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: [email protected] c 1998—2020 Alex F Bielajew c 1998—2020 The University of Michigan May 31, 2020 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, May 31, 2020 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 . 24 2.1.3 Cumulative probability distributions . 27 2.2 Discreterandomvariables ............................ 27 3 Random Number Generation 31 3.1 Linear congruential random number generators. ...... 32 3.2 Long sequence random number generators . .... 36 4 Sampling Theory 41 4.1 Invertible cumulative distribution functions (direct method) . ...... 42 4.2 Rejectionmethod................................. 46 4.3 Mixedmethods .................................. 49 4.4 Examples of sampling techniques . 50 4.4.1 Circularly collimated parallel beam . 50 4.4.2 Point source collimated to a planar circle . 52 4.4.3 Mixedmethodexample.......................... 53 iii iv CONTENTS 4.4.4 Multi-dimensional example . 55 5 Error estimation 59 5.1 Directerrorestimation .............................. 62 5.2 Batch statistics error estimation . .. 63 5.3 Combiningerrorsofindependentruns. .. 64 5.4 Error estimation for binary scoring . 65 2 2 2 2 5.5 Relationships between Sx and sx, Sx and sx .................. 65 5.6 Varianceofthevariance ............................. 68 6 Oddities: Random number and precision problems 71 6.1 Randomnumberartefacts . .. .. 71 6.2 Accumulationerrors ............................... 78 7 Ray tracing and rotations 83 7.1 Displacements................................... 84 7.2 Rotationofcoordinatesystems . 84 7.3 Changesofdirection ............................... 87 7.4 Puttingitalltogether .............................. 88 8 Transport in media, interaction models 93 8.1 Interaction probability in an infinite medium . 93 8.1.1 Uniform, infinite, homogeneous media . 94 8.2 Finitemedia.................................... 95 8.3 Regions of different scattering characteristics . ...... 95 8.4 Obtaining µ frommicroscopiccrosssections . 98 8.5 Compoundsandmixtures ............................ 101 8.6 Branchingratios ................................. 102 8.7 Otherpathlengthschemes . .. .. 102 8.8 Modelinteractions ................................ 103 8.8.1 Isotropicscattering ............................ 103 CONTENTS v 8.8.2 Semi-isotropic or P1 scattering...................... 103 8.8.3 Rutherfordianscattering . 104 8.8.4 Rutherfordian scattering—small angle form . 104 9 Lewis theory 107 9.1 Theformalsolution................................ 108 9.2 Isotropic scattering from uniform atomic targets . ...... 110 10 Geometry 115 10.1Boundarycrossing ................................ 116 10.2 Solutions for simple surfaces . 120 10.2.1 Planes ................................... 120 10.3 General solution for an arbitrary quadric . .... 122 10.3.1 Intercept to an arbitrary quadric surface?. .... 125 10.3.2 Spheres .................................. 130 10.3.3 Circular Cylinders . 131 10.3.4 CircularCones .............................. 133 10.4 Usingsurfacestomakeobjects. 133 10.4.1 Elementalvolumes ............................ 133 10.5 Tracking in an elemental volume . 140 10.6 Usingelementalvolumestomakeobjects . 143 10.6.1 Simply-connected elements . 143 10.6.2 Multiply-connected elements . 148 10.6.3 Combinatorialgeometry . 150 10.7Lawofreflection ................................. 150 11 Monte Carlo and Numerical Quadrature 159 11.1 The dimensionality of deterministic methods . 159 11.2 Convergence of Deterministic Solutions . 162 11.2.1 Onedimension .............................. 162 vi CONTENTS 11.2.2 Twodimensions.............................. 162 11.2.3 dimensions ............................... 163 D 11.3 ConvergenceofMonteCarlosolutions. 164 11.4 Comparison between Monte Carlo and Numerical Quadrature . ...... 164 12 Photon Monte Carlo Simulation 169 12.1 Basicphotoninteractionprocesses. .... 169 12.1.1 Pair production in the nuclear field . 170 12.1.2 The Compton interaction (incoherent scattering) . 173 12.1.3 Photoelectric interaction . 174 12.1.4 Rayleigh (coherent) interaction . 177 12.1.5 Relative importance of various processes . 178 12.2 Photontransportlogic .............................. 178 13 Electron Monte Carlo Simulation 187 13.1 Catastrophicinteractions. 188 13.1.1 Hardbremsstrahlungproduction . 188 13.1.2 Møller (Bhabha) scattering . 188 13.1.3 Positron annihilation . 189 13.2 Statistically grouped interactions . 189 13.2.1 “Continuous”energyloss. 189 13.2.2 Multiple scattering . 190 13.3 Electrontransport“mechanics” . 191 13.3.1 Typicalelectrontracks . 191 13.3.2 Typical multiple scattering substeps . 191 13.4 Examplesofelectrontransport . 192 13.4.1 Effect of physical modeling on a 20 MeV e− depth-dose curve . 192 13.5 Electrontransportlogic ............................. 204 14 Electron step-size artefacts and PRESTA 211 CONTENTS vii 14.1 Electronstep-sizeartefacts. .... 211 14.1.1 What is an electron step-size artefact? . 211 14.1.2 Path-lengthcorrection . 217 14.1.3 Lateraldeflection ............................. 222 14.1.4 Boundarycrossing ............................ 222 14.2PRESTA...................................... 224 14.2.1 TheelementsofPRESTA . 224 14.2.2 ConstraintsoftheMoli`ereTheory. 226 14.2.3 PRESTA’s path-length correction . 231 14.2.4 PRESTA’s lateral correlation algorithm . 234 14.2.5 Accountingforenergyloss . 236 14.2.6 PRESTA’s boundary crossing algorithm . 239 14.2.7 CaveatEmptor .............................. 241 15 Advanced electron transport algorithms 245 15.1 What does condensed history Monte Carlo do? . .... 248 15.1.1 Numerics’ step-size constraints . 248 15.1.2 Physics’ step-size constraints . 251 15.1.3 Boundarystep-sizeconstraints. 252 15.2 The new multiple-scattering theory . 253 15.3 Longitudinal and lateral distributions . 255 15.4 Thefutureofcondensed historyalgorithms . ..... 257 16 Electron Transport in Electric and Magnetic Fields 265 16.1 Equationsofmotioninavacuum . 266 16.1.1 Special cases: E~ =constant, B~ = 0; B~ =constant, E~ =0........ 267 16.2Transportinamedium.............................. 268 16.3 Application to Monte Carlo, Benchmarks . 272 17 Variance reduction techniques 283 viii CONTENTS 17.0.1 Variance reduction or efficiency increase? . 283 17.1 Electron-specificmethods . 285 17.1.1 Geometry interrogation reduction . 285 17.1.2 Discardwithinazone. .. .. 287 17.1.3 PRESTA! ................................. 289 17.1.4 Rangerejection .............................. 289 17.2 Photon-specificmethods . 292 17.2.1 Interactionforcing ............................ 292 17.2.2 Exponential transform, russian roulette, and particle splitting . 295 17.2.3 Exponential transform with interaction forcing . 298 17.3Generalmethods ................................. 299 17.3.1 Secondaryparticleenhancement . 299 17.3.2 Sectioned problems, use of pre-computed results . .... 300 17.3.3 Geometryequivalencetheorem. 301 17.3.4 Useofgeometrysymmetry. 302 18 Code Library 307 18.1Utility/General .................................. 308 18.2 Subroutines forrandomnumber generation . ..... 310 18.3 Subroutines for particle transport and deflection . ....... 329 18.4 Subroutines for modeling interactions . 333 18.5 Subroutinesformodelinggeometry . 336 18.6Testroutines ................................... 343 Chapter 1 What is the Monte Carlo method? The Monte Carlo method is a numerical solution to a problem that models objects inter- acting with other objects or their environment based upon simple object-object or object- environment relationships1. It represents an attempt to model nature through direct sim- ulation of the essential dynamics of the system in question. In this sense the Monte Carlo method is essentially simple in its approach—a solution to a macroscopic system through simulation of its microscopic interactions. A solution is determined by random sampling of the relationships, or the microscopic in- teractions, until the result converges. Thus, the mechanics of executing a solution involves repetitive action or calculation. To the extent that many microscopic interactions can be modelled mathematically, the repetitive solution can be executed on a computer. However, the Monte Carlo method predates the computer (more on this later) and is not essential to carry out a solution although in most cases computers make the determination of a solution much faster. There are many examples of the use of the Monte Carlo method that can be drawn from social science, traffic flow, population growth, finance, genetics, quantum chemistry, radiation sciences, radiotherapy, and radiation dosimetry but our discussion will concentrate on the simulation of neutrons, photons and electrons being transported in condensed materials, gases and vacuum. We will make brief excursions into other