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—2001 Alex F Bielajew

c 1998—2001 The University of Michigan

September 17, 2001

2

Contents 1 What is the Monte Carlo method?

1

1.1

Why is Monte Carlo? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

1.2

Some history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

2 Elementary probability theory 2.1

2.2

Continuous random variables

15 . . . . . . . . . . . . . . . . . . . . . . . . . .

15

2.1.1

One-dimensional probability distributions

. . . . . . . . . . . . . . .

15

2.1.2

Two-dimensional probability distributions . . . . . . . . . . . . . . .

17

2.1.3

Cumulative probability distributions . . . . . . . . . . . . . . . . . .

20

Discrete random variables . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

3 Random Number Generation

25

3.1

Linear congruential random number generators . . . . . . . . . . . . . . . . .

26

3.2

Long sequence random number generators . . . . . . . . . . . . . . . . . . .

30

4 Sampling Theory

35

4.1

Invertible cumulative distribution functions (direct method) . . . . . . . . .

36

4.2

Rejection method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

4.3

Mixed methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

4.4

Examples of sampling techniques . . . . . . . . . . . . . . . . . . . . . . . .

44

4.4.1

Circularly collimated parallel beam . . . . . . . . . . . . . . . . . . .

44

4.4.2

Point source collimated to a planar circle . . . . . . . . . . . . . . . .

46

4.4.3

Mixed method example . . . . . . . . . . . . . . . . . . . . . . . . . .

47

iii

iv

CONTENTS 4.4.4

Multi-dimensional example . . . . . . . . . . . . . . . . . . . . . . . .

5 Error estimation

49 53

5.1

Direct error estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

5.2

Batch statistics error estimation . . . . . . . . . . . . . . . . . . . . . . . . .

57

5.3

Combining errors of independent runs . . . . . . . . . . . . . . . . . . . . . .

58

5.4

Error estimation for binary scoring . . . . . . . . . . . . . . . . . . . . . . .

59

5.5

Relationships between Sx2 and s2x , Sx2 and s2x . . . . . . . . . . . . . . . . . .

59

6 Oddities: Random number and precision problems

63

6.1

Random number artefacts . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63

6.2

Accumulation errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

70

7 Ray tracing and rotations

75

7.1

Displacements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

76

7.2

Rotation of coordinate systems . . . . . . . . . . . . . . . . . . . . . . . . .

76

7.3

Changes of direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

7.4

Putting it all together . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

80

8 Transport in media, interaction models 8.1 Interaction probability in an infinite medium . . . . . . . . . . . . . . . . . . 8.1.1

85 85

Uniform, infinite, homogeneous media . . . . . . . . . . . . . . . . . .

86

8.2

Finite media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87

8.3

Regions of different scattering characteristics . . . . . . . . . . . . . . . . . .

87

8.4

Obtaining µ from microscopic cross sections . . . . . . . . . . . . . . . . . .

90

8.5

Compounds and mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93

8.6

Branching ratios

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

94

8.7

Other pathlength schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . .

94

8.8

Model interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

8.8.1

Isotropic scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

8.8.2

Semi-isotropic or P1 scattering . . . . . . . . . . . . . . . . . . . . . .

95

CONTENTS

v

8.8.3

Rutherfordian scattering . . . . . . . . . . . . . . . . . . . . . . . . .

96

8.8.4

Rutherfordian scattering—small angle form

96

9 Lewis theory

. . . . . . . . . . . . . .

99

9.1

The formal solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

9.2

Isotropic scattering from uniform atomic targets . . . . . . . . . . . . . . . . 102

10 Geometry

107

10.1 Boundary crossing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 10.2 Solutions for simple surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 10.2.1 Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 10.3 General solution for an arbitrary quadric . . . . . . . . . . . . . . . . . . . . 114 10.3.1 Intercept to an arbitrary quadric surface? . . . . . . . . . . . . . . . . 117 10.3.2 Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 10.3.3 Circular Cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 10.3.4 Circular Cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 10.4 Using surfaces to make objects . . . . . . . . . . . . . . . . . . . . . . . . . . 125 10.4.1 Elemental volumes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 10.5 Tracking in an elemental volume . . . . . . . . . . . . . . . . . . . . . . . . . 132 10.6 Using elemental volumes to make objects . . . . . . . . . . . . . . . . . . . . 135 10.6.1 Simply-connected elements . . . . . . . . . . . . . . . . . . . . . . . . 135 10.6.2 Multiply-connected elements . . . . . . . . . . . . . . . . . . . . . . . 140 10.6.3 Combinatorial geometry . . . . . . . . . . . . . . . . . . . . . . . . . 142 10.7 Law of reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 11 Monte Carlo and Numerical Quadrature

151

11.1 The dimensionality of deterministic methods . . . . . . . . . . . . . . . . . . 151 11.2 Convergence of Deterministic Solutions . . . . . . . . . . . . . . . . . . . . . 154 11.2.1 One dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 11.2.2 Two dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

vi

CONTENTS 11.2.3 D dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 11.3 Convergence of Monte Carlo solutions . . . . . . . . . . . . . . . . . . . . . . 156 11.4 Comparison between Monte Carlo and Numerical Quadrature . . . . . . . . 156

12 Photon Monte Carlo Simulation

161

12.1 Basic photon interaction processes . . . . . . . . . . . . . . . . . . . . . . . . 161 12.1.1 Pair production in the nuclear field . . . . . . . . . . . . . . . . . . . 162 12.1.2 The Compton interaction (incoherent scattering)

. . . . . . . . . . . 165

12.1.3 Photoelectric interaction . . . . . . . . . . . . . . . . . . . . . . . . . 166 12.1.4 Rayleigh (coherent) interaction . . . . . . . . . . . . . . . . . . . . . 169 12.1.5 Relative importance of various processes . . . . . . . . . . . . . . . . 170 12.2 Photon transport logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 13 Electron Monte Carlo Simulation

179

13.1 Catastrophic interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 13.1.1 Hard bremsstrahlung production . . . . . . . . . . . . . . . . . . . . 180 13.1.2 Møller (Bhabha) scattering . . . . . . . . . . . . . . . . . . . . . . . . 180 13.1.3 Positron annihilation . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 13.2 Statistically grouped interactions . . . . . . . . . . . . . . . . . . . . . . . . 181 13.2.1 “Continuous” energy loss . . . . . . . . . . . . . . . . . . . . . . . . . 181 13.2.2 Multiple scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 13.3 Electron transport “mechanics” . . . . . . . . . . . . . . . . . . . . . . . . . 183 13.3.1 Typical electron tracks . . . . . . . . . . . . . . . . . . . . . . . . . . 183 13.3.2 Typical multiple scattering substeps . . . . . . . . . . . . . . . . . . . 183 13.4 Examples of electron transport

. . . . . . . . . . . . . . . . . . . . . . . . . 184

13.4.1 Effect of physical modeling on a 20 MeV e− depth-dose curve . . . . 184 13.5 Electron transport logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 14 Electron step-size artefacts and PRESTA

203

14.1 Electron step-size artefacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

CONTENTS

vii

14.1.1 What is an electron step-size artefact? . . . . . . . . . . . . . . . . . 203 14.1.2 Path-length correction . . . . . . . . . . . . . . . . . . . . . . . . . . 209 14.1.3 Lateral deflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 14.1.4 Boundary crossing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 14.2 PRESTA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 14.2.1 The elements of PRESTA . . . . . . . . . . . . . . . . . . . . . . . . 216 14.2.2 Constraints of the Moli`ere Theory . . . . . . . . . . . . . . . . . . . . 218 14.2.3 PRESTA’s path-length correction . . . . . . . . . . . . . . . . . . . . 223 14.2.4 PRESTA’s lateral correlation algorithm . . . . . . . . . . . . . . . . . 226 14.2.5 Accounting for energy loss . . . . . . . . . . . . . . . . . . . . . . . . 228 14.2.6 PRESTA’s boundary crossing algorithm . . . . . . . . . . . . . . . . 231 14.2.7 Caveat Emptor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 15 Advanced electron transport algorithms

237

15.1 What does condensed history Monte Carlo do? . . . . . . . . . . . . . . . . . 240 15.1.1 Numerics’ step-size constraints . . . . . . . . . . . . . . . . . . . . . . 240 15.1.2 Physics’ step-size constraints . . . . . . . . . . . . . . . . . . . . . . . 243 15.1.3 Boundary step-size constraints . . . . . . . . . . . . . . . . . . . . . . 244 15.2 The new multiple-scattering theory . . . . . . . . . . . . . . . . . . . . . . . 245 15.3 Longitudinal and lateral distributions . . . . . . . . . . . . . . . . . . . . . . 247 15.4 The future of condensed history algorithms . . . . . . . . . . . . . . . . . . . 249 16 Electron Transport in Electric and Magnetic Fields

257

16.1 Equations of motion in a vacuum . . . . . . . . . . . . . . . . . . . . . . . . 258 ~ =constant, B ~ = 0; B ~ =constant, E ~ = 0 . . . . . . . . 259 16.1.1 Special cases: E 16.2 Transport in a medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 16.3 Application to Monte Carlo, Benchmarks . . . . . . . . . . . . . . . . . . . . 264 17 Variance reduction techniques

275

17.0.1 Variance reduction or efficiency increase? . . . . . . . . . . . . . . . . 275

viii

CONTENTS 17.1 Electron-specific methods

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

17.1.1 Geometry interrogation reduction . . . . . . . . . . . . . . . . . . . . 277 17.1.2 Discard within a zone . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 17.1.3 PRESTA! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 17.1.4 Range rejection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 17.2 Photon-specific methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 17.2.1 Interaction forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 17.2.2 Exponential transform, russian roulette, and particle splitting . . . . 287 17.2.3 Exponential transform with interaction forcing . . . . . . . . . . . . . 290 17.3 General methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 17.3.1 Secondary particle enhancement . . . . . . . . . . . . . . . . . . . . . 291 17.3.2 Sectioned problems, use of pre-computed results . . . . . . . . . . . . 292 17.3.3 Geometry equivalence theorem . . . . . . . . . . . . . . . . . . . . . . 293 17.3.4 Use of geometry symmetry . . . . . . . . . . . . . . . . . . . . . . . . 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 18.6 Test routines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335

Fundamentals of the Monte Carlo method for neutral ...

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