Bielajew A.F. Fundamentals of the Monte Carlo method for neutral and charged particle transport
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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
10.2.1 Planes ................................... 112
10.3Generalsolutionforanarbitraryquadric.................... 114
10.3.1 Intercepttoanarbitraryquadricsurface?................ 117
10.3.2 Spheres . . ................................ 121
10.3.3 CircularCylinders ............................ 123
10.3.4 CircularCones .............................. 124
10.4Usingsurfacestomakeobjects.......................... 125
10.4.1 Elementalvolumes ............................ 125
10.5Trackinginanelementalvolume......................... 132
10.6Usingelementalvolumestomakeobjects.................... 135
10.6.1 Simply-connectedelements........................ 135
10.6.2 Multiply-connectedelements....................... 140
10.6.3 Combinatorialgeometry ......................... 142
10.7Lawofreflection ................................. 142
11 Monte Carlo and Numerical Quadrature 151
11.1Thedimensionalityofdeterministicmethods.................. 151
11.2ConvergenceofDeterministicSolutions..................... 154
11.2.1 Onedimension .............................. 154
11.2.2 Twodimensions.............................. 154
vi CONTENTS
11.2.3 D dimensions ............................... 155
11.3ConvergenceofMonteCarlosolutions...................... 156
11.4ComparisonbetweenMonteCarloandNumericalQuadrature ........ 156
12 Photon Monte Carlo Simulation 161
12.1Basicphotoninteractionprocesses........................ 161
12.1.1 Pairproductioninthenuclearfield ................... 162
12.1.2 TheComptoninteraction(incoherentscattering) ........... 165
12.1.3 Photoelectricinteraction......................... 166
12.1.4 Rayleigh(coherent)interaction ..................... 169
12.1.5 Relativeimportanceofvariousprocesses ................ 170
12.2Photontransportlogic .............................. 170
13 Electron Monte Carlo Simulation 179
13.1Catastrophicinteractions............................. 180
13.1.1 Hardbremsstrahlungproduction .................... 180
13.1.2 Møller(Bhabha)scattering........................ 180
13.1.3 Positronannihilation........................... 181
13.2Statisticallygroupedinteractions ........................ 181
13.2.1 “Continuous”energyloss......................... 181
13.2.2 Multiplescattering ............................ 182
13.3Electrontransport“mechanics” ......................... 183
13.3.1 Typicalelectrontracks.......................... 183
13.3.2 Typicalmultiplescatteringsubsteps................... 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
14.1.1 Whatisanelectronstep-sizeartefact? ................. 203
14.1.2 Path-lengthcorrection .......................... 209
14.1.3 Lateraldeflection............................. 214
14.1.4 Boundary crossing ............................ 214
14.2PRESTA...................................... 216
14.2.1 TheelementsofPRESTA ........................ 216
14.2.2 Constraints of the Moli`ereTheory.................... 218
14.2.3 PRESTA’spath-lengthcorrection.................... 223
14.2.4 PRESTA’slateralcorrelationalgorithm................. 226
14.2.5 Accountingforenergyloss........................ 228
14.2.6 PRESTA’s boundary crossing algorithm ................ 231
14.2.7 CaveatEmptor .............................. 233
15 Advanced electron transport algorithms 237
15.1WhatdoescondensedhistoryMonteCarlodo?................. 240
15.1.1 Numerics’step-sizeconstraints...................... 240
15.1.2 Physics’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
17.0.1 Variancereductionorefficiencyincrease?................ 275
viii CONTENTS
17.1Electron-specificmethods ............................ 277
17.1.1 Geometry interrogation reduction .................... 277
17.1.2 Discardwithinazone........................... 279
17.1.3 PRESTA! ................................. 281
17.1.4 Rangerejection .............................. 281
17.2Photon-specificmethods ............................. 284
17.2.1 Interactionforcing ............................ 284
17.2.2 Exponentialtransform,russianroulette,andparticlesplitting .... 287
17.2.3 Exponentialtransformwithinteractionforcing............. 290
17.3Generalmethods ................................. 291
17.3.1 Secondaryparticleenhancement..................... 291
17.3.2 Sectionedproblems,useofpre-computedresults............ 292
17.3.3 Geometryequivalencetheorem...................... 293
17.3.4 Useofgeometrysymmetry........................ 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