cavity-theory integrals and the Monte Carlo techniques used to derive the necessary electron spectra in the range of energies commonly used in radiation dosimetry, i.e., photon and electron beams with energies up to 50 MeV.

23.2 FUNDAMENTALS OF STOPPING-POWER RATIOS The determination of the absorbed dose in a medium is based on the Bragg-Gray principle that relates the absorbed dose to the medium, D"., to the absorbed dose in the gas (usually air) filling a cavity of the medium, Dair:

D". = Dair f (23.1)

* Formerly at Seccion de Fisica, Hospital Clinico Universitario, Zaragoza, Spain

485 486 P. Andreo

The proportionality factor was identified by Gray as the ratio of the mass stopping powers in the two materials, S ...... ir (although he considered it to be independent of the velocity of the particle). Its determination constitutes the aim of the cavity theory, the development of which has been reviewed in detail by the NCRpl. Here, a formulation similar to that given by the ICRU2 will be followed.

When a detector is exposed to the same electron fluence as the medium, and electrons are assumed to lose their energy continuously and locally, the stopping power ratio is defined by

(23.2)

where (cI>E .... )p is the primary electron fluence, differential in energy, at the measuring depth, and (%).01 denotes the unrestricted mass collision stopping power of an elec• tron with kinetic energy E in medium m or air. The assumption of continuous and local energy loss ignores the finite ranges of secondary electrons (delta rays) whose energy is assumed to be deposited where they are produced. Thus, these secondaries do not contribute to the electron fluence. This is why only primary electrons are used. Stopping-power ratios obtained in this manner are called Bragg-Gray s-ratios, and their use might be justified when equilibrium conditions are reached due to a complete build• up of the secondary electron spectrum.

At this point, it is important to stress that the electron fluence at the place of interest (primary only, in this case) is the physical quantity that has to be calculated with the Monte Carlo method in a first step, in order to evaluate numerically the so-called cavity integrals given in Eqn. 23.2 and following. Its determination will be considered in detail in the next section.

A very simple approximation for the Bragg-Gray stopping-power ratio has been given by Harders as (!®.) H _ P col,m. sm.,air - (23.3) (SeE) -) , P col ,air where unrestricted mass collision stopping powers are evaluated at the mean energy E of the primary electron spectrum at the point of interest. This is a good approximation when (S / P).01 varies linearly with energy over the range of the primary electron spec• trum, as is the case when the detector and the medium are similar materials. However, the evaluation of E is critical in the case of a gaseous detector material (as in Eq. 23.3) at relativistic energies where the density or polarization effect 1•2 causes s!. .. ir to vary rapidly with E. Fig. 23.1 illustrates this point clearly; in this case, E had been evaluated from the very approximate expression2

E = Eo(1- z/Rp), (23.4)

whereas (PhiE .... ) in Eq. 23.2 was obtained from a Monte Carlo calculation4•5• 23. Stopping-Power Ratios for Dosimetry 487

o o

.~ IX + 1 .L1J ._ I g '...

-.9 >

Figure 23.1. Difference between "Harder formula" ( Eqn. 23.3), evaluated using E from Eq. 23.4, and the "exact evaluation" of the Bragg-Gray water• to-air stopping-power ratio using Eq. 23.2 for 5, 10, 20 and 30 MeV electron beams. (From Nahum6 ). (from ref. 6).

An attempt to account for the experimentally observed deviations of Sm,air from the predictions of the Bragg-Gray theory was made by Spencer and Attix6. They introduced the generation of secondary electrons, and assumed that the detector walls were equivalent to the surrounding medium ("wall-less" detector). Electrons are divided into two groups depending on their energy which is compared with a certain cutoff value, ~, related to the dimensions of the cavity (~ is the energy of an electron whose range is equal to the mean chord length across the cavity). Those electrons with energy greater than ~ are assumed to originate outside the cavity (i.e., wall and surrounding medium), and to deposit their energy locally inside the cavity. The restricted mass collision stopping power (L / p).6. has to be used with the total electron spectrum. The energy interval between ~ and 2~ deserves special attention as it is possible for an electron to drop belo.w ~ where its energy would be dissipated on the spot, which is equivalent to a continuous-loss assumption. Spencer and Attix developed a special stopping-power formula to account for this track-end energy dissipation which becomes increasingly complicated as the electron energy approaches~. Based on the modifications of Burch7 to account for such track-end terms, Nahum4 developed an approximate expression to evaluate Spencer-Attix stopping-power ratios

(23.5) 488 P. Andreo where Cf>E(D.) ... and (S(D.)/p) ... are, respectively, the total electron spectrum and un• restricted stopping power evaluated at energy D.. The product of these two quantities gives, approximately, the number of electrons that drop below D. which, when multiplied by D., gives the total energy dissipated by the track-ends··s.

The merit of the two-group theory of Spencer-Attix is that, unlike the Bragg-Gray expression, it does take account of the size of the cavity through the parameter D.. It is not an exact theory, though, as the cavity is assumed not to disturb the total electron fluence above D.. However, it has had considerable success in accounting for experimentally derived stopping-power ratios.

23.3 THE NEED FOR TRANSPORT CALCULATIONS TO DERIVE ELEC• TRON SPECTRA

Except for the case of calculating primary electron spectra using the continuous slowing-down approximation where the fluence is given simply by the reciprocal of the total stopping power, the determination of electron spectra requires a rather complicated procedure. When secondary-electron production and/or bremsstrahlung-produced pho• tons are included and their spatial diffusion is taken into account, the complexity of the problem increases considerably.

Since the fifties, a great amount of work has been done to compute electron spectra from different sources to be used in dosimetry, radiobiology, and several other appli• cations. The majority of these calculations have been restricted to depth-independent distributions, where the fluence of electrons generated in an irradiated medium, irrespec• tive of their position, was estimated taking into account different interaction processes. In order to include the production of generations of electrons and photons, an integro• differential transport equation has to be solved, normally by some recursion procedure, to yield the complete electron spectra. An excellent review of the literature has been given by Nahum4, and will not be repeated here.

Depth-independent calculations have proven to be useful when the electron spectrum does not change with spatial position, as approximately happens in photon beams be• yond the depth of maximum absorbed dose, (see Fig. 23.2), but they are not valid, for example, for electron beams where a significant depth dependence of the electron spectrum occurs (see Fig. 23.3). The early work of Spencer8•9 took into consideration this variation using the so-called moment-method to solve the electron transport equa• tion in a depth-dependent way. Kessaris10.1l further improved the method obtaining electron spectra at different depths produced by electron beams, his results being used to derive stopping-power ratios for electron dosimetry, as described in the first ICRU report on electron dosimetry12.

An important limitation of these analytical solutions to the transport equations is that electron penetration is based mainly on the continuous-slowing-down approxima• tion. Although it would be possible to overcome this limitation, the increasing degree of sophistication needed would not avoid the restriction to applications where the medium is unbounded and homogeneous. The Monte Carlo method is not restricted in this sense, and its advantages, already described elsewhere in this book, make this method extremely useful for this type of calculation. 23. Stopping-Power Ratios for Dosimetry 489

"j > -~~ ...J U. Z o f- o r e... Wf- U Z 0001 Zw ~g ...JU U.Zz;:: 0- a:Z f-=> Ua: ~W we...

ELECTRON ENERGY I MeV Figure 23.2. Electron fluence, iIlE(z), differential in energy, at depths of 0.5(solid), 3.5(dash), 8(dot) and 20 cm(solid) in water for a 20-MeV monoen• ergetic photon beam; histogram above Tout (0.5 MeV), smooth curve below Tout. The individual points are from a Monte Carlo computation using the Nahum· code. (From Andreo and Nahuml8 ).

23.4 MONTE CARLO CALCULATIONS OF ELECTRON SPECTRA In what follows, we will restrict this presentation to the techniques utilized when the Monte Carlo method is used to derive electron fluence spectra. As already mentioned in Chapter 20 on electron pencil beams, this is derived by adding the total track-length in the different energy intervals within a given volume. Any Monte Carlo code will allow this information to be extracted since the energies at the beginning and end of every single step within the volume of interest are known. Details on how to modify the electron track to account for sub-segments in different regions will not be given here.

A very common and immediate approach to computing the electron spectrum is to score all the electron track-lengths in the energy bin containing the electron kinetic energy at the midpoint of the step, or the average energy along the step. Although for very small steps the scheme is satisfactory, Nahum· has reported systematic artifacts for this method when certain energy interval widths were chosen.

A more accurate approach that does not require a big programming effort is to subdivide the electron track into different energy intervals in a manner similar to the one being used to modify electron steps when several geometrical regions are crossed. Fig. 23.4 shows a typical example of energy and pathlength subdivision where sev• eral configurations are illustrated. Using this procedure, every energy interval between the initial and final energies along the electron step will score a track segment propor• tional to its length within the energy bin crossed. Straight tracks are assumed in this calculation, but this approximation is valid when the step-sizes are short enough. 490 P. Andreo

101~mr~~~~~~~rM~~~~l-'-1

100 1

z/ro = 0-0.05

7 > G) 0 :::IE 10

10-1 X ::::> -l 1J LL 10-2 1 z " 0 3 a:: ~ z/ro=0.45-0.50 (.) 10-3 1 lLI ] -l :l lLI 10-4 100

10-1

z/ro=0.95-1.00

1~5~~~~~~~~~~~~~~~~~~ 10- 3 10-2 10- 1 10 T (MeV)

Figure 23.3. Electron fluence, ~E(Z), differential in energy, at various depths in a water phantom irradiated with a 20-MeV electron beam. Units of flux are MeV - cm-2 - sec-Ii normalization corresponds to an incident current of one electron cm-2 - sec-I. (From Berger and SeltzerI3 ). (from ref. 13). 23. Stopping-Power Ratios for Dosimetry 491

Energy Intervals Tn-t

Case I em III } >- (not possible) n-l th (!) Tj a::: T 1 Tj w Ija:Sft I Z Tn I w Case m. I ;Toy T I (!) I Tj I z I Casen I (f) I ·Toy TTj eToy } nth « .lTb I I I w .lTb I a::: • Tov I (J Tn+t I w I 0 1}a:Sf2 .lTb l Tb } n +1 1h Tn+2 Figure 23.4. Details of spectrum computation regarding the subdivision of a track segment into different energy intervals. (From Nahum4). (from ref. 6).

23.4.1 The Technique of Transport Down to the Monte Carlo Cutoff Plus a CSDA Calculation Spencer-Attix stopping-power-ratio calculations are based on the knowledge of the elec• tron spectrum down to the cutoff value .6. of the Spencer-Attix theory (see section 23.2) which, for common ionization chambers, .6. usually is taken as 10 keY.

If limitations in stopping power and scattering theories at low energies are dis• regarded, in principle it is possible to simulate the transport of electrons down to a Monte Carlo cutoff energy, T.ut , equal to .6., using any Monte Carlo code. As geomet• rical dimensions are usually bigger than the residual electron range at low energies, such a scheme involves the simulation of electron transport in conditions where spa• tial resolution becomes unimportant. Considering that in many of the existing codes, the number of steps needed to decrease a given electron energy to half of its value is approximately constant, and that the computation time is roughly proportional to the number of steps, simulations down to 10 keY generally require a large abount of CPU time. A more efficient solution is to use a depth-independent electron slowing-down spectrum calculation between a given Monte Carlo electron cutoff and the minimum energy required to compute stopping-power ratios. In this manner, higher values of T. ut can be used, and the total computation time will be considerably reduced compared with the standard method. Restrictions on the value chosen for T.ut will be given by the dimensions of the geometry being used (an electron energy whose csda range is about half of the minimum dimension is a reasonable choice).

Such a two-step approach has been described by Berger and Seltzer13 and Nahum" who used the theories of Spencer-Fano and Burch, respectively, to compute the electron spectrum down to energies as low as 100 eV by coupling a Monte Carlo simulation to the depth-independent scheme, executing the latter calculations on completion of the Monte Carlo part. 492 P. Andreo

The input for the depth-independent calculation consists of all the electrons (pri• maries and secondaries) from the Monte Carlo simulation whose energy has dropped be• low Tcul (i.e., histories terminated), and electrons (i.e., delta rays or photon-generated electrons) that, when created, have energies below Tcui. When pedorming the csda calculations, the first group will always start in the first energy bin below T cui , whereas the second group will be located at different lower energy bins depending on their initial energy. The number and energy distribution of delta rays in lower energy bins are com• puted from the M!IllIer cross section and added to the total number of electrons already existing in a given bin. The fluence in each bin is given, to a good approximation, by the number of electrons in the bin divided by the total stopping power. Bremsstrah• lung photons usually are not included in the csda calculation, and the value chosen for T cui should be small enough to make their production practically negligible below the cutoff.

A very detailed description of the above computation has been given by ~ahum', and the method has been included also in our own code. For a 20-MeV electron-beam simulation, the computation time per history for Tcul = 10 keY (i.e., full standard simulation) is about 4 times longer than when Tcul = 1 MeV plus csda calculation down to 10 keY. As shown in Fig. 23.5, there is virtually pedect agreement between the electron spectra at the depth of maximum absorbed dose calculated by the two methods.

10°

'j 1 >Q) 10- ~ N I,E 10-2 .3

w 10-3 (,) z :::> ...J 4 u. 10- z 0 a: I- 10-5 (,) W ...J w 10-6

10-2 10-1 10° 101 ELECTRON ENERGY (MeV) Figure 23.5. Electron spectra at the depth of maximum absorbed dose in water (z/ro = 0.60 - 0.65) for a 20-MeV electron beam computed as a full Monte Carlo simulation down to a cutoff TcuI=10 keY and pedorming a simulation down to Tcut = 1 MeV followed by a csda calculation. Results are so close to each other that it is not possible to distinguish between the two calculations. Computation time is 4 times longer in the first case. 23. Stopping-Power Ratios for Dosimetry 493

23.5 STOPPING-POWER RATIOS FOR ELECTRON BEAMS

Spencer-Attix stopping-power ratios for monoenergetic broad-plane parallel electron beams are shown in Fig. 23.6 as calculated by Berger14 (data given in the AAPM dosimetry protocol) and ourselves. Electron stopping-power data are from ICRU16(or Berger and Seltzer16), in both cases, spectra at different depths being calculated with the respective Monte Carlo codes (ETRAN and our program). The large variation of the electron spectra with depth shown in Fig. 23.3 results in S""IJir being significantly depth dependent. It can be seen that our values are systematically above the Berger values, especially for the highest energies. A similar trend has been obtained by Nahum (see Fig 4.10 in ICRU2), although values given there were based on less than exact Sternheimer-Peierls evaluation of the density-effect correction to the stopping power in water.

- .:.> 51

0 ~ w '"~ ~ C)z a::... 12 III O.

CEPTH•• /em

Figure 23.6. Depth variation of the Spencer-Attix water/air stopping-power ratio, S""IJir, for ~ = 10 keY, derived from Monte Carlo generated electron spectra for monoenergetic, plane-parallel, broad electron beams. Solid lines, data from Berger14j symbols, data calculated with our code. 494 P. Andreo

It is possible to use data from monoenergetic electrons to derive central-axis depth• dose distributions and the corresponding stopping-power ratios at depth for an electron spectrum using a weighted sum of the values for monoenergetic beams (d. Andreo and Nahum 17). For a given spectrum of initial energies, NE , the depth-dose D",(z) will be given by D ( ) _ D",(E,z) NE dE f (23.6) '" z - f NE dE ' where D",(E,z) are the depth-dose distributions for broad monoenergetic beams of energy E. In the same manner, an expression for the (inverse) stopping-power ratio of the spectrum can be derived as (see Andreo and Nahum18 for details):

Sair,,,,(Z) = f fE (Z)Sair,tu (E, z)dE , (23.7) where D",(E, z) NE ( ) (23.8) fE Z = f D",(E,z) NE dE and sair,,,,(E, z) are the stopping-power values for monoenergetic electrons. A computer program (AVEREL) has been described by Andreo and Nahum17 to perform these steps, using Berger and Seltzer16 or our own data. Results for different electron spectra are shown in Fig. 23.7a and 23.7b for depth-doses and stopping-power ratios, respectively.

Conventionally, S""air values for the dosimetry of electron beams are selected as a function of the mean energy at the phantom surface, Eo (determined from the half value depth, R50) and the depth of measurement. s""air(Eo, z) values calculated in this way (Eo = 11.37 and 12.0 MeV for the broad and narrow beams, respectively) have been included in Fig. 23.7 b. It can be seen that Stu,air(Eo, z) increases more rapidly with depth than S""air(Z), implying that the mean energy E. falls more slowly with depth for a beam with a broad initial energy spectrum than for a monoenergetic beam.

Even if this set of stopping-power ratios can be considered state of the art according to the recent advances in stopping-power datal5 , it has to be pointed out that experi• mental results from Mattsson19 have revealed a certain inconsistency in the dosimetry of electron beams based on the selection of s""air(Eo, z) set of values. Absorbed-dose determinations in water for clean electron beams are higher from ionometric (i. e., using Sw,air values to convert to absorbed dose) than from ferrous sulphate measurements, whereas a fairly good agreement is found for conventional beams having a certain en• ergy and angular spread at the phantom surface. We have studied the effect of such spreads on s-ratios by calculating the electron spectra within a lO-cm diameter beam after layers of lead of different thicknesses followed by 1 m of air, using the Monte Carlo code EGS. Such spectra, all having most probable energy Ep ~ MeV, have been used as input to our code, adding an angular spread given by Gaussian distributions with different Orm. angle. Fig. 23.8 shows the influence on stopping-power ratios of the maximum difference in tlE and tlO which corresponds to a layer of 1 mm of lead with Orma = 20 degrees, compared with a monoenergetic beam with no angular spread. The difference in Stu,air at the depth of the maximum of the respective depth-dose curves is about 1.6%. However, if we consider that most clinical electron beams in this energy interval will have Sw,air values lying between the two curves, the difference would be much less, which agrees better with the experiments of Mattsson for conventional clini• call;>eams. These results bring into question the selection procedure for S""air(Z) values described above; further studies will be required to clarify this point. 23. Stopping-Power Ratios for Dosimetry 495

1,

/ ./ .. / .' 1.oa / "1,,' / 1.07 .I cS / .. i= ·NARROW~ « too II: / " II: ./ ./ ~.t 1.37 MeV ;=W 0 -BROAD"__ / ~/ " Q. / / ./ o--~.12 M.V / / 0 / / /.~. 0

/ 0

OL-~~2~~3~~4~~5~~6~~~8 o~,L-~--Ll--~~2--~-*3--~-+4--­ DEPTH 1\1 WA TEA. zJern DEPTH IN WATER. z/cm Figure 23.7. Depth doses and stopping-power ratios for an electron spectrum incident on a water phantom, computed from Monte Carlo derived data for 12 MeV monoenergetic electrons using the weighting procedure described in the text. The triangles and squares correspond to monoenergetic beams with energies at the surface of 11.37 and 12.0 MeV respectively; "broad" and "nar• row" refer to the widths of two different incident energy spectra (From Andreo and Nahum17).

1.15 r------'------~

1.10

ci" 1. 05 f-----..",,""

1. 00

z/cm Figure 23.8. The influence of energy and angular spread on stopping-power ratios for electron beams having Ep Ri 10 MeV. Curves correspond to the maximum difference in l:l.() and l:l.E as described in the text. 0, data for a monoenergetic beam with no angular spread; D, data for a beam after passing I-mm lead and 1 m of air with ()rrru = 20 degrees. The lines indicate the depths of maximum dose, for the two beam qualities, which produce the 1.6% difference in Sw,IJir mentioned in the text. 496 P. Andreo

23.6 STOPPING-POWER RATIOS FOR PHOTON BEAMS

As already pointed out, electron spectra produced by photon beams do not change significantly with depth (see Fig. 23.2). Consequently, the water-to-air ratio shows negligible depth dependence at depths beyond the dose maximum. Fig. 23.9 shows such results for broad, plane-parallel and monoenergetic photon beams.

An averaging procedure similar to the one described for an electron spectrum has been developed by Andreo and Nahum18 to compute depth-dose and stopping-power ratios for a photon spectrum as a weighted sum of the values for monoenergetic photon beams. Results obtained with our program, AVERMV, are compared in Fig. 23.10 with direct calculations using the Monte Carlo method where it can be seen that the inherent statistical noise of a Monte Carlo computation is considerably reduced by the averaging method.

Energy (MeV) 0.1

1 14 ~~ ____~ __~ ____~ ______~~ ______3 1 12 E=~=~=="==~-=-~=~=='~=~-=~==-===--=-==-======~.;

o ~-;;: Il: en Il:t W o II) a. -"

0.98

DEPTH IN WATER, z/cm Figure 23.9. Depth variation of the Spencer-Attix water/air stopping-power ratio, Sw,air, for ~ = 10 keY, derived from Monte Carlo generated electron spectra for monoer.ergetic, plane-parallel, broad photon beams. (From Andreo and Nahum18).

The maximum accelerating potential (nominal "MV") is an inadequate specifica• tion of clinical photon beam quality for dosimetry purposes as the photon spectrum changes with the target and filtration design of the accelerator for a given electron maximum energy. Today, most dosimetry protocols20- 22 recommend the dose or ion• ization ratio at two different depths (10 and 20 cm, for example) to specify the photon beam quality since this ratio is more closely related to the penetration properties of the beam. Unfortunately, the available stopping-power ratios were not related to the quality of the beam, and theoretical calculations of different types were used instead to produce electron fluence spectra, whereas dose ratios were obtained experimentally or with independent calculations (cf. Andreo and Brahme23). The Monte Carlo method has been used by us to compute electron spectra and depth-dose distributions for the same beam, using realistic spectra (both experimental and thick-target calculated data) 23. Stopping-Power Ratios for Dosimetry 497 as input to our code. In this manner, both stopping-power ratios and beam-quality de• scriptors are closely related, as shown in Fig. 23.10 where dose and s-ratios can be obtained at any depth. Using this procedure, a whole set of stopping-power ratios as a function of the photon beam quality has been produced (Andreo and Brahme23 ), re• sults being shown in Fig. 23.11. This new set of data has been included in the new and updated dosimetry protocols published by the IAEA 24, and in the N etherlands25 , Switzerland26, Italy27 and Spain28 .

b... -E u 0;. :>.,'" ::IE "- 01-&° ui u z w :::> .....J Z 0 I- 0 :r: Q. I- Z :::> a:: w Q. w f/) 0 0 1.0 0 w m a:: 0 f/) m <

1.10 0:: + iii 1.09

-0 ~ t08 :,t -o ~ 1.07 f/)« .•• of

DEPTH IN WATER z/cm Figure 23.10. Central-axis depth-dose distributions in water and water/air stopping-power ratio, sw,air(ll = 10keV), for a broad, 40-MV thin-target bremsstrahlung beam, calculated directly by the Monte Carlo method (his• togram), and using AVERMV (circles). (From Andreo and Nahum18.) 498 P. Andreo

The depth dependence of stopping-power ratios is of special interest. In Fig. 23.9, we have shown the typical pattern for monoenergetic photons, which has little value for practical dosimetry. When bremsstrahlung spectra are considered, the effect of the low-energy component of the spectrum will modify the general shape, as is shown in Fig. 23.12 where a gradual decrease of s-ratios with depth can be observed for the highest energies. This can be interpreted as a beam-hardening effect due to the progressive loss of low-energy photons with increasing depth (c.f. Brahme and Andre023). The build-up region also shows the influence of the low-energy photons existing in the beam; Their higher s-ratios produce a maximum value close to the surface. Nevertheless, in clinical bremsstrahlung beams, such low-energy photons are filtered out by thick targets and flattening filters, decreasing the stopping-power ratios near the surface. Furthermore, the existing electron contamination will reverse the trend, as is shown in Fig. 23.13 where the smaller s-ratios of the contaminant electrons compensate the increased value close to the surface. A practical independence of stopping-power ratios with depth for the most commonly used photon beams can be concluded from this data.

1.14

1.12

c j 1.10

1.08

1.06

1.2 1.4 1.6 1.8 2.0 10 TPR 20 Figure 23.11. Spencer-Attix water/air stopping-power ratios as a function of the beam quality for calculated bremsstrahlung spectra. Symbols corre• spond to calculations based on published bremsstrahlung spectra from clinical accelerators as input to our Monte Carlo code. (From Andreo and Brahme23 ). 23. Stopping-Power Ratios for Dosimetry 499

10 TPR 20 1.864 1.743 - 1.610 - 1.552 1J2 - 1.487 - 1.421 c -- 1.390 i en -- "--- 1.343 1.10 ~ 1.314 1.298 ~ 1.282 1.08 1.268

2 4 6 8 10 20 30 40 50 Z (em) Figure 23.12. Depth dependence of the water/air stopping-power ratios for plane-parallel bremsstrahlung spectra as a function of the quality of the beams. (From Andreo and Brahme2S).

1.10

c "$. III

100

80

0~ 60 "-I is 40

20

0

Figure 23.13. The influence of electron contamination in depth-dose and water/air stopping-power ratio for a calculated 20-MV photon beam as obtained by the Monte Carlo method. Histograms correspond to the pure photon beam (P) and the electron contaminating spectrum (E). Full curves correspond to the electron-contaminated pho• ton beam. (From Andreo and Brahme2S). 500 P. Andreo

REFERENCES 1. N CRP (National Council on Radiation Protection and Measurements), "Stop• ping Powers for Use with Cavity Chambers", NCRP Report No. 27 (1961); published as NBS Handbook 79 (1961). 2. ICRU (International Commission on Radiation Units and Measurements), " Ra• diation Dosimetry: Electron Beams with Energies Between 1 and 50 MeV", ICRU Report 35 (1984). 3. D. Harder, "Berechnungder Energiedosis aus lonisationsmessungen bei Sekundar• elektronen-Gleichgewicht", In Symposium on High-Energy Electrons, edited by A. Zuppinger and G. Poretti, (Springer-Verlag, Berlin, 1965) 40. 4. A. E. Nahum, "Calculations of Electron Flux Spectra in Water Irradiated with Megavoltage Electron and Photon Beams with Applications to Dosimetry", The• sis, University of Edinburgh, U.K. (1976); (Available from University Micro• films International, 30-32 Mortimer Street, London WIN 7RA, order number 77-70,006). 5. A. E. Nahum, "Water/Air Mass Stopping Power Ratios for Megavoltage Photon and Electron Beams", Phys. Med. BioI. 23 (1978) 24. 6. 1. V. Spencer and F. H. Attix, "A Theory of Cavity Ionization", Radiat. Res. 3 (1955) 239. 7. P. R. J. Burch, "Comment on Recent Cavity Ionization Theories", Radiat. Res. 6 (1957) 79. 8. 1. V. Spencer, "Theory of Electron Penetration", Phys. Rev. 98 (1955) 1597. 9. 1. V. Spencer, "Energy Dissipation by Fast Electrons", National Bureau of Stan• dards Monograph NBS 1 (1959). 10. N. D. Kessaris, "Penetration of High Energy Electron Beams in Water", Phys. Rev. 145 (1966) 164. 11. N. D. Kessaris, "Absorbed Dose and Cavity Ionization for High Energy Electron Beams", Radiat. Res. 43 (1970) 288. 12. ICRU (International Commission on Radiation Units and Measurements), "Ra• diation Dosimetry: Electrons with Initial Energies Between 1 and 50 MeV", ICRU Report 21 (1972). 13. M. J. Berger and S. M. Seltzer, "Calculation of Energy and Charge Deposition and of the Electron Flux in a Water Medium Bombarded with 20 MeV Elec• trons", Ann. N.Y. Acad. Sci. 161 (1969) 8. 14. Berger, M.J., As cited in AAPM (1983). 15. ICRU (International Commission on Radiation Units and Measurements), "Stop• ping Powers for Electrons and Positrons", ICRU Report 37 (1984). 16. M. J. Berger and S. M. Seltzer, "Stopping Powers and Ranges of Electrons and Positrons", National Bureau of Standards report NBSIR 82-2550 (1982). 17. P. Andreo and A. E. Nahum, "Influence of Initial Energy Spread in Electron Beams on the Depth-Dose Distribution and Stopping-Power Ratios", Proceedings of the XIV ICMBE and VII ICMP, Espoo, Finland (1985) 608. 18. P. Andreo and A. E. Nahum, "Stopping-Power Ratio for a Photon Spectrum as a Weighted Sum of the Values for Monoenergetic Photon Beams", Phys. Med. BioI. 30 (1985) 1055. 19. L. O. Mattsson, "Comparison of Different Protocols for the Dosimetry of High• Energy Photon and Electron Beams", Radiother. OncoI. 4 (1985) 313. 23. Stopping-Power Ratios for Dosimetry 501

20. NACP (Nordic Association of Clinical Physics), "Procedures in External Radia• tion Therapy Between 1 and 50 MeV", Acta Radioi. Oncoi. 19 (1980) 55. 21. AAPM (American Association of Physicists in Medicine)", A Protocol for the De• termination of Absorbed Dose from High-Energy Photon and Electron Beams", Med. Phys. 10 (1983) 741. 22. SEFM (Sociedad Espanola de Fisica Medica), "Procedimientos Recomendados para la Dosimetrfa de Potones y Electrones de Energias Comprendidas Entre 1 MeV y 50 MeV en Radioterapia de Haces Externos", Madrid, Spain, Publication No. 1-1984 (1984). 23. P. Andreo and A. Brahme, "Stopping Power Data for High Energy Photon beams", Phys. Med. BioI. 31 (1986) 839. 24. IAEA (International Atomic Energy Agency), "Absorbed Dose Determination in Photon and Electron Beams: An International Code of Practice", Technical Report Series No. 277 (1987). 25. B. J. Mijnheer, A. H. 1. Aalbers, A. G. Visser and F. W. Wittkamper, "Con• sistency and Simplicity in the Determination of Absorbed Dose in High-Energy Photon Beams: A New Code of Practice", Radiother. Oncoi. 7 (1986) 371. 26. Swiss Society of Radiation Biology and Radiation Physics, "Dosimetry of High Energy Photon and Electron Beams: Recommendations", (1986). 27. AIFB (Italian Association of Biomedical Physicists), "Outline of the Italian Pro• tocol for Photon and Electron Dosimetry in Radiotherapy", Sub-Committee for Basic Dosimetry, Committee for Dosimetry Standardization in Radiotherapy, (1987). 28. SEFM (Sociedad Espanola de Fisica Medica). Suplemento al Documento SEFM no.I-1984: "Procedimientos Recomendados para la Dosimetrfa de Fotones y Elec• trones de Energias Comprendidas Entre 1 MeV y 50 MeV en Radioterapia de Haces Externos", Madrid, Spain, Publication No.2-1987, (1987). 24. Photon Monte Carlo Transport in Radiation Protection

B. Grosswendt

Physikalisch-Technische Bundesanstalt D-3300 Braunschweig Federal Republic of Germany

24.1 INTRODUCTION When the human body is exposed to ionizing radiation, its tissues are traversed by charged particles such as, for example, secondary electrons in the case of photon ra• diation, or heavy ions in the case of neutron radiation. This may lead to a variety of biological effects as a consequence of energy transfer to the human cells by physical radiation interactions. Prevention of detrimental non-stochastic effects and limitation of the probability of stochastic effects to health to levels considered acceptable are now the main objectives of radiation protection. To attain these objectives, it is necessary to specify the degree of irradiation in numerical terms, firstly, to define authorized lev• els and regulatory limits, and secondly, to have the possibility of supervising them. This means that it is necessary to look for a suitable quantity which is proportional to radiation induced detrimental effects to health, and to define a measuring procedure applicable in practical radiation fields.

As radiation effects are caused by energy deposited by charged particles, it is possible to choose the absorbed dose, D, as a relevant quantity. But since different types of radiation differ in their biological effectiveness per unit absorbed dose, it is necessary to weight D by a quality factor, Q, equal to 1 for a reference radiation (conventionally I radiation or hard x-rays with an energy ~ 30 keY) and up to 25 for neutrons, protons and other heavier particlesl . The product H = QD is called the dose equivalent, and is equal to the absorbed dose of the reference radiation which causes the same biological effect as the radiation under consideration.

The International Commission on Radiological Protection (ICRP)2 has now recom• mended that to estimate the total individual detriment resulting from the irradiation of all radiosensitive organs and tissues within the human body, the degree of irradia• tion in radiation protection should be based on dose equivalents in various organs (e.g., the skin, or the lens of the eye) and on the effective dose equivalent HE, which is the weighted sum of dose equivalents HT in the most important organs of an individual:

(24.1 )

503 504 B. Grosswendt

The weighting factors, WT given in Table 25.1, were derived from the mortality risk due to cancerogenic effects assuming a linear dose-risk relationship. In the case of the gonads, the weighting factor is estimated from the risk of genetic effects in the first two generations. The "remainder tissues" are the 5 organs or tissues with the highest dose equivalents, and each is associated with a WT value of 0.06.

Table 24.1. Recommended Average Weighting Factors, wT, for the determination of the effective dose equivalent according to Eqn. 24.1.

Tissue T WT Gonads 0.25 Breast 0.15 Red Bone Marrow 0.12 Lungs 0.12 Bone Surface 0.03 Thyroid 0.03 Remainder 0.30 Total 1.0

Unfortunately, the effective dose equivalent as well as the dose equivalents in the organs are essentially unmeasurable in practical radiation protection, and therefore must be estimated from other measurable quantities as, for example, the air kerma in the human environment, or the dose equivalent in simple phantoms such as a cylinder or a sphere. This, however, is possible only if these measurable quantities can be related to the dose equivalents in the organs within the human body in an unambiguous way. In principle, this means that, for any practical radiation field, the dose equivalent distribution within the individual must be known so that a conversion factor with respect to a measurable quantity can be defined. To get these conversion factors, one has to use a model by which the dose-equivalent distribution within the human body, approximated by a suitable tissue equivalent man or woman phantom, can be determined for a variety of irradiation parameters . This can be done by calculation or by measurement.

As we are concerned with calculations, a mathematical anthropomorphic phantom, well adapted for the determination of organ doses and the effective dose equivalent, and a Monte Carlo photon transport model for calculating absorbed dose distributions, are de• scribed in the following sections, and some results for HE and organ doses are presented. Subsequently, the sphere phantom recommended by the International Commission on Radiological Units and Measurements (ICRU)3 is treated, and some Monte Carlo re• sults concerning dose-equivalent quantities at appropriate locations in the sphere are discussed. These might serve as operational quantities.

24.2 THE ANTHROPOMORPHIC PHANTOM The mathematical human phantom usually used in radiation protection is that defined by the Medical Internal Radiation Dose Committee (MIRD). It has a simple geometrical shape, and approximates the human body as far as major forms are concerned. Twenty• two internal organs are defined as subregions of the phantom; they are considered to be homogeneous in composition and density. An extensive description of the phantom 24. Photon Monte Carlo Transport in Radiation Protection 505 is given by Snyder et a14 • A general view of the main features of the MIRD phantom is given in Fig. 24.1. The length of the phantom is 174 cm and the total weight, 70 kg. These correspond to the data of the reference man according to the ICRps. The phantom consists of an elliptical cylinder of main axes 40 cm and 20 cm including arms, torso and hips. Its legs are truncated cones; head and neck are represented by an elliptical cylinder. The Cartesian coordinate system used in the mathematical description is located at the center of the base of the trunk. Its positive z-axes run vertically from foot to head, its x- and y-axes from the left to the right side of the body and from back to front, respectively.

80

Organs Not Shown : Leg Bones Ovaries Testes Adrenals Thymus Spleen Pancreas Thyroid Morrow Dimensions in Stomach Uterus Skin centimeters

Figure 24.1. General view of the main features of the MIRD phantom.

Fig. 24.1 shows the skeleton. It consists of the leg bones, the arm bones, the pelvis, the spine, the skull, and the ribs. Let us consider the leg bones as an example of the mathematical description of the phantom subregions. Each leg bone is a frustrum of a circular cone, and the space occupied by the left leg bone is given by

8z)2 2 ( 2.5Z)2 ( x-l0-- +y < 3.5+- (24.2) 79.8 - 79.8 for -79.8 S Z S 0, and with x, y, and z in cm. The volume of one bone is 1399.5 cm3 .

The right part of Fig. 24.1 shows the anterior view of the principal organs in the head and the trunk of the phantom. 506 B. Grosswendt

The whole phantom is subdivided into three regions of different density: The skele• tal region (bone plus marrow) has a density of 1.5 g/cm3 , the lungs have a density of 0.3 g/cm3 , and the remainder of the phantom has a density of 1.0 g/cm3 . The ele• mental composition of the different density regions in the phantom is shown in Table 24.2. These values can be taken, for example, from Tipton et al 6 or Rosenstein7. It can be seen from the table that the lung region and the remainder of the phantom are primarily composed of hydrogen, carbon, nitrogen and oxygen, whereas in the skeletal region, additional elements, e.g., calcium and phosphorus, amount to about 17% of the total mass.

Table 24.2. Relative Elemental Composition by weight of Different Regions in the MIRD phantom7.

Element Skeleton Lungs Remainder of the

(p = 1.5 g/cm3 ) (p = 0.3 g/cm3 ) total body % by weight % by weight % by weight

H 7.04 10.21 10.47 C 22.79 10.01 23.02 N 3.87 2.80 2.34 0 48.56 75.96 63.21 Mg 0.11 7.4 X 10-3 0.015 P 6.94 0.081 0.24 S 0.17 0.23 0.22 Ca 9.91 7.0 X 10-3 - Cl 0.14 0.27 0.14 K 0.15 0.20 0.21 Na 0.32 0.19 0.13 Fe 8.0 X 10-3 0.037 6.3 X 10-3 Zn 4.8 X 10-3 1.1 X 10-3 3.1 X 10-3 Zr - - 8.0 X 10-4 Sr 3.2 X 10-3 5.9 X 10-6 3.4 X 10-5 Rb - 3.7 X 10-4 5.7 X 10-4 Pb 1.1 X 10-3 4.1 X 10-5 1.6 X 10-5

24.3 THE MONTE CARLO PHOTON TRANSPORT MODEL In order to obtain organ doses and the effective dose equivalent for the anthropomorphic phantom irradiated by a known photon field, it is necessary to simulate the history of a large number of photons within the phantom, taking its very complicated geometrical structure as well as its different density regions and elemental compositions into account. The spatial distribution of energy absorption caused by physical photon energy transfer processes can then be used to obtain the mean absorbed dose DT , and at the same time the mean dose equivalent HT within the organs (the quality factor for photon radiation is assumed to be = 1). The effective dose equivalent follows from Eqn. 24.1. The main problem in using this equation is the definition of the remainder of the phantom having the weighting factor, WT = 0.3, because one has to seek the maximum and less-than• maximum organ doses in the whole body which might be dependent on the irradiation 24. Photon Monte Carlo Transport in Radiation Protection 507 conditions. A full discussion on the remainder problem is given by Kramer and Drexler8. Bearng this in mind, the photon transport model must include four main elements: (1) a model for the most essential photon interaction processes, (2) a procedure to obtain the correct interaction sites in the case of a very complicated phantom, (3) a bookkeeping method to get the spatial energy absorption distribution, and (4) a set of input data necessary to perform the calculation. These elements are discussed in the next sections.

24.3.1 Photon Interaction Model The energy range of interest in photon radiation protection is 10 keV to 10 MeV. In this energy range, three different photon interaction processes predominate. These are the photoelectric effect, Compton scattering, and electron-positron pair creation in the nuclear electrostatic field (Rayleigh scattering, though quite probable at low energies, can be neglected because only slight changes in photon direction and very small energy losses are associated with this process). In the photoelectric effect, the photon is totally absorbed by an atom, and an electron of kinetic energy equal to that of the photon, less the electron binding energy, is emitted. Fluorescence radiation is usually assumed to be locally absorbed at its point of emission. Compton scattering is usually treated approximately, neglecting electron binding and polarisation effects. The photon direction after scattering is calculated from the Klein-Nishina cross section formula, dO" (E' 2 E E' 1 cos2 iJ) (24.3) -dO ""' -)E (-E' + -E - + ' where E and E' are the photon energies before and after scattering, dO = sin iJdiJd

E'- E (24.4) -1+[E/(moc2 )](1-cosiJ) , where moc2 is the electron energy at rest. The energy of the Compton electron is given by Eel = E - E' , (24.5) and its polar angle iJ.I by

(24.6)

The selection technique to get cos iJ is described in detail by Zerby 9.

In the photon energy range above 1.02 MeV or 2.04 MeV, electron-positron pair creation in the nuclear electrostatic field, or in the electron field, can take place. In general, pair-production events are treated as if they all occur in the nuclear field. The kinetic energies of the electrons and positrons formed in pair creation can be calculated either on the assumption that the energy transferred to one particle is uniformly dis• tributed between 0 and the total energy E - 2moc2 which is available, and that the energy of the second particle completes that of the first one, or by using the Bethe• Heitler expression for pair production, a technique also described by Zerby9. Positron annihilation is treated in such a way that two 0.511-MeV photons are assumed to be emitted isotropically at the place where the positron comes to rest. The transport of 508 B. Grosswendt secondary electrons, such as photoelectrons, Compton electrons and pair particles, is ne• glected in most radiation protection calculations; it is assumed that the particle kinetic energy is locally absorbed at the point at which the interaction generating the particles takes place. The calculated quantity in fact is the kerma, not absorbed dose. It can be assumed, however, that this approximation is satisfactory where charged-particle equi• librium exists; it does not introduce significant errors at energies up to about 3 MeV. For the influence of the electron transport, see Section 24.10.

24.3.2 Interaction Site Model The distance between two successive interaction points and the type of interaction event are conventionally determined using the mass attenuation coefficients, T / p, 0-./ P and IC/ p for photoelectric effect, Compton scattering, and electron-positron pair creation, respectively; their sum is p/ p. In the case of an infinitely extended and homogeneous medium, the distance £ between two successive interaction points is given by

£ = -(In ",)/p , (24.7) if", is a random number uniformly distributed in the interval 0 < '" S; 1. The probability for the occurrence of one of the three predominant types of interaction is given by T / p, 0-./ P and IC/ p.

Starting Point P (x,Y, z) direction {cos tX, cosj1,cosy } distance I; 0

yes

Figure 24.2. Flow diagram of the sampling procedure to determine the in• teraction sites in the case of the anthropomorphic phantom. The quantities PT, T = 1,2... are the total linear attenuation coefficients of the different materials defined within the phantom.

Because of the complicated structure of the anthropomorphic phantom including its regions of differing composition (leading to different mass attenuation coefficients in some of the phantom subregions), the straightforward use of Eqn. 24.7 is very time consuming, and has to be exchanged for a more effective method. A suitable calculation 24. Photon Monte Carlo Transport in Radiation Protection 509 procedure, described in detail by Coleman10, is shown in Fig. 24.2. It gives the photon the correct expectation value for reaching any point on a straight line, regardless of how many boundaries it crosses.

24.3.3 Bookkeeping Model

Using the assumptions of the preceding sections, the history of each photon within the phantom can, in principle, be followed step by step from one interaction point to the other, and the absorbed-dose distribution can be calculated by summing up the locally absorbed energy in the subregions of the phantom followed by a division by their masses. The main disadvantage of this procedure is that it is very time consuming to obtain an energy absorption distribution with small statistical uncertainty, in particular in the case of low-energy photons and large penetration depths. This is due to photoabsorption which is predominant at low energies, and which results in only a small number of photons penetrating to larger depths. One way to overcome this difficulty is the method of using "fractional photons" applied by Snyder et al4 in their investigation on absorbed fractions for monoenergetic photon sources uniformly distributed in various organs of the MIRD phantom. To compensate partially for photoabsorption, each photon is given a weight that is initially set equal to 1. With each photon interaction, this weight is reduced in such a way that it represents the probability of survival, allowing the photon to continue undergoing Compton scattering only. If W n - 1 is the weight of the photon after the (n - l)th interaction, and En - 1 its energy, then the weight after the nth interaction is: (24.8)

This reduction in weight is equal to the probability of Compton scattering in the nth interaction.

The history of a photon is terminated (i) if it escapes from the phantom, (ii) if its energy is less than a cutoff energy of the order of 5 keV, or (iii) if its weight is less than 10-5 . In case (ii) and (iii), it is assumed that the photon energy is absorbed locally. Using this concept of fractional photons, the energy deposition at the point of the nth interaction is given by

(24.9)

Here, the first term represents the part of the photon energy that is deposited in the case of a photoelectric event, the second term the part of energy transferred to a Compton electron, and the third term the kinetic energy of the pair particles. The annihilation of the positron is taken into account by starting a single photon of energy 0.511 MeV with a random orientation at the point of the pair-creation event, simulating its history in the same way as has been done for the primary photons. The initial weight of this photon is set equal to

(24.10)

The factor 2 accounts for only a single photon being followed instead of a pair. 510 B. Grosswendt

24.3.4 The Input Data

Mass attenuation coefficients for the three predominant photon interactions must be known in the energy range between 10 keV and 10 MeV for all different parts of the body. They can be calculated according to their elemental composition given in Table 24.2. The coefficients for H, C, N, and 0 can be taken from Hubbellll, Storm and IsraeP2, Plechaty et al 13, and those for the other elements from Storm and Israel or from Plechaty et al as well. Any differences between the data concerning the same element are not significant for radiation protection purposes.

24.4 RESULTS CONCERNING THE MIRD PHANTOM

To give an impression of some of the results calculated by various authors4.8.14-24of the conversion factors concerning the MIRD phantom, Fig. 24.3 presents the results of Kramer and Drexler8 for the energy dependence of the ratio HEIKa, where HE is the effective dose equivalent, and Ka is the air kerma in free air, for the case of a whole body irradiation with a parallel monoenergetic photon beam and four different irradiation conditions: AlP (anterior-to-posterior) irradiation from the front of the phantom; PIA (posterior-to-anterior) irradiation from the back; LLAT (left lateral) irradiation from the left side of the phantom; ROT (rotational) irradiation uniformly distributed from all directions perpendicular to the phantom axis.

1.5

t

Figure 24.3. Energy dependence of the ratio HEI Ka of the effective dose equivalent HE and the air kerma Ka in free air in the case of a whole body irradiation with a parallel monoenergetic photon beam: AlP irradiation from the front of the phantom; PIA irradiation from the back; LLAT irradiation from the left side of the phantom; ROT irradiation uniformly distributed from all directions perpendicular to the phantom axes. 24. Photon Monte Carlo Transport in Radiation Protection 511

It follows immediately from Fig. 24.3 that the air kerma in free air is not a conser• vative measure of HE, and therefore not well suited for radiation protection purposes because, in the case of AlP-irradiation, HEI Ka > 1. The high values of the ratio HE I Ka are striking. They are due to the relatively large contributions of some frontal organs (testes, breast) to the weighted sum HE which are shadowed by other tissues in the case of PI A- or LLAT-irradiation. This shadowing effect is even more pronounced if one considers single organs. In Fig. 24.4, the energy dependence of the ratio HT I Ka (with HT being the dose equivalent in the testes), is shown for the same irradiation con• ditions as in Fig. 24.3. The curve shape for AlP-irradiation differs substantially from that associated with the other irradiation conditions. The high values of HT I Ka around 75 keY caused by a high degree of multiple scattering, and hence a high backscatter factor, are very striking.

2.0

1.5

0.5

keY 104

Figure 24.4. Energy dependence of the ratio HT I Ka of the dose equivalent HT in the testes and the air kerma Ka in free air in the case of the same irradiation conditions as in Fig. 24.3.

24.5 OPERATIONAL RADIATION PROTECTION QUANTITIES

With the aid of the conversion factors concerning the MIRD phantom, the effective dose equivalent HE, or organ doses HT , in principle can now be determined indirectly by measuring a quantity, such as air kerma (or exposure) free in space. But this procedure works well only in the case of known irradiation fields because of the strong energy dependence of the conversion factors and their different behaviour with regard to the irradiation geometry. One must therefore seek another measurable operational quantity the behaviour of which should be as similar as possible to that of HE. According to the recommendations of the ICRU3 , such a quantity should be defined at appropriate loca• tions in a tissue-equivalent sphere of 30-cm diameter and density of 1 g/cm3 , having a mass composition of 76.2% oxygen, 11.1 % carbon, 10.1% hydrogen, and 2.6% nitrogen. Dosimeters calibrated in terms of dose equivalent in the sphere are recommended for routine operations. For purposes of environmental and area monitoring, two concepts are introduced linking the external radiation field to the effective dose equivalent. First, the ambient dose equivalent HO(d), appropriate for strongly penetrating radiation, and 512 B. Grosswendt secondly the directional dose equivalent H' Cd), suitable for weakly penetrating radia• tion. The ambient dose equivalent H*Cd), at a point in a radiation field, is the dose equivalent that would be produced by an aligned and expanded field in the ICRU sphere at a depth, d, on the radius opposing the direction of the aligned field. (In an aligned field, the fluence is defined to be unidirectional; in an expanded field, the fluence and its angular and energy distribution have the same values throughout the volume of interest as in the actual field at the point of reference.) The directional dose equivalent H'(d), at a point in a radiation field, is the dose equivalent that would be produced by an expanded field in the sphere at a depth, d, on a radius in a specified direction. These recommendations of the ICRU are based on extensive Monte Carlo studies of the photon transport within the sphere phantom performed in recent years. In the next sections, some basic quantities, which can be derived from spatial dose distributions, are defined and details given of geometrical and special computation techniques frequently used in ICRU sphere investigations. Moreover, some results on sphere quantities are discussed and compared with dose equivalents of the MIRD phantom.

24.6 ICRU-SPHERE QUANTITIES

To define dose equivalent quantities in the sphere, let H(r, eo) be the dose equivalent produced by an arbitrary radiation field at the point (r, eo) within the sphere. On the assumption of a fixed coordinate system with the origin at the center of the sphere, r is the distance from the center, and e~ the unit vector of direction. The radiation field can be described by the spectral distribution of photon fluence in angle and energy [8 2

(24.11)

Here, hCE', r, w) is defined such that

If the unit vectors of direction, eo and eo" are expressed by the polar angles 0,0' and the azimuthal angles a, a', in the following form,

eo = {sin 0 cos a, sin 0 sin a, cos O} (24.12) eo, = {sin 0' cos a' , sin 0' sin a', cos O'} , then the angle w is given by

w = arccos [sin 0 sin 0' cos a cos a' + (24.13) sin 0 sin 0' sin a sin a' + cos 0 cos 0'] 24. Photon Monte Carlo Transport in Radiation Protection 513

In the case of unidirectional irradiation of the sphere with B' 0°, the angle w therefore is equal to B, and the ambient dose equivalent H·(d) and the directional dose equivalent H'(d) can be expressed as

E H·(d) = j[::,]h(E',R- d,B = OO)dE' o (24.14) E H'(d) = j [::,] h(E', R - d, B)dE' o where R is the sphere radius.

If one looks for the index quantity H1,d (the maximum value of dose equivalent within the core region of the sphere with a 14 cm radius), Eqn. 24.11 leads to

E H1,d = max{j [::,] h(E', r, B)dE'} , (24.15) o where r ~ 14 cm.

Mean values of the dose equivalent over special regions of the sphere volume, for example, spherical shells of radii Rl and R 2 , and cones of half angles e, can also be determined from Eqn. 24.11 by simple integration over the desired volume:

Ro 6 3 J J H(r, eo)r2 sin BdrdB ------~~~v R 0 H(Rl' R2 , B) = __1 ------,;::--- (24.16) (R~ - R~)[l - cos e)

For e= 71", for example, Egn. 24.16 refers to the volume of a spherical shell between the radii Rl and R2 , for R. = 0 to the volume of a cone of spherical base with a half-opening angle e.

Another quantity suitable for characterizing the irradiation geometry of a radiation field, the so-called geometry factor G, can be derived from

G = H(r, eo) (multidirectional irradiation) (24.17) H(r, eo) (unidirectional irradiation) , if the receptor-free fluence is uniform and has the same magnitude in both cases. The main problem in obtaining quantitative relations between radiation field and sphere quantities therefore is the calculation of h(E, r, 0) from which all other quantities can be numerically derived.

24.7 DOSE DISTRIBUTION GEOMETRY In contrast to the MIRD phantom, the IeRD sphere is assumed to be quite homogeneous with respect to internal structure as well as its elemental composition and density. In 514 B. Grosswendt order to obtain a well resolved spatial dose equivalent distribution, the sphere volume must, however, be subdivided into a number of volume elements. To do this, two different methods were used. The method most frequently used20.25-s6 is shown in the left part of Fig. 24.5. It subdivides the sphere into a surface layer of O.007-cm thickness, followed by a spherical shell up to a depth of 1 cm, which again is subdivided into shells of a typical thickness of 0.2 cm, and a core region subdivided by spherical shells of a typical thickness of 0.5 cm. These spherical shells are now further subdivided into ring-like volume elements around the z-axis of the sphere using an angular grid with typical intervals f1() of 5° with respect to the polar angle, rJ.

1 1

~ r var iable A / II

I cPa: ~z~cml Figure 24.5. Geometry for calculating the spatial dose equivalent distribution within the ICRU sphere phantom. Left part of the figure: subdivision of the sphere into spherical shells; right part of the figure: subdivision of the sphere into circular cylinders.

The main disadvantage of this method is that the subvolumes along the centre line of the sphere are smaller than those of the off-centre regions, such that poor statistics may be obtained. To overcome this difficulty, another method of subdividing the sphere was used by Williams et al S1 . It also is shown in Fig. 24.5. Williams et al subdivided the sphere into circular cylinders around the z-axis, the radii of which were chosen in such a way that the squares of the radii of neighbouring cylinders differed by 1 cm2 • These cylinders were then cut into rings of 0.2-cm thickness using a linear grid along the z-axis. This method provides a very fine spatial resolution at the equator of the sphere and gives a system of subregions of equal volume, except for those elements truncated by the sphere surface.

24.8 SPECIAL CALCULATION TECHNIQUES

The calculation of the spatial dose equivalent distribution is usually performed along the lines described in Section 24.3, either by following up the history of each photon from one interaction point to the other, or by using a fractional photon model similar to that of Snyder et al", either in connection with Eqn. 24.9 or in connection with the procedure used by Williams et al s1, based on the calculation of the kerma for small 24. Photon Monte Carlo Transport in Radiation Protection 515 planar segments defined by the spatial grids described in the last section. In the latter case, the kerma K;, for planar element j, of area A;, associated with the volume element Yj, is given by

Ki = ~ '" WiEi [JL(Ei )/ P]tr , (24.18) A- L IcosO·1 3 i ' where Wi is the photon weight of the ith photon, Ei its energy, [JL(Ei )/ P]tr the corre• sponding energy transfer coefficient for the ICRD tissue, and Oi the angle between the photon direction and the normal of the plane segment j. The summation runs over all photons i that cross the plane element Ai. Special care must be taken for the case Oi ~ 7r leading to I cos Oi I-A -t 00 can be introduced in the calculation. It should be noted that Egn. 24.18 is only a particular form of the common definition of kerma,

(24.19)

where (8'IjJ /8E) is the differential energy fluence around the volume element j, and the integration is performed over the total photon energy spectrum. If, instead of the total kerma, the collision kerma (which is equal to the absorbed dose in the case of charged-particle equilibrium) is used, [JL(E)/ P]tr has to be exchanged with the energy absorption coefficient, [JL(E)/ P]en.

Special calculation techniques can be used to improve the statistical accuracy of the dose equivalent distribution near the sphere surface by path length shortening before the first photon interaction, or to improve the accuracy of the dose distribution along the centre line of the sphere by prejudicing the photon source in such a way that the number of photons incident on the central zone is increased26.

24.9 RESULTS CONCERNING THE ICRU SPHERE

Conversion factors at specific depths with respect to dose equivalent quantities, or the index quantities, have been published by various authors20,21,25,26,29,31,32,34,35,37, where the data with the smallest statistical uncertainties at present are those of Williams et a1 37• The angular dependence of specified depth dose and the influence of the ir• radiation geometry have been studied, for example, by Grosswendt and Hohlfeld28,3o and Williams et a1 37, the conversion factors concerning mean dose equivalents, e.g., by Kramer20 , Grosswendt and Hohlfeld33 and by Williams et a1 37• To give an impres• sion of some of the results for the directional dose equivalent, the conversion factors, H'(0.07 cm,O = 0°)/ Ka (see Section 24.5) and H*(l cm)/ Ka for the ambient dose equivalent, are plotted in Fig. 24.6 and Fig. 24.7 against photon energy E for mo• noenergetic undirectional photons. The agreement of the results of different authors is generally better than 5%. The discrepancies are due to statistical uncertainties and to differences in the calculational models used by the authors. The angular depen• dence H'(l em, 0)/ H'(l cm,O), calculated by Grosswendt and Hohlfeld3o for parallel monoenergetic beams, is shown in Fig. 24.8 with the photon energy E as a parameter. 516 B. Grosswendt 1.8 Sv/Gy T 1.6 1.4 ~T 1--x 1.2 .l. t 1.0 ~ 0--- 0.8 0 CD• E 0.6 ...... u 0 0 c::i 0.4 ":t:: 0.2

0.0 10 50 100 keV 500 E-----..J..... - Figure 24.6. Conversion factor H'(0.07 cm,8 = OO)jKa for monoenergetic photon radiation plotted against photon energy E: the solid line is a fitting curve38; ~,reference 37; x, reference 32; +, reference 29.

1.8 Sv/Gy 1.6 r\~ 1.4 , 1.2 0 1 ~0-Loo & +-ls~ t 1.0 l 0 0 0 ~ 0.8 E-- u 0.6 ~ :::t:: 0.4 1

0.2

0.0 10' 10 2 103 keV 10 4 E .. Figure 24.7. Conversion factor H*(l cm)j J(a for monoenergetic photon radiation plotted against photon energy E: the solid line is a fitting curve (reference 38); <>, reference 20; x, reference 29; ~, reference 34; 0, reference 37; +, reference 32; V, reference 35. 24. Photon Monte Carlo Transport in Radiation Protection 517 1.0~~~~ __

5. 3. H'(1cm,{)) 2. H'(1cm,OO) 1.25

0.662 0.30 0.15 L-__L-_~~:"---=::~L--===='0.096 oo ° 45° 90° 135 ° 180 ° () .. Figure 24.8. Angular dependence of H'(lcm, B)I H'(lcm, 0) for monoenergetic parallel photon beams of energy E.

1.0 /cm=2D r,,+ ~ .--e--e-. L5

to

o { IIIIIIII1 I IIIIIII1 I IIIIIII1 10' 102 [---)----

Figure 24.9. Ratio of the effective dose equivalent HE in the case of whole body AlP irradiation to the dose equivalent H'(d,B = 0°) against photon energy E. The curves are for d = 0.3 cm (+), 1.0 cm (0), and 2.0 cm (V). The irradiation geometry is indicated by the top view of a figure in the inset. The shaded region roughly characterizes the position of tho volume element within the ICRU sphere where H'(d, B) has been calculated. Its actual position depends on the depth, d. 518 B. Grosswendt

To estimate the applicability of the sphere quantities as operational quantities for radiation protection purposes, the sphere quantities must be compared with dose equivalents within the MIRD phantom. This is done in Figs. 24.9 - 24.11. These figures taken from reference 36 show the ratio of the effective dose equivalents HE for A/P• , LLAT-, and P / A-irradiation of the whole body and the directional dose equivalent H'(d,O) for depths of 0.3 cm, 1 cm, and 2 cm and the angle 0 = 0°,90° and 180°. As can be seen from the figures, the directional dose equivalent is a conservative measure of the effective dose equivalent for all three depths, except for P / A-irradiation where a considerable underestimation exists. Except for P / A-irradiation, which should be avoided in radiation protection, the sphere quantities are well suited as operational quantities.

24.10 INFLUENCE OF ELECTRON TRANSPORT In most Monte Carlo calculations concerning radiation protection, the finite range of electrons and positrons has not been taken into account. In the case of the MIRD phantom, the particle ranges generally are small compared with the diameter of most of the organs. The absorbed dose changes abruptly only at boundaries where composition and density changes cause a different interaction density of the photons. The neglect of particle transport only accentuates the change of absorbed dose at such a boundary. Therefore, if the absorbed dose distribution near boundaries is not of special interest, the particle transport can be neglected. In the case of the ICRU sphere, the particle transport is important for higher photon energies if the dose equivalent distribution near the sphere surface is of special interest. This has been discussed e.g., by Dimbylow and Francis26•31, Nelson and Chilton32, and Hollnagel et al 3s • In most cases, however, a kerma approximation combined with a sphere phantom in vacuum should yield dose distributions very similar to those measured in air because electrons set in motion in the air volume surrounding the sphere improve the charged particle equilibrium near the surface in such a way that the kerma approximation works well.

24.11 CONCLUSION In conclusion, it must be said that the present state of radiation protection, in particular when looking from the fundamental point of view and to the definition of quantities, is widely based on investigations performed by Monte Carlo methods. This is true not only in the case of the dose distributions within the human body, but also in the case of a deeper understanding of the physical quantities recommended for practical application. Radiation protection therefore is a typical example of the importance of Monte Carlo simulation techniques in present-day physics. 24. Photon Monte Carlo Transport in Radiation Protection 519

L5 HE (LLATl H'(d, OJ

E .. Figure 24.10. Ratio of HE to H'(d,O = 90°) plotted against photon energy E in the case of LLAT whole body irradiation; for the values of d, see Fig. 24.9.

50 + Idiom. "3 40

\+ 30 \ - t 10-\ \ - Q) HE(P/A) 20 - W(d,B) m-l\ - ,~\ 10 \~~~

0 10' 10 2 103 keV 104

E ~ Figure 24.11. Ratio of HE in the case of P / A whole body irradiation to H'(d,O = 180°) against photon energy E; for the values of d, see Fig. 24.9. 520 B. Grosswendt

REFERENCES 1. ICRU-Report 40, "The Quality Factor in Radiation Protection", (1986). 2. ICRP-Publication 26, "Recommendations of the International Commission on Radiological Protection", (1977). 3. ICRU-Report 39, "Determination of Dose Equivalents Resulting from External Radiation Sources", (1985). 4. W. S. Snyder, M. R. Ford, G. G. Warner, H. 1. Fisher, "Estimates of Absorbed Fractions for Monoenergetic Photon Sources Uniformly Distributed in Various Organs of a Heterogeneous Phantom", J. Nucl. Med. Suppl. 3, Vol. 10 (1969). 5. ICRP-Publication 23, "Report of the Task Group on Reference Man", (1975). 6. I. H. Tipton, W. S. Snyder, M. J. Cook, "Elemental Composition of Standard Man", Health Phys. Div. Ann. Prog. Rept. ORNL-4007 (1966) 241. 7. M. Rosenstein, "Organ Doses in Diagnostic Radiology", HEW Publication (FDA) 76-8030, Bureau of Radiological Health, Rockville, MD. (1976). 8. R. Kramer, G. Drexler, "On the Calculation of the Effective Dose Equivalent", Radiat. Prot. Dosim. 3 (1982) 13. 9. C. D. Zerby, "A Monte Carlo Calculation of the Response of Gamma-Ray Scin• tillation Counters", Meth. Compo Phys. 1 (1963) 89. 10. W. A. Coleman, "Mathematical Verification of a Certain Monte Carlo Sampling Technique and Applications of the Technique to Radiation Transport Problems", Nucl. Sci. Eng. 32 (1968) 76. 11. J. H. Hubbell, "Photon Mass Attenuation and Mass Energy-Absorption Coeffi• cients for H, C, N, 0, Ar, and Seven Mixtures from 0.1 keV to 20 MeV", Radiat. Res. 70 (1977) 58. 12. E. Storm, H. I. Israel, "Photon Cross Sections from 1 keV to 100 MeV for Ele• ments Z = 1 to Z = 100", Nucl. Data Tables A7 (1970) 565. 13. E. F. Plechaty, D. E. Cullen, R. J. Howerton, "Tables and Graphs of Photon Interaction Cross Sections from 1.0 keV to 100 MeV Derived from the LLL Eval• uated Nuclear Data Library", Lawrence Livermore Lab. report UCRL-50400, Vol. 6, Rev. 2 (1978). 14. L. Kolblinger, P. Zarand, "Monte Carlo Calculations on Chest X-Ray Examina• tions for the Determination of the Absorbed Dose and Image Quality", Phys. Med. BioI. 18 (1973) 518. 15. T. D. Jones, J. A. Auxier, W. S. Snyder, G. G. Warner, "Dose to Standard Reference Man from External Sources of Monoenergetic Photons", Health Phys. 24 (1973) 241. 16. J. W. Poston. W. S. Snyder, "A Model for Exposure to a Semi-Infinite Cloud of a Photon Emitter", Health Phys. 26 (1974) 287. 17. R. Kramer, G. Drexler, "Zum Verhiiltnis von Oberfliichen und Korperdosis in der Rontgendiagnostik", Medizin. Phys. Vol. 2 (1977) 683. 18. T .D. Jones, J. W. Poston, "Isotropic and Cloud Source Irradiation by Monoen• ergetic Neutrons and Photons", Health Phys. 34 (1978) 83. 19. K. O'Brien, R. Sanna, "The Effect of the Male-Female Body Size Differences on Absorbed Dose-Rate Distributions in Humans from Natural Gamma Rays", Health Phys. 34 (1978) 107. 24. Photon Monte Carlo Transport in Radiation Protection 521

20. R. Kramer, "Ermittlung von Konversions-Faktoren zwischen Korperdosen und relevanten StrahlungskenngroBen bei externer Rontgen- und Gamma- Bestrahlung" , GSF Report S-556 (1979), Gesellschaft fiir Strahlen- und UmweltforschungmbH, Miinchen. 21. R. Kramer, G. Drexler, "Practical Implications of the Concept of Dose Equivalent Index", in Application of the Dose Limitation System for Radiation Protection, Proceedings of a Seminar, IAEA (1979) 451. 22. R. Kramer, M. Zankl, G. Williams, G. Drexler, "The Calculation of Dose from External Photon Exposures Using Reference Human Phantoms and Monte Carlo Methods. Part 1: The Male (ADAM) and Female (EVE) Adult Mathematical Phantoms", GSF Report 5-885 (1982), Gesellschaft fiir Strahlen- und Umwelt• forschung mbH, Miinchen. 23. G. Williams, M. Zankl, H. Eckerl, G. Drexler, "The Calculation of Dose from External Photon Exposure Using Reference Human Phantoms and Monte Carlo Methods. Part. 2: Organ Doses from Occupational Exposures", GSF Report S-1079 (1984), Gesellschaft fiir Strahlen- und Umweltforschung mbH, Miinchen. 24. G. Drexler, W. Panzer, 1. Widemann, G. Williams, M. Zankl, "The Calculation of Dose from External Photon Exposure Using Reference Human Phantoms and Monte Carlo Methods. Part. 3: Organ Doses in X-Ray Diagnosis", GSF Report S-1026 (1984), Gesellschaft fiir Strahlen- und Umweltforschung mbH, Miinchen. 25. G. Drexler, R. Kramer, "Die GroBe Aquivalentdosisindex und ihre praktische Bedeutung im Strahlenschutz" ,Proc. 12th Annual Meeting of the Fachverband fiir Strahlenschutz (1978) 773. 26. P. J. Dimbylow, T. M. Francis, "A Calculation of the Photon Depth-Dose Dis• tributions in the ICRU Sphere for a Broad Parallel Beam, a Point Source and an Isotropic Field", Report NRPB-R92, National Radiological Protection Board, Chilton, Oxon, U.K. (1979). 27. B. Grosswendt, "Berechnungvon Photonenspektren in verschiedenen Tiefen einer Kugel aus gewebeaquivalentem Material bei Bestrahlung mit parallelen monoer• getischen Photonen", Medizin. Phys. (1979) 85. 28. B. Grosswendt, K. Hohlfeld, "Effect of the Irradiation Geometry on Dose-Equivalent Index", Health Phys. 41 (1981) 657. 29. K. Hohlfeld, B. Grosswendt, "Conversion Factors for Determining Dose Equiva• lent Quantities from Absorbed Dose in Air for Photon Radiation", Radiat. Prot. Dosim. 1 (1981) 277. 30. B. Grosswendt, K. Hohlfeld, "Angular Dependence of Specified Depth Dose Equivalent Quantities in the ICRU Sphere for Photon Radiation", Radiat. Prot. Dosim. 3 (1982) 169. 31. P. J. Dimbylow, T. M. Francis, "The Effect of Photon Scatter and Consequent Electron Build-Up in Air on the Calculation of Dose Equivalent Quantities in the ICRU Sphere for Photon Energies from 0.662 to 10 MeV", Phys. Med. BioI. 28 (1983) 817. 32. R. F. Nelson, A. B. Chilton, "Low-Energy Photon Dose Deposition in Tissue Slab and Spherical Phantoms", Report No. NUREGjCR-3425, Univ. of Illinois (1983). 33. B. Grosswendt, K. Hohlfeld, "Mean values of Dose Equivalent in the ICRU Sphere for Limiting the Effective Dose Equivalent", Radiat. Prot. Dosim. 8 (1984) 177. 522 B. Grosswendt

34. P. J. Dimbylow, T. M. Francis, "The Calculation of Dose Equivalent Quantities in the ICRU Sphere for Photon Energies from 0.01 to 10 MeV", Radiat. Prot. Dosim. 9 (1984) 49. 35. R. Hollnagel, R. Jahr, B. R. 1. Siebert, "Influence of Charged Particle Build-up on Dosimetric Quantities in the Surface Layer of the ICRU Spherical Phantom" , in Digest 6th Internat. Congr. Internat. Radiat. Prot. Assoc. (1984) 970. 36. B. Grosswendt, "Theoretical Studies on a Proposed Operational Quantity for Individual Photon Dosimetry", Radiat. Prot. Dosim. 12 (1985) 135. 37. G. Williams, W. P. Swanson, P. Kragh, G. Drexler, "Calculation and Analysis of Photon Dose Equivalent Distributions in the ICRU Sphere", GSF Report S-958 (1983), Gesellschaft fur Strahlen- und Umweltforschung mbH, Munchen. 38. S. R. Wagner, B. Grosswendt, J. R. Harvey, A. J. Mill, H. J. Selbach, B. R. 1. Siebert, "Unified Conversion Functions for the New ICRU Operational Radiation Protec• tion Quantities", Radiat. Prot. Dosim. 12 (1985) 231. 25. Simulation of Dosimeter Response and Interface Effects

A. E. Nahum

Joint Department of Physics Institute of Cancer Research and Royal Marsden Hospital Sutton, Surrey, SM2 5PT, U.K.

25.1 INTRODUCTION A dosimeter measures the absorbed dose at some specified point in the irradiated medium, Dm. The dosimeter yields a signal, be it charge, light intensity, change in absorbance, etc., which is proportional to the amount of energy absorbed in the sensi• tive element of the dosimeter, the "cavity", Dc. It will be assumed that the relationship between the signal and Dc is known. In general, this sensitive material will differ in its radiation-absorption properties from that of the surrounding medium, and furthermore, it may be surrounded by a wall of yet another material (e.g., a graphite-walled air-filled ionization chamber in water). A factor Im.c is defined thus:

Dm = Im.c Dc. (25.1)

In general, the presence of a detector will disturb the fluence of radiation relative to that present in the undisturbed medium, and the evaluation of Im.c will be a difficult task. However, in the case of photon radiation, there are two situations where theoretical expressions can be given for Im.c. The first of these cases is illustrated by the centre cavity in Fig. 25.1. Cavity B is small compared to the electron ranges, and thus can be said to "sample" the electron fluence set up in the medium (M). For this so-called Bragg-Gray cavity, Im.c is given by the stopping-power ratio sm.c (see Chapter 23). In the case of incident electron radiation, the stopping-power ratio will always be involved, whatever the size of the cavity.

Cavity C is large compared to the electron ranges, but is assumed not to disturb the photon fluence in the medium. In this case, the cavity is acting as a photon detector, and Im.c is given by the ratio of the mass energy absorption coefficients, (P.n/ P)m.c' The intermediate case, A, has been treated to varying degrees of approximation in "general" cavity theories, the subject of a recent review by Horowitz1.

The evaluation of stopping-power ratios can be complex, especially for theimpor• tant case of m = water and c = air for the megavolt age electron and photon beams used in radiotherapy. This is a subject in its own right, and is covered in this volume

523 524 A. E. Nahum by Andreo2 • However, many of the dosimeters in common use fall neither into category B nor C (Fig.25.1); their response thus is not simple to predict. For such cases, Monte Carlo simulation of the detector geometry can yield Dc directly; this is the main subject of this chapter.

B en z 0 ....a: c..> A c W ...J W M u. 0 a: w !D ~ ::::> N z M -.-.-.~.--.-.

1600 RELATIVE DISTANCE FROM CAVITY BOUNDARY Figure 25.1. Schematic illustration of the contributions of medium- and cavity• generated electrons to the total absorbed dose in photon-irradiated cavities. A- intermediate size; B- small (Bragg-Gray) cavity; C-Iarge (photon detector) cavity (adapted from Horowitz1 ).

The question of the absorbed dose in the vicinity of interfaces is extremely relevant to dosimeter response. Case C in Fig.25.1 is simple to treat only because any interface effects are confined to the cavity edges, and make a negligible contribution to the total energy deposition in the cavity. Case A, however, is dominated by interface effects. In the context of dosimeter response, we can define such effects as extending over a distance from an interface up to the range of the secondary particles. These are generally Compton electrons in the case of photon beams, but would be knock-on electrons or 8-rays in the case of electron beams. For moderately to severely mismatched cavities, there will be interface effects due to 8-rays even for Bragg-Gray cavities3,\ though 'B' in Fig. 25.1 does not show this. Clearly then, any Monte Carlo simulation of dosimeter response must be able to handle interface effectsS- s.

25.2 AN INTERFACE BENCHMARK

The interface effects alluded to above are the result of differences in the transport of electrons set in motion by photons in adjacent media. Electron transport is simulated in condensed-history fashion9,1O, except in the case of codes for very specialised appli• cations in microdosimetryll. Consequently, an artificial electron step-size is involved which may not be small compared to the dimensions of the detector. An examination of how the code EGS412, with user code INHOM13,14, handled the dose in very small regions illustrates some of the problems that can arise. This example was carried out prior to embarking on a simulation of ion-chamber response. 25. Simulation of Dosimeter Response and Interface Effects 525

The irradiation geometry was that of a plane-parallel 0.5-MeV electron beam per• pendicularly incident on a semi-infinite water medium. The dose was scored in slabs of varying thickness using the user code INHOM. The EGS parameters AE and ECUT were both 0.521 keV (total energy). The dose scored in a very thin slab of the same material, water, with its centre at a given depth, z = 0.025 cm (ro= 0.14), was found to depend on the slab thickness. This was clearly a Monte Carlo artefact, there being no real inhomogeneity involved. The excess Monte Carlo dose over that expected is plotted in Fig. 25.2.

~ ~ 0 15 Q) (/) 0 0 0 X 0. 0 x a> L- 10 0 0 U ..-...... 0. x 2 a> C 0 0 I 5 ~ u (/) ::2 (/) 0 Q) u x w 0 10- 5 10-4 Thickness of Water Slab (cm) -5

Figure 25.2. The variation of the excess "Monte Carlo" dose with slab thick• ness. EGS3 with user code INHOM; 0.5-MeV electrons on water; z/r = 0.14; AE = ECUT = 521 keV; ESTEPE = 0.04.

The problem begins for slice thicknesses t of 10-3 cm or less. As a reference point, the csda range ro for a lO~keV electron is 2.5 x 10-· g-cm-2 in water, which is also the same order of magnitude as the shortest distance across air cavities in typical ion chambers. The "excess" dose peaked at around 18% greater than the expected dose for t = 10-5 cm. These figures were extracted from 20000-history runs; the trend is not due to statistical fluctuations. In order to gain some insight into the reason for this "interface" artefact, the parameter SMAX, the maximum electron step length allowed, irrespective of energy, was varied in the thin slab only, for t = 10-5 cm. Fig. 25.3 shows that this parameter has a dramatic effect on the excess dose. This quantity decreases essentially to zero for a SMAX value of about 15 times the thickness. However, on further decreasing SMAX, the'excess dose increases again. This latter increase is due to switching off multiple scattering for very short pathlengths, as discussed by Rogersl4 , and Bielajew and Rogersl5. In the particular case of t = 10-5 cm, there is only a very narrow range of transport parameters for which the correct dose is obtained. 526 A. E. Nahum

5 10 20 30 50 100 SMAX/Slob thickness Figure 25.3. The effect on the "excess dose" (see Fig.25.2) of varying SMAX, the maxim.1.!m allowed electron pathlength for a slab of thickness 10-5 cm; other parameters are as noted ill Fig. 25.2.

25.3 ELECTRON STEPLENGTH VARIATION

In general, one can expect artefacts in the calculated dose when the electron steplengths begin to approach the dimensions of the critical scoring region. Most electron transport algorithms in Monte Carlo simulations follow Berger9 in choosing steplengths due to continuous losses on the basis of the energy loss in a step being a fixed fraction of the electron's kinetic energy (e.g., Rogers14 introduced ESTEPE into the EGS code system). When scoring dose in very small regions, the question then arises of what value of ESTEPE or its equivalent to choose in order to ensure that the electron steplengths are sufficiently small. It should be emphasised that we are not now discussing the upper limit of ESTEPE necessary to ensure that no pathlength correction16 is required. Such ESTEPE effects have been extensively investigated by Rogers14.

Fig. 25.4, taken from Bielajew and Rogers16, gives the variation of electron step• length due to "continuous" losses with electron energy for several different ESTEPE values. The broken curve gives the values of the steplengths at which multiple scatter• ing is turned off in PRESTA15.16. From Fig. 25.4, one can see that multiple scattering would be switched off for ·all electrons above 10 keV for t = 10-5 cm. A horizontal line has been drawn on the graph at t = 2.5 X 10-4 cm corresponding to the smallest air cavities in practical ion chambers (i.e., 2.5 x10-4 ~ 2-mm air). An ESTEPE value of 0.1 % would be needed to ensure that electron steps were always smaller than this value of t for electrons below 0.5 MeV, but at such a value, multiple scattering would be switched off for all electrons below about 0.6 MeV! However, this restriction on the maximum step size for such air cavities probably is unnecessarily severe, as Fig. 25.3 suggests, where a step size of 15 times the minimum dimension was sufficiently small. Also, allowing ESTEPE to increase, rather than remain constant (cf Fig 25.4) as the electron energy decreases, could still keep the step size small enough while lowering the energy threshold below which multiple scattering would be switched off. This illustrates the difficulty of choosing transport parameters for scoring dose in very small regions, and emphasizes the limitations of choosing electron step-sizes using a constant ESTEPE. 25. Simulation of Dosimeter Response and Interface Effects 527

10-3 ,------,---r----,----r1

Ek (MeV)

Figure 25.4. Total step-size, t, as a function of electron kinetic energy in water, E,., for various values of E.t."., the maximum fractional energy loss per electron step due to continuous processes. Also shown is the value of the minimum step• size for which multiple-scattering theory holds, tmin, as a function of energy (from Bielajew and Rogers16). A horizontal line representing the size of the smallest air cavity in practical ion chambers has been added.

25.4 ION-CHAMBER RESPONSE 25.4.1 Introduction

Air-filled ionization chambers have long been the instruments of choice where high• precision dosimetry is required, as in external-beam radiotherapy (see for example, Svensson and Brahme17). Although such instruments generally behave as Bragg-Gray cavities (i.e., case B in Fig. 25.1), the appropriate value for 1m,. often is complicated by non-medium-equivalent materials in the walls and central electrode18 and, in the case of electron beams, by the scattering differences due to the very different densities of the medium and the air in the cavity19. However, simulating the response of an air cavity is complicated by its very small size, in units of mass per unit area.

25.4.2 In-Air Kerma Calibration in 60CO Radiation

One situation that is common to megavolt age-radiotherapy dosimetry allover the world is the calibration of a reference chamber by placing it, and its associated build-up cap or sheath, at a point in air in a 6OCo beam where the air-kerma (previously, exposure) rate is known. National and regional standards laboratories routinely carry out such calibrations on behalf of radiotherapy clinics. The basic relation involved in such a calibration is that between the absorbed dose to the cavity gas D" and the collision kerma in air at the geometric centre of the ion chamber J(col,air. This can be written as:

DG nCAV -K = IL- AwalJ, (25.2) col,air 528 A. E. Nahum where RcAV for a chamber with wall, build-up cap, and central electrode made of one material m, is given by (25.3) according to cavity theory2o. Awan is a factor that corrects for the attenuation and scatter of photons caused by the material of the wall, build-up cap and electrode. Its value depends critically on the shape of the chamber.

Nath and Schulz21 carried out a Monte Carlo simulation of ion-chamber response in a 60CO i-ray beam precisely in order to obtain values of Awan for the wide range of commercially available ion chambers; the Awan values they obtained were broadly in line with theoretical and experimental expectation, and were subsequently incorporated into the AAPM dosimetry protocol. The simulations also predicted a significant dependence of chamber response, defined as Dg / Keol,air as in Eq.25.2, on chamber dimensions such as radius, depth of cavity (for plane-parallel chambers), etc. This, however, was at variance both with experimental findings and with theory22. Furthermore, values for RcAV for different wall materials were not consistent with Eq. 25.3, as Fig. 25.5 shows.

t08

E t06 "" ex: 0 .... 1.04 U 00: u. ..J ~ 1.02 ex: • ....UJ 00: :< 1.00 ..J ..J 0.99 ~ 0.98 ex: w til 097 :< 00: J: U 0.95 . O"L0.9" , 60 80 100 120 140 160 MEAN IONIZATION POTENTIAL '/ eV

Figure 25.5. The chamber-waIl-material factor RCAV as a function of the mean-ionization potential!. 8: theoretical expression (Eq. 25.2); • extracted from Nath and Schulz21 (taken from Nahum and Kristensen22 ).

Two groups have since carried out simulations of ion chamber response in 60Co beams23- 25 with independent and very different Monte Carlo codes. Fig. 25.6, taken from the simulation by McEwan and Smyth23, shows that these workers obtained essen• tially no variation of chamber response (defined by Eq. 25.2), with radius, in line with theoretical expectations but in marked contrast to the Nath and Schulz21 simulation, also shown in the figure. The curve in Fig. 25.6, labelled 'theory', calculated according to Eq. 25.2, shows a slight dependence on radius for very large radii, but this is entirely due to the variation in Awan. 25. Simulation of Dosimeter Response and Interface Effects 529

1·02_--...,....-~...,....-r-...... ,...,"'T""--...,...-..,...... ,...-r...,.-,...... T"n

1·00 )( o o o o Q-98 o VI )( Z

1'00 f D.98t------i-___o... 0 i .. o ------__ on o g 0'9 '"on o a:.. 0'94 0

~92~----~~~--~~~~~----~~~--~~~~~ D,' 0·2·3·4 ·5·6 ·8 1·0 2·0 3 Inner Radius (emsl Figure 25.6. Relative response and wall-correction factor of a spherical graphite chamber (wall thickness 0.55 g/cm2 ) exposed to 1.25-MeV photons as a func• tion of internal radius. Response is expressed as dose to air per unit collision kerma in air: 0, Nath and Schulz; x, McEwan and Smyth (Monte Carlo); -, cavity theory, i.e., Eqs. 25.2 and 25.3 (from McEwan and Smyth23).

The other simulation was by Bielajew, Rogers and Nahum24,25 using the EGS code system. A marked dependence of chamber response on ESTEPE was found24 , as Fig. 25.7 illustrates very clearly. Here, chamber response is defined as dose to the air in the cavity (Gy) per unit incident photon fluence (cm-2). A value of ESTEPE of the order of 1% was found to be necessary. For even lower values, switching off multiple scattering in EGS begins to be a problem, as the curve associated with the ordinate on the right-hand scale shows.

The explanation that Bielajew et a1 24 have given for this ESTEPE dependence is illustrated schematically in Fig. 25.8. The electron-transport algorithm in the EGS code includes a factor, PLC, which corrects for the difference between the true "curved" electron path and the straight-line distance between the beginning and'end of an electron step12,15. This PLC implicitly assumes that the whole of the electron step takes place in one medium. However, if part of the step could have taken the electron across a boundary into another medium, in this case from carbon in the wall to air in the cavity (or vice-versa), then the PLC and the whole of the remainder of this electron history is invalidated. 530 A. E. Nahum

)( 10-" 6 1. 50 '"E u 0~ >- - ]1.25 VI !:2 5 OJ 0 u 1.00 ~ c: OJ OJ ..c: .= ~ 4 0.75 c: -OJ -VI c: -0 0 0.50 -"0 0.. OJ VI ..c: OJ 3 u ... ~ 0.25 ':i "0 VI -.e. 2 0 '"I: 1 10 ESTEPE (%) Figure 25.7. The variation with the maximum continuous energy-loss per step, ESTEPE, of A, the total response to GOCo of a carbon pancake chamber to a normally-incident broad parallel beam of GOCo (2-mm inner depth, I-em inner radius, 0.5 g-cm-2 walls) and B, the fraction of times the multiple scattering (MS) is 'switched off' by EGS. The broken horizontal line is the cavity-theory prediction of 5.33 x 10-11 Gy-cm-2 (adapted from Bielajew et a1 24 ).

Region 1 Path a

Calculated path A~------~------~ B Interface

Region 2 Figure 25.8. The calculated path and two possible 'physical' paths near an interface. The dose is correctly calculated for path (a) but is incorrect for path (b). If region 2 is a vacuum, for example, path (b) is not physically possible. These interface artefacts can be avoided by reducing the size of the straight-line path in the simulation (from Bielajew et a[24).

The Monte Carlo underresponse at large ESTEPE thus can be understood as due to the parts of the curved electron path that would have taken electrons into the air cavity from the wall, but were not "allowed" to do so by the algorithm. By reducing ESTEPE, this effect is progressively reduced as the curved nature of the electron paths is more faithfully simulated (see Fig.25.3). 25. Simulation of Dosimeter Response and Interface Effects 531

Details of the BRN simulation, including the particular values of the EGS pa• rameters ESTEPE, SMAX, ECUT, AE, PCUT and AP used, are given in Table 25.1. It was shown24 that for a fixed value of AE = 10 keY, the computing time per incident pho• ton depends only weakly on ECUT for values above about 10 keY; this is partly due to the range-rejection technique employed. A preliminary investigation showed that the chamber response per unit incident fluence did not depend on ECUT for values between 100 and I keY. The value of 10 keY (see Table 25.1) was chosen both on the basis of computing efficiency and because the csda range in air of a lO-keV electron (2.4 mm) is less than the minimum dimension of most ion chambers in practical use. Ideally, how• ever, for more severely mismatched materials than carbon and air, a lower ECUT should be used. Obtaining statistical uncertainties of the order of 1% for chamber response required between 5 and 24 hours of CPU time on a VAX 11/780 with floating-point acceleration.

Table 25.1 also gives details of the Nath/Schulz21 (NS) and McEwan/Smyth23 (McS) ion-chamber response simulations. Neither of these simulations included delta• ray transport. However, this should not have any appreciable effect on the results for closely matched materials such as carbon and air; the Bragg-Gray and Spencer-Attix stopping-power ratios, s~~c and s~:'c, for any.6. ~ 10 keY, are very nearly equal25 . Other differences between the three codes may be of greater relevance to the ion-chamber situation, however. Nath andSchulz21 used the code developed by Bond, Nath and Schulz26 which divides the electron track into fractions of the residual range, "usually 10%". This corresponds approximately to ESTEPE = 0.1, or 10%. This value would not have produced valid results in the BRN simulation, as Fig.25.7 makes clear.

Smyth and McEwan27 did not observe any such dependence of chamber response on fractional electron step-size. However, Smyth28 subsequently made clear an important difference between the McS and BRN(EGS) codes; the McS code does not terminate electron steps at boundaries between different media. Such a procedure is possible only if the pathlength correction is ignored, there being no obvious way of computing this factor when the step crosses from one medium to another. Also, multiple-scattering formalisms assume that only one medium is involved. In the particular case of an air cavity, the extremely low density of the air means that the fluence of primary electrons in the wall will be disturbed only negligibly by the cavity material. Furthermore, the fact that charged-particle equilibrium exists in the wall at the depth of the cavity, due to the wall being of a thickness greater than the maximum electron range, means that this wall electron-fluence spectrum is insensitive to the size of the electron steps, and hence to the electron angular-distribution. Bielajew and Rogers29 demonstrated this by obtaining the same correct value for chamber response when turning off all multiple scattering. Consequently, it is not surprising that McEwan and Smyth saw no dependence of chamber response on fractional electron step-size: their code was bound to yield the correct equilibrium electron fluence in the air cavity.

Recently, a new electron-transport algorithm PRESTA 15.16has been developed for use in Monte Carlo simulation. PRESTA incorporates a variable ESTEPE that is auto• matically adjusted close to medium boundaries in order to avoid the type of artefact discussed above. Furthermore, PRESTA allows very large ESTEPE values away from boundaries since an improved version of the pathlength correction factor has been in• cluded. The execution time for a given precision in the ion-chamber simulation with EGS can significantly be reduced using PRESTA, as the number of electron steps is very much less than for the O.l-ESTEPE value required without PRESTA. A factor of 5.5 times faster was reported by Bielajew and Rogers 16 . 532 A. E. Nahum

Table 25.1. Details of three different Monte Carlo codes used to simulate ion-chamber response in 60Co radiation (1.25-!lleV photons).

FEATURE NS 21•26 McS23•27 BRN24 •25 Code In-house In-house EGS + user code CAVITY Type Coupled photon-electron simulation Delta ray NO NO YES transport e - transport 0.0005 cm 5 keY 10 keY cut-off residual range e - step size Constant fraction fr of residual range - ESTEPE14

algorithm fr ~ 0.1 Variable fr Maximum NO NO YES Step size SMAX = 0.2 cm Pathlength NO NO YES correction (PLC) Fermi-Eyges Energy losses - Entirely csda using- Discrete c-ray and collision stopping power bremss. photon production Continuous losses using stopping power restricted to Do < 10 ke V (= AE - moc2) Energy-loss YES NO YES for straggling losses> 10 keY Multiple Gaussian Moliere Moliere switched off

scattering for t < tmin Wall/air e - step terminated No "interruption" e - step terminated interface at boundary at boundary at boundary

25.4.3 Other lon-Chamber Simulations The effect of a thin non-waIl-equivalent lining. The irradiation geometry of the ion-chamber simulation described in the previous sec• tion was such that cavity theory could provide accurate predictions. However, when the build-up cap, wall, and central electrode in a thimble chamber are made of dissimilar materials, then only approximate theoretical expressions exist17. Nahum, Henry and Ross30 measured the effect on the response of an aluminium-walled thimble chamber with aluminium build-up cap of adding thin layers of dag (graphite) to the inside of the wall. The radiation was a 60Co ,-ray beam. The experimental results are compared to the predictions of a "simple" two-component theory in Fig. 25.9. 25. Simulation of Dosimeter Response and Interface Effects 533

1.12 60CO beam

EGS

1.10 • Oag( p = 0.93)

• Graphite( p = 1.76)

z o

~!::! 1.06 z Q (/J (/J ~ 1.04 w ;::>

o 20 40 60 80 100 150 200 WALL THICKNESS (mg cm-2 ) Figure 25.9. The relative response, expressed as ionization per unit mass of air, of a thimble chamber with an aluminium build-up cap and aluminium wall as a function of the thickness of a dag inner lining. Measurements and "simple theory" taken from Nahum et also; EGS results from Bielajew (priv. comm.) with ECUTKE = 10 keY.

The discrepancy is evident. A simulation using EGS4 and user code CAVITY (Table 25.1) also indicated in the figure, yielded values closer to the simple theory than to experiment. One possible explanation for the difference between the EGS result and the measurements could be the 10-keV cutoff chosen in the simulation. This value may not be low enough to model possible 8-ray scattering effects at the wall-lining-cavity interface, as the range of a 10-keV electron is an appreciable fraction of the cavity diameter, 0.6 cm. In the simulation of the previous section however, only carbon and air were involved which are sufficiently similar in atomic number that the choice of the cut-off was unlikely to be critical.

Wall effects in electron beams. Air-filled ion chambers depart from simple Bragg-Gray behaviour in megavolt age elec• tron beams (where the chamber dimensions are very much less than the electron range) due principally to the marked difference in electron scattering between air and medium, usually water or a plastic such as PMMA (lucite, perspex, etc.), or polystyrene. The in-scattering from the walls is not balanced by an equivalent out-scattering from the air cavity, resulting in too high a signal from the detector19.

Fig. 25.10 shows the result of an electron-transport simulation by MandourS1 of the angular and energy distribution of electrons emerging into a vacuum from a scattering medium through a beam-parallel surface; the increasing obliquity of the electron tracks with depth is precisely the reason for this 'in-scattering' perturbation which has no equivalent in photon beams due to the existence of charged-particle equilibrium. 534 A. E. Nahum

2 e"""m-----,.f--ll o. ,em

Figure 25.10. Angular and energy distributions of the particles from a partial beam of 10-MeV electrons as they emerge from carbon into a vacuum (from Mandour31 ).

The required conversion factor, p, for thimble chambers has been determined experimentally17-19, but currently there is no satisfactory theoretical treatment32. This then is an obvious candidate for Monte Carlo simulation. The irradiation geometry is that of a small cylindrical air-cavity, radius 0.3 cm and length 2.5 cm, placed with its axis perpendicular to the beam direction at varying depths in a phantom irradiated by a broad beam of electrons. It is not difficult to see that the probability of one of these electrons depositing energy in the air cavity is very small. The situation is very different from that of simulating photon irradiation, free in air, of an ion chamber with the cavity at the depth of the build-up thickness.

A feasibility study of such an electron-chamber simulation33 has recently been carried out. The EGS4 system with user code DOSRZ was employed. The geometry is constrained to be cylindrically symmetric. A 20-MeV broad, parallel electron beam was incident from the side on a cylinder of water, radius 5 cm, height 8 cm. The air cavity, placed along the axis of the water cylinder, was I cm in diameter and 1 cm in length, with a carbon wall of thickness, 0.273 cm, density, 1.76 g-cm-3 • Table 25.2 gives the details of some simulations in which several EGS parameters were varied. The most important quantity is the estimated CPU time to reach 0.1 % statistics, i. e., 0.1% (one S.D.) on the dose to the air cavity. Thishas been estimated from percentage error E ex: 1/../N, where N is the number of histories.

The data in the table illustrate that even using PRESTA (see Section 25.4.2 and refs. 15 and 16), the goal of 0.1% statistics currently is not feasible unless variance• reduction techniques yield a substantial saving. Range rejection immediately suggests itself; this was used to good effect by Bielajew et a1 24•25 (see section 25.4.2) in the case of photon-generated electrons. A further possibility is that of correlated sampling 25. Simulation of Dosimeter Response and Interface Effects 535 since what is actually required is the ratio of dose in the air cavity (or primary-electron fluence) to that in the undisturbed medium at an equivalent depth. One could track the electrons until they arrive at the cavity boundary and then continue for two separate geometries, one with air in the cavity and the other with water (ignoring, for the present, the non-water wall). The respective cavity doses for these two electrons with a common pre-cavity history would then be scored. The variance per history on this ratio can be expected to be considerably less than that for independent runs for the two different cavity materials.

Table 25.2. Feasibility study of simulating the response of a small ion chamber at 5-cm depth in a phantom in a 20-MeV electron beam; EGS4 with user code DOSRZ (verA) on VAX Il/780 with floating-point acceleration.

Step length ECUTKE AE (Kinetic Energy) Time (sec) CPU (hours) Algorithm (MeV) (MeV) per for ± 0.1% (one S.D.)

History in Dair Phantom Walls + Cavity ESTEPE = 0.01 0.010 1.000 0.010 0.35 1.6 X 10' 0.010 0.010 0.010 0.96 1.7 X 10' PRESTA 0.010 1.000 O.OlD 0.16 9.9 X lOs 0.002 1.000 0.002 0.19 8.3 X lOs

25.5 DOSE DISTRIBUTIONS AT INTERFACES 25.5.1 A Benchmarking Situation

The dose distribution at interfaces irradiated by photons has been the subject of exten• sive experimental studyS-1. It has been possible to explain quantitatively the effects observed in terms of differences in electron scattering and stopping on either side of the interface. Webbs4 simulated a water/aluminium interface in a monoenergetic 60CO beam (1.25-MeV photons). This result is shown, together with that of Rogers and Bielajew3S for the EGS and CYLTRAN codes, in Figs. 25.11a and 25.11b. Both calcu• lations predict similar dose ratios Dw / DAI at the interface, 1.28 ± .02 in Fig. 25.11a, and 1.23 ± .02 in Fig. 25.11 b. According to cavity theory, this must be equal to the ratio of mass stopping powers, as the electron fluence exactly at the interface must be the same in both materials; Scol(E)/P)water/(Scol(E)/p)AI ~ 1.28 for E = 0.3 MeV. How• ever, there the similarity ends. The experimentally verified increase in dose very near the interface on the water side is totally absent in the Webb simulation. This increase is due to electron backscatter from the aluminium being greater than the backscatter contribution in uniform water. The reason for the marked discrepancy between Figs. 25.11a and 25.11b becomes readily understandable on reading in Webbs, that "elec• trons are assumed to deposit their energy along their range according to the Bragg curve in the material", i.e., electron scattering was not simulated. Rogers and Bielajew3S report that such interface calculations are very sensitive to ESTEPE; the value used in the EGS simulation in Fig. 25.11 b was 1%, which is comparable with the value required to obtain valid results in an air cavity (see section 25.4.2). 536 A. E. Nahum

1.0 r-""--r:::::!:~;:;:~~-.---'--r-""---'-I I ~ 'c 0.8 I ">. I ~ 4- WATER+ALUMINIUM- :c ; o.e

w Ul g 0.4 C W m II: g02 m <

10 14 18 22 DEPTH OF BEAM PENETRATION (mm) Figure 25.11a. The distribution of absorbed dose for a 10-cm x10-cm 6OCO beam incident on a water-aluminium interface (from Webb34). e ep to) c ep H O Al E 2 u.. 7 • 'E- epN :2 to) E c to) - >- Cl e -·c 0 ::l- CDO Ul ... -EGS • • • g- 5 • CYLTRAN "0 ~ 0 Ul .c c( 4 1.0 1.1 1.2 1.3 1.4 DEPTH (em) Figure 25.11 b. Comparison of dose deposition at an aluminium-water interface irradiated by 6OCo as calculated by EGS and CYLTRAN (ETRAN). These types of interface calculations are very sensitive to ESTEPE, which was 1% in this case (from Rogers and Bielajew35).

25.5.2 Interface Simulations Involving LiF Some work has been done to date on simulating the response of other types of dosimeters where cavity theory has been demonstrated to be inadequate. Solid-state detectors fall into this category. Thermoluminescent dosimeters (TLDs) have been used extensively in both photon and electron fields. In fact, the whole subject of 'general'- cavity theory, i.e., for cavities intermediate between cases Band C in Fig. 25.1, was developed largely in order to interpret the reading of TLDs in radiation qualities other than that for which they were calibrated. The Monte Carlo work done in this area has aimed at providing more physical insight into what happens at interfaces in order to refine the current general-cavity theories1,36, rather than simply computing the cavity dose. 25. Simulation of Dosimeter Response and Interface Effects 537

Very careful measurements of the dose-deposition profile in a LiF thermoluminescent detector have been made by Ogunleye et a1 37• The geometry was a Pb-LiF-Pb sand• wich, O.044-cm Pb followed by 7 x O.038-cm slabs of lithium fluoride backed by a thick Pb slab; the beam was broad, parallel, and perpendicularly incident. Figs. 25.12a and 25.12b are taken from a simulation of this experiment by Bielajew and Rogers38. Both the EGS and ETRAN codes predicted the experimental data acceptably well, with neither code being favoured. Bielajew and Rogers found that simulating a realistic 6°Co-photon spectrum (as opposed to monoenergetic 1.25-MeV photons) significantly influenced the calculated results. They concluded that details of the incident spectrum should be known if a detailed comparison of theory with experiment is to be made.

calculated dose vs. TL response to 60CO '", 9.0 I --r·--.--'--,----r-- o x - 8.0 EGS4 (PREST A) tPHOTO e - ANGLE SELECT ION "'E u MEASURED TL RESPONSE >- " ~ 7.0

Cl> U C 6 0 Cl> ::J

..... 50 c Cl> TI 'M 4.0 __ .t---- U .~ "• Cl> Ul 3.0 '--_---'--__L-_---'-- __~ _ _'______'I. __ a TI 0,000 0.038 0.076 0.114 0.152 0.190 0.228 0.266

<--upstream-----depth in LiF (em) -----downstream-->

Figure 25.12a. Comparison of measured TL response with EGS and ETRAN dose distributions across the detector; the EGS calculation included photo• electron angular distribution. The dil.ferences between the calculations are 6 and 7 percent at the front and back walls, respectively (from I3ielajew and Rogers38 ).

electron fluence in the LiF o 20 --T--r - ,--,- I w u near back wall C ::: 0 w Ie :J ~ .-. 15 ~ 0 L '+-- Ie ~ C 0 ~ ~ ...,L n u 0 10 near front wall 8 .-.W w w > 0.05 +' CO r" .-. W L _-L-_L...-"-__.L.---,--...l __ -,---.l----. 0.00 0.0 0.2 0.4 0.6 0.8 10 1.2 electron energy (MeV) Figure 25.12b. Electron-fluence spectra, derived from the EGS simulation, in the LiF near the front and back walls and in the middle. Note the high• energy photoelectron contribution in the "front wall" case and the low-energy enhancement in the "back wall" case (from Bielajew and Rogers38 ). 538 A. E. Nahum

Fig. 25.12b shows how the electron-fluence spectrum varies with depth in the LiF. This type of detailed information is virtually impossible to obtain by any other means. It will be required in order to refine theoretical cavity models.

Horowitz et a1 39 have performed studies involving 6OCo ,-ray irradiation of LiF / Al and LiF /Pb interfaces in order to gain more insight into recent developments in general• cavity theory by Kearsley36 and others. The contributions to the total electron fluence of electrons originating in one material, but depositing their energy in the adjacent material, were scored separately. The code CYLTRAN, a descendant of ETRAN for cylindrically symmetric geometry, was used. The authors emphasize the problem of computing the highly space-resolved electron fluences and energy deposition near mate• rial interfaces. Ultra-thin geometrical zones are required. In order to avoid prohibitive execution times, the PHESCE biasing scheme was exploited. PHESCE allows biased secondary-electron (i.e., photon-generated) production along any arbitrary small pho• ton pathlength independent of the occurrence of an explicit photon interaction. Ap• propriate weighting of the electrons ensures unbiased results. The authors verified for one or two cases that using PHESCE yielded the same results as unbiased calculations. The geometry was broad beam to simulate accurately the Ogunleye et a1 37 experiment. The transport of all generations of electrons was simulated down to 0.001 MeV. Char• acteristic x-rays and Auger electrons were included. Fig. 25.13, taken from their paper, contains a great deal of information on the behaviour of the different contribution to the total electron fluence; the coefficient f3 for attenuation in the cavity of electrons generated in the wall material is an essential ingredient in general-cavity theory. .. ~ 500 Pb/LiF Interface LiF/Pb Interface ~ 5, 5,

.;:OJ" Electron Production ; 200 .. in LiF Only Cl. \ ~ c: 100 Electron Production ""U in Lead Only 5' 50 ;;; LiF 0- C 0 6OCo "5 20 LEAD y "".~ 6 10 u "c 5 ~ Ii:" c ~ U 2 OJ ijj

0 2.64 Thickness (9 -em 2) Figure 2.5.13. Electron-fluence distribution as a function of material thickness for the Pb/LiF /Pb configuration irradiated by a 60Co gamma-ray beam. Curve c shows the buildup of the electron fluence in LiF as a function of penetration depth (. LiF only, 0, Pb/LiF, for gamma interactions in LiF only). The slight difference between the two curves does not significantly affect the cavity-theory calculations. Note the significant reduction in the electron fluence in lead as one approaches the Pb/LiF interface. The solid lines are least-squares fits to ·the Monte Carlo data points (from Horowitz et aZS9 ). 25. Simulation of Dosimeter Response and Interface Effects 539

25.5.3 Aluminium/Gold

In order to be able to predict the effects of photon radiation on microelectronic devices, Garth et al

Monochromatic Photon Source 1.4 (1.25 MeV) " Interface Dose -POEM 1.2~ x Experiment ILlen o o ILl > 1.0 ~ -.J ILl a:: 0.8

o. 6 ~---''------'-_--L..._...L.-_.1....----' o 0.1 0.2 0.3 DISTANCE (g/cm2) Figure 25.14a. Calculated dose profile in aluminium downstream of gold (from Garth et al

Garth et al

Co6 0Spectrum (calculated) with 1.4 Forward and Backscattered Components

...... Interface Dose 1.2 w -POEM en 0 x Experiment 0 w > 1.0 I- c::r X --.l >S

0.6 0 0.1 0.2 0.3 DISTANCE (g/cm2 ) Figure 25.14b. Calculated profile in aluminum downstream of gold using estimated 60Co spectrum including scattered components (from Garth et al 40 ). (© 1980 IEEE).

Adadurov and Lazurik8 have also reported a Monte Carlo simulation of a Aul Al interface irradiated by 1.25-MeV photons. Few details are given of their code, but 8-ray production is mentioned. Their paper includes a benchmark simulation of an All Al "interface", with no anomalies in the dose distribution for slab thicknesses of about .008g cm (estimated from their graph). Fig. 25.15, taken from their paper, indicates excellent agreement between their simulation and the same experiment to which Garth et al 40 compared their calculations.

In particular, the very steep rise in the dose in aluminium very close to the interface is predicted, despite the use of monoenergetic photons, in contrast to the findings of Garth et al, and Seltzer and Berger42 discussed above. The reason for this difference could be connected with electron step-size effects (see section 25.4.2), though insufficient detail is given in the work of either Garth et al or Adadurov and Lazurik to enable this hypothesis to be tested.

25.5.4 Electron Beams

Rogers14 has investigated the dependence of the results of interface simulations in elec• tron beams on electron step-size. He chose aluminium followed by gold in a 2-MeV electron beam, and found that when using EGS, an ESTEPE value of 0.005 was neces• sary to achieve good agreement with ETRAN in gold near the interface (Fig. 25.16). 25. Simulation of Dosimeter Response and Interface Effects 541

Au A{I

c: 0 +' 0 6 .s::: c- O> N ---E u >- Q) ::E 5 N I a ..-i

>- c.!l et:: 4 I.L.I Z I.L.I Cl I.L.I co et:: a U) co 3

2L-~ __~ __L-~L-~--~~~~ 0.3 0.2 0.1 0 0.1 0.2 0.3 DISTANCE FORM BOUNDRY (g/cm2) Figure 25.15. Distribution of absorbed energy in an AI-Au target for varying order of the layers with respect to the beam direction, indicated by an arrow; • experiment41 ; - Monte Carlo simulation, this work (from Adadurov and Lazurik8 ).

2 MeV e - incident on Al followed by Au 10

• ESTEPE - 41 8

C\J • OEF AUL T (21) E u 8 o ESTEPE - 0.51 >- (!) - ETRAN 0 ... 4 I 0 Al ...-1

2 Au

0 0.0 0.2 0.4 0.6 0.8 1.0 DEPTH (g/cm2) Figure 25.16. Sensitivity to step-size parameters and comparison to ETRAN for energy deposition by electrons crossing an aluminium/gold interface (AE = 521 keY, ECUT - 551 keY). Taken from Rogers14. 542 A. E. Nahum

Lockwood et a1 43 have carried out an extensive series of electron-beam Monte Carlo dose-distribution benchmarks against experiment. The geometry was equiva• lent to broad beam, with a highly monoenergetic and monodirectional electron source. The dose was determined by very precise calorimetry in thin layers of material. The Monte Carlo code used was TIGER44, which uses the same electron transport scheme as ETRAN. Two of these comparisons, for I-MeV electrons, are shown here in Figs. 25.I7a and 25.I7b. The second figure also includes the results of a calculation us• ing EGS4 with PRESTA. The agreement with other codes is outstanding in the alu• minium/gold/aluminium case, but less good in the carbon/gold/carbon example. Pos• sible reasons for the discrepancy are not discussed by the authors. The physics behind these distributions must involve electron scattering effects, though this has not yet received as much attention as interfaces irradiated by photon beams.

C Au C CAR BON I GOLD I CAR BON 4.0 o 1. 0 MeV 0°

C CALORIMETER THICKNESS 1.561 X 10- 2 g/cm 2 I--l N- 3.0 Au CALORIMETER THICKNESS E ~ I.399X 10- 2 g/cm 2 0"' ::; H '" !S o EXPERIMENT :z 1.. THEORY ~ !:: 2.0 VI 0 a.. w 0 >- '-' 5 ~ 1.0

FR ACT! ON OF A MEAN RANGE Figure 25.I7a. Comparison of experimental and theoretical energy-deposition profiles in a carbon/gold/carbon configuration for l.O-MeV electrons incident at an angle of 00 (from Lockwood et aI 43 ).

25.6 SUMMARY AND CONCLUSIONS Precise dosimetry requires improvements in existing cavity theories, as these are ade• quate only in rather ideal situations which are seldom fulfilled in practice. In principle, the response of any dosimeter can be simulated by Monte Carlo methods. This review has deliberately concentrated on the technicalities involved in doing such simulations, rather than on the specific results obtained, as is appropriate in a book on Monte Carlo. It has been emphasized that one can never rule out artefacts whenever the artificial electron step-sizes inherent in the "macroscopic" condensed-history electron transport-simulation approach are of the same order as the dimensions of critical dose• scoring regions. In the case of a very important class of dosimeter, the small air-filled ionization chamber used in radiotherapy photon and electron beams, the above prob• lem is inevitable. The example of three different codes correctly predicting one quantity 25. Simulation of Dosimeter Response and Interface Effects 543

AwalI, but producing different results for another, absolute, response in the case of in-air 60CO ,-ray irradiation of carbon-walled ion chambers, has been used to highlight these difficulties. Furthermore, such simulations presently require extremely large amounts of CPU time on the VAX machines on which they have been run. Considerable ingenuity therefore is required in increasing the efficiency of such simulations. But it appears that we are still some way from being able to achieve precisions on the order of 0.1 %, especially for ion chambers at depth in electron beams, a situation where analytical theories for factors correcting for departures from ideal Bragg-Gray conditions are far from satisfactory19.32.

Studies of the dose distributions extremely close to interfaces between widely dis• similar materials do not differ in principle from cavity response studies. Here also, careful attention must be paid to the fine detail of the electron simulation. Examples have been given of differences in the prediction from different codes that are easy to understand (Webb vs EGS/ETRAN), and also examples that are puzzling (Adadurov and Lazurik vs Garth et al).

A few concluding remarks may be in order. The field of dosimeter-response sim• ulation is in its infancy. It is fair to say that very little really new information has yet been furnished by the modest amount of work done so far. However, there is no lack of problems awaiting the attention of Monte Carlo codes. As well as the response of solid-state detectors (e.g., LiF TLD) in photon radiation, the behaviour of such de• tectors (TLD, semiconductor diodes) in electron radiation is improperly understood, as the literature attests17• Bragg-Gray theory, directly applicable only to air-filled ion chambers, certainly was refined and extended in its range of applicability by Spencer and Attix17•19, but very little experimental and theoretical work has been done to verify this much-used theory, especially in electron beams. However, it is likely that current "macroscopic" electron-transport codes will need a substantial influx of new physics in the form of a single-scattering option for electron step-sizes where multiple scattering ceases to be valid, and improved electron-electron cross-sections and restricted stop• ping powers at energies comparable to the binding energies in high-Z materials, if the response of air cavities with walls of very non-air-equivalent materials (e.g., AI, Pb) is to be predicted correctly. This is just one of the challenges that lie ahead. 544 A. E. Nahum

1 MeV ekectrons on Al-Au-Al

o Lockwood et al TIGER 4 EGS4(PRESTA)

..... 2 u c::: AA ...... AI Q) III o lu -0

0~~~~~--~--~--~--~--~--~~~--~ 0.0 0.2 0.4 0.6 0.8 1.0 depth (fmr) Figure 25.I7b. A comparison of Monte Carlo calculated and experimentally measured dose deposition in aluminium/gold/aluminium slab geometry irra• diated normally by a broad beam of 1.0-MeV electrons. The depth scale is in fractions of the mean range for a I-MeV electron in the appropriate element (adapted from Lockwood et al 43 by Rogers(5). 25. Simulation of Dosimeter Response and Interface Effects 545

REFERENCES 1. Y. S. Horowitz, "Photon General Cavity Theory", Radiat. Prot. Dosimetry 9 (1984) 5. 2. P. Andreo, "Stopping-Power Ratios for Dosimetry", Chapter 23, this volume. 3. L. V. Spencer and F. H. Attix, "A theory of Cavity Ionization", Rad. Res. 3 (1955) 239. 4. A. E. Nahum, W. H. Henry and C. K. Ross, "Response of Carbon- and Aluminium• Walled Thimble Chambers in 60CO and 20-MeV Electron Beams", Med. and BioI. Eng. and Computing 23 SuppI. Part I, Proceedings of the VII ICMP, Espoo (1985) 612. 5. J. Dutreix, A. Dutreix and M. Bernard, "Etude de la dose au voisinage de l'interface entre deux milieux de composition atomique differente exposes aux rayonnement, du 60CO", Phys. Med. BioI. 7 (1962) 69. 6. G. AIm Carlsson, "Dosimetry at Interfaces: Theoretical Analysis and Measure• ments by Means of Thermoluminescent LiF", Acta RadioI. SuppI. (1973) 332. 7. 1. de Freitas, Chapter 4 in Inhomogeneity Corrections in Radiation Treatment Planning, Thesis, University of London (1982). 8. A. F. Adadurov and V. T. Lazurik, "Distribution of Absorbed Energy in a Lay• ered Target Irradiated with ,-rays". Translated from A. Energ. 43 (1977) 57. 9. M. J. Berger, "Monte-Carlo Calculations of the Penetration and Diffusion of Fast Charged Particles", in Methods in Computational Physics, Vol. 1, edited by B. Alder, S. Fernbach and M. Rotenberg, (Academic Press, New York, 1963) 135. 10. A. E. Nahum, "Overview of Photon and Electron Monte Carlo", Chapter 1, this volume. 11. A. Ito, "Electron Track Simulation for Microdosimetry", Chapter 16, this volume. 12. W. R. Nelson, H. Hirayama and D. W. O. Rogers, "The EGS4 Code System", Stanford Linear Accelerator Center report SLAC-265 (1985). 13. D. W. O. Rogers, "Fluence to Dose Equivalent Conversion Factors Calculated with EGS3 for Electrons from 100 keY to 20 GeV and Photons from 11 keY to 20 GeV", Health Phys. 46 (1984) 891. 14. D. W. O. Rogers, "Low Energy Electron Transport with EGS", NucI. Instr. Meth. A 227 (1984) 535. 15. A. F. Bielajew. "Electron Step-Size Artefacts and PRESTA" Chapter 5, this volume. 16. A. F. Bielajew and D. W. O. Rogers, "PRESTA: The Parameter Reduced Electron• Step Transport Algorithm for Electron Monte Carlo Transport", NucI. Instr. Meth. B18 (1987) 165. 17. H. Svensson and A. Brahme. "Recent Advances in Electron and Photon Dosime• try" in Radiation Dosimetry - Physical and Biological Aspects, edited by C. G. Or• ton, (Plenum Press, New York, 1986) 87. 18. AAPM (American Association of Physicists in Medicine). "A Protocol for the Determination of Absorbed Dose from High-Energy Photon and Electron Beams", Med. Phys. 10 (1983) 741. 546 A. E. Nahum

19. International Commission on Radiation Units and Measurements, "Radiation Dosimetry: Electron Beams with Energies between I and 50 MeV", ICRU Report 35 (1984). 20. A. F. Bielajew, "Ionisation Cavity Theory: A Formal Derivation of Perturbation Factors for Thick-Walled Ion Chambers in Photon Beams", Phys.in Med. BioI. 31 (1986) 161. 21. R. Nath and R. J. Schulz, "Calculated Response and Wall Correction Factors for Ionization Chambers Exposed to 60CO Gamma Rays", Med. Phys. 8 (1981) 85. 22. A. E. Nahum and M. Kristensen, "Calculated Response and Wall Correction Factors for Ionization Chambers Exposed to 60Co Gamma Rays" , Med. Phys. 9 (1982) 925. 23. A. C. McEwan and V. G. Smyth, "Comments on Calculated Response and Wall Correction Factors for Ionization Chambers Exposed to 60Co Gamma-Rays", Med. Phys. 11 (1984) 216. 24. A. F. Bielajew, D. W. O. Rogers and A. E. Nahum, "The Monte Carlo Simulation of Ion Chamber Response to 60Co - Resolution of Anomalies Associated with Interfaces", Phys. in Med. BioI. 30 (1985) 419. 25. D. W. O. Rogers, A. F. Bielajew, A. E. Nahum, "Ion Chamber Response and Awall Correction Factors in a Co-60 Beam by Monte Carlo Simulation" ,Phys. in Med. BioI. 30 (1985) 429. 26. J. E. Bond, Ravinder Nath and R. I. Schulz, "Monte Carlo Calculation of the Wall Correction Factors for Ionization Chambers and A.q for 60CO 8-Rays", Med. Phys. 5 (1978) 422. 27. V. G. Smyth and A. G. McEwan, "Interface Artefacts in Monte Carlo Calcula• tions", letter in Phys. in Med. and BioI. 31 (1986) 299. 28. V. G. Smyth, "Interface Effects in the Monte Carlo Simulation of Electron Tracks", Med. Phys. 13 (1986) 196. 29. A. F. Bielajew and D. W. O. Rogers, "Interface Artefacts in Monte Carlo Cal• culations", letter in Phys. in Med. and BioI. 31 (1986) 301. 30. A. E. Nahum, W. H. Henry and C. K. Ross, "Response of Carbon and Aluminium• Walled Thimble Chambers in 60Co and 20-MeV Electron Beams", Med. and BioI. Eng. and Computing 23 (1985) SuppI. Part 1, p.612. (Proceedings of the VII ICMP, Espoo, 1985). 31. M. A. Mandour, "The Influence of Cylindrical Air Cavities on Fluence Distribu• tions of High Energy Electrons", NucI. Instr. Meth. A254 (1987) 434. 32. 1. Olofsson and A. E. Nahum, "An Improved Theoretical Evaluation of the Perturbation Factors for Ionisation Chambers in Electron Beams", Med. and BioI. Eng. and Computing 23 (1985) SuppI.Part 1, p.619. (Proceedings of the VII ICMP, Espoo, 1985). 33. A. F. Bielajew and A. E. Nahum, unpublished. 34. S. Webb, "The Absorbed Dose in the Vicinity of an Interface Between Two Media Irradiated by a 60Co Source", Brit. J. of Radiol. 52 (1979) 962. 35. D. W. O. Rogers and A. F. Bielajew, "The Use of EGS for Monte Carlo Calcu• lations in Medical Physics", report PXNR-2692 (National Research Council of Canada) 1984. 25. Simulation of Dosimeter Response and Interface Effects 547

36. E. Kearsley, "A New General Cavity Theory", Phys. in Med. BioI. 29 (1984) 1179. 37. O. T. Ogunleye, F. H. Attix and B. R. Paliwal, "Comparison of Burlin Cavity Theory with LiF TLD Measurements for Cobalt-60 Gamma-Rays", Phys. in Med. BioI. 25 (1980) 203. 38. A. F. Bielajew and D. W. O. Rogers, "A Comparison of Experiment with Monte Carlo Calculations of TLD Response" , presented at AAPM annual meeting, 1986 (unpublished). 39. Y. S. Horowitz, M. Moscovitch, J. M. Mack, H. Hsu and E. Kearsley, "Incor• poration of Monte Carlo Electron Interface Studies into Photon General· Cavity Theory", Nucl. Sci. Eng. 94 (1986) 233. 40. J. C. Garth, E. A. Burke and S. Woolf, "The Role of Scattered Radiation in the Dosimetry of Small Device Structures", IEEE Trans. on Nucl. Sci. NS-27 (1980) 1459. 41. J. A. Wall and E. A. Burke, "Gamma Dose Distributions At or Near the Interface of Different Materials", IEEE Trans. on Nucl. Sci. NS-17 (1970) 305. 42. S. M. Seltzer and M. J. Berger, "Energy Deposition by Electron, Bremsstrahlung, and Co-60 Gamma-Ray Beams in Multi-Layer Media", Int. J. Appi. Radiat. and Isot. 38 (1987) 349. 43. G. J. Lockwood, 1. E. Ruggles, G. H. Miller and J. A. Halbleib, "Calorimetric Measurement of Electron Energy Deposition in Extended Media - Theory vs Experiment", Sandia Laboratories report SAND79-0414 (1980). 44. J. A. Halbleib, "Structure and Operation of the ITS Code System", Chapter 10, this volume. 45. D. W. O. Rogers, Private communication (1987). 26. Dose Calculations for Radiation Treatment Planning

Radhe Mohan

Memorial Sloan-Kettering Cancer Center 1275 York Avenue New York, New York 10021, USA

26.1 INTRODUCTION The most common method of treating cancer patients with radiation is externally ap• plied beams of photons generated by linear accelerators, or by high-intensity 60Co ra• dioactive sources. Electron beams generated by linear accelerators also are employed, but not as often as photon beams. Other particles, such as protons, neutrons, pions and heavy ions, have been experimented with, but their use is limited to a few research facilities. For certain types of tumors, internally applied radiation, alone or in combi• nation with external radiation, is suitable. In this chapter, we will confine ourselves to externally applied photon and electron beams only.

Prior to the delivery of radiation treatment, a team consisting of radiation thera• pists and radiotherapy physicists develops a treatment plan. The objective of treatment planning is to devise treatment techniques and arrangements of radiation beams that will result in regions of high and uniform dose which conform to the shape of the tumor, in order to eradicate the tumor while minimizing damage to healthy organs. One of the most important steps in the radiation treatment planning process is the determination of radiation dose distributions in the body. The probabilities of local control and of complications are sensitive functions of dose delivered. Therefore, accurate knowledge of dose at all critical points within the body is essential. It is not possible, in general, to measure non-invasively radiation dose inside a patient. The dose distributions therefore must be obtained by computer calculations.

Because of the essentially infinite variety of situations encountered in radiation treatment planning with respect to the sizes and shapes of patients, target region and anatomic structures, customized treatment plans for each patient must be designed. The design process must be rapid and interactive to permit development and evaluation of alternative treatment strategies.

Ideally, the most accurate means of dose calculation is by the use of Monte Carlo techniques. However, it is not possible to use Monte Carlo techniques for routine treatment planning. In order to calculate dose distributions in a patient using Monte Carlo techniques, we must obtain an accurate estimate of electron and photon spectra, and their angular distributions at the entrance surface. These data also can be obtained

549 550 R. Mohan by Monte Carlo simulations of the radiation-treatment machine geometry. Also, we must acquire the point-by-point density and elemental composition of the patient from a series of closely spaced CT scans. Assuming that all relevant cross-sections are available, in principle one can follow histories of a large enough number of particles to produce a dose-distribution pattern in a patient. On the order of one hundred million histories may need to be followed to obtain sufficiently low statistical uncertainty. This process may have to be repeated for a number of beams in a treatment plan, and may require several months of CPU time on a computer like the VAX 11/780. Even on the fastest computers available, it is not practical to carry out such calculations for each patient. Thus, in order to perform dose-distribution calculations rapidly and interactively, we must resort to approximate methods.

While Monte Carlo techniques are not practical for routine dose calculations, they are extremely valuable in developing approximate models of dose calculations. Monte Carlo techniques also provide an invaluable tool for assessing the validity of many assumptions and approximations that are made in creating a practical model. Further• more, Monte Carlo techniques provide an excellent method for performing "theoretical experiments" to verify the results of calculations. Often, this is necessary when experi• mental techniques have not advanced sufficiently to yield accurate results, or when the experiments are too difficult to perform. An example of the former may be measurement of dose in a region where electron disequilibrium exists.

In the next section, we will describe generally but briefly some of the conventional methods of dose calculations employed in computerized treatment planning systems in current use, and highlight their limitations. In the succeeding sections, we will describe the use of Monte Carlo techniques to develop more accurate methods of dose calculations.

26.2 CONVENTIONAL METHODS OF DOSE CALCULATIONS

Over the last three decades, a number of approximate dose-computation models have evolved. Essentially they are empirical in nature. The assumptions and approximations made in each of these models vary widely, as does their impact on accuracy.

One basic and easily justifiable assumption made by all models is that the human body is essentially water equivalent. Surface curvature and internal inhomogeneities, such as lungs, bones, etc., are treated as perturbations to a cuboid-shaped body of water, the "water phantom". Radiation dose distributions for rectangular beams for a range of sizes are mapped in a water phantom. Beam-shaping devices (wedges, blocks, compensators, etc.) also affect the dose distributions in a non-trivial manner. For most commonly used beam-shaping devices, such as standard beam blocks or wedges, measured data in a water phantom can be acquired. For arbitrarily shaped custom designed blocks and compensators however, approximate calculation methods must be employed.

Most of the conventional methods of dose calculations fall into two broad categories, namely, the equivalent-pathlength methods and the scatter-integration methods. They are briefly described in the following paragraphs. 26. Dose Calculations for Radiation Treatment Planning 551

26.2.1 Equivalent-Pathlength Methods

The underlying assumption in these methods is that the dose is affected only by the material lying in the path of rays joining the computation point and the source of radiation. The dose at a point in the patient due to a single beam is calculated using the tissue-equivalent radiological pathlength from the surface of the patient to the computation point along the ray joining the radiation source with the computation point. If any custom-made beam shaping devices are present, the dose is reduced using approximate relationships which utilize empirically determined effective density to linearly attenuate the beam. The use of equivalent pathlength to compute dose accounts only approximately for dose perturbations due to the inhomogeneous structure, irregular surface, varying composition of the body and the beam-shaping devices. The variation of dose due to the transport of scattered photons and electrons in the body is neglected.

While equivalent-pathlength models have a number of weaknesses, their great• est strength is their simplicity, which permits rapid computer calculations. Even sys• tems which employ more accurate but significantly slower computation methods use an equivalent-pathlength method as an alternative rapid method, especially when calcu• lating the effect of inhomogeneous internal structure on dose.

26.2.2 Scatter-Integration Models

In these types of models1- s, the dose is separated into primary and scatter components. The primary dose component is assumed to result from the primary photons (originating at the source) at the point of their first interaction within the medium. The remainder of the dose is attributed to scattered photons. The amount of scatter contribution to the dose at a point on the central axis of a beam is a function of the field size. It is assumed that as the size of the field approaches zero, the scatter contribution to dose at the points on the central axis approaches zero. This assumption provides us with the means of obtaining primary dose. One can measure depth dose for a number of square or circular field sizes, and obtain the primary dose by extrapolation. The scatter component of dose at the central axis of a beam is obtained by subtracting zero-area field dose from the finite-area field dose. The scatter component of dose may further be subdivided into differential scatter functions with respect to radius and depth. The differential scatter functions represent contributions to dose on the central axis due to small volume elements located within the radiation field.

The primary component of dose in an inhomogeneous medium is the primary dose at the same point in space, but at the density-weighted equivalent depth in the ho• mogeneous medium. To calculate the scatter contribution of dose, the body is divided into volume elements. Total scatter contribution is calculated by integrating differen• tial scatter functions, weighted by primary photon intensity, over all volume elements surrounding the point of computation. Inhomogeneities in the path of scattered ra• diation are approximately taken into account by using the water or tissue-equivalent pathlengths computed in the manner described in the previous subsection, and then using these pathlengths to obtain differential scatter functions.

A problem common to all existing models of photon dose computations is the as• sumption that the energy deposited at a point is proportional to the kinetic energy 552 R. Mohan released (KERMA) by the photons in collisions at that point. In other words, KERMA is assumed to be equal to dose. This approximation is valid only for low-energy pho• tons. Electrons ejected by high-energy photons may travel several centimeters in unit• density material, and much further in low-density material such as the lung. Therefore, the equivalent-pathlength methods and the scatter-integration methods, which consider transport of photons only, are likely to produce erroneous results in regions where elec• tron equilibrium does not exist. These regions include points near the boundaries of beams defined by blocks and collimators, in and near inhomogeneities and cavities, and at entrance and exit surfaces4- 6 •

26.2.3 Electron Beams While treatment with photons is the dominant mode, there are a number of situations that call for the use of electron beams. A special characteristic of electron beams is that the dose rises steadily up to a certain depth, and then falls off rapidly. This behavior of electron beams provides radiation therapists with a tool to irradiate tumors located near the surface without irradiating sensitive organs behind them.

Many of the assumptions of the equivalent-pathlength method fail for electron beams. Furthermore, since electrons are continuously undergoing scattering, it is not possible to define primary and scatter components for electron beams. Nonetheless, attempts have been made to extend the photon models, with minor modifications, to electron beams.

A substantial improvement in the accuracy of calculated electron beam dose has resulted from the use of the so-called pencil-beam methods7 . In methods of this type, a broad beam of arbitrary shape is divided into narrow beams (or pencils) of electrons incident on the patient's surface. The dose at a point of interest is computed by summing the contributions from individual pencils. It has been demonstrated that pencil-beam methods are well suited for calculating dose for irregularly shaped beams, and for taking into account the surface curvature.

In most of the currently implemented pencil-beam models of electron dose calcula• tions, analytic functional forms of pencil beams, derived with the aid of the Fermi-Eyges theory, are employed. These models are limited in accuracy when calculating dose at larger depths in inhomogeneous media and in the beam boundary regions.

Realizing the limitations of the conventional photon and electron dose calculation methods, attempts have been made to learn more about the energy-deposition phenom• ena using Monte Carlo techniques, and to formulate improved methods of calculations. The new methods are approximate as well. Often, there is a compromise between speed of calculations and accuracy of results. The objective of the continuing investigations, however, is to determine what approximations can be made to enable us to perform dose-distribution calculations within a reasonable time with affordable computers with• out significantly compromising accuracy. In the following sections, some of the recent efforts in this regard are discussed.

26.3 PENCIL-BEAM-CONVOLUTION METHOD OF DOSE CALCULATION

This method8 addresses only the issue of dose calculations for beams with shaping devices such as blocks, compensators, etc. The problem of calculating dose distributions for irregularly shaped and compensated fields is separated from the whole problem of 26. Dose Calculations for Radiation Treatment Planning 553 dose calculations. It is assumed that correction for inhomogeneities and curvature can be performed with the aid of one of the conventional techniques. Pencil-beam (point mono-directional beam) dose distributions are convolved, using Fast Fourier Transforms (FFTs), with the relative primary-fluence distributions of the modified (shaped and compensated) field to yield the modified-field dose distributions for a flat homogeneous phantom. Let the dose at a point in a patient for a specified rectangular field with arbitrary beam modifiers be represented by

D(patient) = Do(phantom) X em x es , (26.1 ) where D(patient) is the dose in the patient, including the effects of beam modifiers, inho• mogeneities and surface curvature. Do(phantom) is the dose at the same point in a flat, homogeneous, tissue-equivalent phantom for the open rectangular field of the same size, and incident normally on the phantom. Do(phantom) is obtained by table look-up and interpolation of measured data. em is the correction factor due to beam modifiers. es is the correction factor for inhomogeneities and surface irregularities which can be calcu• lated using one of the several approximate methods described in the literature, and will not be discussed here any further. For the case of distributions in a flat homogeneous phantom due to a beam with modifiers, Eqn. 26.1 reduces to

Dm(phantom) = Do(phantom) X em , (26.2) where Dm(phantom) is the dose in a flat, homogeneous, unit-density phantom, includ• ing the effects of beam modifiers. The variation of relative fluence due to the beam modifiers, including the effects of changing the energy spectrum across the beam pro• file, the formation of the penumbra at the edges of the irregularly shaped block due to finite size of the source, and the effects of changes in the contribution of scattered pho• tons and electrons, are all incorporated into em. em is approximated by the ratio of the modified- and the open field-dose values calculated from first principles, but under somewhat idealized conditions. Formally,

em = Dm,c(phantom)/ Do,.(phantom) , (26.3) where subscript c denotes calculated dose values in the flat, homogeneous phantom. In the calculation of Dm,. and Do,. values, some simplifying approximations have to be made. For example, it is assumed that the relative primary fluence in a plane through the point of computation and perpendicular to the central ray remains constant within the open beam. Further, it is assumed that the energy spectrum and relative fluence (normalized to the central axis) do not vary with depth. These approximations are justified since em is the ratio of two calculated dose values, and so the resulting errors in em will cancel, to a large extent. In addition, the dose value Do(phantom) of Eqn. 26.1 and Eqn. 26.2 already includes the effects of the variation of primary spectra and relative fluence with depth and lateral distance.

Dm,. and Do,. are calculated at a given depth by convolving the relative primary• fluence distribution with the profile of the pencil-beam distribution at the same depth. The dose for an open, or a modified, field is given by

D.(x, y, d) = JJ (a, b) K(x - a, y - b, d) da db , (26.4) 554 R. Mohan where Dc denotes either Dm,c or Do,c . Quantities x, y, a and b are lateral distances from the central ray. ~ represents the relative fluence distribution for either the open or the modified field. I<, the convolution kernel, is the two-dimensional cross-sectional profile of the pencil-beam dose distribution at depth d.

The two-dimensional integral in Eqn. 26.4 may be evaluated expeditiously by means of Fast Fourier Transforms on an array processor. Equation 26.4 can be written in terms of Fourier Transforms as

F{Dc(x,y,d)} = F{O(x,y)} x F{I«x,y,d)}, (26.5) where F signifies the Fourier transform of the quantity in the following braces. To evaluate the integral, two-dimensional Fourier Transforms of the kernel I< and of the relative fluence distribution are taken. The Fourier Transforms are multiplied term-by• term, and the two-dimensional inverse Fourier transform of the product is taken.

In order to calculate the relative fluence ~, a two-dimensional fluence matrix of the open field and a point source is created. Since, as stated above, the term Do(phantom) of Eqn. 26.2 includes the effects of fluence and spectral variation, etc., and em is only a correction factor, the initial point-source fluence matrix is approximated by the relative fluence of unity at all points inside the open beam, and by collimator transmission outside the collimator boundary. Another point-source fluence matrix (for the numerator term of Eqn. 26.3) is then created in which all the values in the first fluence matrix have been attenuated exponentially according to the pathlength of rays originating from the point source and passing through the compensators and blocks, taking into account the spectral variation across the beam. In order to accomplish this, the energy spectra must be known as a function of radial distance from the central axis, and can be calculated for any linear accelerator by Monte Carlo techniques9• The effect of the finite size of the source on both these matrices also must be included. This effect is especially important for 6OCo machines. It is assumed that the source can approximately be represented by a circular disc. The source distribution is projected through a "pinhole" onto a plane at the isocenter level. The projection is convolved with the point-source relative fluence matrices. The pinhole is positioned at the level of the blocking tray.

The pencil beam kernel, the quantity I< of Eqn. 26.4, incorporates the transport of scattered photons and secondary electrons in the phantom material. It is equal to the cross-sectional profile of a pencil beam at the specified depth. Pencil-beam dose distributions in water or another tissue-equivalent material are obtained with Monte Carlo calculations. Fig. 26.1 shows an example of calculations performed using the EGS Monte Carlo codelO • The incident pencil beam was assumed to have an energy spectrum predicted by previously performed Monte Carlo calculations. Because of axial symmetry, scoring was done in cylindrical geometry, as shown in Fig. 26.1a. The convolution kernels for the problem under consideration are simply the profiles of the pencil beams at selected depths. Depending upon the energy, typically between 5 to S depths are selected. Two-dimensional kernel matrices for each of these depths are obtained from the pencil-beam distributions by interpolation. Fig. 26.1b shows the profile of the pencil-beam kernel at a depth of 5 cm for an IS-MV photon beam of a Therac-20 linear accelerator. 26. Dose Calculations for Radiation Treatment Planning 555

(a) PENCIL BEAM SCORING GEOMETRY

INCIDENT PENCil BEAM DEPTHS I RADII- f--i-i-+--E-~r--T-+-+--l 0 d,

(b) 18 MV PENCIL BEAM PROFILE AT DEPTH OF 5 em

Figure 26.1. (a) The cylindrical geometry used for scoring pencil-beam dose distributions. (b) Two-dimensional representation of a typical pencil-beam kernel. This plot is for a kernel at a depth of 5 cm generated using an IS-MV spectrum with the EGS Monte Carlo program. The grid points are 0.25 mm apart.

A consideration in the evaluation of the convolution integral is that the kernel must remain invariant as a function of the lateral position of the pencil relative to the central axis of the beam. This is not strictly true. For example, the energy spectrum and the angular distributions of photons vary radially9. Thus, the dose distributions of pencils at or near the central axis is not exactly the same as those for pencils off the axis. However, experience indica.tes that pencil-beam dose distributions are relatively insensitive to such spectral variations. It can be assumed that the pencil beam calculated for a spectrum averaged over a median field size, such as a 20-cm X 20-cm field, can be applied to all fields and all regions of a field without any significant error.

Fig. 26.2 summarizes the calculation process. Fig. 26.2a and b represent the point-source fluence matrices for an open beam and the same beam with a central block with a narrow-beam transmission factor of 0.01. Fig. 26.2c and d show results of convolving Fig. 25.2a and b with a finite size source. Figure 26.2e and f are the dose distributions resulting from the convolution of Fig. 26.2c and d with a Monte Carlo generated pencil-beam kernel. The values of the correction factor em are given by the ra.tios of Fig. 26.2f and e. 556 R. Mohan

INTENSITY INTENSITY

~ I I (a) ~ n (b) o n,o LATERAL DISTANCE LATERAL DISTANCE

INTENSITY INTENSITY

(e) (d) ~ I o \ ~ 0o 0 LATERAL DISTANCE LATERAL DISTANCE DOSE

(e) (I) o UULo LATERAL DISTANCE LATERAL DISTANCE Figure 26.2. Calculation of correction factor Cm.. Profile through the central axis of the point-source relative fluence distribution for a 15-cm X 15-cm 60Co (a) open beam, and (b) with a 7.6-cm X 7.6-cm central block of cerrobend. ( c) and (d) are the relative fluence distributions, including the effect of finite size of the source obtained by convolving distributions of (a) and (b), the source distribution matrix. (e) and (f) are dose distributions obtained by convolving finite-source relative fluence distributions of (c) and d with the Monte Carlo generated pencil beam kernel. The correction factor Cm values are given by the ratios of the dose values calculated in this manner for the blocked and compensated field (figure (f)) to the corresponding dose values for the OPen field (figure (e)).

26.4 EXAMPLES

As the first illustration of the application of this model of dose calculation, Fig. 26.3 shows the beam profile data at a depth of 5 cm for an IS-MV, 15-cm X 15-cm beam of a Therac-20 linear accelerator with an SA-cm X SA-cm central block of cerrobend. Fig. 26.3a is the profile of the transverse scan, and Fig. 26.3b is for the diagonal scan. For comparison, dose calculated with the equivalent-pathlength (or linear-attenuation) approach is also plotted. The measured data of Fig. 26.3a have been replotted along with calculations done with the scatter-integration approach.

As another example, Fig. 26.5 shows measured and calculated transverse beam profiles at a depth of 10 cm for the 6-MV 30-cm X 30-cm blocked beam of a Clinac-6 machine. Both the calculated and the measured dose distributions were normalized to the open 30-cm x 30-cmfield at the depth of maximum dose.

These data demonstrate the improvement in the accuracy of calculated dose result• ing from the inclusion of the transport of scattered photons and secondary electrons. 26. Dose Calculations for Radiation Treatment Planning 557

(a) 18 MV TRANSVERSE SCAN, DE PTH = 5 em (b) 18 MV DIAGONAL SCAN. DEPTH .. 5 em _ MEASUREMENTS. CO CONVOLUTION, X X LINEAR ATTENUATION _ MEASUREMENTS. CO CONVOLUTION, X X LINEAR ATTENUATION 1.0

0.8

0.6 w en 0 0 0.4

0.2

0 o 10 -16 o 16 DISTANCE FROM THE CENTRAL AXIS (em) DISTANCE FROM THE CENTRAL AXIS (em) Figure 26.3. Comparison of calculations with measurements (a) along the transverse axis and (b) along the diagonal for a Therac-20 IS-MV I5-cm X I5-cm beam with a S.4-cm X S.4-cm central block cerrobend. Solid line: Measurements., Squares: Calculations with the new method. Crosses: Calcu• lations with the linear-attenuation method.

18 MV TRANSVERSE SCAN, DEPTH = 5 em - MEASUREMENTS, 00 SCATTER INTEGRATION 1.0

0.8

o o 0.6 w en 0 0 0.4

o

0.2

o 0 'boo 0 0 0 0 0 0 0 0 0 0 0 DOd' o 10 DISTANCE FROM THE CENTRAL AXIS (em)

Figure 26.4. Comparison of measurements shown in Fig. 26.3a with calculations done with a scatter-integration program. 558 R. Mohan

1.0 (e) 6 MV TRANSVERSE SCAN. DEPTH . 10 em .." (a) 6 MV TRANSVERSE SCAN . DEPTH ~ .0 em 00 MEASUREMENTS. - CONVOLUTION o MEASUREMENTS. - CONVOLUTION oe 0.8

w 06 0_6 w

0.2 0.2

0 ·20 -.0 0 '0 20 .10 0 10 DISTANCE FROM THE CENTRAL AXIS (em) DISTANCE FROM THE CENTRAL AXIS (em)

Ib)S MV TRANSVERSE SCAN. DEPTH . 10 em 00 MEASUREMENTS. - . CONVOLUTION SCAN POSITIONS 0.8

O.S a-4------f.oa w or> 0 0 0.' b~-----+----~~b

0.2 c c

-." .0 20 DISTANCE FROM THE" CENTRAL AXIS lem) Figure 26.5. Comparison of calculations and measurements of beam profiles for blocked 6-MV beam. Figures (a), (b) and (c) are the plots of the profiles at positions marked on the drawing showing the arrangment of blocks on the tray. Measurements: circles; calculations with FFT method: solid lines.

26.5 "DIFFERENTIAL PENCIL-BEAM" AND "DOSE-SPREAD-ARRAY" MOD• ELS

The model discussed above properly takes into account the transport of electrons set in motion by an arbitrarily shaped beam of photons in a flat homogeneous medium. In order to predict dose more accurately for inhomogeneous internal structure and irregular surfaces, we must go a step further. Two fundamentally similar models have been proposed by Mohan et aZ 5 and by Mackie et aZ 4 • These models approximately include the three-dimensional nature of photon and electron transport and the relevant energy-deposition phenomena, and thereby offer significant improvements in accuracy of computed dose distributions for almost all situations of clinical interest. These models are especially suited for high-energy photon beams. In this section, the Differential Pencil-Beam (DPB) model of Mohan et aZ will be discussed in detail. Differences with the Dose-Spread-Array model of Mackie et aZ will'be described at the end.

The Differential Pencil Beam (DPB) is defined as that fraction of photons in a pencil beam that have their first collisions within an infinitessimal volume element at a specified point on the path of the pencil beam. The DPB may be monoenergetic, or may be composed of photons with an energy spectrum. The dose distribution relative 26. Dose Calculations for Radiation Treatment Planning 559 to the specified point (the first collision) per unit collision density in an infinite homo• geneous medium of unit density is called the DPB dose distribution. Since, in real life, photon collisions at a point cannot be isolated from collisions at points before and after the point, differential pencil-beam dose distributions cannot be measured. They can, however, be obtained by Monte Carlo calculations, and their legitimacy can be verified indirectly by computing broad-beam dose distributions and comparing them with ex• perimental data. For generation of DPB dose distributions, each history is considered to be initiated in an infinite medium by a first collision of an incident photon. The resulting particle cascade is followed until the energies of all secondary electrons and photons have fallen below a cutoff value, at which point the residual particle energies are considered to be absorbed locally. The DPB dose values are scored in a spherical coordinate system whose origin corresponds to the position of the first photon collision. Polar angles are measured with respect to the incoming primary photon direction. The choice of a spherical coordinate system is dictated by the expected symmetries in the dose deposition tables.

26.5.1 Characteristics of Differential Pencil Beams

The following are some observations regarding characteristics of DPB dose distribu• tions. 1. DPB dose distributions calculated as a function of angle and radial distance for several energies ranging from 1.25 MeV (60 Co) to 20 MeV in water are shown in Figs. 26.6a through d as semilogarithmic plots of energy deposition per unit mass in volume elements along radii emanating at various angles from the point of first collision (origin). The geometric factor is removed from the tabulated values by multiplying each by the corresponding r2. Thus, the plotted curves show the net effect of material attenuation and scattering. The plot of dose x r2 shows a rapid fall-off near the origin, with a more gradual decrease at larger distances. The rapid fall-off region corresponds to energy deposition associated with the electrons knocked out in the initial photon collisions. The region increases in size radially with increasing photon energy, extending almost to 10 cm for incident 20-MeV photons. Energy deposition is sharply forward peaked, particularly at high energies. 2. Beyond the rapid fall-off region, the dose is due to contributions in which the transport has closely followed the path of the first scattered photon. This is consistent with the observation that the energies of further scattered photons and electrons, and hence their additional contributions to flight paths, decrease very rapidly. The significance of this observation is that it is the material along the path from the point of first collision to the point of dose computation which is of greatest importance in determining the material effect on dose transport. Conversely, the effect of an inhomogeneity on the DPB will be important only for those directions for which the radii pass through that inhomogeneity. The contribution of higher order scattering to dose is small enough so that the distance dependence of dose x r2 in the DPB tables remains approximately exponential for distances of significance. At the same time, the contribution of multiple scattering to dose is sufficiently large so that the effective attenuation coefficient for dose is less than that for a photon with the energy corresponding to single scattering in the same direction. As is evident from Fig. 26.6a through d, dominance of the first scattered photons over multiply scattered photons in the energy deposition process, and the persistance of the exponential nature 560 R. Mohan

of distance dependence of dose x r2 (for distances greater than the range of electrons), are valid for the entire range of energies of therapeutic interest. 3. The importance of electron transport in DPB distributions depends strongly on the initial photon energy. Electrons ejected by higher energy photons may travel a significant distance. In the application of DPB distributions in calculation of dose, it is assumed that secondary electrons moving away from the collision point travel in a straight line and remain within the particular conical shell in which they were emitted. In fact, multiple scattering of the electrons causes the trajectories to spread laterally. The actual spatial dependence of dose due to the electron component as shown in Fig. 26.6 is due to multiple scattering. Electrons ejected forward scatter sideways so that, at greater angles and small distances, the electron dose x r2 increases with distance as electrons initially ejected at small angles scatter into the larger angular ranges. Thus, while the "straight-ahead" approximation is valid for multiply scattered photons, its va• lidity is questionable for electrons ejected in the first collision. However, it has been found that the inclusion of electron transport, even with the "straight• ahead" approximation, results in considerable improvement in accuracy over the conventional models.

(e) 10 MeV DPS Dose Distribution (a) Co·60 DPS Dose Distribution

Point of first Collision I r 0-: I t I 0° <9< 2.56° 0° <9< 2.56° 25.8° <9< 31.8° 25.8° <9< 31.8° 45.5° <9< 49.5°

10 15 20 0 10 15 20 r (em) r (em) (d) 20 MeV DPS Dose Distribution \ (b) 5 MeV DPS Dose Distribution ~

u ~ 0° <9< 2.56° o c 0° <9< 2.56° 125.8° <9< 31.8° 45.5° <9< 49.5° 45.5° <9< 49.5° O.10~ ':;:::::::;:::::=?:="-'~-=~-~-'-:1~5~~~200 \-- ~~~~~-'-~1J,.0~~-'-""15""""~-'-'-'20 r (em) Figure 26.6. Differential Pencil-Beam Dose Distributions for various energies.

4. The DPB dose is observed to decrease rapidly as the angle and the distance from the point of collision increases. Fig. 26.7a and b are the DPB dose distribu• tions for 60 Co and 5-MeV mono-energetic beams of photons, and demonstrate the forward-peaked nature of dose deposition for high-energy photons. For 5- MeV photons, for example, dose deposited in the forward direction within the first millimeter of the point of first collision is almost 100 times the dose de• posited in the 45-degree direction. A significant fraction of energy, however, is 26. Dose Calculations for Radiation Treatment Planning 561

transported laterally, and contributes to the diffuseness of beam boundaries. For high energies, most of the lateral transport is carried out by secondary electrons. As the energy of the incident photons increases, so does the range of secondary electrons, and hence the diffuseness of beam boundaries.

(a) Co·SO DPB Dose Distribution (b) 5 MeV DPB Dose Distribution

10' point of First Collision I ~r 8-1 I I

r = 0.1 em

·1

Figure 26.7. DPB dose distributions plotted as a function of cos H.

5. Fig. 26.S depicts the DPB dose distributions for 10-MeV photons in a water• equivalent medium of density 0.4 (dashed lines). The dose is plotted against the radial distance scaled by density. The data for unit density water also are plotted for comparison (solid lines) and show, which might be apparent intuitively, that the DPB dose distribution in non-unit-density homogeneous material can be obtained from the DPB dose distribution in unit-density material of the same composition by simply scaling the distances by density. It should be emphasized that the same is not true in general for media of different composition.

10 MeV DPB Dose Distributions --- Water, p = 1.0 gm/cm3 ------Water, p = 0.4 gm/cm3

25.80 <8< 31.80

10 15 20 r • P (gm/cm2) Figure 26.S. This figure demonstrates that the DPB dose distributions in wa• ter of density other than unity can be obtained by simply scaling the distance r from the point of first collision by density. 562 R. Mohan

6. Fig. 26.9 shows the DPB dose X r2 for 60Co and for 20-MeV mono-energetic photons for bone (density 1.85, electron density 1.71) as a function of density X distance from the collision point. Data appropriately scaled for water are also plotted for comparison. Simple scaling of distance dependence of water data by bone density to represent bone data fails because radiation transport is a function of electron density and elemental composition of the medium, and not just of physical density. One might consider scaling by electron density, as is commonly done. For 6OCO, there is agreement between bone and water data scaled by electron density since Compton scattering is the sole dominant interaction for both primary photons and all secondary radiations. However, as shown in Fig. 26.9b, that approximation breaks down for 20-MeV photons, for which pair production is important. It should be noted that the bremsstrahlung spectrum of a typical high-energy photon beam used in radiation therapy has a relatively small high-energy component. The mean energy of the spectrum is between 25 to 30% of the maximum energy. The use of a 20-MeV monoenergetic beam exaggerates the effect of pair production.

(a) Co-60 DPB Dose Distributions -- Bone 1()3 ------Water scaled by Electron Density Ratio (1.71)

1(J2 ~

1~~--~--~--7---~~~~--~--~~ o 2 4 6 8 10 r (cm) in Bone

(b) 20 MeV DPB Dose Distributions -- Bone Water scaled by Electron Density Ratio (1.71) .~"- ~ ~~ ~~ • 1(J2 "'···1 5l ... --~ ~··-i 8 '-.--. L_-1 10

1~~_~ __~~~~_~ __~~~~ __~ o 2 4 6 8 10 r (cm) in Bone Figure 26.9. DPB dose distributions in bone for 60Co and 20-MeV mono• energetic photons plotted as functions of density X r2. Data for water are also plotted for comparison. For 60Co, Compton scattering dominates, and scaling the bone DPB data with electron density of bone will produce a good match with the water DPB data. For 20-MeV photons, pair-production contribution is significant and use of electron density is not satisfactory. An "effective" density of 1.85 gives accurate results. 26. Dose Calculations for Radiation Treatment Planning 563

26.5.2 Dose Computations with Differential Pencil Beams

Dose at a point is calculated by volume integration of Differential Pencil Beams as fol• lows. Coordinate transformation is carried out so that the point of dose computation is at the origin, and the zero degree axis corresponds to the line joining the source of radiation and the point of computation. A spherical grid with the same spacings as the one for the DPB dose table is established around the point of computation. The dose contribution from each spherical volume element to the point of computation is the product of the DPB dose value (obtained from the table) and the number of collisions in the volume element. The collision density in the volume element is given by the product of the photon fluence· of the primary beam in the medium, mass attenuation coefficients of the medium, and the density of the volume element. Relative photon flu• ence and density can be approximated by the values at or near the center of the volume element. The photon fluence at a point in the medium is computed by exponentially attenuating photon fluence at the same point in air. Photon fluence "in air" for the un• collimated beam is obtained from measured beam profiles of the largest possible fields. For polyenergetic beams, differential photon fluence with respect to energy is obtained using the energy spectrum. The energy spectra and angular distributions are generated using Monte Carlo techniques9• The total dose at a point is the summation of contribu• tions from all volume elements. Inhomogeneities in the beam path between the source and a scattering element are incorporated using the total linear attenuation coefficient averaged over the beam path. The mathematical expression used in dose calculations is given by (see Fig. 26.10)

Dp = f f ~Q(E) e-,O,E)xdVQ dE (26.6)

In this expression, ~Q(E) is the differential fluence with respect to energy E at point Q in the absence of the patient, and < J.L(E) > is the linear attenuation coefficient averaged over tissues of various densities and composition along the ray joining the source and the volume element at Q. The next two terms are the density and the mass attenuation coefficient respectively. < P'1f > is the average "effective" density along the line joining P to the volume element Q.

dVa P

Dp =f f ~a(E) x expl-' 8, E) x dVa dE

Figure 26.10. Computation of dose using Differential Pencil Beams. 564 R. Mohan

26.5.3 Examples

Central-axis depth dose. Figs. 26.11a and b show the results of calculation with the DPB model of Tissue Maximum Ratios (TMR) and Tissue Phantom Ratio (TPR) for 10-cm x 10-cm and 30-cm X 30-cm beams of 60CO and 15-MV photons incident normally on a semi-infinite phantom. The photon energy spectrum required for the 15-MV beam dose computa• tions was obtained using the EGS Monte Carlo code. For comparison, measured data are also plotted. The two agree remarkably well in regions beyond buildup. Since the calculations were done assuming no electron contamination in the incident beam, we do not expect good agreement in the buildup regions. In actual practice, the DPB model is used to compute correction factors to be applied to measured data. There• fore, calculations for patients in fact do include the dose deposition due to electron contamination.

(a) Co-50

0.8

a: ~ 0.5

0.4 x x x Measurements (Normalized at Depth of 0.5 cm) 0.2

o 10 Depth (cm) 20 30

(b) 15 Mv Photons

~ 0.6 -- DPB Calculations 10 X 10 cm2 0.4 x x x Measurements (Normalized at Depth of 5 em) 0.2

o 10 Depth (cm) 20 30 Figure 26.11. Measured and calculated Tissue Maximum Ratios (TMR) for 60Co and Tissue Phantom Ratios (TPR) for Clinac-20 15-MV photons.

Curved surfaces. To illustrate the validity of the DPB model for media that have curved irregular sur• faces, let us consider an extreme example of what may be called a "quarter-infinite" medium. Fig. 26.12a shows a comparison of measured data with the DPB calculations for 15-MV photons. Measurements were performed with a small (0.25 cc.) parallel• plate chamber held parallel to the vertical face of the phantom. Up to a lateral distance of approximately 2 cm from the vertical face, electron equilibrium does not exist, and the factors to convert ionization to dose are not known. Fig. 26.12b shows results of 26. Dose Calculations for Radiation Treatment Planning 565 calculations using the DPB model and the Monte Carlo calculation of the beam pro• files in water for a parallel beam of 5-MeV mono-energetic photons incident, in the same way as above, on the "quarter-infinite" medium. In this idealized case, we have eliminated uncertainties associated with measurements as well as with scattering from the beam-defining system, and therefore this case should serve as a good example for ascertaining the accuracy of the DPB model.

(a) __ DPB Calculations 1.0 0.8, x x. Measurements 15MV Photons ~ 0.6 ~ c 1~~~~2Ocm 0.4 A' , Ir .12.5 cm po: 0.2

o 5 10 14 X(cm)

(b) Depth = 2-4 cm

Monte Carlo Calculations --- or· .. · DPB Calculations

5 MeV Photons Depth =22-26cm

0.4 l~~~20cm Air 0.2

o 0.5 XlHalf-width Figure 26.12. Use of DPB model to compute dose distributions in a "quarter infinite" phantom_ In figure (a), 15-MV dose values measured with an ioniza• tio~ chamber are compared with the calculations done with the DPB model. .In figure (b), DPB calculations for a 5-MeV parallel beam are compared with the corresponding Monte Carlo calculations.

Inhomogenei ties. Computation of dose using the DPB model is a two-step process. In the first step, the primary photon fluence in the medium at the scattering volume element is calcu• lated by exponentially attenuating primary fluence in free space. The composition of the intervening tissues is employed for this purpose. There is no approximation involved in the first step_ In the second step, the contribution of each scattering volume element to the point of computation is modified by scaling the pathlength between the two by an "effective" density, as discussed above.

As an example of inhomogeneity correction, Fig. 26.13a shows the central-axis depth dose in a 0.4 density water slab sandwiched between slabs of unit-density wa• ter. Calculations were done with both the Monte Carlo code and the DPB model. A small (3-cm x 3-cm) incident beam was chosen because there are significant electron non-equilibrium effects at the interfaces between media of different densities for fields whose sizes are of the same order of magnitude as the range of electrons. Because of 566 R. Mohan statistical fluctuations in the Monte Carlo calculations, it is difficult to assess the degree of agreement between the two sets of calculations. Fig. 26.13b shows comparison results of depth-dose calculations with the DPB model and ion chamber measurements for an experimental set-up similar to that for Fig. 26.13a, with the exception that 0.4 density water has been replaced with an air gap of equivalent size. It appears that the DPB model slightly over-estimates dose in the region of re-buildup following the air gap. The magnitude of the difference between the measured and calculated values is not clinically significant.

15 MV photons (a) 3x3 1.0

0.8 Dose normalized to w 5 em depth ~ 0.6 c ..n.. EGS 0.4 DPB Conventional Models 0.2 P =1.0 P =0.4 P =1.0

0 5 10 15 20 25 30 15 MV photons (b) 3~ 10 ~ Poly . 5 em Air Dose normalized to • w 5 em depth 20!L3 ~0.6 ...... 1 c x Measurements 0.4 -DPB Conventional Models 0.2 Poly Air Poly 0 5 10 15 20 25 30 DEPTH (em) Figure 26.13. Use of DPB model for inhomogeneity corrections.

26.5.4 Dose-Spread-Array Model

Mackie et al 4 developed the Dose-Spread-Array (DSA) model which is similar to the dif• ferential pencil-beam model, and was developed basically for the same purpose, namely, to take into account the transport of charged particles set in motion by photons in or• der to predict dose more accurately in regions of electron disequilibrium. Their work will not be described here. However, differences between the DSA model and the DPB model will be highlighted. 26. Dose Calculations for Radiation Treatment Planning 567

As in the case of the DPB model, the dose distibutions due to first collisions were calculated using Monte Carlo techniques. Mackie et ai, however, recorded the dose distributions due to primary and scattered photons separately. They utilized a rectilinear coordinate system as opposed to the polar coordinate system for the DPB model. They employed a Monte Carlo code, named MOCA and developed by them, to map the spatial distribution of charged-particle energy from the point of interaction of primary photons. This distribution is called the "Primary Dose-Spread Array" and may be convolved with kinetic energy released at all points within the medium to yield primary dose distribution. Fig. 26.14 shows volume elements of two primary dose• spread arrays for 15-MV x-rays interacting in homogeneous water phantoms, the first for a density of 1 gm/ cc, and the second for 0.2 gm/ cc. The incident beam is assumed to have the same cross-sectional area as the volume elements. The photons are forced to interact in the volume element with bold borders.

4i 4i o 2 o 2 3 ~ 5 e

-.0001 -1 -.0001

.3250 .0110 0 .1120 .0051 .0001

.2340 .02311 .0004 .0fI08 .0101 .0008 .0001

.08117 .01110 .0008 2 .0438 .0104 .0013 .0003 p- 0.2 g/em3

.0178 .00811 .0007 3 .02~8 .0087 .0018 .0004 .0001

.0047 .0027 .0007 ~ .01C3 .0083 .0015 .0004 .0001

1 em} .000fI .0008 .0001 5 .0080 .0043 .0012 .0004 .0002 .0001 t---I 1 em 6 .0048 .0031 .0011 .0004 .0002 .0001 p - 1.0 g/em3 7 .0031 .0023 .0010 .0008 .0002 .0001 4k 8 .0020 .0016 .0008 .0005 .0002 .0001 .0001

II .001~ .0012 .0007 .0005 .0002 .0001 .0001

10 .0010 .000fI .0008 .0004 .0002 .0002 .0001

11 .0007 .0007 .0005 .0003 .0002 .0001 .0001

12 .0005 .0008 .0004 .0002 .0002 .0001 .0001

13 .0003 .0003 .0003 .0002 .0001 .0001

1~ .0002 .0002 .0002 .0002 .0001 .0001

15 .0001 .0001 .0001 .0001 .0001

16 .0001 .0001 .0001 .0001 .0001 I1 em 17 .0001 .0001 I--l 1em Figure 26.14. Primary dose-spread arrays. The numbers represent dose de• posited in the voxel normalized to the collision KERMA in the interaction volume element (bold borders). (Courtsey T. R. Mackie4). 568 R. Mohan

The same Monte Carlo program is used to follow the scattered photons and the charged particles set in motion by the scattered photons. The portion of the first scattered dose that is deposited in the proximity of the site of primary interaction is scored separately, and is placed in a truncated first-scatter (TFS) dose-spread array. The remainder of the first-scatter dose and higher order multiple scatter are deposited in the RFMS dose-spread array. Fig. 26.15 and Fig. 26.16 illustrate examples of TFS and RFMS dose-spread arrays. They are displayed in isodose format, and are normalized in the same manner as the primary dose-spread arrays, namely, to the collision KERMA in the voxel with bold borders (Fig. 26.14). Any first scatter photons escaping the TFS region are accounted for in the RFMS dose-spread array.

Ai 0123456 -2 f""o. .. ~ ~ -1;; ~ ~ r\. " \. 0"'" ~ t\\ '\ "'\ 1\ 1 ,\\1\ :\ \ \ 2 \ \ i\ 3-i 1\ \ 4 1 \ , 5-7 6 - I 7 I 8 ...... , Ak 9 I I 10 / 11 II I 12 / I 13 1 x 10-4 I 14 IJ I 15 I II 16 / ~ 17 / J

Figure 26.15. Truncated first-scatter (TFS) dose-spread array in isodose for• mat. Medium density = 1 gm/cc.

A major reason for separation of dose deposition into three distinct arrays is the differing distance dependence of each component. The primary dose varies most rapidly as a function of distance from the point of interaction, whereas the multiple-scatter dose varies most weakly as a function of distance. Each component may be scored in a matrix of optimum grid size to provide accurate dose calculations by table look-up and interpolation at the maximum possible speed. A second reason for this separation 26. Dose Calculations for Radiation Treatment Planning 569 is that studying dose deposition due to each component separately affords a better understanding of the physical phenomena with regard to the relative importance of each component in different clinical situations. Another reason is that this type of separation can be used as the basis for making the model of dose calculation progressively more accurate as higher speed computers become available.

Dose calculation with dose-spread arrays is carried out basically in the same manner as with differential pencil beams. Inhomogeneities in the medium are also corrected in essentially the same fashion.

Ai 0123456 -2 ...... t'... -1 --0" "- "-" \ o '\ \ "\ \ 2 , \ ,

2 1 X 10-3 3

4 I 5 6 / / I I .,'/ Ak 7 1/ I J 8 J I I I 9 )/ I I I 5 x 10-4 ; 10 1/ 11 If / 12 / / I I 13 -2 X 10-4 I I 14 II I II 15 / / J 16 ... 1 X 10-4 iL / 17 i -, J V 15cm 5 x 10- 5 1--1 5cm Figure 26.16. Residual first- and multiple-scatter (RFMS) dose-spread array in isodose format. Medium density = 1 gm/cc. (Courtsey T. R. Mackie4).

26.5.5 Electron Beams During the past several years, pencil-beam models have been applied with considerable success to calculations of patient dose due to clinical electron beams. As indicated in Section 26.2, in methods of this type a broad beam of arbitrary shape is divided into narrow beams (or pencils) of electrons incident on the patient's surface, and dose at the point of interest is computed by summing the contributions from individual pencils. In the development of many of these models, the Fermi-Eyges theory has been applied to characterize the pencil beams. The Fermi-Eyges approach has an advantage 570 R. Mohan in that it yields a relatively simple, analytic model, and therefore qualitative estimates of physical parameters can easily be derived. However, the only interaction which is treated explicitly is Coulomb scattering from nuclei, and further, only small angle scattering is taken into account. The consequences of these approximations are that the calculated dose at larger depths and near the beam boundaries usually is not sufficiently accurate.

An alternative to the use of the Fermi-Eyges approach to characterizing pencil beams is the use of the Monte Carlo technique to generate pencil-beam dose distribu• tions for monoenergetic electrons normally incident on a semi-infinite slab of waterll. Quantities which may be used to characterize the incident broad beam are the planar energy fiuence, energy spectrum, angular spread, and divergence angle of electrons at every point on the patient surface. These quantities may be determined experimentally. Taking into account these characteristics, the pencil beams can be integrated over the cross-section of an arbitrarily shaped field to yield an approximate broad-beam dose distribution in homogeneous tissue.

As compared to photon dose calculations, the process of correcting for the presence of inhomogeneities beneath the surface is more complicated due to the fact that, by the time the electron pencil beam has penetrated to a certain depth, it no longer is monodirectional as it was assumed to be on the surface. Furthermore, energy spectra at depth are vastly different than at the surface. The effect of varying density and elemental composition on such beams is difficult to compute. One can, however, approximately correct for inhomogeneous structure by scaling the depth and lateral distance from the pencil linearly according to the electron densities of the intervening tissues.

26.6 SUMMARY In this chapter, the potential of using Monte Carlo techniques for developing methods for calculating radiation dose distribution required for planning of radiation treatments of cancer patients has been discussed. It is not practical to use Monte Carlo techniques directly for routine treatment planning. In order to perform dose distribution calcula• tions rapidly and interactively, we must resort to approximate methods. Conventional models are inadequate for many clinical situations. Monte Carlo techniques may be employed in developing more accurate models. Monte Carlo techniques also provide an invaluable tool for assessing the validity of many assumptions and approximations that are made in creating practical models. With the increasing speed of computers and the availability of array processors and other specialized hardware, these methods should become practical in the not-too-distant future. 26. Dose Calculations for Radiation Treatment Planning 571

REFERENCES 1. J. R. Cunningham, "Scatter-air Ratios", Phys. Med. BioI., 11 15 (1976). 2. J. Cunningham, P. Shrivastava, J. Wilkinson, "Program IRREG - Calculation of Dose from Irregularly Shaped Radiation Beams" , Comput. Programs in Biomed. 2 192 (1972). 3. K. B. Larson and S. C. Prasad, "Absorbed-dose Computations for Inhomoge• neous Media in Radiation-Treatment Planning Using Differential Scatter-air Ra• tios", IEEE Proceedings of the Second Annual Symposium on Computer Appli• cation in Medical Care (1978) 93. 4. T. R. Mackie, J. W. Scrimger, J. J. Battista, "A Convolution Method of Calcu• lating Dose for 15-MV X-rays", Med. Phys. 12 (1985) 188. 5. R. Mohan, C. Chui, L. Lidofsky, "Differential Pencil Beam Dose Computation Model for Photons", Med. Phys. 13 (1986) 64. 6. R. Mohan and C. Chui, "Validity of the Concept of Separating Primary and Scatter Dose", Med. Phys. 12 (1985) 726. 7. K. R. Hogstrom, M. M. Mills and P. R. Almond, "Electron beam dose Calcula• tions", Phys. Med. BioI. 26 (1981) 445 . 8. R. Mohan and C. Chui, "Use of Fast Fourier Transforms in Calculating Dose Dis• tributions for Irregularly Shaped Fields for Three-dimensional Treatment Plan• ning", Med. Phys. 14 (1987) 70. 9. R. Mohan, C. Chui, L. Lidofsky, " Energy and Angular Distributions of Photons from Medical Linear Accelerators", Med. Phys. 12 (1985) 592. 10. R. L. Ford and W. R. Nelson, "The EGS Code. Computer Programs for the Monte Carlo Simulation of Electromagnetic Cascade Showers", SLAC-210 (1978) 11. R. Mohan, R. Riley and J. S. Laughlin, "Electron Beam Treatment Planning: A Review of Dose Computation Methods" , in Computer Tomography in Radiation Therapy, edited by C. C. Ling, C. C. Rogers and R. J. Morton, (Raven Press, New York, 1983) 229. 27. Three-Dimensional Dose Calculation for Total Body Irradiation

Akira Ito

Cyclotron Laboratory The Institute of Medical Science The University of Tokyo 4-6-1 Shirokanedai, Minato-ku, Tokyo, 108 Japan

27.1 INTRODUCTION

Bone Marrow Transplant (BMT) therapy has been a big success in the treatment of leukemia and other haematopoietic diseasesl . Prior to BMT, total body irradiation (TBI) is given to the patient for the purpose of (1) killing leukemia cells in bone marrow, as well as in the whole body, and (2) producing immuno-suppressive status in the patient so that the donor's marrow cells will be transplanted without rejection. TBI employs a very large field photon beam to irradiate the whole body of the patient. A uniform dose distribution over the entire body is the treatment goal. To prevent the occurrence of a serious side effect (interstitial pneumonia), the lung dose should not exceed a certain level. This novel technique poses various new radiological physics problems. The accurate assessment of dose and dose distribution in the patient is essential. Physical and dosimetric problems associated with TBI are reviewed elsewhere2,s.

Conventional dose calculation methods for TBI, however, do not adequately ac• count for the complicated three-dimensional structure of a patient, inhomogeneity cor• rections, and especially the lack of secondary electron equilibrium in and around the lung, as reviewed by Cunningham4 • Mohan's chapter on Radiation Treatment Planning (see Chapter 26) also describes this subject well. The discussions of those aspects will not repeated here.

Monte Carlo calculations have the capability of overcoming the inadequacies of con• ventional dose calculation methods. The Monte Carlo calculation of three-dimensional dose distributions for external photon beams was pioneered by Webb5,6, and has been studied in recent years by Ito7, Kijewski et als, Mohan et al g , Han et aIle, and Williamsonll. Since the EGS code system was refined for medical physics applications by Nelson, Hirayama and Rogers12, the Monte Carlo method is becoming popular (see Chapter 1).

In the case of 6OCo gamma rays, the physical processes are relatively simple: the major photon interaction is Compton scattering, and the range of the secondary electron

573 574 A. Ito is short (around 2 mm in water). Also, three-dimensional patient-density data, as the basis for the three-dimensional dose calculation, can be obtained accurately from multi• slice x-ray CT scans.

VAX computers for running Monte Carlo programs are now available to medical physicists. However, the Monte Carlo method in its most analog manner demands an enormous amount of computing time in order to obtain enough accuracy (2 to 3%) for three-dimensional dose-distribution calculations. Therefore, it is generally believed that the Monte Carlo method is impractical for routine treatment planning dose calculations in the clinical field (see Chapter 1, and Wong et aPS). Nevertheless, the Monte Carlo method is becoming feasible. The introduction of careful variance-reduction techniques, as opposed to a full analog Monte Carlo method, increases the efficiency of computation. Also, the continuing increase in computer power (mini-Super computers and parallel processing computer systems) holds much promise.

Consequently, this chapter demonstrates the feasibility of the Monte Carlo method in clinical three-dimensional dose-distribution calculations for TBI patients.

27.2 PHOTON-TRANSPORT MONTE CARLO MODEL

The photon-transport problem relevant to medical physics can be solved very well by the Monte Carlo method. The very first model considered is the pencil-beam geometry in a water phantom, as depicted in Fig. 27.1.

Sea tlered Don ~ .• ~1n~c~id~e~nl~~~7-____~ ___. -'__ ~ __~ ____ Pdury IDoee I

SI de Sea Iter Figure 27.1. Schematic illustration of the photon pencil-beam method. Incident photons enter the cylindrical water phantom. Photons are classified as pene• tration, absorption and escape. Primary and scattered doses can be treated separately.

The incident photon pencil beam. impinges on the center of a coaxial water phan• tom of radius R (cm), and thickness Z (cm). Some of the incident photons penetrate the phantom without interacting. In the energy region between 30 keV and 20 MeV, Compton scatter dominates the physical processes. The Compton electron from the first collision is the source of the primary dose. Scattered photons of various energies and angular directions generate the scatter dose. Scattered photons either escape from the phantom, or are absorbed by photoelectric interaction. The escaping photons may be classified into forward scatter, side scatter, or backward scatter. 27. Three-Dimensional Dose Calculation for Total Body Irradiation 575

The photon-interaction cross-section data for water in this energy region is known to an absolute accuracy of 1-2% 14. Fig. 27.2 is a plot of the photon cross-section data compiled by Hubbell. The mass attenuation coefficient (p,/ p, in units of cm2/g) is the macroscopic expression of the total cross section (customarily in units of barns/atom). The reciprocal of the mass attenuation coefficient is the mean free path (in g/cm2 ), i.e., the average distance between interactions. For the purposes of Monte carlo calculations, it is convenient to prepare a fractional photon cross-section table as shown in Fig. 27.3. In a homogeneous water phantom, the interaction distance can be generated by an exponential random number whose mean value is equal to the mean free path shown in Fig. 27.2. Also, the type of interaction can be determined by comparing a uniform random number (between 0 and 1), with the fractional cross-section table in Fig. 27.3.

Il85/r---:----r----,-_---1h-___-:r ___ --.- ___-,-..;(lg1
ATlON COB FICIENTS J HATER

a.--... ]'18''t--\---t-\:+----;,---+~---4--_;_-I~~:..'T~'''2.· '!....--+__fIg;;'' co Cn pto" ~ ... Pt'loID - .lec.l r c ...... Polr-pr-odUi t ~ .2 .c ~( 18~,r_--.Il--t-+__,~_t_-_'c-__t---=::a.,o:_1f-----I--rIg-,3J~ u ., o" .....'"

., o ~

Figure 27.2. Mass attenuation coefficients in water compiled by Hubbell14. The ordinate scale is changed for photoelectric effect below 10 keY. The re• ciprocal of mass attenuation coefficient is the mean free path (g/cm2 ), which is used to generate the random interaction distance.

The angular distribution of the Compton-scattered photon is expressed by the Klein-Nishina equation. For Monte Carlo applications, generation of the random Comp• ton angle is customarily computed by Kahn's sampling algorithm. Alternatively, it is useful to prepare the Compton angular-distribution data for all photon energies, as shown in Fig. 27.4. The integral of the differential Compton cross section over the solid angle (-180 to +180°) is computed, and its normalized integral values (0-1) are stored on the computer memory for each photon energy. Then the Compton angle can be sam• pled by uniform random numbers. By interpolating both photon energy and sampled angle, the random Compton angle can be generated very efficiently. This technique gives fast sampling, though it requires large two-dimensional data arrays. It is suitable for computers with large main memories. 576 A. Ito

100

90

- 60 !::5 ,pO -;:;- ~60.. e50 u '040

0" :: 30 u ..0 lJ.. 20

10

~ O Z 10 3 lOS 106 10' Phot On Ene~9Y (eV)

Figure 27.3. Fractional photon cross section in water. Incoherent (Compton) scattering is dominant between 30 keV and 20 MeV. This table is used directly in the Monte Carlo program.

100' 90 ' 80'

10'

60 SO 40 3D 20 10 0 10 20 30 40 SO 60 70 OJ fferen I J 0 1 CI""OSS Sec t \ on (.b/ SI"" / el Figure 27.4. Angular distribution of the scattered photons from the Compton effect as calculated from the Klein-Nishina equation. An incident photon of a given energy interacts with a free electron at the center of this graph. The intensity of the scattered photon at a given angle is plotted. 27. Three-Dimensional Dose Calculation for Total Body Irradiation 577

Now let us consider the more difficult part: transport of the secondary electron. If we restrict ourselves to the transport of 6OCo gamma rays in water, the story becomes fairly simple. The maximum energy of the Compton electron is about 1.2 MeV, which is a range of only 0.6 g/cm2• Bremsstrahlung can be ignored. Secondary electrons can be handled very precisely on one hand, or they can be treated very simply on the other. Fig. 27.5 illustrates an example of the track structure of a Compton electron from a 6OCo gamma ray, as calculated by the ETRACK program15. This two-dimensionally projected electron track structure with many delta rays is computed in an interaction• by-interaction manner. Every ionization event is plotted with a dot. (Excitation events are not plotted.) This is a very precise picture of a secondary electron. But do we need such detailed information for macroscopic dose calculations? The answer may be 'No'. The next accurate method is to use an electron pencil beam. The radial and depth-dose distribution along an electron track of a given energy d(r, z, E) is calculated from many histories of the electron track structures shown in Fig. 27.5. Its distribution looks more or less like a water drop in shape. By placing this electron dose distribution along the electron path, one can get the secondary electron dose with good accuracy.

MMl: C060. ORT Z-i PLANE (,uml 0.0 100.0 200.0 300.0 ijOO.O 500.0 600.0 700.0 800.0 900.0 1000.

Figure 27.5. Two-dimensional plot of a secondary (Compton) electron track in water from a 6OCo gamma ray, calculated with the ETRACK program. The electron started at the left and ended at the right. A dot represents an individual ionization event. Many delta rays were created along the electron track.

At the other extreme, it is possible to ignore completely the transport of the sec• ondaryelectrons. One can compute the KERMA (kinetic energy released in matter), instead of the electron dose, when secondary electron equilibrium is satisfied. This is true for a homogeneous water phantom. However, this oversimplification is invalid for dose calculations in the air-tissue or bone-tissue interface region, as well as in an inhomogeneous region such as the lung. The next simplification of secondary electron transport is the adoption of a "range-energy" curve as shown in Fig. 27.6. This plot is derived from the csda (continuous-slowing-down approximation) model calculated by Berger and Seltzer16• Any electron is assumed to travel in a straight path, and to stop at this distance. It is reasonable to use this oversimplification of the calculation of dose 578 A. Ito due to secondary electrons from 6O,CO gamma rays with a spatial resolution of about 0.5 cm, the size of a volume element (voxel) for three-dimensional calculations. Most secondary electrons deposit their energy within the voxel where the electron was set in motion or, in the neighboring voxel when density is unity. In a lung, however, where the density varies from 0.1 to 0.4 g/cm3 , electrons travel through several voxels, depositing part of their kinetic energy along the path.

Electron Rcnge in Wcter Berger and Seltzer (1962) 0.6 N E o "- 0>

w 0.4 (.!) z <[ a:: 0.2

O~~~~~~~-L~~~~~~ o 0.5 1.0 1.5 ELECTRON ENERGY (MeV) Figure 27.6. Electron range in water from Berger and Seltzerl6. The absorbed dose of the secondary electrons from 60CO gamma rays can be calculated with this "range-energy curve".

27.3 60CO GAMMA-RAY PENCIL-BEAM CALCULATION

A photon pencil beam impinging on a water phantom generates a first collision (or primary) dose along the beam axis, as well as a higher collision (or multiple-scattered) dose around the axis. This can clearly be visualized in Fig. 27.7a. A Monte Carlo program (PTRACK) generated the picture, for 100 60Co gamma rays entering a 30 cm thick water phantom. Dotted lines are the scattered photon tracks. The short-range secondary electrons are observed around the beam axis.

Fig. 27.7b illustrates both primary and scattered electron tracks from 10,000 in• cident 6OCO gamma rays. In this picture, photon tracks are excluded. The primary dose is sharply distributed along the beam axis within a cylinder of radius of 1-2 mm. Electrons escaping at the exit of the phantom also can be observed.

By contrast, the scattered dose is distributed widely over the water phantom. If we look carefully into the density of electron tracks (i.e., scattered dose) in the picture, we find that the scattered dose is higher within a cylindrical region of several cm in radius, and at a depth between about 5 and 20 cm. Also, the electron tracks are longer than those at the outer regions; this means that first-scattered photons, generating higher 27. Three-Dimensional Dose Calculation for Total Body Irradiation 579 energy Compton electrons, are dominant. In the outer region beyond several cm in radius, the electron tracks are more sparse, and their density decreases gradually; the multiply scattered photons, generating lower energy Compton electrons, are dominant.

J • / PHOTDN TRRCK IN WATER PHANTOM ~. , \

TrockI!

Figure 27.7a. Plot of a 6OCO gamma-ray pencil beam in a water phantom. Dotted lines are the tracks of the scattered photons.

Trocklll

Figure 27.7b. Plot of the tracks of secondary electrons generated by a 6OCo gamma-ray pencil beam. The dense line along the central axis represents the primary dose. 580 A. Ito

Radial dose distributions for different generations of scattered photons are shown in Fig. 27.8. In this plot, the radial dose distributions d(r, z), in units of Gy/photon, at a depth of z = 4 to 5 cm, are shown. The total scatter dose has a solid line (denoted as T); the first scatter is a long dashed line (denoted as 1), the second scatter is a short dashed line (denoted as 2), and the third (plus higher) scatters are dotted lines (denoted as 3+). The radial profile in the total scattered dose seems to have two slopes; the first component decreases steeply to about 5 cm, and then the second component decreases more gradually beyond several cm in radius. The first-scatter dose distribution d1(r, z) could be calculated analytically by integrating the fractional dose contribution along the central axis computed from the Klein-Nishina equation of the differential Compton cross section, considering the attenuation of the secondary photon. However, the dose from a higher order scattering cannot easily be determined analytically.

The radial dose distribution around a pencil beam is the basic information needed to derive depth and lateral dose distributions for various beam sizes in a homogeneous water phantom. Mohan et a1 9 also have calculated the pencil-beam dose distribution using the EGS code. Further, they have proposed the differential pencil-beam (DPB) method for calculating three-dimensional dose distributions in a generalized manner (see Chapter 26). However, this method is accurate only in a homogenous phantom where the basic distributions were derived.

Radial Distribution of Scatter Dose Co~ 60 7' ray Pencil beam at 0epth-4-5cm 10- 13

c 0 "0 ..s::: Cl. "- >- 10- 14 ~

w (f) 0 0 10- 15

5 10 15 LATERAL DISTANCE (em) Figure 27.8. Radial distribution of scattered dose from a 60Co gamma-ray pencil beam in a water phantom. The first, second, third plus higher scatter components are plotted separately.

Now let us consider what happens inside a cylindrical water phantom, with a radius of 15 cm and a thickness of 30 cm, where the 60Co gamma-ray pencil beam has impinged. Some of the incident photons penetrate the water phantom without any interaction, depending on thickness. The first collision takes place along the central 27. Three-Dimensional Dose Calculation for Total Body Irradiation 581

axis giving the primary dose. Some or all of the kinetic energy is given to the secondary electron which is eventually absorbed in water. A Compton photon is predominantly created, and scatters inside the phantom. Some scattered photons undergo multiple scattering before being absorbed due to the photoelectric effect, or escape out of the phantom.

Table 27.1 summarizes the overall balance of photon interactions and their energy deposition. It is worth noting that 94% of the incident photons eventually escape from the phantom, only 6% being completely terminated by the photoelectric effect. However,62% of the initial kinetic energy is transferred to the water phantom through the Compton effect; 42% of the energy is given to the primary dose, and 20% to the scattered dose, of which only 0.3% is due to the photoelectric effect.

Table 27.1. Summary of interactions of a 6OCo gamma-ray pencil beam in a cylindrical water phantom (thickness 30 cm, radius 15 cm).

Number Energy/ Average Interaction (%) Dose (%) Energy (keV) Incident 100 100 1254 Escape 94 38 (photon) *Penetration 15 16 1253 *Forward 21 12 693 *Side 45 8 292 * Backward 12 2 187 Absorption 6.0 62 (electron) * Primary (85) 42 589 a. Photoelectric 0 0 - b. Compton (85) 42 589 c. Pair Production 0.03 0.01 288 *Scatters (248) 20 98 a. Photoelectric 6 0.3 60 b. Compton (242) 20 99 c. Pair Production 0.00 0.00 212

Note: Numbers include round-off errors.

The numbers listed in Table 27.1 are subject to change when the thickness and radius of the water phantom are different from the standard model (30-cm thick with 15-cm radius). A semi-infinite water slab is used as the preferred calculational model, especially for the purpose of radiation shielding. Also, a cubic water phantom with 30-cm sides is the preferred choice for a dosimetry standard in medical physics. This chapter covers the TBI of an actual patient whose body thickness may change from 10 to 30 cm, and where the radiation field is generally very large - one that fully covers the patient. Therefore, we must consider the scatter effects caused by the phantom thickness and radius. Considerations of the actual patient will be discussed in the following section. 582 A. Ito

Fig. 27.9 shows the energy balance of a 6OCo gamma-ray pencil beam in a cylindri• cal water phantom as a function of the phantom thickness. The incident photon energy fluence is normalized to 100%. As the thickness increases, photons which penetrate without undergoing collisions decrease. However, energy absorption by both primary and scatter dose increases. Also, the ratio of scatter-to-primary dose increases from 0.27 at 10 cm to 0.51 at 30 cm.

188.------,------,------,------r------~ Radius of Cylindrical PhantoD = 15 (co) 98 F r a 88 c Penetration (No Coil ision) t i 78 0 n 68 0 f 58 (scatter) E ~----I n e 48 r « 38 y 28 Absorbed (Pri.ary)

18 28 38 58 Thickness of Cylindrical Phantol (ca) Figure 27.9. Energy balance of a 60CO gamma-ray pencil beam in a cylindrical water phantom as a function of thickness. The incident photon energy fluence is divided into penetration, absorption (primary and scattered), and escape to various directions. As the phantom thickness increases, scattered dose fraction increases.

Fig. 27.10 illustrates the 60Co gamma-ray pencil-beam energy balance for a cylin• drical water phantom 30-cm thick as a function of the phantom radius. In this example, the fraction of penetrating photons and the primary doses are constant. 'When the phan• tom radius increases, both the scattered dose and the photons escaping in the forward direction increase. The ratio of the scattered~to-primary dose increases from 0.43 at 10- cm radius to 0.63 at 30-cm radius. Beyond a 30-cm radius, the scatter dose is almost saturated because very few scattered photons reach such large distances.

The energy deposition balance as described above must always be kept in mind when geometrically sophisticated problems are solved by the Monte Carlo method.

27.4 CALCULATION OF TISSUE AIR RATIO (TAR) FOR 6OCo GAMMA RAYS

Once a pencil-beam dose distribution for a given phantom size is calculated, it is easy to calculate the Tissue Air Ratio (TAR) at depth z (em) and field area A (cm2). The algorithm for calculating TAR(z,A) from a pencil-beam dose distribution d(z,r) is depicted in Fig. 27.11. TAR(z, A) is divided into two parts: the primary dose Dp(z), which is independent of field size, and the Scatter Air Ratio SAR(z, A). SAR(z, A) is calculated by integrating the pencil-beam dose distribution d(z, r) over the field area A. 27. Three-Dimensional Dose Calculation for Total Body Irradiation 583

188 Phanto. T~ickness = 38 c. 98 - Pene tra t i on (No Coil ision) - F r a 88 - c Escape ( Forward Sca tter) t i 78 - Escape (Side Scatt~ 0 ------Escape (Backward Scatter _ n 68 0 f 58 Absorption (Sca tter) - E n 48 ~ e r « 38 - - y 28 - Absorption (Pri.ary) - n

~ 18 - -

I I I I 18 28 38 58 Radius of Cylindrical Phanto. (c.)

Figure 27.10. % Energy balance of a 60CO gamma-ray pencil beam in a cylin• drical water phantom as a function of radius. The incident photon energy fluence is divided into penetration, absorption (primary and scattered) and escape in various directions. The scattered dose fraction tends to saturate beyond 30 cm in radius.

R

TAR(z,A) = Dp(z) + SAR(z,A) Dp(z); Pri.ary Dose SAR(z,A) Scattered Dose R SAR(z,A) f d(z,r)dA r=9 d(z,r) Dose Distribution Depth Z Field Size A of Penci I Bea.

Figure 27.11. A model for calculating the TAR from a pencil-beam dose distri• bution. The central-axis dose is divided into primary and scattered dose. The scattered dose is calculated by integrating the pencil-beam dose distribution over the irradiated field size.

Some results of the TAR calculation and comparison with measured measured data are presented in Fig. 27.12 which shows the TAR depth-dose curves for a 10-cm X 10-cm field. The solid lines are from calculations showing scattered, primary and total dose distributions separately. The dashed line is the data evaluated in BJR S1717. 584 A. Ito

Both curves agree well within one percent error at depths greater than 7 cm. However, some discrepancy exists at the surface region. Measured data give about a 2% higher dose than calculated. Han et al lO recently calculated the actual photon spectrum of a 60Co treatment unit, and found a considerable number of scattered photons from the 60Co source and field defining devices. Thus, the higher dose in the measured data at the surface region can be attributed to the incident scattered dose. The Monte Carlo calculation assumes only 1.17-MeV and 1.33-MeV photons, and does not include such scattered photons.

188~~~---.------r------.------'------.------~

98 TAR Calculated ITO 1986 88

Prillary

38

28 Sea tter 18

18 28 39 Depth in Cylindrical Phanton (ca) Figure 27.12. Comparison of 60Co TAR depth-dose curve for a 10-cm x 10- cm field between the Monte Carlo calculation and the measured TAR table given in BJR 81717. Both agree well at depths deeper than 7 cm. The small discrepancies at the surface region are attributed to the scattered component from the 60Co treatment unit.

Fig. 27.13 shows the 60Co TAR at a 5-cm depth as a function of the rectangular field length. The TAR from the Monte Carlo calculation is shown as solid lines. Contri• butions of primary, first scatter and higher scatters are also illustrated separately. The dashed line is the evaluated TAR value in BJR 817. BJR data show about 2% higher results than Monte Carlo calculation. This difference can also be attributed to scattered photons from the 60Co unit. Table 27.2 tabulates the differences in 60Co TAR values (in %) between the BJR table and the Monte Carlo calculated table for all field sizes ranging from zero to 75 cm x 75 cm and for depths from 0.5 to 30 cm. It is remarkable that the calculated TAR agrees to better than 1% for such a wide range of both field size and depth. As was discussed, the TAR table of the BJR has a consistently higher dose (about 2%) near the surface.

These results indicate that the Monte Carlo pencil-beam dose distribution is very accurate, as the computed TAR values agree very well with the BJR table of measured TAR values, at least in the case of a homogeneous water phantom. Thus, we can have confidence in the Monte Carlo calculation, as well as in the basic physical data behind the calculation. 27. Three-Dimensional Dose Calculation for Total Body Irradiation 585

98 Higher Scatter

88 Firs t Sea tter

Priury

Depth at 5 CI 38 ------TAR Measured BJR S17 1983 --- TAR Calculated ITO 1986 28

18

18 28 38 48 58 68 78 Len~th of rectan~ular field (CI) Figure 27.13. Comparison of 60CO TAR values at a depth of 5 cm as a function of field size between the Monte Carlo calculation and the measured TAR table given in BJR 81717. Fractional dose from primary, first scatter, and higher• order scatters are also given. Measured TAR gives a consistently higher value by a few percent which is attributed to the scattered component from the saCo treatment unit.

27.5 CALCULATION OF THREE-DIMENSIONAL DOSE DISTRIBUTIONS IN PATIENTS

Invention of x-ray computerized tomography (CT) has had a great impact, not only on diagnostic radiology, but also on radiation-therapy procedures. Delineation of patient, organs or tumor, as well as the availability of density information for accurate dose calculation, are the major consequences of CT scans for radiation therapy treatment planning. In order to calculate the three-dimensional dose distribution in a patient, multi-slice CT data is indispensable. Conventionally, CT data are stored on magnetic media (tape or disk), and thus are transferable to a computer for running a Monte Carlo program. However, differences in the data format and data-compression method among different CT machines sometimes preclude the free exchange of CT data. To overcome these difficulties, a standardized data format, known as the "Uppsala format" for radiation therapy procedure, has been proposed18 • In the Uppsala format, CT data are stored as a 256 x 256 matrix. For a three-dimensional dose calculation, the matrix size has been reduced to 64 slices, with a 64 x 64 matrix. The elementary volume (voxel) size is about 0.5 x 0.5 cm, with a slice thickness of 0.5 cm or 1.0 em. The CT numbers are averaged over each voxel and converted into density relative to water (p = 1.0). The CT number is linearly related to the electron density in soft tissue as well as in bone19. A patient is regarded as a collection of three-dimensional voxels with the electron-density information in each voxel.

Fig. 27.14 illustrates the three-dimensional coordinate system of the patient and CT slices for dose calculation. This notation will be used throughout the chapter. 586 A. Ito

Table 27.2. Percentage differences in 6OCo TAR tables between measured values (BJR S17) and those calculated by the Monte Carlo method for field sizes between zero and 75 cm x 75 cm, and depths between 0.5 and 30 cm. In most cases, both agree very well (within 1-2%). In the surface region below 5 cm, the measured TAR values are 2-3% higher than the calculated ones due to scattering from the 6OCo gamma-ray source and beam defining devices.

d:A 8 4 5 6 7 8 9 18 12 15 28 25 38 35 48 58 68 75 8.5 2.8 8.9 8.9 1.8 1.8 8.9 1.1 1.1 1.2 1.4 1.5 1.4 1.2 1.1 1.8 8.8 8.7 8.9 1.8 3.5 1.4 1.6 1.9 2.1 1.9 2.1 2.2 2.3 2.5 2.4 2.3 2.2 2.8 2.1 2.2 2.4 2.9 2.8 2.6 8.7 1.1 1.5 1.7 1.5 1.6 1.8 2.8 2.2 2.1 2.8 1.9 1.7 1.6 1.6 1.8 2.1 3.8 2.6 8.8 1.1 1.5 1.7 1.3 1.6 1.8 2.1 2.3 2.5 2.1 2.1 1.8 1.6 1.7 2.8 2.4 4.8 2.1 8.8 1.8 1.31.51.1 1.3 1.5 1.9 2.2 2.2 1.9 1.9 1.7 1.7 1.7 2.1 2.5 5.8 2.8 8.8 1.8 1.2 1.4 8.9 1.1 1.31.6 2.8 2.8 1.8 1.7 1.4 1.3 1.6 1.9 2.4 6.8 1.2 8.1 8.3 8.5 8.7 8.8 8.3 8.6 8.9 1.4 1.2 1.8 8.9 8.7 8.5 8.6 1.1 1.6 7.8 8.9 8.8 8.3 8.4 8.6-8.28.1 8.38.7 1.2 1.8 8.7 8.8 8.5 8.4 8.6 1.8 1.2 8.8 8.9 8.3 8.3 8.3 8.5-8.4-8.2 8.8 8.4 8.9 8.9 8.7 8.8 8.4 8.3 8.6 1.8 1.3 9.8 8.8-8.5-8.38.8 8.1-1.1-8.9-8.5-8.1 8.38.38.2 8.2 8.8-8.2 0.8 8.4 8.7 18.8 1.8 8.3 8.8 8.8 1.8-8.1 8.2 8.3 8.6 1.1 8.9 8.8 8.8 8.6 8.5 8.8 1.8 1.2 11.8 8.70.38.4 8.7 8.7-8.5-8.8-8.2 0.1 B.5 8.5 8.3 8.4 8.2 8.1 8.4 8.7 1.1 12.8 -8.3-8.7-8.5-8.5-8.5-1.7-1.5-1.2-1.8-8.6-8.8-8.9-8.8-1.8-1.1-8.7-8.5-8.1 13.8 8.3 8.8 8.2 8.2 8.3-8.9-8.8-8.5-8.2 8.2 8.2 8.8 8.2 8.1 8.8 8.3 8.6 8.9 14.8 8.2-8.1 8.8 8.8 8.2-1.2-1.1-8.9-8.6-8.4-8.3-8.4-8.3-8.4-8.4-8.1 8.1 8.2 15.8 8.1 8.1 8.28.4 8.5-8.9-8.9-8.6-8.2 8.1 8.1 8.8 8.2 8.1 8.8 8.4 8.5 8.7 16.8 8.8 8.2 8.2 8.1 8.1-1.1-8.8-8.7-8.2 8.8-8.1-8.2 8.8-8.1 8.8 8.3 8.4 8.1 17.8 -8.2-8.2 8.1 8.2 8.2-1.1-8.8-8.6-8.5-8.1-8.3-8.4-8.3-8.3-8.3 8.2 8.1 8.8 18.8 -8.3 8.1 8.2 8.4 8.5-8.7-8.6-8.4-8.3 8.1-8.1-8.3-8.2-8.1-8.2 8.2 8.2 8.1 19.8 8.1 B.l 8.1 8.38.4-8.7-8.5-8.3-8.28.1-8.2-8.4-8.1-8.2-8.1 8.4 8.5 8.1 28.8 -8.7-8.5-8.5-8.4-8.3-1.4-1.3-1.1-8.9-8.6-8.8-1.1-8.9-8.8-8.7-8.3-8.3-8.3 22.8 -8.5-8.1-8.2-8.1 8.8-1.1-8.9-8.8-8.5-8.2-8.4-8.6-8.3-8.2-8.3-8.1 8.1 8.3 24.8 -8.2 8.8 8.1 8.28.3-8.7-8.6-8.4-8.28.2-8.1-8.3-8.1 8.8 8.1 8.58.58.5 26.8 -8.8-8.3-8.4-8.3-8.2-1.2-8.9-8.7-8.5-8.1-8.4-8.7-8.5-8.3-8.28.1 8.1-8.1 28.8 -8.5-8.1-8.1 8.8 8.1-8.7-8.5-8.3-8.2 8.1 8.1-8.1 8.28.38.4 8.6 8.6 8.3 38.8 -8.6-8.2-8.1 8.88.2-8.5-8.4-8.4-8.1 8.3 8.1 8.2 8.5 8.6 8.fi 8.7 8.7 8.7

y

X-ray CT

x Figure 27.14. Coordinate system for three-dimensional dose calculations in a patient. A multi-slice CT matrix, consisting of up to 64 x 64 X 64 volume elements (voxels), is used. 27. Three-Dimensional Dose Calculation for Total Body Irradiation 587

Photon transport in a three-dimensional patient is solved by the Monte Carlo method as depicted in Fig. 27.15. An incident photon, impinging on the three• dimensional voxel, is given a free path generated by an exponential random number. The intersecting length dr and the density p in each voxel are examined along the pho• ton passage. When the integral path length reaches the given free path, an interaction takes place. In the present case, the Compton effect is predominant. The secondary (Compton) electron is traced until it is absorbed. The energy deposition in each voxel is recorded and later used to calculate the absorbed dose. Next, the scattered photon is traced in a voxel-by-voxelmanner. This photon history is finally terminated when it is completely absorbed by the photoelectric effect, or when it escapes out of the patient body.

v

Incident Photon

~ Volume Elements

Electron Collision

Intergral Density

Figure 27.15. Transport of a photon in a three-dimensional patient. Density and intersecting length in a voxel are examined along the photon or electron path until it is completely absorbed by the photoelectric effect, or it escapes out of the patient.

This voxel-by-voxel tracing is performed repeatedly in the voxel geometry routine. Fig. 27.16 illustrates a single-voxel geometry. A photon or an electron moving from a point P(x, y, z) may intersect one of the neighboring planes, depending on its direction cosines. Fig. 27.17 shows the flow chart of the voxel geometry-processing routine. The Monte Carlo program spends most of its computing time in this repetitive routine. 588 A. Ito

27.6 VARIANCE-REDUCTION TECHNIQUES

In a typical case, the thorax region of the patient contains as much as 3 x 104 or more voxels. To obtain a reasonable dose accuracy of 2-3%, each voxel must be traversed by between 1000 to 2500 photon histories. The total number of histories required is between 3 and 8 x 107• The computing time for each history is about 30 ms in a full analog Monte Carlo calculation with the dedicated three-dimensional Monte Carlo program (TBI3DMC) on a VAX-ll/750 computer with a floating point accelerator option. Thus, total computing (CPU) time amounts to as much as 1 to 3 X 106 seconds, or 300 to 1000 hours, i.e., 2 to 6 weeks! This lengthy computation time to date has made it unfeasible to use the Monte Carlo method for patient dose calculations for treatment planning (see Chapter 1). However, a reduction in computing time by a factor of 100 can be achieved with the use of a "mini-Supercomputer" and a series of variance-reduction techniques, as discussed below. Actual inhomogeneous three-dimensional problems can be solved in only 3 to 10 hours!

z

Figure 27.16. The voxel geometry planes with which a particle can intersect. A particle moving from point P(x,y,z) with direction cosines (Cx,Cy,Cz) will intersect one of six planes. The intersection length d". and voxel density p are used to determine the subsequent behavior of the particle.

Table 27.3 summarizes the effects of the different variance-reduction techniques on the accuracy of dose calculations and on the computing time for a typical patient in the thorax region. They are grouped into four categories: physics approximations, geometry considerations, weighting techniques and computer performance. 27. Three-Dimensional Dose Calculation for Total Body Irradiation 589

(Entry) Particle moving from (x,y,z) toward (Cx,Cy,Cz) direction t travels Free-Path (g/cm2) Path Len=O.O I - I

Which direction is it moving along? Cx(X-axis) I Cy(Y-axis) I Cz (Z-axis) 1 1 I Same as Y I Isame as yl

Find crossing surface to next voxel +-X (Y-Z plane) I +-Y(X-Z plane) I +-Z(X-Y plane) ~ l I Same as ±Y I Isame as ±Y I

\It dr (path length increment) rho=densty (I, J, K) new (x,y,z) next J=J+l/-l

Yes Terminate ? -< Particle Terminated 1 NO I Path_Len=Path_Len + rho*drl

1 Yes Path Len Free Path ? Next Interaction) - > NO

No Yes Escape ? " Escape from Patient \.

Figure 27.17. Geometry routine for three-dimensional voxels: The particle is followed in a voxel-by-voxel manner. (See text for details.)

Physics approximations include three items: a. Rayleigh-scatter cutoff, b. multiple• scatter cutoff and c. replacing electron transport by KERMA. These radiation transport approximations were introduced in order to shorten the computing time. If the dose calculation accuracy is not compromised and the savings in computing time is signifi• cant, it is adopted. Ignoring coherent (Rayleigh) scattering has an insignificant effect on the dose calculation for MeV photons. The saving of computing time, however, is only 1%. The cutoff of the multiple Compton scattering has direct influence on both dose accuracy and computing time. Several Monte Carlo trials were made to observe the effect of cutting off scattering after different numbers of scatters. 590 A. Ito

Table 27.3. Variance-reduction techniques and their effects on dose accuracy and com• puting time for a typical patient (thorax).

Effect on Reduction in Variance-Reduction Technique Dose Computing Accuracy Time 1. Physics Approximation a. Rayleigh-scatter cutoff none 0.99 b. Multiple-scatter cutoff 0.90-0.994 0.29-0.89 c. Electron dose or KERMA large' 0.86 2. Geometry Consideration

d. Side scatt~r 0.97c 0.72 e. Full or patient matrix none" 0.77 3. Weighting Technique f. Separate primary/scatter none 0.4" g. Voxel density(lung) none 0.31 4. Computer Performance h. FPS M64/30 vs. VAX-ll/750 none 0.13

4 Depends on the cutoff level of multiple scattering # (1 to 4). , Especially if interface or inhomogeneous boundary is involved. c Poor at top and bottom ends of patient. " For simple geometry. • When 80% is primary and 20% is scatter in dose. 1 When lung (p = 0.3) is involved.

The average dose over the thorax region was computed as a function of the number of multiple scatters. Table 27.4 summarizes the numerical results. Also, Fig. 27.18 is a plot of the effects on both dose accuracy and computing time. If we cut off all the scatters, i.e., only the primary dose computed, the dose is about 19% lower than the full-scatter dose, though the gain in computing time is about 4.5. The error is unacceptably large. The accuracy increases when first, second and third scattered photons are included, at the expense of computing time .. Multiple scattering beynd the fourth scatter does not take place frequently, and the contribution to dose is small. Thus, including up to the fourth scatter, we can maintain the dose accuracy to better than 1% while the savings in computing time are about 10% compared to the full-scatter case.

The use of the KERMA approximation, i.e., depositing all the secondary electron kinetic energy at the point of creation instead of distributing the energy along the tracks of the secondary electrons, introduces little error for a homogeneous phantom where secondary electron equilibrium is satisfied. In the case of an inhomogeneous patient, however, it causes large errors in the interface region between air and tissue or between tissue and bone, as well as in and around the lung. One of the main advantages of using the Monte Carlo method may be lost by the KERMA approximation. Also, savings in computing time are not large. Therefore, this approximation should not be adopted. 27. Three-Dimensional Dose Calculation for Total Body Irradiation 591

Table 27.4. The photon multiple-scatter cutoff and its effect on the accuracy of doses in the thorax region and computing time for GOCo photons.

Number of Number per first Energy of Absorbed Computing Scatters Collision(% ) Scatter (ke V) Dose(%) Time(%) Primary 100 - 100 100 1 49 (49) 231 19.2 (119.2) 167 (267) 2 28 (77) 99 (183) 4.6 (123.8) 76 (343) 3 17 (94) 50 (159) 1.4 (125.2) 52 (385) 4 11 (105) 25 (145) 0.7 (125.9) 25 (410) 5 8 (113) 18 (136) 0.3 (126.2) 16 (426) 6-10 15 (128) 17 (122) 0.3 (126.5) 28 (454) 11-00 3 (131) 9 (119) 0.0 (126.5) 4 (458)

Note: Numbers in ( ) are integrated.

O~~--~--L-~--~~ o 2 3 4 5 10 00 NI.rnber of Scatters included Figure 27.18. Effect of the cutoff number for multiply scattered photons on the absorbed-dose accuracy, and also on the computing time. As the higher• order scatters are included, the dose accuracy increases, and computing time increases. The optimized cutoff level includes up to fourth-order scatter.

In general, there seems to be little scope for using physics approximations to shorten computing time while keeping the dose accuracy within a few percent.

The next group of variance-reduction techniques involves considerations of the patient geometry. First, we are interested in the calculating the dose in the whole body, especially in the thorax region where the lung is included. In the actual calculation, CT slices from the lung region only are used instead of from the whole body. This reduces 592 A. Ito the memory and time requirements. However, it poses the problem of missing the body at both the top and bottom ends of the thorax region from whence no scattered photons can then come. It is possible to assume "mirror symmetry" at both ends. The scattered photons going out through both ends are forced to reflect back to the thorax region as if they were the scattered photons from the rest of the body. In so doing, we ignore the contribution to the scatter dose amounting to about 3%, though the computing time is reduced to 72%. Thus, we do not adopt this cutoff of side scatters.

In the CT matrix, there are "empty" voxels which represent air. In the case of a simple patient contour such as an ellipse, these empty voxels need not be processed; when a photon encounters the air, it can be terminated. If this is the case, the computing time spent in the geometry routine can be reduced to 77%, for example. But for a very complicated geometry, e.g., a patient with arms placed in front of the chest wall to prevent lung overdose, this simplification may be erroneous.

Group III includes weighting techniques. An analog Monte Carlo scheme follows all photon histories, regardless of the precision of calculation and computing time. Investigation into the statistical nature of photon interactions in patient voxels reveals some bottlenecks in the calculation. First, multiple Compton scatters take place after the first collision. The number of such multiple scatters is about the same as that of primary collisions. The computing time required for multiple scattering is, however, about three times greater than for the primary collision, whereas the fraction of absorbed dose due to multiple scatter is only 20% of the total dose. It is obvious that excessive computing time is spent on the less important portion of the final results, i.e., the total absorbed dose.

Fig. 27.19 illustrates this situation. If multiple scattering is sampled by weighting by a certain factor (W= 0.2, the dose fraction due to multiple scatters), the computing time is reduced. Only a fraction W of the scattered photons are followed, but the deposited energy is multiplied by I/W, resulting in the same dose. The statistical precision deteriorates for multiple scatters. However, the overall dose accuracy does not change because the primaty dose dominates the total dose. This sampling technique reduces the total computing time to 40%.

Another important variance-reduction technique utilizes forced photon interac• tions. This has a significant effect in the case of low-density regions. Fig. 27.20 illustrates this technique. When the photon fluences are the same, the actual number of photon interactions in a voxel is linear with respect to the density of the voxel. A low-density voxel, such as the lung, may have only one-third the number of interactions as the unit-density case. Thus, dose precision is worse in the lung than in normal tissue. To increase the precision, more histories are needed. However, if the total number of histories is increased, the computing time increases by factor 3 in this case, and an unnecessarily large number of photons traverse the normal (i.e., unit-density) tissue. Alternatively, it is possible to force an appropriate number of primary-photon interac• tions to occur in any voxel throughout the patient. The number of forced interactions in a voxel should be linear with respect to the photon fluence at each voxel, so that statistical errors are equal at any voxel in the patient. The number of interactions are weighted (up) by the factor of 1/ p, whereas the deposited energy is weighted (down) by the factor p. In this manner, the computing time required to achieve a given precision can be reduced to about 0.3 when the lung is involved. 27. Three-Dimensional Dose Calculation for Total Body Irradiation 593

Number of P rim a r y S c a t t e r s Interactions I 1 1.3 Np Ns

Absorbed P r i m a r y cat t e r s Dose 0.8 Is 0.2

Dp Ds

Computing Primary F u 1 1 Sea t t e r s Time (Analogue) 1 4.5

tp ts

Computing Sampled Primary N's=NS(DS/Dp) Time (Sampled) Scatter 1 1

tp t's Figure 27.19. A diagram for explaining the variance-reduction technique of sampled scattering.

Finally, use of a faster computer reduces the computing time. At present, main• frame computers, such as the IBM 3083, are about 10 to 20 times faster in scientific calculations than mini-computers, such as the VAX-ll/750. Mainframes are shared among many users, hence, very long Monte Carlo computations generally are not fea• sible. Super computers are very fast when the calculation algorithm is vectorized. Otherwise, scalar operation is only a few times faster than the fastest mainframe com• puters. The photon Monte Carlo method is far from a vectorized algorithm; hence, super computers may not be appropriate for Monte Carlo calculations (see Chapter 1). Recently, a new type of computer called "mini-Supercomputer" became available for fast scientific calculations. We now use such a new computer for Monte Carlo calcu• lations; the M64/30 computer manufactured by Floating Point Systems (USA). This machine is attached to the host computer (VAX), and the same FORTRAN program on the VAX runs at a maximum speed of 12 MFLOPS (million floating point operations per second); about 20 times faster than the VAX. For Monte Carlo calculations, it runs about eight times faster than the VAX. This machine can be used solely for Monte Carlo calculations. Thus, reduction in computing time by factor 8 has been obtained. This is the biggest single factor among all the variance-reduction techniques.

The variance-reduction techniques mentioned above are almost independent of each other so that a total reduction in computing time of the order of a factor of 100 is possible. The time required to obtain 1000 histories in a voxel (total 3 X 107) is now reduced to only six hours, as compared to about 600 hours for an analog Monte Carlo run on a VAX-ll/750. An overnight calculation (15 hours) produces a three-dimensional dose calculation of a patient with a 2% accuracy. 594 A. Ito

Normal Density(1.0) low Density (0.3) A) 8)

c) 1. Weight factor W=1/0.3

2. Force the Interaction to occur by factor W

3. Reduce dose in voxel by factor 1 / W

4. Statistical ~cision increases by factor ,fW

5. Interactions/ voxel is constant over volume Figure 27.20. Variance-reduction technique with forced photon interaction. In a low-density voxel such as the lung, the number of interactions is less by a factor of three (B) as compared to the normal-density voxel (A). This results in poor statistics. A weight factor W = 3 is used to force to interactions to occur more frequently (C). The deposited energy, however, is reduced by factor W, resulting in the same dose with better precision.

27.7 THREE-DIMENSIONAL DOSE DISTRIBUTION IN A PATIENT FOR TBI

The three-dimensional dose distribution in a patient is calculated by the Monte Carlo program (TBI3DMC) described above. Some calculation results are presented here, and their significance discussed. Fig. 27.21 shows a plot of the CT numbers (normalized to 100 at density = 1) at the middle lung slice. In this patient, the heart is located at the middle, and the lung density is very low (about 0.15 g/cm3 ). The CT numbers outside the patient are set to -1, so that any particles going out of patient are terminated (escape).

Statistical fluctuations are overcome by the use of a series of variance-reduction techniques on the mini-Supercomputer. Fig. 27.22 illustrates the dose distribution for a precise Monte Carlo run. Five thousand primary photons including A-P and P-A directions are generated in a voxel. The total number of histories amounted to as many as 150 million in this run. It took about 30 hours on a FPS M64/30 computer. The dose distribution is normalized to 100% at the center of body. The central region of the thorax has a uniform dose around 100%. Both lungs, however, receive more than a 120% dose, as the density of lung is very low. Also, at both sides of the body more than a 120% dose is received. The A and B arrows are the A-P lines where depth-dose curves are plotted. 27. Three-Dimensional Dose Calculation for Total Body Irradiation 595

-I ., -I ., -I .J ·1 -I ., -I ., ·1 ·1 ·1 ·1 -I ·t -I ., ·1 0' ., ·1 "I ", "I -I ., ·1 ., -I -I -I -t "I .~ -I . , .1 ., ·1 .1 ·1 ·1 ., ·1 ·1 ·1 oJ ·1 ·1 ·1 .J -I ·1 ., ·1 .' ., ·1 . ; -1-1-1-"'-1,,-1-1-1-1-'-1-' -1-,-1 -'-1" -1-1-"1-1-"1-1-, -1-'-I-'·I·\·l·'·I·'·I·1 ·'·1-\-1-\-1-1-1-1-1-1-1·,·1-1-'" ·1-"

Figure 27.21. A plot of a CT slice in the middle of the lungs of a patient. The three-dimensional Monte Carlo calculation was performed on this complicated geometry.

Fig. 27.23 shows the linear plots of the dose distribution along the patient A-P direction at the center (along arrow A in Fig. 27.22) and at the right lung (along arrow B in Fig. 27.22). First of all, the buildup of the dose is clearly seen at the entrance surfaces. Both dose distributions are fairly flat. However, the dose is about 25% higher in the right lung.

The calculated three-dimensional dose distributions are best observed with a three• dimensional color-graphics system. With three-dimensional color graphics, multi-slice x-ray CT data are superimposed on the dose distribution, displaying dose in color, and patient density in brightness. By showing each axial slice for a certain time, i.e., "movie" -mode display, one can observe the three-dimensional anatomical and dose in• formation from such a color display.

Fig. 27.24 shows the computer network system on which this work has been done. The conventional dose calculation for treatment planning is done on a TP-ll system (Atomic Energy of Canada Limited) and on a PDP-ll/34 computer. When a three• dimensional Monte Carlo calculation is needed, preprocessing of the patient's x-ray CT data is done on a VAX-ll/750 computer. The time-consuming Monte Carlo run is executed on a FPS M64/30 computer. The result of the calculation is post processed on a VAX, and the generated three-dimensional pictures are sent to a PDP for color display. In this manner, we can generate a few three-dimensional dose distributions by the Monte Carlo method for up to four TBI patients a day. The time has come when Monte Carlo dose calculations are feasible for clinical use. 596 A. Ito

B A t +

Figure 27.22. TAR dose distribution for a 5000-historiesfvoxel Monte Carlo run. The statistical precision is 1.4%. The 60CO gamma-ray beam is incident from the A-P and P-A directions. The dose distribution in the central region is uniform at 100%. Both lungs, however, receive about a 125% dose when there is no lung shielding.

Dose(%) Right Lung ( 125% ) 130 r, i, ,., r, : : ,., r ____ r-r.J : ,.r'\.!'L~ LrJ -l.f~J L __ •. f" L __ r~ I\''' -, r- I I I "0 : Center ( '00%) I I I 100r- _r---

90

80

70 - :r J o 5 10 15 Anterior Depth ( em ) Posterior

Figure 27.23. Plot of the TAR distribution of opposed A-P and P-A irra• diations along a patient. A-P direction at center (A) and right lung (B) of the patient as indicated in Fig. 27.22. Both show a flat dose distribution. However, overdose in the lung area is apparent. 27. Three-Dimensional Dose Calculation for Total Body Irradiation 597

Super Mini- Mini- (VAxIVMS) LAN Mini-Super

FPS M64/30 8MB Mamory

Figure 27.24. Computer network system for preprocessing, running a Monte Carlo program and postprocessing the results. The three computers are connected by ethernet cabling and use DECnet soft• ware. Data from each computer can freely be accessed.

Acknowledgement: I would like to thank Dr. Yoshio Onai of Cancer Insti• tute Tokyo for his encouragement, Ms. Sayuri Noguchi for preparation of the prelininary manuscript and Mr. Hisanori Takasaki for his drafting work. 598 A. Ito

REFERENCES 1. E. D. Thomas, "Marrow Transplantation for Malignant Diseases" , Jour. Clin• ical Oncology 1 (1983) 519. 2. G. P. Glasgow, "The Dosimetry of Fixed, Single Source Hemibody and Total Body Irradiators", Med. Phys. 9 (1982) 311. 3. J. J. Broerse and A. Dutreix, "Physical Aspects of Total Body Irradi• ation", J. Eur. Radiother. 3, (1982) 187. 4. J. R. Cunningham, Progress in Medical Radiation Physics, Vol. 1, edited by C. G. Orton, (Plenum Press, 1982) 103. 5. S. Webb and R. P. Parker, "A Monte Carlo Study of the Interaction of External Beam X-radiation with Inhomogeneous Media", Phys. Med. BioI. 23 (1978) 1043. 6. S. Webb and R. A. Fox, "Verification by Monte Carlo Methods of a Power Law Tissue-air Ratio Algorithm for Inhomogeneity Corrections in Photon Beam Dose Calculation", Phys. Med. BioI. 25 (1980) 225. 7. A. Ito, "Three Dimensional Dose Calculation by the Monte Carlo Method" , Radiat. Therapy Sys. Res. 2 (SI) (1985) 25. 8. P. K. Kijewski, B. E. Bjarngard and P. L. Petti, "Monte Carlo Calcula• tions of Scatter Dose for Small Field Sizes in a 6OCo Beam" , Med. Phys. 13 (1986) 74. 9. R. Mohan, C. Chui and 1. Lidofsky, "Differential Pencil Beam Dose Computation Model for Photons", Med. Phys. 13 (1981) 64. 10. K. Han, D. Ballon, C. Chui and R. Mohan, "Monte Carlo Simulation of a Cobalt-60 Beam", Med. Phys. 14 (1987) 414. 11. J. F. Williamson, "Monte Carlo Evaluation of Kerma at a Point for Photon Transport Problems", Med. Phys. 14 (1987) 567. 12. W. R. Nelson, H. Hirayama and D. W. O. Rogers, "The EGS4 Code System", Stanford Linear Accelerator report SLAC-265 (1985). 13. J. W. Wong, R. M. Henkelman, A. Fenster and H. E. Johns, "Second Scatter Contribution to Dose in a Cobalt-60 Beam", Med. Phys. 8, (1981) 775. 14. J. H. Hubbell, "Photon Mass Attenuation and Mass Energy-Absorption Coefficients for H, C, N, 0, Ar and Seven Mixtures from 0.1 keV to 20 Mev", Radiat. Res. 70 (1977) 58. 15. A. Ito, Chapter 16, this book. 16. M. J. Berger and S. M. Seltzer, "Stopping Powers and Ranges of Elec• trons and Positrons" , NBSIR 82-2550-A (1982). 17. M. J. Day et ai, "Central Axis Depth Dose Data for Use in Radiother• apy", Brit. J. RadioI., SuppI. 17 (1983). 18. P. Ekstrom, H. Dahlin and B. Hongstrom, "A Generalized Format Spec• ification for Use of CT Information in Radiotherapy Dose Calculation" , in Proc. of the Seventh lCCR, Tokyo, Japan, Sept. 22-26, 1980, edited by Y. Umegaki, p. 491. 19. P. W. Henson and R. A. Fox, "The Electron Density of Bone for Inho• mogeneity Correction in Radiotherapy Planning Using CT Numbers", Phys. Med. BioI. 29, (1984) 351. 28. High-Energy Physics Applications of EGS

A. Del Guerra* and Walter R. Nelsont

Department of Physics, University of Pisa, Piazza Torricelli 2, 1-56100, Pisa, Italy

28.1 INTRODUCTION When a shower takes place in a large detector mass, most of the incident energy appears as ionization or excitation in the medium. The energy of the initiating particle can be determined to a reasonable degree by sampling the energy deposited in the device• hence, the name "calorimeter". Calorimetry has become an essential tool for high• energy physics experiments. With the new colliders-e.g., LEP and LHC at CERN, HERA at DESY, SLC at SLAC, and the proposed SSC in the U.S.- calorimeters have almost completely replaced conventional spectrometers for particle identification and the measurement of their energy. Furthermore, the particle physics concept of jets has shifted the instrumentation emphasis from individual particle measurement to precise determination of the energy balance of multiparticle systems-i.e., towards calorimetry. Unfortunately, the complexity of these devices increases tremendously in the hundred-GeV energy range, and it is therefore important to have reliable simulation tools available for designing calorimeters and evaluating their performance with specific experiments in mind.

In this chapter, we will present the use of the EGS4 codel in electromagnetic calorimetry. After a short review of the physics of the electromagnetic cascade itself, and of the various types of shower counters that exploit it, several comparisons of EGS simulations with relevant experiments will be presented. As a specific example, the application of EGS4 to the design of a lead-glass drift calorimeter will be discussed. Finally, we will present a series of other applications of the EGS code in high-energy physics, including accelerator design, health physics, and radiation damage in general.

* Present address: Department of Physics, University of Napoli Napoli, Italy t Radiation Physics Group Stanford Linear Accelerator Center Stanford, California 94309, U.S.A.

599 600 A. Del Guerra and W. R. Nelson

28.2 THE EGS CODE IN ELECTROMAGNETIC CALORIMETRY 28.2.1 The electromagnetic cascade shower.

When a high-energy electron or photon enters a material, it produces an electromagnetic (EM) shower with the following properties: - The fraction of the energy, E, of the incident particle which is absorbed by the material is distributed among a large number of secondary particles; the thickness necessary for total absorption increases as In E. - The entire shower is well collimated; hence, the shower axis is a fairly good representation of the direction of the incident particle. - The inherent fluctuation in energy deposition of the shower, both in magnitude and in position, depends upon the material in which the shower develops, and limits intrinsically the energy resolution and the position determination.

To describe an EM shower, several quantities are of general use for every material:

• The radiation length Xo (g/cm2), given by2

(28.1)

where Z and A are the atomic number and weight, respectively, and f accounts for the Coulomb correction according to:

f = 1.202x - 1.0369x2 + 1.008x3 /(1 + x) , (28.2)

with (28.3)

• The photon absorption length, A (g/cm2), given by

A = 1/Jl ::::: 9Xo/7 , (28.4)

where Jl is the attenuation coefficient.

• The Moliere radius, rm (g/cm2), which characterizes (or scales) the lateral spread of the shower, and is given by the approximate formula

21 rm::::: -Xo , (28.5) e

where e ::::: 800/(Z + 1.2) is the critical energy (MeV)-i.e., the energy at which the collision energy loss is equal to the bremsstrahlung loss.

Since EM shower development is built upon the repetition of a few physical pro• cesses, Monte Carlo simulation is well suited as an application tool. Among the many codes which have been implemented3- 7, the EGS4 code by Nelson, Hirayama, and Rogersl is now most widely used. However, while the detailed shower development is accurately described by the Monte Carlo tables and histograms, the gross features of 28. High-Energy Physics Applications of EGS 601 the cascade can be described by "phenomenological" parameters8, such as the median depth, i.e., the depth within which one half of the energy is contained,

tmed(XO) = In(E/e) + 1.22 ~ tmaz(XO) + 1.0 , (28.6) where tmaz(XO) is the depth at which the shower reaches its maximum. The shower is 98% longitudinally contained for a length of '" 3tmed, whereas it is 95% laterally contained for a size of 1.5 to 2rm8.

28.2.2 Electromagnetic calorimeters.

Electromagnetic calorimeters can be subdivided into two main categories: homogeneous and sampling (see Fig. 28.1).

81

Absorber 8nd 8ctille m8teri81

bl

~/----8bsorber Figure 28.1. Schematic of: a) homogeneous calorimeter, b) sampling calorime• ter.

In the first category, the absorbing block is also the active material in the detection process. Since all the energy lost by the shower is dissipated in the active material, in principle this type of detector gives the best energy resolution, limited only by the cutoff energy (i.e., the energy below which the detector is not responding), by the detector inhomogeneity, and by the intrinsic resolution of the detector. However, homogeneous calorimeters tend to be expensive, and their size often places limitations on the spatial resolution that can be obtained.

With sampling calorimeters, the cascade develops in a dense material, and the energy lost, is usually sampled in slices of active material interspaced between the absorbers (Fig. 28.1b). In this case, the detector is cheaper and its geometry is more flexible. If an adequate number of readout channels are used, the intrinsic spatial 602 A. Del Guerra and W. R. Nelson resolution can be excellent. However, the energy resolution is degraded by the sampling fluctuations.

Homogeneous calorimeter. Table 28.1 shows the main properties of the most common scintillators and the en• ergy resolution, in the GeV range, obtained for homogeneous calorimeters9. At higher energies, the imperfections and the inhomogeneity of the larger fiducial volume for the shower dominate, and the energy resolution deteriorates.

Table 28.1. Properties of Most Common Scintillators and Intrinsic Energy Resolution for Homogeneous Calorimeters (E in GeV)9.

Crystal NaI(Ti) BGO Pb-Glass (SF-6) Density (g/cm3 ) 3.7 7.1 5.2 Xo (cm) 2.59 1.12 1.69 u(E)/E rv 0.3%/ El/2 rv 0.4%/ El/2 3.6%/E1/2

NaI(TQ is a standard scintillator, and its performance is well knownlO-it has a high scintillation efficiency (5 times larger than plastic scintillators), a short radiation length (Xo rv 2.6 cm), and a low cutoff energy. The main problems arise from the long decay time of its scintillation (230 ns for the fast component), which may cause pile-up problems. In addition, it is extremely hygroscopic and is not very resistent to radiation.

BGO is the most dense scintillator material and has the shortest radiation length (Xo = 1.12 cm). More than 50% of its light is emitted within 60 ns, and it is very resistent to radiation. Unfortunately is quite expensive. Other crystals have been proposed (e.g., CsI(Tl) and BaF2 ). BaF2 scintillates in the UV region and needs a special readout, such as a gas detector doped with TMAE which absorbs UV light and produces photoelectronsll. Liquid argon and "warm" liquids have also been used as active materials.

Lead glass is a transparent glass containing PbO (50% to 70% by weight), with a typical radiation length of a few cm. In this case, the energy lost by the shower is measured by collecting the Cerenkov light produced by fast particles and transmitted through the glass. The energy resolution is mainly limited by the cutoff energy for light emission (Ee rv 0.5 MeV). Although the resolution is not as good as for NaI(Tl), lead glass is superior in terms of cost, flexibility, and count rate capability.

Sampling calorimeter. In a sampling calorimeter, the main contribution to the energy resolution comes from the sampling fluctuations. The energy deposited in the active part of a sampling calorimeter will fluctuate statistically, and the energy resolution will depend on the thickness of the absorber. It has been experimentally shown8 that the energy reso• lution follows the general law of E-l/2 , with an additional dependence on the thick• ness, t (g/cm2), of the absorber slab and on the critical energy, c (MeV). Namely,

(28.7) 28. High-Energy Physics Applications of EGS 603 where E is in GeV. This rough dependence has to be refined in real calorimeters12. Typical experimental values are9 :

u(E) 17%0 for scintillator or liquid argon, (28.8) E ..fE

u(E) 25%0 for gaseous detector. (28.9) E ..fE

In a gaseous detector, the low density leads to an increase in the length of charged particle tracks. Together with an enhancement of Landau fluctuations, the net result is that energetic delta rays, traveling along the sampling gap, can deposit atypically large amounts of energy in a single cell. At very high energy, where u / E becomes of the order of a few percent, the instrumental noise and the intercalibration become as important as the sampling resolution, or even the most significant contribution, as for homogenous calorimeters.

28.2.3 EGS4 simulation of EM calorimeters in general.

As described in the manuaP, the user communicates with the EGS4 code by means of subroutines HOWFAR (to specify the geometry) and AUSGAB (to score and output the results). Initialization for these routines is done in MAIN.

For homogeneous calorimeters, the simulation is very simple because the geometry usually consists of a narrow beam impinging on a single block (Fig. 28.1a) of NaI, BGO, etc., and the scoring is done by keeping track of the total energy deposited in the active material (plus the lateral and longitudinal leakage if full containment is not achieved).

Sampling calorimeters are usually described by a series of absorbing slabs with active detector gaps interspaced to form what is commonly called a sandwich counter. The shower pictures shown in Fig. 28.2 and Fig. 28.3 represent a simulation* of the multi-wire proportional chamber (MWPC) array that was used in SLAC experiment E-13713 . This shower calorimeter was made up of eight modules (see closeup Fig. 28.4), each consisting of: 8.9-cm air gap; - Wire chamber (O.6-cm AI, l.6-cm gas, O.6-cm AI); 15.2-cm air gap; l.O-cm plastic scintillator; 1.6-cm air gap; - 8.3-cm aluminum converter, for detectors located in the (downbeam) modules.

* The EGS4 User Code: UPE137. 604 A. Del Guerra and W. R. Nelson

:. -.-! . . ' ' f+;' or . ~ "' ~ ,- /' ,r::

. Figure 28.2. Single 5-GeV photon entering at 20° (all particles shown).

\1odule , :::: :~ /. , ~ V ,I ~ -"" ~ . ~ ~ 1\ "'" V

Figure 28.3. Single 5-GeV photon entering at 20° (charged particles only).

Shower leakage (and other) fluctuations. Both longitudinal and lateral shower leakage are clearly shown in Figs. 28.2-28.3. As stated earlier, fluctuations caused by shower leakage result in a decrease in the overall energy resolution of the calorimeter. Fluctuations can also be caused by charged par• ticle tracks having lengths greater than the active detector gap width, such as those shown in the closeup view (Fig. 28.4) of one of the MWPC gas regions-identified in the figure as the region bounded on either side by a pair of the closest parallel planes. Of the 11 particles crossing the gap, the 9 in the middle traverse a gas distance slightly larger than the size of the gap itself (as a result of the 20° angle). However, the upper and lower tracks are 8 to 11 times longer, and the bottom one actually scatters back into the gas volume where it deposits even more energy. Pictures of this type aid in explaining how "channeling" within the detector gap-i.e., adding transverse absorbers around the wires in the gap to stop delta rays-can be an effective means of improving the overall energy resolution of sampling calorimeters. 28. High-Energy Physics Applications of EGS 605

1----- Module ------1 A,r PloslIC

Figure 28.4. Expanded view of a portion of the previous figure, showing 11 charged particle tracks crossing a MWPC gas region (the wire chamber walls are shaded). The top and bottom particles travel much farther in the gas; the bottom one actually scatters back into it.

Detector response. One of the most critical parameters in the simulation is the energy cutoffs-i.e., the energy at which the radiation transport is terminated, and the remaining energy is de• posited locally. Obviously, the photon and charged particle cutoffs should be set below the intrinsic energy response level of the detector if realistic simulations are to be made. A well known problem with EGS3, as a result of a 1-MeV (K.E.) cutoff limitation for charged particles, was the appearance of "ghostly" multipeaks in the energy distribu• tion plots. With EGS4, this problem no longer exists since calculations can now be performed down to 10 keV.

Simulation efficiency. EGS is primarily an analog Monte Carlo code-i.e., each and every particle is followed to some completion throughout the simulation. With the release of EGS4, however, one is able to take advantage of various biasing and weighting schemes in order to speed up the simulation appreciably. In the case of EM calorimeter design, for example, a relatively simple Mortran3 macro14 can be introduced at the beginning of the User Code in order to select preferentially the leading particle of an interaction-i.e., the particle that has the highest energy. In other words, by biasing the calculation in favor of events that are the most significant to the development of the cascade, the overall efficiency increases. However, in order to "play the game fairly", some events must also be chosen randomly to represent the counterpart in the interaction. Furthermore, an appropriate weight factor must be assigned to each particle-to be carried along by the progeny-and any scoring must take the final particle weights into consideration (see the EGS4 manuaP for a complete description of leading-particle biasing).

In Fig. 28.5, we have plotted the fraction of energy that leaks out of a gas sampling calorimeter as a function of the thickness. Somewhat simplier than the MWPC shown 606 A. Del Guerra and W. R. Nelson in the previous figures, this semi-infinite sandwich counter consists of alternate layers of PbD (10 mm, p = 6.2 g/cm3 (see Table 28.2)) and gas (1 mm at 10 atm). The solid curve represents a normal (i.e., unbiased) EGS4 calculation and was obtained in six 20 minute runs at 10 GeV (8.62 sec/case). The closed circles were done using leading-particle biasing (0.031 sec/case), resulting in a factor of 277 increase in speed! Even though the agreement between the two is quite good, the error bars can further be reduced at large depths by allowing the leading-particle biasing to be applied only in the important region. That is, if we turn the scheme on during the initial 11 radiation lengths where 90% of the energy is deposited, and off for the remainder, we obtain the results shown as open circles (0.045 sec/case) in Fig. 28.5.

~ a:: -2 ~ 10 ILl

10 20 30 DISTANCE (radiation length) Figure 28.5. Longitudinal energy fraction leakage versus detector length for 10-GeV incident electron beam (solid curve: no biasing; closed circles: total biasing; open circles: limited biasing).

Efficiency in a Monte Carlo calculation can be measured by the inverse of the product of variance and calculation time (e.g., see Chapter 18). In the present example, we have taken advantage of our "pre-knowledge" of the physics of EM cascades to select the most efficient calculation in determining the size of the calorimeter. To be specific, to determine the 99% shower containment depth of the calorimeter, it took about two hours of IBM-3081 computer time using the normal (unbiased) code, whereas the same results were obtained in about two minutes using limited leading-particle biasing. 28. High-Energy Physics Applications of EGS 607

Detector-resolution calculations. One can use leading-particle biasing in order to save computer time when designing the overall size of a calorimeter-i.e., determining a predefined containment level to min• imize fluctuations caused by leakage. However, importance sampling methods should not be used in resolution calculations themselves since the biasing and weighting will significantly distort the true statistical behavior. In other words, importance sam• pling can be exploited when determining a quantity involving the average behavior of a shower, but simulations involving event-by-event scoring must be done in an unbi• ased way. Although resolution calculations usually require lots of computer time, a calorimeter design method has been described by Hirayama et al. 15in which most of the time-consuming calculations are done at the lower (faster) energies.

28.2.4 EGS4 design of a lead-glass drift calorimeter.

Principle of operation of a high-density time-projection calorimeter. Drift-Collection Calorimeters16 and High-Density Projection Chambers17 have been proposed as a method of achieving fine granularity in gas sampling calorimeters while minimizing the number of readout channels. This is done by drifting the ionization produced in a gas-sampled radiator over a fairly large distance to a separate detecting wire plane. Digitization of the drift time allows the reading out of a complete "image" of the shower with a high degree of segmentation via a modest number of wires.

The construction schemes for these devices usually consist of plates of high-Z mate• rial alternating with gas sampling regions, with various drift field shaping arrangements. The field shaping arrangement is of great importance in avoiding loss of electrons while drifting in long narrow spaces, and numerous schemes have been proposed. EGS4 has been used extensively* to simulate a drift calorimeter15 in which the radiator and drift field-shaping structures are combined in the form of high-density (,...., 6 g/ cm3 ) lead-glass tubing18 . A highly resistive layer of metallic lead, which acts as the continuous field shaping electrode, is formed by surface reduction of the lead oxide (Fig. 28.6). The tubes are fused together into a "honeycomb" structure with their axes perpendicular to the direction of the incident radiation. A prototype, consisting of lead-glass tubing of 40-cm drift length fused together to form a total longitudinal dimension of ,...., 20Xo, has been constructed, and currently is being tested19.

Simulating the honeycomb structure with a planar geometry. Figure 28.7 is a schematic of the proposed calorimeter. For the Monte Carlo simu• lation, the tube geometry was approximated by alternating semi-infinite slabs of lead glass and gas regions, with the effective dimensions chosen so as to give the same av• erage cross-sectional area of gas and solid material when viewed from the edge of the slabs (or tubes) (see appendix in Hirayama et al. 15 ).

* The EGS4 User Code: UCCAL2DW. 608 A. Del Guerra and W. R. Nelson

electric field lines conduct ive surface , '"' ...w.---- -Vo x

-· 6V(x )= Vo~ d

surf~ e continous resistIvity d - x voltage ,o"- 10'·!l. /a divider .,.,...._5

• • • • • • electron• • detection • • region I MW V() Figure 28.6. Single element schematic of a high-density drift structure (i.e., lead-glass tube) equipped with a conventional MWPC.

E <.> ...... o> o It) ! t..

Figure 28.7. Overall schematic of the high-density drift calorimeter with lead• glass radiator (drift tube) structure and wire-chamber readout. 28. High-Energy Physics Applications of EGS 609

The track-length restriction that normally would be imposed by the tubes in the direction transverse to the tube axes was retained by appropriately limiting the energy deposition along the track. This was done in SUBROUTINE AUSGAB where, in addition, an algorithm was included in order to sample Landau fluctuations of the energy deposited in the gas by the charged particles. The effect of varying the inner diameter of the tube (i. e., the length of gas seen by the charged particles) is shown in Fig. 28.8 for the case of 1 GeV, for a gas pressure of 1 atmosphere, and for a total length of 15Xo. The four curves correspond to the four combinations of applying (or not) the track length restriction and Landau sampling algorithms. The actual detector situation is given by the curve labeled "TR/Landau".

50

40

+-' c 30 (]) u (]) s- o. 20 ...... w b TR only 10

o 5 10 15 20 INNER DIAMETER (mm) Figure 28.8. Fractional energy resolution as a function of the inner diameter of the tubes (1 GeV /15Xo/l mm wall/l atm.), for various combinations of track length restriction and Landau sampling.

Optimization of the design of the calorimeter. The wire-plane readout for a drift collection calorimeter may be operated either in the standard proportional mode or in one of the modes in which the pulse is saturated (e.g., the self-quenching streamer mode20). In the former, the signal collected from each wire is proportional to the amount of energy deposited in the tubes sampled by that wire. The resolution of a calorimeter operating in this mode is degraded by the Landau fluctuations in the energy deposited by each track. In the saturatt;d modes, one uses digital sampling in which the energy is assumed to be proportional to the number of tracks counted. This eliminates the deleterious effects of Landau fluctuations, but can cause the energy response of the calorimeter to saturate at high energies due to overlapping tracks being registered as a single particle traversing a calorimeter cell.

In order to model the two modes of operation, the Monte Carlo scoring was done in two ways. When the proportional mode was being simulated, the energy deposition in the gas regions was scored, and the width of the distribution gave the energy resolution 610 A. Del Guerra and W. R. Nelson which would be achieved in an ideal (i.e., no-electron-loss-during-drift) calorimeter, read out by an ideal proportional chamber. In simulating the digital sampling mode, the number of charged particles in the gas regions was scored. When more than one charged particle traversed a calorimeter cell, just one track was scored in order to mimic the effects of overlapping tracks at high energy.

Three tube geometries were studied with inner diameter/wall thicknesses of: 10/1, 5/1 and 5/2 mm. Each was filled with a gas mixture of 70% argon - 30% methane at pressures of 1, 2, 3, 5 and 10 atm. The 5/1 case was also examined with the gas filling replaced with liquid argon. These configurations were chosen to give reasonable drift efficiency, a modest number of readout channels, and a reasonably compact structure. In all cases, the simulated calorimeter was taken to be 20Xo in depth, and infinite in the transverse directions (x and y) to the beam direction (note: according to Fig. 28.5, 20Xo is sufficient for 99% shower confinement for incident energies up to 10 GeV15). Table 28.2 provides a summary of the characteristics of each configuration.

Table 28.2. Characteristics of Various Geometry Configurations Simulated.

70% Argon - 30% Methane Liquid Argon Inner Diameter (mm) 10 5 5 5 Wall Thickness (mm) 1 1 2 1 Sampling Medium: p (g/cmS). 0.00148P 0.OO148P 0.OO148P 1.4 Xo (em)" 14500/P 14500/P 14500/P 13.96 Radiator: p (g/cmS) 6.2 6.2 6.2 6.2 Xo (em) 1.28 1.28 1.28 1.28 Effective Xo (em)·· 4.20 2.62 1.86 2.34 Sampling Thickness (em) Xo/4.5 Xo/4.8 Xo/2.6 Xo/4.4 Equivalent Average p (g/cmS) 1.9 3.0 4.3 3.8

• P is the absolute pressure (atm.) . •• Effective radiation length is the physical length of the device that is equivalent to one radiation length in the longitudinal shower development.

Figure 28.9 shows the fractional energy resolution «(j / E) at an incident electron energy of 1 Gev, as a function of gas pressure, for the three tube geometries. Results for both proportional and digital sampling are presented (solid lines drawn merely to guide the eye). The errors on these points were obtained by dividing the EGS runs, for each case, into five runs of approximately 200 incident showers each. Each data point and error bar correspond to the mean and standard deviation of the mean for the five runs for that case. 28. High-Energy Physics Applications of EGS 611

25 25

~---i ii ---~ proportional c _/a 20 .~20.. i~ Q.o o G. ::::'15 -~---/1--i-- ~ III 1a/1~ ~ b digital I .. ---.--_/a -~--. -~~ e ~f~-===~::j1O

1 2 3 4 5 6 7 8910 PRESSURE (atml Figure 28.9. Fractional energy resolution at 1 GeV as a function of pressure for three tube geometries (both proportional and digital sampling).

In the proportional mode cases, the fractional resolution, as expected, shows im• provement as the pressure is increased. This improvement is due to an effective increase in the amount of active gas region, which both diminishes the relative track fluctuations and causes a reduction in size of the Landau fluctuations.

The resolution achieved using digital sampling is clearly superior at this energy. The pressure independence of these resolution curves is a clear reflection of the effects of eliminating Landau fluctuations. The very good resolution given by digital sam• pling at 1 GeV makes this method an ideal choice for such applications as tracking calorimetry in nuclear-decay experiments. At higher energies the superiority of digital over proportional sampling begins to deteriorate as the effects of saturation become relevant.

We also modelled the response of the lead-glass tube structure filled with liquid argon. A fractional energy resolution of 5% at 1 GeV was predicted (5 mm diam• eter/I mm wall), which is in good agreement with experimental results obtained by Hitlin21 in finely sampled liquid argon calorimeters.

Comparison with experimental results. In constructing the calorimeter, a glass different than that assumed in the calcula• tions was used*. In order to havea compact calorimeter, tubes of 5 mm inner diameter, and 1 mm thickness were chosen (which also allowed for a reasonably long drift dis• tance of 40 cm). Figure 28.10 shows the fractional energy resolution expected for this calorimeter as a function of pressure at 1 GeV for 20 radiation lengths, which implies full containment (i.e., 98-99%) of the shower. The energy dependence obtained from Monte Carlo simulation at 2 atm. was a / E '" (14.8 ± 0.3)%/ VB). Preliminary experimental data22 obtained on a'11.2 Xo prototype at 2 atm. showed a alE of (18 ± 2)%/VB in the energy range 2-5 GeV (Fig. 28.11). This experimental value is higher than the Monte Carlo prediction, and is probably due to longitudinal leakage. In fact, a Monte Carlo simulation of the 11.2 Xo prototype response gives a value of (16.2 ± 0.6)%/VB at 1 GeV and 2 atm., which compares fairly well with the experimental results.

* Schott RS-520, 71% PbO (by weight), p = 5.2 g/cm3 , Xo = 1.66 cm. 612 A. Del Guerra and W. R. Nelson

alE (%) 20

15

10

5

1 2 10 pressure 5 (atm) Figure 28.10. Fractional energy resolution at 1 GeV for a 20 - Xo calorime• ter (currently being tested) operating in proportional mode, as a function of pressure. The solid line is drawn to guide the eye through the data.

0.3

0.2

0.1 I

0.0 1 2 ~ 4 5 6 ENERGY (GeV) Figure 28.11. Energy resolution (u IvE) as a function of the incident energy. Experimental data (with error bars) measured with a 11.2 - Xo prototype (solid line is a least-squares fit). 28. High-Energy Physics Applications of EGS 613

28.3 COUPLING EGS WITH HADRONIC CASCADE PROGRAMS A subtantial fraction of the energy deposited in a hadron cascade is a direct result of EM cascades produced by the decay of '11"0 mesons (mean life'" 10-16 sec) into two photons. EM showers, as we have discussed previously, are best measured longitudinally in terms of radiation length units, whereas the scale for the development of hadronic showers is given by the nuclear absorption (interaction) length Al (obtained from the inelastic cross section). Experimental values of Al for materials suitable for calorimetry of hadronic showers range from 12 cm (uranium) to 34 cm (carbon)2s. The corresponding Xo values are 0.32 cm (uranium) and 18.8 cm (carbon), so that the spatial distribution of the energy deposition along the direction of the beam is not really controlled by the EM cascade, but rather by the longitudinal development of the hadronic cascade itself.

We have also stated earlier that the lateral spread of EM cascades is conveniently measured in terms of the Moliere radius, which is a manifestation of the dominant process involved-i.e., the multiple Coulomb scattering of low-energy electrons. The rms angle of multiple scattering is given approximately by

(28.10) where the transverse and longitudinal momentum of the charged particle, ]JT and p, respectively, are in MeV Ic, and the distance, t, is in radiation lengths. However, since bremsstrahlung and pair production are the primary interaction processes involved in the development of the EM shower, one can let t ~ 1 and obtain

]JT(EM) ~ 15 (MeV Ic) (28.11) for the effective transverse momentum associated with EM cascades. The transverse momentum in hadron collisions, on the other hand, is determined, on average, by the "size" of hadrons, with the well known (for over 25 years) result2(

]JT(hadron) ~ 350 (MeV Ic). (28.12)

In other words, the lateral spread of energy deposition is also not controlled by the EM cascade, but by the hadron shower itself.

As a result of the above analysis, there is a large class of hadronic cascade prob• lems that does not require a very sophisticated approach for handling the EM cas• cade component-e.g., target and dump heating, induced radioactivity, and even some hadron calorimetry studies. In these cases, a parameterization scheme can be employed to account for the EM shower, as has been done in the older versions of the FLUKA hadron cascade code. In the current versions of FLUKA26,26 , however, the user has the option to select a parameterization scheme for the EM cascade, or simply to couple EGS4 directly into the hadronic shower simulation27•

28.3.1 Hadron calorimetry.

With the recent coupling of EGS4 with FLUKA8726- 27, one has the capability of running a hadron calorimetry problem in which the EM cascade part is done by EGS4. A 614 A. Del Guerra and W. R. Nelson calorimetry "option" can be selected such that the built-in histogramming package produces plots for direct read-out of energy resolution. Furthermore, one can run a pure EM cascade-i.e., one initiated solely by electrons or photons. As a result, the calorimetry ("signature") of photons (or electrons) and hadrons can be studied relative to one another using the built-in geometry package of FLUKA87, and thereby avoid any problems related to geometry/material normalization, etc.

EGS has also been used rather extensively for hadronic calorimetry in connection with the hadronic cascade code developed at the Oak Ridge National Laboratory called HETC28. A recent example of the use of EGS with HETC is provided in the study by Alsmiller et a1 29•

28.3.2 Photohadron production with FLUKA87/EGS4.

Recently, the calculation of high-energy hadron cascades induced by electron and photon beams in the GeV energy range has been made possible3o through the coupling of EGS4 with FLUKA8725- 27. The most important source of high-energy hadrons around a multi-GeV electron accelerator is the hadronic interaction of real photons that are part of the EM cascade generated by the electron (or photon) beam. Using the EGS4 computer code, high-energy photons were allowed to interact hadronically according to the vector meson dominance (VMD) model, facilitated by a Monte Carlo version of the dual multi string fragmentation model (i. e., "quarks") used in the FL UKA87 cascade code. .

The results of this calculation compare very favorably with experimental data on hadron production in photon-proton collisions and, even more importantly, with data on hadron production by electron beams on extended targets-i.e., with measured yields in secondary beam lines. In Fig. 28.12, a comparison is made between the coupled FLUKA87/EGS4 code and experimental data that was taken over 18 years ago when SLAC was first started up. The comparison is an absolute one, and the first of its kind!

This technique has already found use in the design of secondary beam lines at SLAC, and is expected to be of further use in the determination of high-energy hadron source terms for shielding purposes, and in the estimation of induced radioactivity in targets, collimators, and beam dumps.

28.4 ACCELERATOR DESIGN APPLICATIONS

In the following sections, we will touch lightly on some recent applications of EGS4 to the design of accelerators, most notably the SLAC Linear Collider (SLC) (for a comprehensive introduction to the SLC itself, a recent paper by Fischer31 should be of general interest). Most of the work described here has been reported elsewhere32, and we will not go into much detail. Instead, we will simply emphasize some of the uses to which EGS has been put. 28. High-Energy Physics Applications of EGS 615

18 GeV e- - 0.3 R.L. Be - 7T+

10-2 (a) ~: Experiment - Calculation 10-3

104 u c

10-5 >-'" OJ (!).... .'!!'" u 10-6

0 Q. (b) ~: Experiment 10- 3 - Calculation

Nzl~ "0 "OQ. "0

104

10-5

106

I 0 2 4 6 e 10 p (GeV/c) Figure 28.12. Comparison of 11'+ yields from a 0.3Xo beryllium target hit by an 18-GeV electron beam.

28.4.1 Positron target design.

Secondary particle production is generally important around high-energy accelerator facilities, and the SLC at Stanford University is no exception. Indeed, successful oper• ation of the SLC depends on the production of low-energy (2 to 20-MeV) positrons by 33-GeV electrons incident upon a positron production target.

The basic scheme behind the SLC project is to produce and collide a beam of 50-GeV positrons with a beam of 5O-GeV electrons*, the purpose of which is to observe and measure intermediate vector bosons (ZO, W+, W-) at the center-of-mass energy of 100 GeV. Production of low-energy positrons is accomplished by directing a 33- GeV electron beam into a high-Z target at the two-thirds point of the SLAC two-mile accelerator. These positrons are then "collected", re-injected into the machine, and accelerated, along with another beam of electrons, to the required energy of 50-GeV.

EGS4 was used in order to determine the size and nature of the target necessary to accomplish the task. As it turned out, a 6Xo high-Z target (90% Ta, 10% W) was selected on the basis that it produced enough positrons in the energy range of interest, as well as satisfying various engineering needs. Figure 28.13 shows an EGS4 shower generated by a single 33-GeV electron striking the 6Xo cylindrical target, where

* 5 X 1010 e+ and e-/pulse at 180 pulses/sec. 616 A. Del Guerra and W. R. Nelson photons are shown as dots and charged particles as solid lines. Figure 28.14 is for the same statistical run, but this time only the positrons are shown .

.... .

Figure 28.13. Shower produced in 6Xo cylindrical target struck by a single 33-GeV electron (all particles shown).

Figure 28.14. Shower produced in 6Xo cylindrical target struck by a single 33-GeV electron (positrons only shown).

Clearly, not only are a lot of positrons created by a single electron, but there is a central core of energy deposition that can lead to serious engineering problems, par• ticularly target heating and melting. This core is primarily due to the production of bremsstrahlung followed by pair production. The small size of the core is determined by the characteristic angle of these processes-i.e., the ratio of the electron rest mass to the incident energy-convoluted with the charged particle multiple-scattering angle given above (Eqn. 28.10). The core dimension turns out to be very small at the high energies involved. Furthermore, the incident beam spot is Gaussian and of the order of 50 microns (0"", = 0"1/ = 50 X 10-4. cm), so that even with a single pulse of 5 x 1010 e-, the temperature rise is extremely high-i.e., in excess of 500°C/pulse (note: the melting temperature of the Ta-W target is 3035°C). The target wheel, inside a vacuum chamber, is rotated at 2 Hz to distribute the power from the beam operating at 180 pps, and the target is cooled by water passages in the drive shaft. The resulting stress from 28. High-Energy Physics Applications of EGS 617 each thermal pulse is ~ 32,000 psi, and the real concern involves a scenerio in which cracks develop due to material fatigue, leading to discontinuities in the heat transport path, and ultimately a melt down33 . Therefore, EGS has been useful in determining not only the optimum positron yield, but also to point out some of the difficulties involved in running small beam spots into thick targets. In addition, since the positron target is a water-cooled device spinning in a vacuum, a number of components are made of mate• rials that are subject to radiation damage, and EGS has been very useful in estimating their mean life to failure.

28.4.2 Heating of beam pipes and other components.

Although targets are usually cooled and the size of the beam spot can be purposely made larger to help alleviate problems, the same is not generally true of beam pipes, beam position monitors, and many other components. Under normal operations of the SLC, for example, both the positron and electron beams are designed to travel down the center of the beam pipe. EGS4 was used to forecast the difficulties that would arise should a beam inadvertantly impinge upon the pipe at a small grazing angle ("-' few milliradians)34. Typically, we found temperature rises of 50 to 100°C/pulse for aluminum (Tmelt = 660°C), and 300 to 700°C/pulse for copper (Tme/t = 1083°C), depending upon the grazing angle. From these studies, it was determined that the temperature rise in copper precluded its use, and aluminum was chosen instead as the SLC beam pipe material.

Similarly, an EGS4 study was made to find out if the beam position monitors (BPMs), which are located outside the beam pipe radius, would suffer in any way due to shower leakage from beams inadvertantly striking the beam pipe upstream at a glancing angle-i.e., shower "punch through" 35 . The study showed that, for BPMs made out of stainless steel, the temperature rise could increase to the extent that they could lose their calibration. Obviously this was undesirable, so it was recommended that the BPMs be made out of aluminum instead.

28.4.3 Synchrotron radiation.

The spectral distribution of synchrotron radiation has adequately been described by many authors, starting from the original derivation by Schwinger36. In 1975, Nelson et al 31 pointed out that a large fraction of the synchrotron radiation associated with the PEP storage ring at SLAC would scatter out of the beam pipe and cause significant radiation damage to coil windings, plastic hoses, etc. In addition, the production of ozone in trapped layers could result in the creation of nitric acid, leading to further difficulties with the passage of time.

As an aid to understanding the overall problem, a series of EGS calculations have been performed over the years, not only for the PEP facility, but also for LEP at CERN38 , the SLC at SLAC39, as well as other high-energy accelerator facilities4o. Our intention is not to discuss any of the details involved in these problems at this time, but merely to point out that methods have been developed for sampling a synchrotron radiation spectrum for direct input into EGS. Of particular interest is an application* to the production of photoneutrons in the materials surrounding the beam pipe of LEp38,41. 618 A. Del Guerra and W. R. Nelson

28.5 SIMULATION OF A HYDROGEN BUBBLE CHAMBER In closing, we would like to illustrate the use of EGS4 for radiation transport in magnetic fields. To some extent, this has been described in the EGS4 manuaJ1 for a problem involving low-energy electrons. Reference is also made in the EGS4 manual to what has become an award-winning paper by Rawlinson, Bielajew, Galbraith, and Munr042 in which EGS4 was also used to transport radiation in a strong electric field.

Recently, there has been a fair amount of interest at SLAC in using EGS4 to simulate the production and transport of background radiation in the large detectors associated with the SLC project. To provide an example of how one can accomplish this, an EGS4 User Code was created** to simulate a one-meter hydrogen bubble chamber. Figure 32.15 is a 1-GeV photon-initiated EM cascade in the SLAC 40-inch (one meter) LH2 bubble chamber, consisting of an iron cylinder 2.5-cm thick containing liquid hy• drogen, and with a 0.5Xo Pb slab at the center. The photon enters the chamber from the bottom where it strikes the plate at 90 0 • The magnetic field strength is 20 kGauss, and is applied along the cylinder axis. Bremsstrahlung is produced rather dramatically into the forward direction. One can easily identify the following interactions in this pic• ture: bremsstrahlung, pair production, Compton scattering, both Bhabha and· Mfliller delta rays, and ionization loss.

The same set of statistics are provided in Fig. 28.16, but this time only the charged particles are visible-which is what one sees in real bubble chamber pictures.

* The EGS4 User Code: UCGAMNSR. ** The EGS4 User Code: UPBUBBLE. 28. High-Energy Physics Applications of EGS 619

Figure 28.15. Hydrogen bubble chamber: A single 1-GeV photon strikes a 0.5Xo Pb slab from the bottom at 90°. All charged particles (solid) and photons (dots) are shown.

Figure 28.16. Hydrogen bubble chamber, but only charged particles (solid) are shown. 620 A. Del Guerra and W. R. Nelson

REFERENCES 1. W. R. Nelson, H. Hirayama and D. W. O. Rogers, "The EGS4 Code System", Stanford Linear Accelerator Center report SLAC-265 (1985). 2. Y. S. Tsai, "Pair Production and Bremsstrahlung of Charged Leptons", Rev. Mod. Phys. 46 (1974) 815. 3. D. F. Crawford and H. Messel, "Energy Distribution in Low-Energy Electron• Photon Showers in Lead Absorbers", Phys. Rev. 128 (1962) 2352. 4. H. H. Nagel, "Die Berechnung von Elektron-Photon-Kaskaden in Blei mit Hilfe der Monte-Carlo Methode", Inaugural-Dissertation zur Erlangung des Doktor• grades der Hohen Mathematich-Naturwissenschaftlichen Fakultat der Rhein• ischen Freidrich-Wilhelms-Universitat zu Bonn, 1964; "Elektron-Photon-Kaska• den in Blei", Z. Phys. 186 (1965) 319. 5. U. Volkel, "Elektron-Photon-Kaskaden in Blei fur Primarteilchen der Energie 6 GeV", DESY report DESY 65/6 (1965); "A Monte-Carlo Calculation of Cascade Showers in Copper Due to Primary Photons of 1 Ge V, 3 Ge V, and 6 Ge V, and 6-GeV Bremsstrahlung Spectrum", DESY report DESY 67/16 (1967). 6. H. Messel and D. F. Crawford, Electron-Photon Shower Distribution Function, (Pergamon Press, Oxford, 1970). 7. R. L. Ford and W. R. Nelson, "The EGS Code System: Computer Programs for the Monte Carlo Simulation of Electromagnetic Cascade Showers (Version 3)", Stanford Linear Accelerator Center report SLAC-21O (1978). 8. S. Iwata, "Calorimeter", Nagoya University Department of Physics report DPNU- 13-80 (1980). 9. J. Engler, "Perspectives in Calorimetry", Nucl. Instr. Meth. 235 (1985) 301. 10. P. Blum, H. Guigas, H. Kock, M. Meyer, H. Poth, U. Raid, B. Richter, G. Backen• stoss, M. Hasinoff, P. Pavlopoulos, J. Repond, 1. Tauscher, D. Troster, 1. Adiels, I. Bergstrom, K. Fransson, A. Kerek, M. Suffert and K. Zioutas, "A Modular NaI(Tl) Detector for 20-1000 MeV Photons", Nucl. Instr. Meth. 213 (1983) 251. 11. D. F. Anderson, G. Charpak, W. Kusmierz, P. Pavlopoulos and M. Suffert, "Test Results of a BaF2 Calorimeter Shower with Wire Chamber Readout", Nucl. Instr. Meth. 228 (1984) 33. 12. U. Amaldi, "Fluctuations in Calorimetry Measurements", Physica Scripta 23 (1981) 409. 13. A. Abashian, J. Bjorken, C. Church, S. Ecklund, L. Mo, W. R. Nelson, T. Nuna• maker, P. Rassman and D. Scherer, "Search for Neutral, Penetrating, Metastable Particles Produced in the SLAC Beam Dump", presented at the Fourth Moriond Workshop on Massive Neutrinos in Particle and Astrophysics, LaPlagne, France (15-21 January 1984). 14. A. J. Cook, "Mortran3 User's Guide", SLAC Computation Research Group tech• nical memorandum CGTM 209 (1983). 15. H. Hirayama, W. R. Nelson, A. Del Guerra, T. Mulera and V. Perez-Mendez, "Monte Carlo Studies for the Design of a Lead-glass Drift Calorimeter", Nucl. Instr. Meth. 220 (1984) 327. 16. 1. E. Price, "Drift-Collection Calorimeter", Physica Scripta 23 (1980) 685. 17. H. G. Fisher and O. Ullaland, "A High Density Projection Chamber", IEEE Trans. Nucl. Sci. NS-27 (1980) 38. 18. T. Mulera and V. Perez-Mendez, "Observation of Large Saturated Pulses in Wire 28. High-Energy Physics Applications of EGS 621

Chambers Filled With Ar-C02 Mixtures", Nucl. Instr. Meth. 203 (1982) 609; (see references therein). 19. M. Conti, A. Del Guerra, R. Habel, T. Mulera, V. Perez-Mendez, G. Schwartz, "Use of a High Lead Glass Tubing Projection Chamber in Positron Emission Tomography and in High Energy Physics", Nucl. Instr. Meth., A225 (1987) 207. 20. T. Mulera, V. Perez-Mendez, H. Hirayama, W. R. Nelson, R. Bellazzini, A. Del Guerra, M. M. Massai, "Drift Collection Calorimetry Using a Combined Radiator and Field Shaping Structure of Lead Glass Tubing", IEEE Trans. Nucl. Sci. NS-31 (1984) 64. 21. D. Hitlin, J. F. Martin, C. C. Morehouse, G. S. Abrams, D. Briggs, W. Carithers, S. Cooper, R. Devoe, C. Friedberg, D. Marsh, S. Shannon, E. Vella and J. S. Whitaker, "Test of a Lead-Liquid Argon Electromagnetic Shower Detec• tor", Nucl. Instr. Meth. 137 (1976) 225. 22. A. Del Guerra, M. Conti, G. Gorini, P. Lauriola, P. Maiano and C. Rizzo, "En• ergy Resolution Measurements of a Lead Glass Drift Calorimeter Prototype", presented at the IEEE Nuclear Science Symposium, San Francisco, 21-23 Octo• ber 1987. 23. K. Kleinknecht, Detectors for Particle Radiation, (Cambridge University Press, Cambridge, 1986). 24. D. H. Perkins, Introduction to High Energy Physics (Addison-Wesley Publishing Co., Menlo Park, CA, 1987). 25. P. A. Aarnio, A. Fasso, H. J. Moehring, J. Ranft and G. R. Stevenson, "FLUKA86 User's Guide", CERN Divisional report TIS-RP /168 (1986). 26. P. A. Aarnio, J. Lindgren, J. Ranft, A. Fasso and G. R. Stevenson, "Enhance• ments to the FLUKA86 Program (FLUKA87)", CERN Divisional report TIS• RP /190 (1987). 27. J. Ranft, H. J. Mohring, T. M. Jenkins and W. R. Nelson, "The Hadron Cascade Code FLUKA82: Setup and Coupling with EGS4 at SLAC", Stanford Linear Accelerator Center report SLAC-TN-86-3 (1986). 28. K. C. Chandler and T. W. Armstrong, "Operating Instructions for the High• Energy Nucleon-Meson Transport Code, HETC", Oak Ridge National Labora• tory report ORNL-4744 (1972). 29. F. S. Alsmiller, T. A. Gabriel, R. G. Alsmiller, "Hadron-Lepton Cascade Calcu• lations (1-20 GeV) for a Pb-AI-Lucite Calorimeter", Oak Ridge National Labo• ratory report ORNL/TM-9153 (1984). 30. J. Ranft and W. R. Nelson, "Hadron Cascades Induced by Electron and Photon Beams in the GeV Energy Range", Nucl. Instr. Meth. A257 (1987) 177. 31. G. E. Fischer (for the SLAC Staff), "SLC-Status and Development", Stanford Linear Accelerator Center report SLAC-PUB-4012 (1986). 32. T. M. Jenkins and W. R. Nelson, "Unique Radiation Problems Associated with the SLAC Linear Collider", Stanford Linear Accelerator Center report SLAC• PUB-4179 (1986); CONF 8602106, Invited talk at the Midyear Symposium of the Health Physics Society (Reno, Nevada, February 8-12, 1987). 622 A. Del Guerra and W. R. Nelson

33. S. Ecklund and W. R. Nelson, "Energy Deposition and Thermal Heating in Materials Due to Low Emittance Electron Beams", Stanford Linear Accelerator Center collider note CN-135 (1981). 34. W. R. Nelson and T. M. Jenkins, "Temperature Rise Calculations for the Beam Pipe in the SLC Arcs", Stanford Linear Accelerator Center collider note CN-235 (1983). 35. W. R. Nelson and T. M. Jenkins, "Temperature Rise in Iron Beam Position Monitors", Stanford Linear Accelerator Center collider note CN-276 (1984). 36. J. Schwinger, "On the Classical Radiation of Accelerated Electrons", Phys. Rev. 75 (1949) 1912. 37. W. R. Nelson, G. J. Warren and R. L. Ford, "The Radiation Dose to the Coil Windings and the Production of Nitric Acid and Ozone from PEP Synchrotron Radiation", Stanford Linear Accelerator Center PEP note PEP-109 (1975). 38. W. R. Nelson and J. W. N. Tuyn, "Neutron Production by LEP Synchrotron Radiation Using EGS", CERN internal report CERN-HS-RP /037 (1979); also distributed as LEP Note 187 (1979). 39. T. M. Jenkins and W. R. Nelson, "Synchrotron Radiation in the Collider Arcs", Stanford Linear Accelerator Center collider note CN-69 (1981). 40. C. Yamaguchi, "Absorbed Dose and Energy Deposition Calculation Due to Syn• chrotron Radiation from PETRA, HERA and LEP", DESY Internal report DESY D3/38 (1981). 41. T. M. Jenkins and M. Hofert, "The Effects of Synchrotron Radiation from the Wigglers", CERN internal report CERN-TIS-RP /IR/85-23 (1985). 42. J. A. Rawlinson, A. F. Bielajew, D. M. Galbraith and P. Munro, "Theoretical and Experimental Investigation of Dose Enhancement Due to Charge Storage in Electron-Irradiated Phantoms", Med. Phys. 11 (1984) 814. INDEX

A at rest, 171, 300, 341, 471 in-flight, 171, 177, 300, 341, 470, 478 accelerator (simulation), non-collinear, 473,482 beam monitor, 455, 461, 462 point-spread function, 474 beam pipe, 617 array processor, 554, 570 secondary beam, 614 artefact (see step-size) shielding, 402 atmosphere (modeling), 272 target, 455-459, 615-617 atomic binding (see binding) therapy machine, Chapter 21 attenuation coefficient (see coeffi- Varian Clinac-20, 457-459 cient) Varian Clinac-35, 456, 459-461 Auger electron, 154, 155, 171, 173, ACCEPT (see codes) 177,251,300,336,363-377,538 adjoint (see Monte Carlo) AUSGAB (see scoring, EGS4) albedo, 155, 264 AVEREL (see codes) analog (see Monte Carlo) AVERMV (see codes) angle, Awall (see wall effects) bremsstrahlung, 170, 340, 616 characteristic (screening) angle, 24, 25, 158-160 B correlation, 123 back-projection algorithm, 481,482 cutoff,35-37, 42,43,53, 159,440,441, backscatter, 135, 310, 414, 462 590 electron, 44, 136, 185, 195, 196, 242, Compton, 590-592 243, 264, 265, 271-274, 535 Rayleigh, 590, 591 photon, 99, 100, 229-232, 266, 337, distribution, 574, 581 Compton, 154, 575, 576 Barkas-Bloch corrections, 63 electron, Chapter 2, 6, 7, 140, 156- beam, 163, 170-172, 187, 196, 197, 328- collimator, 74, 415, 416, 453-467,552, 330, 337, 340, 365, 478, 494, 495 554 photon, 154-156, 177,213,340,478, electron filter, 467 555 hardening, 148, 454, 457, 499 sampling, 32, 33, 46, 337 pinching, -267-269 scoring, 174, 457, 459 position monitor (BPM), 617 therapy machine, Chapter 21 shaping, 316, 453-461, 499, 550, 552- mean deflection, 157 558 multiple scattering, 116, 117, 120, correction factor, 553-556 123, 616 benchmark, 8, 21, 127, 129, 421, 427, pair production, 154, 340, 616 430-433 photoelectric, 337, 537 EGS4, Chapter 13, 287, 289, 293-297 scaled, 32 electron beam, 542 annihilation, 153, 222-225, 234, 235, external fields, 427-433 288, 310, 441, 463, 469, 478, 508, ETRAN, Chapter 8 509 interface, 524-526, 535-542 623 624 INDEX

benchmark (continued) graphics, 618, 619 ITS, Chapter 11 positron, 99 pencil beam, 450 production, 187, 265, 267, 270, 272, POEM, 539 281, 315-317, 488 transport equation, 423 reflection, 99, 100 treatment planning, 450 sampling, 170 BETA (see codes) spectrum, 177, 213-216, 221-235, beta ray, 242-244, 361, 368-372, 375- 315-319, 340, 440, 499, 500, 562 378 thick target, 213-216, 315-319, 497, absorber, 308-310 499 lineal energy, 368, 374 thin-target, 498 RBE value, 361, 378 transmission, 99, 100 spectrum, 362, 368-371, 374,477 yield, 99, 100, 241, 318 track structure, 367-373 broad beam (also see geometry), 6-8, Bethe-Heitler (see cross section) 17, 145, 149, 168, 169, 175, 176, Bhabha (see cross section) 186, 204, 223, 224, 230, 234, 235, biasing (see variance reduction) 308, 312, 315, 317, 330-336, 339 binary-encounter theory (see colli- bubble chamber (see detector, simula• sion, binary) tion) binding, 44-53, 154, 191, 251, 543 buildup, 146, 265, 412, 444, 457, 467, atomic, 69, 103-105, 127, 154, 155, 486, 499, 534, 538, 564, 566 171, 177, 331, 335, 336 cap, 527, 528, 532, 533 chemical, 67, 68 correction (form factor), 154, 177 c molecular, 31, 60, 106 block (see beam, shaping) calorimeter (see detector, simulation) Blunck-Leisegang (see energy, loss) cascade shower (see electromagnetic Blunck-Westphal (see energy, loss) cascade or hadron cascade) Born approximation, 22, 23, 47, 48, cavity, 7, 74, 410, 485-488, 552 63, 71, 76, 81-85, 89, 103, 107, theory, 311, 485, 486, 523 108 Bragg-Gray, 73, 485-487, 523, 524, Borsch-Supan (see energy, loss) 527, 531, 533-535, 543 boundary crossing (see geometry) Spencer-Attix, 73, 118, 487, 488, Bragg additivity, 66 491, 493, 496, 499, 524, 531 Bragg-Gray (see cavity, theory or stop• wall (see wall effects) ping power, ratio) Cerenkov (see radiation) bremsstrahlung, 6, 135, 139-146, 153, channeling, 173, 604 169-171,175,177,183,238,240, characteristic x-ray (see fluorescence) 310, 324, 325, 330-338, 409-412, charge deposition, 172, 174, 187, 199, 417, 424, 438, 440, 454, 455, 463, 200, 259, 266, 269, 278, 323 492, 532, 577, 613 charged particle equilibrium, 118, (also see cross section, bremsstr• 265,410,457,486, .519, 531, 533, ahlung) 539, 552, 564-566, 573, 577, 591 angle, 170, 340, 616 Chechin-Ermilova (see energy, loss, angular distribution, 177 straggling) beam, 241, 242, 499 chord length (mean), 114,487 benchmark, 90-96 Class I (or II) (see electron transport, converter, 241, 242, 271, 279 algorithm) efficiency, 100, 102 Co-60, 10, 74, 461-465, 554,559-564 EGS4, 300 collimator, 416, 462, 463 energy deposition, 330, 335, 336 electron contamination, 416 escape, 222, 223 filter, 416 INDEX 625

Co-60 (continued) 271, 280 simulation, 415, 537-540 ACCEPTP, 252 teletherapy unit, 453-455, 461-464, applications, Chapter 11 467, 549, 554, 556 coupling, 272 codes, CYLTRAN, 6,8,142,153,250-254, AVEREL, 494 264-269, 272, 275-277, 327-340, AVERMV, 496,498 404, 535-538 BETA,260 CYLTRANM, 250-252, 266-273, coupling, 272, 404, 614 279-281 EGS4, Chapter 12, 5-8, 12, 17, 21, 32, CYLTRANP, 250-252 167, 139-142, 169, 201, 202, 260, EZTRAN, 249,250 265, 316, 408-412, 422, 429-431, SPHEM, 250, 251 450, 455, 494, 524-526, 529-537, SPHERE, 250, 251 540, 543, 554, 555, 564, 573, 580, TIGER, Chapter 10, Chapter 11,8, 599 21, 153, 188, 402-404, 539, 542 applications, TIGERP, 250-252, 265, 271 high-energy physics, Chapter 28 MCEF, 440 positron emission tomography MCNP, 260 (PET), Chapter 22 Messel and Crawford shower code, 287 benchmark, Chapter 6, Chapter 13, MFK, 233, 234 293-299 MOCA, 567 Distribution Tape, 289-292, 297, MORSE, 301, 389, 397, 398, 402, 403 301, 393-397 ONETRAN, 260 ETRAN comparison, Chapter 14 PTRACK, Chapter 16,578 FLUKA87 (coupling), 404,614 SANDYL, 153, 251, 260 GEOMAUX, 393 SHIELDOSE, 235, 237 PEGS4, 289-293, 299, 301 SHOWER, 288, 289 SHOWGRAF,302 TBI3DMC, 588-597 ETRACK, Chapter 16, 577 UGS77, 302, 479 ETRAN, Chapter 7, 1, 5-8, 24, 32- VPHOT,14 46, 59, 60, 73, 98-100, 140, 142, Zerby-Moran (ORNL) shower code, 250, 251, 260, 263, 265, 310, 316, 288, 296-299 402-404, 440, 450, 493, 536-538, coefficient, 540-543 attenuation, 57, 154, 471, 473, 508- applications, Chapter 9 510, 563, 575, 600 benchmark, Chapter 8 effective, 559 COMBIXD, 173 electron, 538 DATAPAC, 173 energy absorption, 454, 523 EGS comparison, Chapter 14 energy transfer, 515 ZTRAN, 174, 176, 187, 212, 238- homogeneity, 234 243,539 inelastic scattering, 51 FLUKA, 389, 396, 397, 402, 404, 613, parentage, 106 614 reflection, 187, 198 HETC, 614 transmission, 60,187-190 HFI, 233, 234 coherent scattering (see scattering) ITS (Integrated TIGER collimator (see beam) Series), Chapter 10, 5, 142, 153, collision, 307, 323, 327, 336-341, 402-404, binary, 103, 104 422, 430, 431 close, 103, 105 ACCEPT, 153, 188, 237-240, 250- density, 563 254, 257, 268-273, 397, 404 distant, 103-105 ACCEPTM, 250-252,257,258,270, elastic, 4, 421, 424, 425, 429 626 INDEX

collision, (continued) UNIVAC, 252 first, 551, 558-561, 567-569 VAX, 5, 14-16, 177, 252, 254, 292, hard, 47, 363, 365, 366 338-340, 543, 574, 594, 596 inelastic, 421, 424, 427-429 VAX 11/750,367,589,591,594,596 knock-on, 103,139-146,153,171-177, VAX 11/780, 5, 15, 16, 415, 531, 324-326, 338, 524 535,550 number, 21, 37-39, 153, 156 VAX 880,15 soft, 47, 363, 365, 366 parallel processing, 574 combinatorial geometry (see geome- performance rating (Dongara), 177 try) processor (see processor) COMBIXD (see codes, ETRAN) scaler, 15 command processor (see processor) single board (SBC), 16 compensator (see beam, shaping) super, 594 Compton (see scattering, Compton or time, 5, 155, 156, 170, 176-177, 237, cross section, Compton) 238, 260, 338-340, 378, 509, 574, computer, 5, 10, 14-17, 121, 129, 135, 575, 588-594, 596 173,174,289,292,395,439,454, transputer, 16 467, 482, 491, 492, 531, 534, 535, vector, 14-16,593 538, 543, 550-552, 569, 570, 605- condensed history (see Monte Carlo) 607 contamination (electron), 307, 311, codes (see codes) 312, 412, 416, 453, 454, 457-461, mini-super, 574, 589, 594, 595 499, 500, 564 models, continuous-slowing-down approxi- ALLIANT, 252 mation (see slowing down, csda) AMDAHL, 252 conversion, APOLLO, 252 efficiency, 293, 294, 474-479 CDC, 251, 252, 254, 255 factor, 504, 511, 515 CDC-7600, 14 fluence-to-dose, 315, 415, 504, 511, CRAY, 14-16,252,254 515, 516 CRAY XMP/2, 14 ionization-to-dose, 307, 313, 454, 564 CRAY-2,15 convolution, 187, 209-211, 221, 226, CYBER, 177, 205, 237 231-234, 243, 244, 444, 453, 454, CYBER-205, 14 552-556, 567, 616 DECnet, 598 Trombka, 222, 226 Edinburgh Concurrent Supercom- coupling (see codes) puter, 16 CPE (see charged particle equilibrium) ELEXSI, 252 critical energy (see energy) FPS-264, 15 cross section, FPS M64/30, 591, 595, 596 Bethe-Heitler, 81-83, 85, 88, 89, 98, HP (Hewlett-Packard), 252 99, 154, 169, 170, 241, 242, 340, IBM, 5, 15, 173, 177, 252, 254, 389 351, 508 IBM 3081, 173, 177, 238, 482 Bhabha, 61, 164, 171, 177, 300, 341, IBM-3083, 594 618, 619 IBM 3090-200, 15 bremsstrahlung, Chapter 4, 57, 58, IBM-370/195, 15 169, 173, 184, 337, 341, 440 ICL,5 DBMO theory, 82, 83, 87, 91, 99, MicroVAX, 15, 16 169 MULTIFLO/TRACE, 252 electron-electron, Chapter 4, 169 NAS, 252 electron-nucleus, Chapter 4, 169 NxFPS-264, 15 Hang, 89, 169 PDP-11/34, 367, 378, 596 integrated, 96-98, 102 T800 (transputer), 16 Jabbur-Pratt, 84-86 INDEX 627

cross section, photon, 57, 575 bremsstrahlung (continued) fractional, 575, 576 positron-electron, 82 Rutherford, 22, 23, 26, 103, 157, 158, positron-nucleus, 99-102 163 Schiff,91 scattering (Spencer), 264 Tseng-Pratt, 82, 83, 86, 89-98, 169 single scattering, 24, 34, 35, 41-43, uncertainty, 88, 90, 100 158, 161, 165 CEPX, 260 tables, 57 characteristic x-ray (see fluorescence) total (shell-by-shell), 103 close collision, 103, 104 transport, 159, 160, 167 coherent (Rayleigh), 57, 155, 177 crystal structure, 31, 32, 70 Compton scattering, 336, 575, 580 csda (see slowing down) distant collision, 104, 105 cutoff angle (see angle) elastic scattering, 22-35, 42, 53, 57, cutoff energy (see energy) 81, 159-163, 171, 173, 177, 251, CYLTRAN (see codes, ITS) 265, 364 electron, 362, 363 o electron-electron, 83 electron collision, 362-366, 370, 378 DATAPAC (see codes, ETRAN) electron-impact, Chapter 4, 103-109, DBMO (see cross section, bremsstrah• 177, 350 lung) excitation, 350-352, 3.56 deep penetration (see variance reduc• ionization, 103-109, 153, 177, 350- tion) 352 delta ray (see secondary electron) energy loss (also see stopping power), density effect, 10, 45, 61-63,69-74,77, 60, 163-168 105, 163, 184, 300 excitation, 48, 49, 350-352, 364 Sternheimer, 69, 70 incoherent, 57, 154, 177 Sternheimer-Peierls, 184, 251, 493 inelastic scattering, 35-38, 51, 57, 59, density (effective), 551,562-565 75, 310, 335, 341, 362, 363, 613 depth dose (see dose) inner-shell, 103, 105, 106 depth straggling (see straggling) ionization, Chapter 4,48-50,103-109, detector, 350-352, 363, 364 (also see cross efficiency, 221-224, 228,235, 308-310, section, photoionization) 471 Klein-Nishina, 4, 154, 507, 575, 576, properties, 473, 602 580 resolution, 221-229, 234, 295, 296 knock-on collision, 103 intrinsic, 211, 473, 482 Moliere, 23-25, 32-35, 39-42, 44, 126- simulation, Chapter 9, Chapter 25, 128, 159-163, 300 187, 209-211, 308-311 Moliere-Mott, 159 BGO scintillator, 265 Mjilller, 49, 50, 53, 61, 104-108, 163, bubble chamber, 119, 618, 619 171, 300, 341, 440, 492 calorimeter (shower), 394, 601-614 collision, 362, 365 calorimeter (thermal), 542 graphics, 618, 619 diode, 209-210, 266-270, 279, 310 momentum transfer, 159 Ge, 229-235, 308 Mott, 22, 23, 29, 103, 158-163, 351 ion chamber, 73, 118-122, 125,311, multigroup, 260 410, 412, 527-535 pair production, 57, 184 liquid scintillator, 242-244 photoelectric, 47, 57, 64, 184 NaI, 211, 221-228, 295, 296, 308- photoionization, 104-105, 177 310 molecular, 106 PET, Chapter 22 total, 106-108 plastic scintillator, 293, 294 628 INDEX

detector, directional, 512, 513, 515, 519 simulation (continued) distribution, 504, 515, 519 proportional chamber, 374 spatial, 512-515 specular reflection, 268 effective, 503-507, 510-512, 517, thin film, 264 519 TLD, 265-269, 275, 279, 337, 537, index, 513, 515 538 maximum, 513 wire chamber (MWPC), 473-482, mean, 506, 513, 515 602-611 path length, 551 wall (see wall effects) weighting factor, 503, 506 detour factor (see range) fluence-to-dose, 315, 415, 504, 511, dielectric, 515,516 constant, 105 ionization-to-dose, 307, 313, 454, 564 polarization, 105 isodose contour, 437, 438, 441-446 response, 47, 65, 69, 74, 77 low level, 376 diffraction, 31-34, 38, 173 maximum, 488, 492, 496 diffusion, 6, 488 mean, 485, 506 equation, 450 peak,415 free radical, 375, 377 ratio, 496, 497 gamma ray, 4 RFMS, 568, 569 diode (see detector, simulation) risk (linear), 504 dipole, scatter integration method, 551, 552 moment, 89 scoring, 6, 414, 417, 525, 526, 535, oscillator, 47, 64, 65, 69 539,542 discrete ordinate methods, 260 surface, 312, 317, 333, 335, 412-415, dissociation (molecular), 348 4.53,459 DNA,375-378 TFS, 568 dose, Chapter 25, Chapter 27, 6, 10, 59- treatment planning, Chapter 26 62,71-77,375,377,438,453,454, dose spread array (DSA), 554, 558, 457-460, 467, 485, 494, 503, 519 566-569 average, 361, 373 dosimeter response, Chapter 25 (also boundary, 453, 540-542 see detector, simulation) central axis, 417, 438, 445, 450, 453, double strand break (dsb), 375-378 463, 551, 564, 565 drift chamber (see detector, simula• conversion factor, 485 tion (wire chamber» depth, 7-10, 17,59,60, 135, 142-145, DSA (see dose spread array) 167-169,172,175,176,185,188, 203-208, 235-239, 242, 307, 316, E 327, 346, 414-417, 437-445, 449, 450, 453, 454, 460, 467, 494-497, effective dose equivalent, 503-507, 500,577 510-512, 517, 519 electron, 314, 315, 330-335, 339 efficiency, lCRU sphere, 512, 515, 519 computer (see computer, time) photon, 311-314 conversion, 474-479 distribution, 361, 437, 439, 441-444, detector, 221-224, 228, 235, 308-310, 535, 536, 541 471 radial, 368, 370, 372, 442-445, 449, photopeak, 222, 229, 230, 234, 266, 454, 504, 509, 553-558, 563, 565, 308,473 577,580 scintillator, 474 equivalent, 503, 504, 511, 512, 515, EGS4 (see codes) 519 EGS4MAC (see Mortran3, macro) ambient, 512, 513, 515 electric field (see electron transport) INDEX 629

electro disintegration, 173 energy, electromagnetic cascade, 59, 183, absorption coefficient (see coefficient) 201, 202, 265, 277, 287-289, 296, critical, 58, 167, 600, 602 297,302,394,404,407,440,599- cutoff, 6-8, 52, 53, 73, 105, 132-135, 607, 610, 613-618 140-149, 154, 155, 167, 170-174, central core, 616 288-293, 300, 337-340, 363-366, fluctuations, 296, 600-604, 607 370, 409, 410, 440, 441, 487, 491, lateral spread, 600, 603, 604, 613 492, 525, 531-535, 541, 559, 601, leakage, 295, 296, 601-607, 610, 611, 602,605 617 AE,6,7,142, 147,324,325,329,410, longitudinal spread, 601-606,610-613 525, 531, 532, 535, 541 maximum, 601 AP, 141, 147,410,531 median depth, 601 ECUT, 6, 7, 140, 145, 147, 149, 337, parameterization of, 613 339, 34~ 410, 525, 531, 541 radioactivity, 613, 614 ECUTKE,533,535 shower book (Messel and Crawford); PCUT, 410, 531 287 deposition, 9, 21, 51, 59, 75,174,187, shower counter, 288, 289 222, 242, 259, 264-281, 296, 323, temperature rise, 613, 616, 617 324, 333, 337, 361, 368, 373, 378, electron-impact (see excitation or ion• 409, 410, 441, 442, 485, 503, 509, ization) 551, 552, 558, 559 electron transport, bremsstrahlung, 330, 335, 336 algorithm, Chapter 5, Chapter 16, local, 440, 441, 486, 487, 559 156-173, 308, 311, 324-327, 348- near interface, 538, 541, 542 350, 518, 526, 529-532, 535 radial,441 Class I, 140, 324, 329, 338 scoring, 171, 172, 175,293,294,297, Class II, 140, 142, 324, 328, 329, 439,538 338,340 spectrum, 221-223, 231, 361, 374 electric field, Chapter 19, 251, 257, stochastic, 362 266, 268, 272, 273, 281, 301, 348, excitation (see mean excitation en• 408,474,618 ergy) constant, 423, 430, 431 loss, (also see stopping power), Chap• strong, 425, 427, 432 ter 3,21, 139, 143, 156-158, 163- time-dependent, 421 174, 409, 411, 42~ 426, 431 field-free, 421, 425-427 continuous, 6, 300 lateral, 429, 430 distribution, 44-46, 51-53, 71, 164- magnetic field, Chapter 19, 18, 251, 166,169-172,177,191 257, 267-273, 280, 301, 348, 408, fluctuations ( see energy, loss, strag- 474,618 gling) constant, 421, 423, 430, 431 fraction, 156 spiral, 424, 432 inelastic collision, 421, 428 strong, 425, 427, 429 large, 165, 185, 187 time-dependent, 421 maximum, 167 path-segment model, 156 mean, 45, 139, 141, 145, 148, 163- microdosimetry, Chapter 16 168, 326, 327, 444, 445, 448, 449 transport equation, 16, 172, 173, 425, most probable, 46, 71, 164, 165, 191, 488 192 vacuum, 421-423, 427, 428, 430, 431 multiple scattering, 131-133 weak-field limit (WFL), 423 PRESTA, 131-133 electronic equilibrium (see charged radiative, 96-102, 140-142, 169, particle equilibrium) 170,327 Elwert factor, 83, 85, 300, 316, 337 rate, 421 630 INDEX energy, F loss (continuecl) sampling, 46, 49, 51 factorization, 22, 26, 27, 35, 41 straggling, Section 7.2.2, 6, 21, 44- relative error, 22, 23 47, 51, 52, 59, 60, 77, 103, 140, Fano, 142, 145-148, 155, 175, 192-194, factor, 356 308, 310, 323-327, 330, 333-338, energy dependence, 356, 357 370, 440, 441, 459, 566 plot, 48, 351, 352 Blunck-Leisegang, 44-46, 51, 142, Spencer-Fano theory, 352 164-166,170-172,326,441 theorem, 118 Blunck-Westphal, 45, 46, 52, 164, FFT (see Fourier transform) 191 field flattener (see beam, shaping) Bohr, 163 filter (see beam, hardening or beam, Borsch-Supan, 45, 164, 167 shaping) Chechin-Ermilova, 46,47,51 Fischer (see photoelectric effect) Landau, Section 7.2.2, 6, 44-46, 51, flattener (see beam, shaping) 52, 142, 185, 326, 441, 603, 609, fluctuations (also see energy, loss, 611 straggling) radiative, 169 electromagnetic cascade, 296, 600, restricted, 441 602, 604, 607 Williams, 163 fluence, 145-147,174,444,563,565 Vavilov, 52 electron, 450, 486, 488 mean, 145, 148, 444, 445, 448, 449, disruption, 74, 485, 488, 523, 531, 486,494 538 most probable, 148, 444 distribution, 538 spectrum, 187, 195, 197, 553-555, 559, spectrum, 485, 489, 490, 537, 538 562-564, 570 matrix, 554, 555 scoring, 456, 457 planar energy, 570 therapy machine, Chapter 21 photon (disruption), 523 transfer, 103, 163, 165, 171 relative, 553-556, 563 coefficient ( see coefficient) spectra, 155, 323 distribution, 346-348 scoring, 171, 172 large, 89, 103, 163, 351 FLUKA (see codes) maximum, 141, 159, 163 fluorescence, 153-155, 171, 173, 177, small, 89, 163, 351 222, 223, 226-234, 251, 271, 300, ESTEPE (see track length, restriction) 336, 507, 538, 539 ETRACK (see codes) efficiency, 155, 171 ETRAN (see codes) form factor, 83, 154, 169, 177 event-by-event (see Monte Carlo) Fourier transform, 553, 554, 558 excitation, (also see cross section, exci- Fowler equation, 348, 352, 353 tation), Chapter 3, Section 7.2.2, fractional photon method, 509 50, 51, 81, 106, 109, 163 fragmentation model, 614 free radical, 361, 368, 375-377 electron-impact, Chapter 4, 350-352 energy (see mean excitation energy) chemical modifier, 377, 378 decay, 377 event,577 diffusion, 375, 377 molecular, Chapter 16, 49, 64, 67, 348 scavenger, 377 threshold energy, 60 expansion parameter, 33, 41 exponential transform (see variance G reduction) geometry, Chapter 17, 21, 188, 229, EZTRAN (see codes, ITS) 235, 236, 238 INDEX 631

geometry, (continued) 186, 229, 310, 409, 416, 463, 489, avoiding subprogram calls, 408 574, 577-584 beam, function, 444 PET, Chapter 22 boundary, 389, 392, 397, 400, 408- plane, 385, 386, 409, 412, 417, 418 410, 519, 591 semi-infinite, 416 crossing, 21, 123-126, 133, 134, 171, slab, 301 337-340, 408, 409 reduced interrogation (see variance re- box, 394 duction) broad beam, 230, 339, 415, 417, 439, reflecting planes, 417, 418 444, 449, 467, 493-497, 530, 534, shortest distance, 411 537, 538, 542, 552, 559, 569, 570 slab, 174, 186, 221, 238, 313, 315, 394, Cartesian, 396, 404 402, 409, 412, 414, 455, 525, 537, cone, 268, 301,404, 417,431,455,456 539, 565 conic surface, 386 semi-infinite, 415, 417 combinatorial, 273, 301, 323, 397-404 SMAX, 116-120, 525, 526, 531, 532 cylinder, 174, 186, 221, 288, 298, 301, spacecraft, 235-240 323, 389, 403, 404, 410, 417, 418, sphere, 239, 240, 301, 313, 361, 368, 431-433, 438, 456, 459, 534, 538, 373, 374, 404, 417, 444, 513-.514 554,570 sphere-cylinder, 514 cylinder-slab, 297 symmetry, 271, 416-418, (also see EGS4, User Code, U CCELL) auxiliary macros and routines, thin slab, 139-147 Chapter 17, 301 unit cell (see User Code, UCCELL) DELCYL, 389 voxel,396,441,567,568,578,585-597 DNEAR, 390, 409 GEOMAUX, 393 GEOMAUX, 393 Goudsmit-Saunderson (see scatter• HDWFAR,290-293,301,302,389-396, ing, multiple) 409 graphics, 251, 254, 257, 273, 290, 301, initialization, 389 302,479-481, 603, 604, 616, 618, equivalence (see variance reduction, color, 596 reciprocity) movie mode, 596 ETRAN, Section 7.3.2 SHOWGRAF, 302 factor, 513 tracks and events, 431-433,577,578 honeycomb structure, 475, 479, 607 Unified Graphics (UGS77), 302, 479 ICRU sphere, 505 inhomogeneity, 416, 417 H interface (see interface) interrogation reduction, 408, 409 hadron, irradiation, 510-520 cascade, 404, 613, 614 lattice, 417, 418 production, 173, 270, 614, 618 medical accelerator head, 455 hardening (see beam, hardening 01' ra- micrometer sphere, 361, 368, 373, 374 diation, hardening) narrow beam, (also see geometry, pen- Harder formula, 486, 487 cil beam), 230, 415, 437-439, 441, Hartree-Fock (see potential) 444, 449, 454, 494, 552, 555, 569 Haug (see cross section, bremsstrah- package, 456 lung) EGS4, 389-402, 479 HETC (see codes) FLUKA, 404,614 HFI ( see codes) flattening filter, 455 HISPET (see detector, simulation ITS, 323 (wire chamber)) patient, Chapter 27 homogeneity coefficient (see coeffi• pencil beam, Chapter 20, Chapter 26, cient) 632 INDEX HOWFAR (see geometry, EGS4) J Jabbur-Pratt (see cross section, brem• sstrahlung) ICRU, sphere, 313, 504, 512-515, 519 K tissue, 515 impact parameter, 47, 103-105 kerma, 515, 519, 552, 567, 568, 577, importance sampling (see variance 590, 591 reduction) air, 10, 345, 504, 510, 511, 527-532 inhomogeneity, 10, 70, 71, 77, 315, Klein-Nishina (see cross section) 316, 416, 417, 525, 550-553, 558, 559, 565, 566, 569, 570, 573, 577, L 589, 591 Landau (see energy, loss, straggling) boundary, 591 lateral, detector, 601 correlation, 123, 130, 133, 134 inner-shell, deflection, 120-134, 187 cross section, 103, 105, 106 displacement, 123, 130-133 ionization, 103, 105, 106 scattering, 145 Integrated TIGER Series (see codes ITS) , spread, 145 transport, 429, 430 interaction forcing (see variance re• leading-particle biasing (see variance duction) reduction) interface, 410, 565, 591 (also see wall leakage (see electromagnetic cascade) effects), Lindhard electron-gas model 63 65 air-tissue, 577, 591 liquid scintillator (see detecto;, bone-tissue, 577, 591 si~u- lation) . effect, Chapter 25, 31, 32 mterference effect, 31,32 ~on. ch~mber (see detector, simulation) M IOnIZatIOn, Chapter 3, Section 7.2.2, macro (see Mortran3) 47-51 magnetic field (see electron transport) cross section, Chapter 4, 48-50, 103- MCEF (see codes) 109 M CNP (see codes) electron-impact, Chapter 4, 103-109, mean excitation energy, 10, 45-51, 153, 177, 350-352 61-69,73-77,163,165,173 event, 577 assignment scheme (Thompson) 67 inner-shell, 103-106 68 ' , K-shell, 336, 338 compound, 66-68, 77 molecular, Chapter 16, 48-.51, 347 table, 62, 63 number distribution, 346-350, 354 MFK (see codes) ratio, 496 micro dosimetry, Chapter 16 51 stopping power, 106 mICrometer. sphere (see geometry)' threshold energy, 60, 75, 106, 348-352 MIKY (see random number generation) yield, 345-357 MIRD (see phantom, MIRD) fluctuation, 348, 356 MOCA (see codes) Spencer-Fano theory, 352 Moliere (see cross section or scattering, isodose (see dose) multiple) ITS ( see codes) characteristic (screening) angle, 24, I-value (see mean excitation energy) 25, 158-160 radius (or length), 600, 613 INDEX 633

Moliere (continued) angle, 154, 340, 616 screening approximation, 264 cross section, 57, 184 Moliere-Mott (see cross section) electron field (triplet), 154, 155 moment (central), 347,349,354-357 graphics, 618, 619 moments (method of), 24, 206, 208, partial-wave analysis, 22, 25-30, 37, 488 41, 43, 82, 83, 86, 99, 101 Monte Carlo, particle-in-cell (PIC) code, 270, 273 adjoint, 237, 260, 272 path length, Section 7.2.2, 24, 32-35, analog, 4,340, 407, 574,589, 593, 594, 39-47, 51-53, 139, 141, 146, 370, 605 371, 422, 426-429, 525 condensed history, 4, 21, 24, 33, 38, biasing (see variance reduction, expo• 115, 123, 124, 135, 143, 156, 183, nential transform) 324, 325, 408, 422, 425, 427, 524, correction, Chapter 5, 311, 338, 427, 542 526, 531, 532 event-by-event, Chapter 16 Berger, 121, 123, 130 microscopic, 425 curved, Cbapter 5 multigroup, 260 equivalent, 550-5.'52, 556 random walk, 21, 33, 38, 153, 156, maximum, 526 157, 183 mean, 444 termination of history, 335-337 photon, 538 vectorization, 14-16, 593 shortening (see variance reduction, ex• MORSE (see codes) ponential transform) Mortran3 straggling, 46, 51, 59, 370 macro, 291,292,297,301,393-396 stretching (see variance reduction, ex• EGS4MAC, 292, 297, 301, 395 ponential transform) pattern (template), 292,301 subdivision, 489 replacement, 292 path-segment model (see Monte string processor, 291, 292, 389, 393, Carlo, condensed history) 395 PEGS4 (see codes, EGS4) M~ller (see cross section) pencil beam, Chapter 20, Chapter 26, Mott (see cross section) 578-582, (also see geometry) muffin-tin model, 26, 29, 30 perturbation, 421, 425-427, 431, 432, multiple scattering (see scattering) 444, 485, 550, 551 MWPC (see detector, simulation (wire PET, Chapter 22 chamber» phantom, 8, 10, 440, 444, 553, 55/J, 564 MIRD, 50,1-514, 519 N quarter infinite, 565 sphere (see ICRU , sphere) N aI (see detector, simulation) surface, 494 narrow beam (see geometry) tissue equivalent, 553 neutron (see hadron) water, 446, 490, 4%, 550, 567, 574- 58.5, 591 o semi-infinite, 445 phase, ONETRAN (see codes) condensed, 65, 67, 68 oscillator strength (see dipole) shift, 22, 23, 82, 87, 88, 159-161, 163, 169, 177 p PHESCE (see variance reduction) photoelectric effect, 47, 57, 64, 86, pair production, 57, 84, 153-155, 177, 153-155, 177,222,300,336,362, 222, 300, 440, 507-509, 562, 581, 440, 507, 508, 537, 539, 574, 575, 613 581, 588 634 INDEX

photoelectric effect, (continued) string (Mortran3), 291, 292, 301, 389, angular distribution, 337, 537 393,395 Fischer distribution, 155, 336 protocol (dosimetry), 485 Sauter distribution, 155, 336, 337 AAPM, 73, 313, 493, 528 photoionization (see cross section) IAEA,497 photon transport, NACP, 313 algorithm, 154-156, 308, 506-509, proton (see hadron) 574-578 proximity function, 368, 372-376 time-dependent (CYLTRAN), 272, PTRACK (see codes) 273,281 photoneutron (see hadron) Q photopeak efficiency, 222, 229, 230, 234,308 quality factor, 503, 506 pi-zero decay, 302, 613 quark model, 614 PLC (see path length, correction) plural scattering (see scattering) R polarization, 25, 62, 69 dielectric, 105 radiation, of medium (see density effect) Cerenkov,272,602 positron, damage, 264, 270, 617 bremsstrahlung, 99 effect, 361 energy loss, 43, 61, 62 electronics, 250, 269, 539 scattering, 25-29, 33, 42 hardening, 269 target, 615-617 length, 600, 610, 613 positron-electron, processing, 241 difference, 81, 102, 164, 171, 288, 310, safety, Chapter 24, 270 341 shielding, 581 stopping power, 61,62 detector, 174 positron emission tomography lung, 597 (PET), Chapter 22 space, 235-241 potential, space, 222-228, 235-241 Coulomb, 22-26, 29, 159, 160 synchrotron, 617-618 correction, 82-84, 87, 169, 600 transition, 272 Maxon-Corman, 89 radiative cross section (see cross sec• Hartree-Fock, 24-32, 38, 42, 43, 83, tion, bremsstrahlung) 106, 161, 169 radiative stopping power (see stop• repulsive, 27 ping power) solid state, 25, 29, 30, 38 radioactivity, spherically symmetric, 26, 31 decay,224 static, 22, 25 electromagnetic cascade, 613, 614 Thomas-Fermi, 23-32, 37, 39-43 hadron cascade, 614 PRESTA, natural, 234 step-size, Chapter 5, 8, 311, 314, 331, random number generation, 3, 4, 332, 335-340, 409, 410, 526, 531, 339, 362, 363, 366, 367 534,535 MIKY, 367,378-380 variance reduction, 409, 410 random walk (see Monte Carlo) processor, range, 6-9, 60-62, 73-76, 156, 172, 175, array, 554, 570 184, 243, 273, 310, 361 370-372, command, 409-413, 438, 439, 450, 487, 519, CDC UPDATE, 251, 255 523, 524, 532-535, 573, 577, 578 EGS4PL,302 csda, 184, 185,339,370,410,441,442, UPEML, 251-253 445, 449, 491, 525, 531 INDEX 635

range, (continued) 349,352 detour factor, 172, 337 SANDYL (see codes) effective, 467 Sauter (see photoelectric) heavy particle, 63, 64, 67 Scatter Air Ratio (SAR), 583 maximum, 410, 531 scattering, mean, 370 coherent (Rayleigh), 57, 155, 177,300, positron, 470, 474, 482 337, 507, 590, 591 practical, 333, 334, 337, 339, 438 Compton, 153, 154, 230, 300, 336, projected, 370 362, 440, 454, 463-466, 472, 480, rejection (see variance reduction) 482, 507-509, 524, 562, 573-581, residual, 172, 337, 491, 531, 532 588 secondary electron, 372, 574, 578 continuum, 224, 226, 229, 231 straggling, 370, 372, 450 multiple, 578-581, 590-594 straight, 370 graphics, 618, 619 table (curve), 60-62,75,76,368,577, incoherent, 57 578 factor, 89, 169 Rayleigh (see scattering, coherent or large angle, 426, 429 cross section, coherent) molecular, 31 RBE, 361, 377-378 multiple, Chapter 2, 59, 71, 153, 156, reciprocity (see variance reduction) 157, 183, 187, 308-311, 368, 408, reflection (see backscatter) 438, 4'10, 441, 444, 531, 543, 559, reflection coefficient (see coefficient) 560, 568, 569, 613 relativity effect, 22-26,32-34,41, 158, angle, 116, 117, 120, 123, 616 159 atomic electron, 329, 330 resolution, Bethe, 117, 120, 126, 328 detector, 221-229, 234, 295, 296 Compton, 578-581, 590-591 energy,600-604,607-614 EGS4 vs. ETRAN, Chapter 14 PET, 472, 473 electric field, Chapter 19 intrinsic, 211, 473, 482 Fermi-Eyges, 120, 123, 450, 532, spatial, 119, 491, 600, 601 552, 569, 570 PET, 469, 471-474, 479-482 Gaussian, 6, 338, 441 time (PET), 471, 472 Goudsmit-Saunderson, 6, 32-37, response, Chapter 25, (also see detec• 41-44, 120, 123, 126, 158-161, tor, simulation) 172, 328-332, 338 retarding force, 424-426, 428, 429, 431 magnetic field, Chapter 19 Russian roulette (see variance reduc• Moliere, 4, 6, 22-25, 32-35, 39-42, tion) 44, 117, 120-134, 158-160, 32- Rutherford (see cross section) 332, 441, 532 PRESTA, 124-136 s step-size artefacts, Chapter 5 turn off, 525-532 sampling, 153-156,166-171,177,408- water-slab benchmark, Chapter 6 414, nuclear, 440 angular distribution, 32, 33, 46 plural, 24, 29-32 Compton angle (Kahn), 575 Rutherford, 429 digital sampling, 609-611 single, 24, 34, 35, 41-43,1.56,165,169, EGS4 vs. ETRAN, 338 426, 429, 543 energy loss, 46, 49, 51 energy loss, 45 importance, 302 Spencer, 264 multiple (Compton) scattering, 593, Schiff (see cross section, bremsstrah• 594 lung) techniques, 4, 153-157, 167-170, 338, scintillator (see detector, simulation) 636 INDEX

scoring, 115, 116, 128, 130, 133, 154- 129, 135, 136, 139-141, 145, 156, 173, 410, 411, 414, 554, 555, 559, 158, 172, 175, 206, 208, 330-336, 568 348, 370, 371, 410, 440-442, 445, dose, 414, 417, 525, 526, 535, 539, 542 449, 488, 491, 492, 525, 531, 532, EGS4 (AUSGAB), 290-293, 298, 302, 577 389, 392, 609 inelastic, 427 energy spectrum, 456, 457 spectrum, 368, 371, 491 fiuence, 538 space radiation (see radiation) function, 413 specular reflection (see detector, sim• plane, 456-459, 463 ulation) region, 408, 411 speed (see computer, time) screening, 22-26, 29-31, 35, 81-83, 89, Spencer-Attix (see cavity, theory or 158-160, 169, 264, 477 stopping power, ratio) characteristic angle, 24, 25, 158-160 Spencer-Fano (see ionization, yield) nuclear, 89 Spencer scattering, 264 parameter, 36, 160 SPHEM (see codes, ITS) Wheeler-Lamb,89 SPHERE (see codes, ITS) secondary electron, 365, 554-562, spin effect, 22, 23, 26, 32-34, 41, 158, 573, 577, 578, 588, 591 159 creation, 324-330, 333, 338 splitting (see variance reduction) delta ray, 135, 136,361-368, 371-373, s-ratio (see stopping power, ratio) 432, 492, 524, 531-533, 540, 577, step-size, Chapter 5, 4, 6-8, 21, 38, 39, 603 42,44,46, 52, 156, 157, 162, 164, cross section, 440 169-173,177,187,188,331,332, graphics, 618, 619 337, 340, 408, 438, 440, 489, 491 dose, 577, 578, 591 artefact, Chapter 5 energy, 363, 366, 581 dosimeter response, Chapter 25 equilibrium (see charged particle equi- EGS4 vs. ETRAN, 337, 338 librium) electric field, 421-436 escape, 222, 223 interface effects, Chapter 25 mean energy, 365 magnetic field, 421-436 production, 362, 368, 486-488 major step, 156, 157, 164, 172, 173, spectrum, 106, 108, 351, 362, 365, 486 177 track, 368, 577, 579 PRESTA, Chapter 5, 8, 311, 314, 331, shaping (see beam) 332, 335-340, 409, 410, 526, 531, shell correction, 62-64 534, 5;)5 Bichsel, 63, 64 SMAX, 116-120, 525, 526, 531, 532 Bonderup, 63, 64 sub-step, 157, 158, 169-173 Walske,63 sterilization (see radiation, processing) shielding (see radiation) Sternheimer (see density effect) SHIELDOSE (see codes) stopping power, Chapter 3,10,34,45- shower (see electromagnetic cascade or 53, 58, 75, 76, 103, 106, 131, 135, hadron cascade) 139,141,156,163,169-175,184, SHOWER (see codes) 251, 310, 324-327, 335, 341, 350, SHOWGRAF, 302 374,441,471 single scattering (see scattering) (also see energy, loss) single strand break (ssb), 375-377 heavy particle, 63, 64, 67 skin dose (see dose, surface) mass (definition), 58 slowing down, 363, 409, 427, 432, (also radiative, 58-61, 96-102, 156, 171, see range, csda) 184, 300, 325, 326, 330 Bethe-Bloch, 425 positron, 100-102 csda, 34, 37, 38, 57-60, 100, 115, 126- INDEX 637

stopping power, (continued) 457, 458, 564 ratio, Chapter 23, 8, 10, 62, 73, 74, 77, Tissue Phantom Ratio (TPR), 564 203, 311, 313, 444, 454, 523, 531, TLD (see detector, simulation) 532, 535 tomography, 5 Bragg-Gray,486,487 CT, 454, 550, 574, 585-587, 592-596, inverse, 494 599 Spencer-Attix, 487, 488, 491, 493, PET, Chapter 22 496,499 single-ring, 471, 472, 474 restricted, 141, 300, 324-326, 440, SPECT, 469 441, 487, 543 TOFPET, 472 tables, 60-62, 75, 76 Total Body Irradiation (TBI), 573, uncertainty, 62, 63, 70, 76, 77 574, 581, 589, 595, 596 water vapor, 109 track, straggling, 265, 440 beta ray, 367-373 depth, 145 ETRACK (see codes) energy loss, Section 7.2.2, 6, 21, 44- graphics, 431-433, 577, 578 47, 51, 52, 58-60, 77, 140, 142, intertrack (and intra-) effect, 374-377 145-147, 175, 176, 192-194, 308, PTRACK (see codes) 310, 370, 440, 441 segment (see step-size) Bohr, 163 structure, Chapter 16, 577-579 restricted, 441 end,487,488 Williams, 163 track length, 73, 368, 604 path length, 46, 51, 59, 370 distribution, 73, 155, 171 straight-ahead approximation, 175, per unit volume, 444, 489 560 restriction, 609 string processor (see processor) ESTEPE,6, 7,115-123,127-131,135, subzoning (see geometry) 143, 331-340, 410, 477, 525, 526, superposition, 444, 446 529-532, 535, 536, 540 surface curvature, 10, 550, 552, 553, scoring, 171, 298, 299, 489 558, 564 transition radiation (see radiation) surface dose (see dose) transmission coefficient (see coeffi• symmetry (see geometry) cient) synchrotron radiation (see radiation) transport equation (see electron transport) T transputer (see computer) treatment planning, Chapter 26, 437, TBI3DMC (see codes) 440, 450 termination of history, 335-337 triplet production (see pair produc• therapy machine (see accelerator) tion, electron field) thin film (see detector, simulation) Tseng-Pratt (see cross section, brems• Thomas-Fermi (see potential) strahlung) TIGER (see codes, ITS) time (see computer) u tissue (see ICRU) Tissue Air Ratio (TAR), 463, 466, UGS77 (see codes or graphics) 582-586, 597 User Code (EGS4), 291, 292, 323, tissue-equivalent, 551-554 324, 336, 389, 390, 404 A-150 DOSRZ, 12, 534, 535 man, 504 INHOM, 525 plastic, 67, 71 UCBEND,18 sphere, 512 UCCAL2DW, 607 Tissue Maximum Ratio (TMR), UCCELL, 474-477, 480 638 INDEX

User Code, (continued) wire chamber (see detector, simula• UCCONEFl, 293 tion) UCEI37, 603 W-value, Chapter 15, 50, 51, 73, 74 UCEDGE, 300 definition, 345 UCGAMNSR, 618 energy dependence, 345, 352, 353 UCH20AL, 297 energy transfer distribution, 346 UCPET, 474, 477-482 ionization number distribution, 346 UCSAMPCG, 301,389,397 literature survey, 352 UCTESTSR, 290, 301 reciprocal, 356 UPBUBBLE, 618 XYZWRN, 396 x v x-ray, 362, 368, 373-378 characteristic (see fluorescence) variance reduction, Chapter 18, 155, electron track, 369, 372, 373 170,302,340,509,515,574,588- emission, 103 594, 605-607 escape, 222-230 deep penetration, 409, 412, 414 extraction efficiency, 271 DNEAR, 390, 409 filtration, 231, 234 exponential transform (path-length flash, 250, 267, 269, 272 biasing), 302, 412-414 RBE value, 378 geometric symmetry, 271, 417, 418 importance sampling, 302 z interaction forcing, 411, 412 leading-particle biasing, 302, 605-607 zone discard (see variance reduction) PHESCE, 538 ZTRAN (see codes, ETRAN) PRESTA, 409, 410, 531, 534 range rejection, 410, 411, 531, 534 reciprocity, 416, 417, 439 reduced interroga.tion of geometry, 409,410 Russian roulette, 302, 412-414 splitting, 302, 412-414 zone discard, 409 Vavilov (see energy, loss, straggling) vectorization (see Monte Carlo) voxel (see geometry) VPHOT (see codes) w wall effects, 10, 118, 243, 244, 295, 296, 311, 410, 487 (also see Chapter 12) wave function (see partial-wave analy- sis) wedge (see beam, shaping) weighting (see variance reduction) weighting factor (dose equivalent), 503,506 Weizsacker-Williams, 47-50, 81, 103, 107, 108