University of New Mexico UNM Digital Repository Nuclear Engineering ETDs Engineering ETDs 8-27-2009 Analysis of moment-preserving methods for charged particle transport Paul Smith Follow this and additional works at: https://digitalrepository.unm.edu/ne_etds Recommended Citation Smith, Paul. "Analysis of moment-preserving methods for charged particle transport." (2009). https://digitalrepository.unm.edu/ ne_etds/47 This Thesis is brought to you for free and open access by the Engineering ETDs at UNM Digital Repository. It has been accepted for inclusion in Nuclear Engineering ETDs by an authorized administrator of UNM Digital Repository. For more information, please contact [email protected]. thesis Analysis of Moment-Preserving Methods for Charged Particle Transport by Paul H. Smith B.S. Nuclear Engineering University of New Mexico, 2007 THESIS Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science Nuclear Engineering The University of New Mexico Albuquerque, New Mexico August, 2009 c 2009, Paul H. Smith iii Dedication I dedicate this work to my wife, Angelina, and two wonderful children, Elijah and Jenae, without whom I would not be myself. iv Acknowledgments I acknowledge the hard work of my advisor and mentor Anil Prinja. His ideas made this work possible and his guidance and wisdom kept me on track. I would like to thank him for the opportunities that he has given me, and I only hope that I’ve lived up to his expectations. I would also like to thank my friend and colleague, Lee Harding. He helped me countless times through this work, and it was the parallels to his own work that helped me realize my errors. In addition, I thank the entire UNM Chemical and Nuclear Engineering faculty, notably Bob Busch, Gary Cooper, Mohammed El-Genk, and Taro Ueki. I spent my undergraduate and graduate years learning from them, and their lessons won’t be forgotten soon. Finally, I would like to thank my mom, dad, and two brothers for their continued encouragement. Who know’s where I would be without them. v Analysis of Moment-Preserving Methods for Charged Particle Transport by Paul H. Smith ABSTRACT OF THESIS Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science Nuclear Engineering The University of New Mexico Albuquerque, New Mexico August, 2009 Analysis of Moment-Preserving Methods for Charged Particle Transport by Paul H. Smith B.S. Nuclear Engineering University of New Mexico, 2007 M.S., Nuclear Engineering, University of New Mexico, 2009 Abstract The simulation of electrons, protons, and other charged particles can be an expen- sive computation. On the one hand, the small mean free path of a charged particle makes direct simulations very costly. On the other hand, the cross sections are very singular making a deterministic calculation difficult to impossible. Nevertheless, the accurate computation of high energy charged particle densities is an important task for many engineering calculations. In this thesis, a variety of methods for computing these densities and distributions are assessed. Among these methods are the well-known Fokker-Planck and Fermi approximations. These methods are known to be particularly bad for electrons. It is shown in this thesis that they are in fact very poor for all charged particle transport. Specifically, it will be seen that the Fokker-Planck method and the Fermi vii approximation that can be derived from it allow for higher order spatial and angular modes to creep into the solutions for shallow and deep penetrations alike. On the other hand, the true physics only requires a few modes at deeper penetrations. Also, for shallow penetrations, most of the modes of the actual solution to the transport equation are incorrectly approximated by the Fokker-Planck approximation. It has been realized in recent years that the Fokker-Planck approximation is simply a leading order term in an asymptotic approximation. This realization with some clever manipulation of mathematical operators allows one to obtain higher order approximations to the scattering operator. The Fokker-Planck operator is then simply a specific case of this general framework. This general framework is explained in great detail. The inability of these higher order approximations to capture the true physics in some situations is assessed. Finally, hybrid methods will be introduced with parallels made to the well known Boltzmann Fokker-Planck approximation. It is seen that the higher order approx- imations used in place of the Fokker-Planck approximation can make the method more robust, more accurate, and more efficient. Also, parallels between the hybrid methodology and naturally occuring physics that is associated with very heavy ions incident on light targets will be seen. A general framework for approximating the solutions to these types of equations by using a lab frame analysis of the relevant physics will be introduced. viii Contents List of Figures xiii List of Tables xx Glossary xxi 1 Introduction 1 2 An Introduction to Transport Theory and Methods 4 2.1 The Transport Equation . 4 2.2 Analog Monte Carlo Methods . 9 2.3 The Condensed History Algorithm . 14 2.4 Moment Preserving Research Overview . 17 3 Interactions of High Energy Charged Particles 19 3.1 Elastic Scattering of Charged Particles off Atomic Nuclei . 22 3.2 Inelastic Scattering of Charged Particles off Atomic Electrons . 27 ix Contents 3.3 Transport Equation for Charged Particles . 31 3.4 Characterizing a “Forward Peaked” Differential Cross Section . 39 4 Fokker-Planck and Fermi Approximations 45 4.1 Notes on the Transport Cross Section . 46 4.2 The Fokker-Planck Operator . 49 4.2.1 Derivation of the Fokker-Planck Operator for Forward-Peaked Transport . 50 4.2.2 Accuracy of the Fokker-Planck Approximation . 52 4.2.3 Delta Function Representation of the Fokker-Planck Operator 56 4.2.4 Exponential Representation of the Fokker-Planck Operator . 60 4.2.5 Arbitrary Approximate Representations of the Fokker - Planck Operator . 62 4.3 Gaussian Distributions and the Fermi approximation . 65 4.3.1 Derivation of Fermi’s Approximation from the Fokker-Planck Equation . 66 4.3.2 Rossi’s Formulation of the Gaussian Approximation . 68 4.3.3 Accuracy of the Fermi Approximation . 71 4.4 Preliminary Conclusions . 79 5 Regularization and Moment Preserving Methods 82 5.1 Higher Order Fokker-Planck Expansion . 84 x Contents 5.2 Boltzmann Operator Regularization Procedures . 87 5.2.1 Regularization of Larsen and Leakes . 88 5.2.2 Generalized Fermi Expansion . 91 5.3 Moment Preserving Methods . 92 5.3.1 The Exponential and Discrete Models . 93 5.3.2 Relationship between Moment Preserving Methods and Renormalization of the DCS . 102 5.4 Numerical Experiments with Moment Preserving Methods . 107 5.4.1 Inelastic Scattering with Straight-Ahead Transport . 109 5.4.2 Monoenergetic Elastic Scattering . 114 5.4.3 Results for Elastic and Inelastic Combined Scattering . 123 5.5 Moment Preserving Summary . 126 6 Hybrid Methods 128 6.1 DCS Decomposition Methods . 129 6.1.1 Boltzmann Fokker-Planck Approximation . 130 6.1.2 Generalized Boltzmann Fokker-Planck Approximation . 132 6.1.3 A Smooth Decomposition of the DCS . 134 6.1.4 Spectral Analysis of the Singular DCS . 137 xi Contents 6.2 Numerical Experiments with Hybrid Methods . 141 6.3 Heavy on Light Ion Interactions . 149 6.4 A Look at Dose in Higher Dimensions . 155 6.5 Hybrid Methods Summary . 162 7 Conclusions and Future Work 165 A Differential Cross Section Derivation 167 A.1 Full Classical Treatment . 168 A.2 Nonrelativistic Quantum Mechanics . 172 A.2.1 Coulomb Interaction and Matter Waves . 172 A.2.2 General Potentials and the Partial Wave Expansion . 175 A.3 Relativistic Quantum Mechanics . 177 Appendices 167 References 181 xii List of Figures 2.1 A possible realization of an analog Monte Carlo particle history. 13 2.2 A typical type I condensed history energy step with a set of six substeps for sampling angular redistributions. 16 3.1 Macroscopic DCS for elastic scattering off of water molecules for high energy neutrons and protons. 20 3.2 Lab frame scattering cosine as a function of the center of mass frame scattering cosine for various α = Mi/Mt................. 25 3.3 Ratio of the Mott DCS σM to the Rutherford DCS σR for carbon and lead [1]. 27 3.4 Screened Rutherford parameters K(E) and η(E) for charged particles on tungsten metal . 33 3.5 Total cross section and transport cross section for particles incident on tungsten metal. 35 3.6 Comparison of the elastic scattering DCS given by Rutherford to the first four Legendre polynomials . 37 xiii List of Figures 3.7 Probability of a particle having at least on scattering event with θ > θ∗ for 1700 MeV protons incident on tungsten metal. 42 3.8 Variation of the probability p for fixed µ∗ for the relatively large angle scatters and problem specification given in Fig.(3.7) . 43 4.1 Moments of the elastic and inelastic DCS for protons incident on tungsten at various energies. 53 4.2 Energy dependence of the first four moments and the total cross section of the elastic and inelastic DCS for protons incident on tungsten. 54 4.3 Leading order term for the error in using a Fokker-Planck approxi- mation in place of screened Rutherford scattering. 56 ∗ 4.4 Fokker-Planck convergence with λ for Σtr = 1/(1000L) and Σtr = 1/(10000L). ............................... 59 4.5 Comparison of the results obtained from the delta function and ex- ponential representations of the Fokker-Planck operator . 62 4.6 Graphical depiction of θx to the geometry of the transport problem. 69 4.7 Angular distributions for the Fokker-Planck approximation and the −5 Fermi approximation with varying step sizes for Σtr = 2.0 × 10 .