Fundamentals of the monte carlo method for neutral and charged particle transport

Fundamentals of the Monte Carlo method for neutral and charged particle transport Alex F Bielajew The University of Michigan Department of Nuclear Engineering and Radiological Sciences 2927 Cooley Building (North Campus) 2355 Bonisteel Boulevard Ann Arbor, Michigan 48109-2104 U. S. A. Tel: 734 764 6364 Fax: 734 763 4540 email: bielajew@umich.edu c  1998 Alex F Bielajew c  1998 The University of Michigan February 11, 2000 2 Preface This book arises out of a course I am teaching for a two-credit (26 hour) graduate-level course Monte Carlo Methods being taught at the Department of Nuclear Engineering and Radiological Sciences at the University of Michigan. AFB, February 11, 2000 i ii Contents 1 What is the Monte Carlo method? 1 1.1 WhyisMonteCarlo? 7 1.2 Somehistory 11 2 Elementary probability theory 15 2.1 Continuousrandomvariables 15 2.1.1 One-dimensional probability distributions 15 2.1.2 Two-dimensional probability distributions 17 2.1.3 Cumulative probability distributions 20 2.2 Discreterandomvariables 21 3 Random Number Generation 25 3.1 Linearcongruentialrandomnumbergenerators 26 3.2 Longsequencerandomnumbergenerators 30 4 Sampling Theory 35 4.1 Invertiblecumulativedistributionfunctions(directmethod) 36 4.2 Rejectionmethod 40 4.3 Mixedmethods 43 4.4 Examplesofsamplingtechniques 44 4.4.1 Circularly collimated parallel beam 44 4.4.2 Point source collimated to a planar circle 46 4.4.3 Mixedmethodexample 47 iii iv CONTENTS 4.4.4 Multi-dimensionalexample 49 5 Error estimation 53 5.1 Directerrorestimation 56 5.2 Batchstatisticserrorestimation 57 5.3 Combiningerrorsofindependentruns 58 5.4 Errorestimationforbinaryscoring 59 5.5 Relationships between S 2 x and s 2 x , S 2 x and s 2 x 59 6 Oddities: Random number and precision problems 63 6.1 Randomnumberartefacts 63 6.2 Accumulationerrors 70 7 Ray tracing and rotations 75 7.1 Displacements 76 7.2 Rotationofcoordinatesystems 76 7.3 Changesofdirection 79 7.4 Puttingitalltogether 80 8 Transport in media, interaction models 85 8.1 Interaction probability in an infinite medium 85 8.1.1 Uniform, infinite, homogeneous media 86 8.2 Finitemedia 87 8.3 Regionsofdifferentscatteringcharacteristics 87 8.4 Obtaining µ frommicroscopiccrosssections 90 8.5 Compounds and mixtures 93 8.6 Branchingratios 94 8.7 Otherpathlengthschemes 94 8.8 Modelinteractions 95 8.8.1 Isotropicscattering 95 8.8.2 Semi-isotropic or P 1 scattering 95 CONTENTS v 8.8.3 Rutherfordianscattering 96 8.8.4 Rutherfordianscattering—smallangleform 96 9 Lewis theory 99 9.1 Theformalsolution 100 9.2 Isotropicscatteringfromuniformatomictargets 102 10 Geometry 107 10.1 Boundary crossing 108 10.2Solutionsforsimplesurfaces 112 10.2.1Planes 112 10.3Generalsolutionforanarbitraryquadric 114 10.3.1Intercepttoanarbitraryquadricsurface? 117 10.3.2 Spheres . . 121 10.3.3CircularCylinders 123 10.3.4CircularCones 124 10.4Usingsurfacestomakeobjects 125 10.4.1Elementalvolumes 125 10.5Trackinginanelementalvolume 132 10.6Usingelementalvolumestomakeobjects 135 10.6.1Simply-connectedelements 135 10.6.2Multiply-connectedelements 140 10.6.3Combinatorialgeometry 142 10.7Lawofreflection 142 11 Monte Carlo and Numerical Quadrature 151 11.1Thedimensionalityofdeterministicmethods 151 11.2ConvergenceofDeterministicSolutions 154 11.2.1Onedimension 154 11.2.2Twodimensions 154 vi CONTENTS 11.2.3 D dimensions 155 11.3ConvergenceofMonteCarlosolutions 156 11.4ComparisonbetweenMonteCarloandNumericalQuadrature 156 12 Photon Monte Carlo Simulation 161 12.1Basicphotoninteractionprocesses 161 12.1.1Pairproductioninthenuclearfield 162 12.1.2TheComptoninteraction(incoherentscattering) 165 12.1.3Photoelectricinteraction 166 12.1.4Rayleigh(coherent)interaction 169 12.1.5Relativeimportanceofvariousprocesses 170 12.2Photontransportlogic 170 13 Electron Monte Carlo Simulation 179 13.1Catastrophicinteractions 180 13.1.1Hardbremsstrahlungproduction 180 13.1.2Møller(Bhabha)scattering 180 13.1.3Positronannihilation 181 13.2Statisticallygroupedinteractions 181 13.2.1“Continuous”energyloss 181 13.2.2Multiplescattering 182 13.3Electrontransport“mechanics” 183 13.3.1Typicalelectrontracks 183 13.3.2Typicalmultiplescatteringsubsteps 183 13.4Examplesofelectrontransport 184 13.4.1 Effect of physical modeling on a 20 MeV e − depth-dosecurve 184 13.5Electrontransportlogic 196 14 Electron step-size artefacts and PRESTA 203 14.1Electronstep-sizeartefacts 203 CONTENTS vii 14.1.1Whatisanelectronstep-sizeartefact? 203 14.1.2Path-lengthcorrection 209 14.1.3Lateraldeflection 214 14.1.4 Boundary crossing 214 14.2PRESTA 216 14.2.1TheelementsofPRESTA 216 14.2.2 Constraints of the Moli`ereTheory 218 14.2.3PRESTA’spath-lengthcorrection 223 14.2.4PRESTA’slateralcorrelationalgorithm 226 14.2.5Accountingforenergyloss 228 14.2.6 PRESTA’s boundary crossing algorithm 231 14.2.7CaveatEmptor 233 15 Advanced electron transport algorithms 237 15.1WhatdoescondensedhistoryMonteCarlodo? 240 15.1.1Numerics’step-sizeconstraints 240 15.1.2Physics’step-sizeconstraints 243 15.1.3 Boundary step-size constraints . 244 15.2Thenewmultiple-scatteringtheory 245 15.3Longitudinalandlateraldistributions 247 15.4Thefutureofcondensedhistoryalgorithms 249 16 Electron Transport in Electric and Magnetic Fields 257 16.1Equationsofmotioninavacuum 258 16.1.1 Special cases:  E =constant,  B =0;  B =constant,  E =0 259 16.2Transportinamedium 260 16.3ApplicationtoMonteCarlo,Benchmarks 264 17 Variance reduction techniques 275 17.0.1Variancereductionorefficiencyincrease? 275 viii CONTENTS 17.1Electron-specificmethods 277 17.1.1 Geometry interrogation reduction 277 17.1.2Discardwithinazone 279 17.1.3PRESTA! 281 17.1.4Rangerejection 281 17.2Photon-specificmethods 284 17.2.1Interactionforcing 284 17.2.2Exponentialtransform,russianroulette,andparticlesplitting 287 17.2.3Exponentialtransformwithinteractionforcing 290 17.3Generalmethods 291 17.3.1Secondaryparticleenhancement 291 17.3.2Sectionedproblems,useofpre-computedresults 292 17.3.3Geometryequivalencetheorem 293 17.3.4Useofgeometrysymmetry 294 18 Code Library 299 18.1 Utility/General . . 300 18.2 Subroutines for random number generation 302 18.3 Subroutines for particle transport and deflection 321 18.4 Subroutines for modeling interactions . 325 18.5 Subroutines for modeling geometry . . 328 18.6Testroutines 335 [...]... a Monte Carlo calculation of the seating patterns of the members of an audience in an auditorium may 1 This presupposes that all uses of the Monte Carlo are for the purposes of understanding physical phenomena There are others uses of the Monte Carlo method for purely mathematical reasons, such as the determination of multi-dimensional integrals, a topic that will be discussed later in Chapter 2 Often... the value of π, albeit slowly Several other historical uses of Monte Carlo predating computers 12 CHAPTER 1 WHAT IS THE MONTE CARLO METHOD? are cited by Kalos and Whitlock [KW86] The modern Monte Carlo age was ushered in by von Neumann and Ulam during the initial development of thermonuclear weapons6 Ulam and von Neumann coined the phrase Monte Carlo and were pioneers in the development of the Monte. .. the physics of electron and photon transport This use of the Monte Carlo method is depicted in Figure 1.6 In this case, theory can not provide a sufficiently precise and entire mathematical description of the microscopic and macroscopic physics Theory can, however, provide intuition for the design of the measurement Monte Carlo methods are an adjunct to this process as well, serving in the analysis of. .. the products of both basic and applied science are dependent upon the trinity of measurement, theory and Monte Carlo Monte Carlo is often seen as a “competitor” to other methods of macroscopic calculation, which we will call deterministic and/ or analytic methods Although the proponents of either method sometimes approach a level of fanaticism in their debates, a practitioner of science should first ask,... constraint is what makes the mathematical solution difficult but is easy to simulate using Monte Carlo methods.) The important role that Monte Carlo methods have to play in this sort of study is illustrated in Figure 1.2 Basic science attempts to understand the basic working mechanisms of a phenomenon The “theory” is a set of assumptions (with perhaps a mathematical formulation of these assumptions) that... remarks of this section seem to favor the Monte Carlo approach, a point made previously should be re-iterated Analytic theory development and its realizations in terms of deterministic calculations are our only way of making theories regarding the behaviour of macroscopic fields, and our only way of modelling particle fluences in a symbolic way Monte Carlo is simply another tool in the theoretician’s or the. .. analysis of the experiment and verifying or invalidating the design 1.1 WHY IS MONTE CARLO? 7 Applied Science EXPERIMENT sis aly int uit i on an intuition MONTE CARLO THEORY verification Practical results Figure 1.6: The role of Monte Carlo methods in applied science 1.1 Why is Monte Carlo? If Monte Carlo did not exist there would be strong motivation to invent it! As argued previously, the products of both... Simplify in the case that x and y are independent 23 24 BIBLIOGRAPHY Chapter 3 Random Number Generation Anyone who considers arithmetical methods of producing random digits is, of course, in a state of sin John von Neumann (1951) The “pseudo” random number generator (RNG) is the “soul” or “heartbeat” of a Monte Carlo simulation It is what generates the pseudo-random nature of Monte Carlo simulations thereby... Basic understanding Figure 1.2: The role of Monte Carlo methods in basic science possible, as in the example of Figure 1.1 where two people can not occupy the same seat, a Monte Carlo simulation enters the picture in a useful way and can serve a two-fold purpose It can either provide a small correction to an otherwise useful theory or it can be employed directly to verify or disprove the theory of microscopic... other phenomena such as: a) for some type of performances, people arrive predominantly in pairs, b) audience members prefer an unobstructed view of the stage, c) audience members prefer to sit in the middle, close to the front, etc Each one of these assumptions could then be tested through measurement and then refined The Monte Carlo method in this case is an adjunct to the basic theory, providing a mechanism . Fundamentals of the Monte Carlo method for neutral and charged particle transport Alex F Bielajew The University of Michigan Department of Nuclear Engineering and Radiological Sciences 2927. may 1 This presupposes that all uses of the Monte Carlo are for the purposes of understanding physical phe- nomena. There are others uses of the Monte Carlo method for purely mathematical reasons,. in favor of Monte Carlo is that the Monte Carlo technique is one based upon a minimum amount of data and a maximum amount of floating-point operation. Deterministic calculations are often maximum