DOCSLIB.ORG

The Henryk Niewodniczariski Institute of Nuclear Physics Krakow, Poland

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Henryk Niewodniczariski Institute of Nuclear Physics Krakow, Poland PL0000843 The Henryk Niewodniczariski Institute of Nuclear Physics Krakow, Poland REPORT No 1838/PN THERMAL NEUTRON DIFFUSION PARAMETERS DEPENDENT ON THE FLUX ENERGY DISTRIBUTION IN FINITE HYDROGENOUS MEDIA Krzysztof Drozdowicz Address: Main site: High Energy Departments, ul. Radzikowskiego 152, ul. Kawiory 26 A,, 31-342 Krakow, Poland 30-055 Krakow, Poland e-mail: [email protected] 3 1-11 e-mail: [email protected] The Henryk Niewodniczahski Institute of Nuclear Physics Krakow, Poland REPORT No 1838/PN THERMAL NEUTRON DIFFUSION PARAMETERS DEPENDENT ON THE FLUX ENERGY DISTRIBUTION IN FINITE HYDROGENOUS MEDIA Krzysztof Drozdowicz Address: ul. Radzikowskiego 152, PL- 31-342 Krakow, Poland e-mail: [email protected] Krakow 1999 PRINTED AT THE HENRYK NIEWODNICZANSKI INSTITUTE OF NUCLEAR PHYSICS KRAK6W, UL. RADZIKOWSKIEGO 152 Xerocopy: INP Krakow 27 copies. In memory of Professor Jan A. Czubek Report INP No. 1838/PN THERMAL NEUTRON DIFFUSION PARAMETERS DEPENDENT ON THE FLUX ENERGY DISTRIBUTION IN FINITE HYDROGENOUS MEDIA Krzysztof Drozdowicz The Henryk Niewodniczanski Institute of Nuclear Physics ul. Radzikowskiego 152, PL- 31-342 Krak6w, Poland ABSTRACT Macroscopic parameters for a description of the thermal neutron transport in finite volumes are considered. A very good correspondence between the theoretical and experimental parameters of hydrogenous media is attained. Thermal neutrons in the medium possess an energy distribution, which is dependent on the size (characterized by the geometric buckling) and on the neutron transport properties of the medium. In a hydrogenous material the thermal neutron transport is dominated by the scattering cross section which is strongly dependent on energy. A monoenergetic treatment of the thermal neutron group (admissible for other materials) leads in this case to a discrepancy between theoretical and experimental results. In the present paper the theoretical definitions of the pulsed thermal neutron parameters (the absorption rate, the diffusion coefficient, and the diffusion cooling coefficient) are based on Nelkin's analysis of the decay of a neutron pulse. Problems of the experimental determination of these parameters for a hydrogenous medium are discussed. A theoretical calculation of the pulsed parameters requires knowledge of the scattering kernel. For thermal neutrons it is individual for each hydrogenous material because neutron scattering on hydrogen nuclei bound in a molecule is affected by the molecular dynamics (characterized with internal energy modes which are comparable to the incident neutron energy). Granada's synthetic model for slow-neutron scattering is used. The complete up-dated formalism of calculation of the energy transfer scattering kernel after this model is presented in the paper. An influence of some minor variants within the model on the calculated differential and integral neutron parameters is shown. The theoretical energy-dependent scattering cross section (of Plexiglas) is compared to experimental results. A particular attention is paid to the calculation of the diffusion cooling coefficient. A solution of an equation, which determines the cooling function, is presented. On this basis the diffusion cooling coefficient is calculated for two hydrogenous materials (Plexiglas, polyethylene). A method is presented to calculate the thermal neutron energy-averaged (stationary and pulsed) parameters of a hydrogenous medium for which the energy dependence of the scattering cross section is unknown. New effective scattering cross sections of hydrogen are determined for this purpose. ACKNOWLEDGEMENTS This work was inspired by Prof. Jan A. Czubek (1935 - 1995). I am indebted to many people for their help or assistance at my activities that have been the basis for this paper. First, I want to mention the three institutions, in which I had the opportunity to carry on my work: my mother institute — The Henryk Niewodniczanski Institute of Nuclear Physics [Instytut Fizyki Ja.drowej im. Henryka Niewodniczanskiego] in Krakow, Poland; Department of Reactor Physics at Chalmers University of Technology [Institutionen for reaktorfysik, Chalmers tekniska hogskola] in Goteborg, Sweden; and the Neutron Division at the Bariloche Atomic Centre [Centro Atomico Bariloche] in S. C. Bariloche, Argentina. Many thanks to all my colleagues from the Neutron Transport Physics Laboratory (Dept. of Environmental and Radiation Transport Physics at the INP) for a friendly atmosphere of work, and especially to B.Sc. Jacek Burda, M.Sc. Barbara Gabanska, M.Sc. Andrzej Igielski, Mr. Tadeusz Kowalik, and Dr. Ewa Krynicka, for their participation in pulsed neutron experiments. I wish to express my particular gratitude to Assoc. Prof. Urszula Woznicka, Head of the Lab, for the fruitful collaboration and for many helpful discussions on preparation of this paper. I thank Prof. Imre Pazsit from the Chalmers University for the possibility I had to stay as the guest researcher at the Dept. of Reactor Physics. I am very indebted to Prof. Nils Goran Sjostrand from that Department for his help to organize my experiment in Studsvik (Sweden) and for the contact to physicists in the Bariloche Atomic Centre. I am grateful to Prof. J. Rolando Granada and Assoc. Prof. Victor H. Gillette from the Bariloche Atomic Centre for the close co-operation (including mutual scientific visits) and numerous discussions on details of the synthetic scattering model for slow neutrons. I thank Prof. Jerzy Loskiewicz, Head of the Department, for laborious overviewing of my typescript and advices how to make it more clear for the reader. VI CONTENTS Abstract V Acknowledgements VI Contents VII List of main symbols IX Introduction XIII 1. Basic relations for the thermal neutron transport 1 2. Thermal neutron parameters for a medium by experiment and by theory 5 2.1. The pulsed neutron experiment 5 2.2. Outline of the analysis of the decaying neutron pulse 8 2.3. Other definitions of the diffusion and diffusion cooling coefficients 13 3. Experiments on hydrogenous moderators: example of Plexiglas 16 3.1. The pulsed neutron experiment for Plexiglas 16 3.2. Measurement of the total cross section with the transmission method using monoenergetic neutrons 23 3.3. The total cross section for Plexiglas by the transmission experiment with the time-of-flight method 28 4. Approximations in the calculation of the thermal neutron parameters 29 4.1. Common approximations 31 4.1.1. Absorption 31 4.1.2. Scattering 32 4.1.3 The average cosine of the scattering angle and the transport cross section . 34 4.2. New effective scattering cross sections of hydrogen for a calculation of the energy-averaged thermal neutron parameters of hydrogenous mixtures .... 36 5. Solution of the equation for the function (J>2(£^ 42 6. Scattering kernel 48 6.1. Introduction 48 6.2. Principles of Granada's synthetic scattering model 50 6.3. The energy transfer kernel by the synthetic model 53 6.3.1. Basic characteristics of the model 54 6.3.2. The n-order terms of the scattering kernel 57 6.3.3. Calculation of the zeroth- and first-order energy transfer kernels 60 — Principal term of the zeroth-order scattering kernel 60 — Principal term of the first-order scattering kernel 61 — Phononic terms in the zeroth-order scattering kernel 62 — Phononic terms in the first-order scattering kernel 63 — The lowest frequency modes of the zeroth-order scattering kernel 64 — The lowest frequency modes of the first-order scattering kernel 65 VII 6.4. Total scattering cross section 66 — The lowest frequency modes of the total scattering cross section 67 6.5. Average cosine of the scattering angle 68 — Transport cross section 69 — Higher-order average cosines 70 7. Calculations using Granada's synthetic scattering model 71 7.1. Sensitivity of calculated thermal neutron parameters to variants in Granada's model 71 7.1.1. Variants of the energy transfer scattering kernel in the frame of the model 71 7.1.2. Calculated thermal neutron parameters on example of Plexiglas 73 — The scattering cross section and the diffusion constant 74 - The diffusion cooling coefficient 78 7.1.3. Conclusions 81 7.2. Diffusion cooling coefficient for thermal neutrons in Plexiglas 82 7.3. Diffusion cooling coefficient for thermal neutrons in polyethylene 86 8. Conclusions 90 Appendix: Free gas model as a limit case of Granada's synthetic model XIX References XXI VIII LIST OF MAIN SYMBOLS The brackets [ ] contain the number of the chapter or section in which the symbol is explained. The abbreviation GSM = Granada's synthetic scattering model. (x) = (x(E)) — general symbol for the Maxwellian-weighted average, Eq. (2.23) A — relative-to-neutron atomic mass ^moi ~ relative molecular mass fl2 - geometric buckling [Ch.l], [Sec.3.1] fl2 _ dynamic buckling [Ch.l] B2 — material buckling [Ch.l] m b(E) — average cosine of the scattering angle [Sec.2.2], [Ch.6] b — approximate value of the average cosine of the scattering angle in a hydrogenous medium C — thermal neutron diffusion cooling coefficient [Ch.2]; different theoretical definitions of the diffusion cooling coefficient: CB - Eq. (2.35) CD - Eq. (2.30) CK - Eq. (2.39) C, — from the theoretical calculation using the fit of Eq.(2.15) [Sec.2.3] C — transport correction to the diffusion cooling coefficient, Eq. (2.32) C, C, C, C — values of the diffusion cooling coefficient C according to variants of the GSM [Ch.7] C^, C — coefficients in the phononic terms of the scattering kernel and of the scattering cross section (GSM) [Ch.6] D, D(E) — thermal neutron
Load more

Recommended publications
  • Lecture 6: Spectroscopy and Photochemistry II
    Lecture 6: Spectroscopy and Photochemistry II Required Reading: FP Chapter 3 Suggested Reading: SP Chapter 3 Atmospheric Chemistry CHEM-5151 / ATOC-5151 Spring 2005 Prof. Jose-Luis Jimenez Outline of Lecture • The Sun as a radiation source • Attenuation from the atmosphere – Scattering by gases & aerosols – Absorption by gases • Beer-Lamber law • Atmospheric photochemistry – Calculation of photolysis rates – Radiation fluxes – Radiation models 1 Reminder of EM Spectrum Blackbody Radiation Linear Scale Log Scale From R.P. Turco, Earth Under Siege: From Air Pollution to Global Change, Oxford UP, 2002. 2 Solar & Earth Radiation Spectra • Sun is a radiation source with an effective blackbody temperature of about 5800 K • Earth receives circa 1368 W/m2 of energy from solar radiation From Turco From S. Nidkorodov • Question: are relative vertical scales ok in right plot? Solar Radiation Spectrum II From F-P&P •Solar spectrum is strongly modulated by atmospheric scattering and absorption From Turco 3 Solar Radiation Spectrum III UV Photon Energy ↑ C B A From Turco Solar Radiation Spectrum IV • Solar spectrum is strongly O3 modulated by atmospheric absorptions O 2 • Remember that UV photons have most energy –O2 absorbs extreme UV in mesosphere; O3 absorbs most UV in stratosphere – Chemistry of those regions partially driven by those absorptions – Only light with λ>290 nm penetrates into the lower troposphere – Biomolecules have same bonds (e.g. C-H), bonds can break with UV absorption => damage to life • Importance of protection From F-P&P provided by O3 layer 4 Solar Radiation Spectrum vs. altitude From F-P&P • Very high energy photons are depleted high up in the atmosphere • Some photochemistry is possible in stratosphere but not in troposphere • Only λ > 290 nm in trop.
    [Show full text]
  • ICANS XXI Dawn of High Power Neutron Sources and Science Applications
    Book of Abstracts ICANS XXI Dawn of high power neutron sources and science applications 29 Sep - 3 Oct 2014, JAPAN Ibaraki Prefectural Culture Center Update : 12 Oct. 2014 Best photography in 7th Oarai Town Photo Contest. WELCOME TO ICANS XXI ICANS (International Collaboration on Advanced Neutron Sources) is a network for scientists who are involved in developing pulsed neutron sources and accelerator based spallation neutron sources. Since 1st ICANS meetings was held in 1977 at Argonne National Laboratory in the day of dawn of spallation neutron technique, ICANS has been continuously held already 20 times somewhere in the world. Now we are extremely happy to announce that the ICANS, the 21st meeting, will be held at Mito hosted by J-PARC this autumn. We have a large number of topics to be discussed, there are twelve topics, such as futuristic idea of neutron source, rapid progress in facilities, integration issues in target-moderator-development, etc. The details can be found in the agenda. The meeting has a two layered structure, one is plenary session and another is workshop. Two of them are complementary and tightly cooperate each other. In the meeting we would like to enhance "workshop style", which is an original and traditional way of ICANS. Actually 2/3 of topics will be discussed in the workshop sessions. It also will be essentially organized/ lead by the workshop chairs. Plenary session shows overall issues in a relevant workshop, whose details should be talked/discussed in the workshop. The venue for the meeting is Mito city, where the 2nd Shogun Family lived for a long period of time during Edo era from 17th to 19th century, when the Tokugawa shogunate ruled the country.
    [Show full text]
  • Φ()Edeat∆≡ Nedeavt () [ ∆ ] (2)
    22.54 Neutron Interactions and Applications (Spring 2002) Lecture Notes on Neutronics of Multiplying Media This is a set of notes for the four lectures on neutron interactions in a reactor system in which a neutron population can change as a result of capture and fission reactions, in addition to scattering. We will discuss the concept of criticality as a way of describing any self-sustaining chain reaction which, in principle, can be just as well chemical in nature, and focus on the fundamental understanding of neutron diffusion and slowing down, and the interplay between materials constants (expressed through cross sections) and geometric factors (system size and shape) in determining criticality. The four lectures are: Lec 14. Neutronics of Multiplying Media – criticality, moderation, leakage Lec 15. Elements of Neutron Diffusion Theory Lec 16. Neutron Slowing Down Lec.17. The Two-Group Two-Region Reactor As a start the following references are recommended for supplemental reading: J. Lemarsh, Introduction to Nuclear Reactor Theory K. M. Case, F. deHoffmann, G. Placzek, Introduction to The Theory of Neutron Diffusion, vol. 1, LASL, 1953. I. Some Basic Notions of Neutron Multiplication Consider a homogeneous beam of neutrons having a distribution of energies. Define distribution function n(E) such that n(E)dE = expected no, of neutrons with energies in dE about E per cm3 (1) Suppose we ask how many neutrons in a certain energy range will cross an area during a time interval ∆t. Call this number φ()EdEAt∆≡ nEdEAvt () [ ∆ ] (2) since all the neutrons in a certain volume Av ∆t will cross during ∆t.
    [Show full text]
  • Photon Cross Sections, Attenuation Coefficients, and Energy Absorption Coefficients from 10 Kev to 100 Gev*
    1 of Stanaaros National Bureau Mmin. Bids- r'' Library. Ml gEP 2 5 1969 NSRDS-NBS 29 . A111D1 ^67174 tioton Cross Sections, i NBS Attenuation Coefficients, and & TECH RTC. 1 NATL INST OF STANDARDS _nergy Absorption Coefficients From 10 keV to 100 GeV U.S. DEPARTMENT OF COMMERCE NATIONAL BUREAU OF STANDARDS T X J ". j NATIONAL BUREAU OF STANDARDS 1 The National Bureau of Standards was established by an act of Congress March 3, 1901. Today, in addition to serving as the Nation’s central measurement laboratory, the Bureau is a principal focal point in the Federal Government for assuring maximum application of the physical and engineering sciences to the advancement of technology in industry and commerce. To this end the Bureau conducts research and provides central national services in four broad program areas. These are: (1) basic measurements and standards, (2) materials measurements and standards, (3) technological measurements and standards, and (4) transfer of technology. The Bureau comprises the Institute for Basic Standards, the Institute for Materials Research, the Institute for Applied Technology, the Center for Radiation Research, the Center for Computer Sciences and Technology, and the Office for Information Programs. THE INSTITUTE FOR BASIC STANDARDS provides the central basis within the United States of a complete and consistent system of physical measurement; coordinates that system with measurement systems of other nations; and furnishes essential services leading to accurate and uniform physical measurements throughout the Nation’s scientific community, industry, and com- merce. The Institute consists of an Office of Measurement Services and the following technical divisions: Applied Mathematics—Electricity—Metrology—Mechanics—Heat—Atomic and Molec- ular Physics—Radio Physics -—Radio Engineering -—Time and Frequency -—Astro- physics -—Cryogenics.
    [Show full text]
  • Conceptual Design Report Jülich High
    General Allgemeines ual Design Report ual Design Report Concept Jülich High Brilliance Neutron Source Source Jülich High Brilliance Neutron 8 Conceptual Design Report Jülich High Brilliance Neutron Source (HBS) T. Brückel, T. Gutberlet (Eds.) J. Baggemann, S. Böhm, P. Doege, J. Fenske, M. Feygenson, A. Glavic, O. Holderer, S. Jaksch, M. Jentschel, S. Kleefisch, H. Kleines, J. Li, K. Lieutenant,P . Mastinu, E. Mauerhofer, O. Meusel, S. Pasini, H. Podlech, M. Rimmler, U. Rücker, T. Schrader, W. Schweika, M. Strobl, E. Vezhlev, J. Voigt, P. Zakalek, O. Zimmer Allgemeines / General Allgemeines / General Band / Volume 8 Band / Volume 8 ISBN 978-3-95806-501-7 ISBN 978-3-95806-501-7 T. Brückel, T. Gutberlet (Eds.) Gutberlet T. Brückel, T. Jülich High Brilliance Neutron Source (HBS) 1 100 mA proton ion source 2 70 MeV linear accelerator 5 3 Proton beam multiplexer system 5 4 Individual neutron target stations 4 5 Various instruments in the experimental halls 3 5 4 2 1 5 5 5 5 4 3 5 4 5 5 Schriften des Forschungszentrums Jülich Reihe Allgemeines / General Band / Volume 8 CONTENT I. Executive summary 7 II. Foreword 11 III. Rationale 13 1. Neutron provision 13 1.1 Reactor based fission neutron sources 14 1.2 Spallation neutron sources 15 1.3 Accelerator driven neutron sources 15 2. Neutron landscape 16 3. Baseline design 18 3.1 Comparison to existing sources 19 IV. Science case 21 1. Chemistry 24 2. Geoscience 25 3. Environment 26 4. Engineering 27 5. Information and quantum technologies 28 6. Nanotechnology 29 7. Energy technology 30 8.
    [Show full text]
  • THE SPALLATION NEUTRON SOURCE (SNS) PROJECT Jose Alonso
    THE SPALLATION NEUTRON SOURCE (SNS) PROJECT Jose Alonso The SNS is currently the US Department of Energy's activation and residual-radiation levels in the tunnels are largest accelerator construction project, and is being low enough to allow quick access and hands-on carried out by a partnership of five DOE National maintenance, and is driven by the very high reliability and Laboratories for building the world's most powerful availability specifications associated with the strong user- accelerator-based pulse spallation source. At its planned orientation of the SNS facility. Areas where unusable 2 MW operating level, it will produce neutron fluxes at beam is diverted (so-called "controlled" beam-loss least a factor of ten greater than Rutherford Appleton points), such as collimators, scrapers and dumps, must be Laboratory’s ISIS, currently the world’s leading spallation designed to collect these particles in a way that contribute source. The current design of the SNS, shown in Figure only minimally to background levels in the tunnels where 1, calls for a full-energy (1 GeV) linac delivering 1 ms access is required. Normal loss mechanisms must receive pulses to an Accumulator Ring which compresses these particular attention to meet this specification. This into 600 ns proton pulses that strike a liquid mercury implies very tight control over beam emittance, and target at a 60 Hz rate. Room-temperature and cryogenic emittance growth; designing for the largest-possible stay- moderators produce beams of slow neutrons suitable for clear apertures in linac, transport and ring structures; materials research with an initial suite of at least 10 understanding of space-charge and halo effects in linac and ring; ensuring highest-possible vacuum in areas neutron-scattering instruments.
    [Show full text]
  • Nist Center for Neutron Research Accomplishments and Opportunities
    RXC NATL INST. OF STAND & TECH WIST PUBLICATIONS A111QS 6=11773- m ON THE COVER Part of the 913-detector array of the new Disk Chopper Spectrometer (DCS) that will be available to users in 2000. To view the detectors as found in the instrument, the picture should be rotated 90°. Information about the DCS and other instruments at NCNR is available at http://www.ncnr.nist.gov/. Photo: Mark Heifer NCNR 1999 NIST CENTER FOR NEUTRON RESEARCH ACCOMPLISHMENTS AND OPPORTUNITIES NIST Special Publication 944 J. Michael Rowe, Director Ronald L. Cappelletti, Editor Linda K. Clutter, Assistant Editor January 2000 U.S. DEPARTMENT OF COMMERCE William M. Daley, Secretary Technology Administration Dr. Cheryl L. Shavers, Under Secretary of Commerce for Technology National Institute of Standards and Technology Raymond G. Kammer, Director DISCLAIMER Certain commercial equipment, instruments, or materials are identified in this report to foster understanding. Such identification does not imply recommendation or endorsement by the National Institute of Standards and Technology, nor does it imply that the materials or equipment identified are necessarily the best available for the purpose. National Institute of Standards and Technology Special Publication 944 Natl. Inst. Stand. Technol. Spec. Publ. 944, 80 pages (January 2000) CODEN: NSPUE2 U.S. GOVERNMENT PRINTING OFFICE - WASHINGTON: 2000 For Sale by the Superintendent of Documents, U.S. Government Printing Office Washington, DC 20402-9325 CONTENTS Foreword iv Introduction to the NIST Center for Neutron Research
    [Show full text]
  • Massively Parallel Cartesian Discrete Ordinates Method for Neutron Transport Simulation Salli Moustafa
    Massively Parallel Cartesian Discrete Ordinates Method for Neutron Transport Simulation Salli Moustafa To cite this version: Salli Moustafa. Massively Parallel Cartesian Discrete Ordinates Method for Neutron Transport Sim- ulation. Other [cs.OH]. Université de Bordeaux, 2015. English. NNT : 2015BORD0408. tel- 01379686v2 HAL Id: tel-01379686 https://tel.archives-ouvertes.fr/tel-01379686v2 Submitted on 14 Oct 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. THÈSE PRÉSENTÉE À L’UNIVERSITÉ DE BORDEAUX ÉCOLE DOCTORALE DE MATHÉMATIQUES ET D’INFORMATIQUE Par Salli MOUSTAFA POUR OBTENIR LE GRADE DE DOCTEUR SPÉCIALITÉ : INFORMATIQUE Massively Parallel Cartesian Discrete Ordinates Method for Neutron Transport Simulation Soutenue le : 15 décembre 2015 Après avis des rapporteurs : Michael A. Heroux Senior scientist, Sandia National Laboratories Alain Hébert . Professor, École Polytechnique de Montréal Devant la commission d’examen composée de : Michael A. Heroux Senior scientist, Sandia National Laboratories Examinateur Alain Hébert . Professor, Ecole Polytechnique
    [Show full text]
  • 1 a Numerical Solution of the Time-Dependent Neutron
    A NUMERICAL SOLUTION OF THE TIME-DEPENDENT NEUTRON TRANSPORT EQUATION USING THE CHARACTERISTIC METHOD. APPLICATIONS TO ICF AND TO HYBRID FISSION-FUSION SYSTEMS Dalton Ellery Girão Barroso Military Institute of Engineering Department of Nuclear Engineering Emails: [email protected], [email protected] ABSTRACT In this work we present a solution of the one-dimensional spherical symmetric time- dependent neutron transport equation (written for a moving system in lagrangian coordinates) by using the characteristic method. One of the objectives is to overcome the negative flux problem that arises when the system is very opaque and the angular neutron flux can become negative when it is extrapolated in spatial meshes as, for example, in diamond scheme adopted in many codes. Although there are recipes to overcome this problem, it can completely degrade the numerical solution if repeated many times. In most common time discretization of the time-dependent neutron transport equation, there is a term, 1/vt, summed to total (or transport) cross sections that can increase enormously the opacity of the medium. This certainly gets worse when there is a combination of lower neutron velocity (in a moderated system) and a very small time-step a very common situation in ICF (inertial confinement fusion). The solution presented here can be easily coupled to radiation-hydrodynamics equations, but it is necessary an additional term to maintain neutron conservation in a moving system in lagrangian coordinates. Energy multigroup method and a former SN method to deal with the angular variable are used, with the assumption of isotropic scattering and the transport cross sections approximation.
    [Show full text]
  • Review and History of Photon Cross Section Calculations*
    INSTITUTE OF PHYSICS PUBLISHING PHYSICS IN MEDICINE AND BIOLOGY Phys. Med. Biol. 51 (2006) R245–R262 doi:10.1088/0031-9155/51/13/R15 REVIEW Review and history of photon cross section calculations* J H Hubbell National Institute of Standards and Technology, Ionizing Radiation Division, Mail Stop 8463, 100 Bureau Drive, Gaithersburg, MD 20899-8463, USA E-mail: [email protected] Received 22 February 2006, in final form 12 April 2006 Published 20 June 2006 Online at stacks.iop.org/PMB/51/R245 Abstract Photon (x-ray, gamma-ray, bremsstrahlung) mass attenuation coefficients, µ/ρ, are among the most widely used physical parameters employed in medical diagnostic and therapy computations, as well as in diverse applications in other fields such as nuclear power plant shielding, health physics and industrial irradiation and monitoring, and in x-ray crystallography. This review traces the evolution of this data base from its empirical beginnings totally derived from measurements beginning in 1907 by Barkla and Sadler and continuing up through the 1935 Allen compilation (published virtually unchanged in all editions up through 1971–1972 of the Chemical Rubber Handbook), to the 1949 semi-empirical compilation of Victoreen, as our theoretical understanding of the constituent Compton scattering, photoabsorption and pair production interactions of photons with atoms became more quantitative. The 1950s saw the advent of completely theoretical (guided by available measured data) systematic compilations such as in the works of Davisson and Evans, and by White-Grodstein under the direction of Fano, using mostly theory developed in the 1930s (pre-World War II) by Sauter, Bethe, Heitler and others.
    [Show full text]
  • Module 6: Neutron Diffusion Dr. John H. Bickel
    Fundamentals of Nuclear Engineering Module 6: Neutron Diffusion Dr. John H. Bickel 1 2 Objectives: 1. Understand how Neutron Diffusion explains reactor neutron flux distribution 2. Understand origin, limitations of Neutron Diffusion from: • Boltzmann Transport Equation, • Ficke’s Law 3. Solution of One-Group Neutron Diffusion Equation for: • Cubical, • Cylindrical geometries (via separation of variables technique) 4. Identify Eigenvalues of Neutron Diffusion Equation (Buckling, Diffusion Length) related to physical properties 5. Understand refinements from Multi-Group Diffusion Model applied to Reflectors, flux depression near strong neutron absorbers, and sources 3 Boltzmann’s Transport Equation • Originates from Statistical Mechanics • Used to understand kinetic theory of gasses • Full equation expressed as integral-differential equation • In simplest terms it is a conservation equation d r N(r,v,t)d 3r = S(r,t)d 3r − N(r,v,t)vΣ (r,v,t)d 3r − J (r,v,t) • nrdA ∫ ∫ ∫∫c dt V V V 4 Boltzmann’s Transport Equation • First simplification: Apply Divergence Theorem r r r ∫ J (r,v,t) • nrdA = ∫∇ • J (r,v,t)d 3r V • Remove volume integration: d r r N(r,v,t) = S(r,t) − N(r,v,t)vΣ (r,v,t) − ∇ • J (r,v,t) dt c • Divergence of Local Neutron Current is simplified by making “Diffusion Approximation” • Alternate solution approach is via Monte Carlo Method • Diffusion Approximation is based upon Fick’s Law 5 Diffusion Approximation Assumes: • Uniform relatively infinite medium thus: Σc(r) ~ Σc • No strong “point” neutron sources in medium • Scattering
    [Show full text]
  • Analytical Benchmarks for Nuclear Engineering Applications
    Data Bank ISBN 978-92-64-99056-2 NEA/DB/DOC(2008)1 Analytical Benchmarks for Nuclear Engineering Applications Case Studies in Neutron Transport Theory B.D. Ganapol Department of Aerospace and Mechanical Engineering University of Arizona with Forewords by Paul F. Zweifel Richard Sanchez and Norman J. McCormick OECD © 2008 NEA No. 6292 NUCLEAR ENERGY AGENCY ORGANISATION FOR ECONOMIC CO-OPERATION AND DEVELOPMENT ORGANISATION FOR ECONOMIC CO-OPERATION AND DEVELOPMENT The OECD is a unique forum where the governments of 30 democracies work together to address the economic, social and environmental challenges of globalisation. The OECD is also at the forefront of efforts to understand and to help governments respond to new developments and concerns, such as corporate governance, the information economy and the challenges of an ageing population. The Organisation provides a setting where governments can compare policy experiences, seek answers to common problems, identify good practice and work to co-ordinate domestic and international policies. The OECD member countries are: Australia, Austria, Belgium, Canada, the Czech Republic, Denmark, Finland, France, Germany, Greece, Hungary, Iceland, Ireland, Italy, Japan, Korea, Luxembourg, Mexico, the Netherlands, New Zealand, Norway, Poland, Portugal, the Slovak Republic, Spain, Sweden, Switzerland, Turkey, the United Kingdom and the United States. The Commission of the European Communities takes part in the work of the OECD. OECD Publishing disseminates widely the results of the Organisation’s statistics gathering and research on economic, social and environmental issues, as well as the conventions, guidelines and standards agreed by its members. * * * This work is published on the responsibility of the Secretary-General of the OECD.
    [Show full text]