Monte Carlo Particle Transport Methods 9781315895734

Table of contents :
Cover......Page 1
Title Page
......Page 2
Copyright Page
......Page 3
PREFACE......Page 4
ACKNOWLEDGMENTS......Page 6
Table of Contents
......Page 7
Chapter 1: Scope and Structure of the Book......Page 14
References......Page 15
Chapter 2: Introduction......Page 18
I. Sampling Probability Distributions......Page 19
A. The Inverse Distribution Method......Page 21
C. The Rejection Techniques......Page 22
D. The Table Lookup Method......Page 25
E. Selection from Power Functions......Page 26
F. Sampling from the Normal Distribution......Page 27
H. The Use of the First Derivative of the Probability Density Function......Page 29
I. Selecting Random Vectors......Page 32
J. Selecting Two- and Three-Dimensional Random Orientations......Page 33
II. Basic Physical Quantities......Page 35
B . The Particle Sources......Page 36
C. Flux-Type Quantities......Page 37
D. Elementary Interactions of Particles with Matter......Page 38
F. Collision Densities......Page 40
G. Quantities to be Determined: Reaction Rates, Responses, Scores......Page 42
References......Page 43
Chapter 3: Direct Simulation of the Physical Processes......Page 46
A. Selection of Source Parameters......Page 47
B. Path Length Selection......Page 52
C. Collisions- in General......Page 54
D. Interactions of Photons with Matter......Page 56
E. Interactions of Neutrons with Matter......Page 60
G. Scoring......Page 67
II. Plausible Modifications of the Analog Game......Page 68
A. Replacement of Absorption and Leakage by Statistical Weight Reduction......Page 69
C. Russian Roulette and Splitting......Page 71
D. Expected Values in Scoring......Page 72
III. Statistical Considerations......Page 75
B. The Actual Computations......Page 76
Appendix 3A: Energy Selection from the Klein-Nishina Formula......Page 78
A. Combination of the Direct Sampling and Rejection Techniques......Page 79
B. The Carlson Method......Page 81
A. Selection from the Maxwellian Distribution......Page 82
B. New Energy Selection from the Differential Thermal Neutron Cross-Section......Page 83
Appendix 3C: Fission Neutron Energy Selection......Page 84
Appendix 3D: Angle Selection for Anisotropic Scatterings......Page 86
B. Sampling from Linear Anisotropic Angular Distribution......Page 88
D. Selection of Discrete Angles from the Legendre Expansion......Page 90
References......Page 92
A. Two Basic Ways for Solving One-Dimensional Integrals......Page 94
C. Integration Domains of Complicated Shape......Page 96
D. Convergence of Numerical Integration Methods......Page 98
A. Mean and Variance in Straightforward Sampling......Page 99
B. Importance Sampling......Page 100
C. Systematic Sampling......Page 102
D. Quota Sampling......Page 103
E. Use of Expected Values......Page 104
F. Correlated Sampling......Page 105
A. Introduction......Page 106
C. Expansion into Neumann Series and Solution by Monte Carlo......Page 107
D. Kernel Distortion, Importance Sampling......Page 109
IV. Collision Density Equations......Page 111
A. Definition of the Collision Densities......Page 112
B. Definition of the Transition and Collision Kernels......Page 113
C. The Equations Connecting the Collision Densities......Page 114
D. The Theory of the Step-By-Step Solution of the Collision Density Equations......Page 116
E. Normalizations of the Transition and Collision Kernels......Page 117
F. Termination of the Monte Carlo Cycle......Page 120
B. Estimation of More than One Response......Page 121
C. Monte Carlo Estimation of the Responses......Page 122
D. Examples of Pay-Off Functions......Page 123
A. Path Stretching (Exponential Transformation)......Page 132
B. Perturbation Monte Carlo......Page 134
C. Criticality Studies......Page 138
VII. Adjoint Monte Carlo......Page 139
A. The Value Equations......Page 140
B. Solution of the Value Equations (Adjoint Monte Carlo)......Page 142
C. Sampling the Adjoint Source......Page 143
D. The Collision Kernel of the Value Equation......Page 144
E. Scoring in the Adjoint Monte Carlo......Page 147
F. Contributions of the Uncollided Particles......Page 150
VIII. Variances......Page 151
A. Variance Estimates by the Moment Equations......Page 152
References......Page 154
I. Introductory Remarks......Page 156
A. Relation of the Expected Score to the Adjoint Collision Density......Page 158
B. Conditions of Existence and Uniqueness......Page 159
C. Analog and Nonanolog Simulation......Page 162
D. Definitions and Notations......Page 164
E. Heuristic Interpretation of the Moment Equations......Page 168
A. Score Probability Equations......Page 171
B. Moment of a General Score Function......Page 173
C. Special Cases: Expectation and Second Moment of the Score......Page 176
D. An Analytical Example......Page 179
III. Extension to Multiplying Games......Page 182
A. Score Probability Equation......Page 183
B. Expectation and Second Moment......Page 184
C. An Equivalent Nonmultiplying Game......Page 186
D. Splitting: When a Nonmultiplying Game is Played as a Multiplying One......Page 191
E. Alternative Forms of the Collision Kernel......Page 195
IV. Further Generalizations......Page 196
A. Interruption and Restart of a Free Flight......Page 197
B. Geometrical Splitting......Page 199
C. Score Probabilities in a General Time-Independent Game......Page 205
D. Inclusion of Time Dependence......Page 206
V. Analysis of the First-Moment Equation......Page 209
A. Unbiased Estimators......Page 210
B. Weight Generation Rules......Page 212
C. A Nonanolog Game Without Statistical Weights: Importance Sampling......Page 216
D. Generalized Exponential Transformation......Page 220
E. Path Stretching......Page 225
F. Computing Time and Number of Events per History......Page 226
G. Feasibility of a Nonanolog Game......Page 229
H. Delta Scattering......Page 235
VI. Partially Unbiased Estimators......Page 239
A. Transformation Theorems......Page 241
B. Commonly Used Estimators......Page 244
C. Analysis of Variances in the Straight-Ahead Scattering Model......Page 249
VII. Approximate Solutions of the Moment Equations......Page 252
A. The Simplified Model......Page 253
B. The Separation Assumption......Page 254
C. On the Quality of the Approximation......Page 257
D. Effect of Surroundings......Page 259
VIII. Analysis of Second Moment Equations......Page 262
A. Zero-Variance Schemes......Page 263
B. On the Roundedness of the Variance......Page 271
C. Sufficient Conditions of Variance Reduction by Nonanolog Games......Page 273
D. Examples: Survival Biasing and ELP and MELP Methods......Page 276
E. Variance and Efficiency of the Equivalent Nonmultiplying Game......Page 278
F. Zero-Variance Partially Unbiased Estimators: The Minimum-Variance Composed Estimator......Page 284
G. Relative Merits of the Common Estimators......Page 288
H. The Self-Improving Estimator......Page 293
I. Variance Versus Efficiency in a Nonanalog Game......Page 296
J. Optimization of Source Distribution......Page 297
IX. Miscellaneous Specific Moment Equations......Page 299
A. Estimation of Bilinear Forms......Page 300
B. Correlation of Estimators......Page 302
D. Coupled Multiparticle Simulation......Page 303
Appendix 5A: Solution of the Moment Equations in the Forward/Backward Model......Page 307
Appendix 5B: Second Moments of Multiple Convolutions......Page 310
Appendix 5C: Solution of the Moment Equations in the Straight-Ahead Scattering Model......Page 313
References......Page 314
I. Correlated Monte Carlo: Perturbation Calculations......Page 318
A. Correlated Moment Equations......Page 320
B. Feasibility of a Correlated Game......Page 324
C. Correlated Difference Estimators......Page 327
D. Variance of the Correlated Score Difference......Page 328
E. Examples and Special Techniques......Page 334
F. Perturbation Source Method......Page 337
G. Parametric Perturbations: Integral Monte Carlo......Page 340
II. Differential Monte Carlo: Sensitivity Analysis......Page 341
A. Estimation of First-Order Derivatives......Page 342
B. Discussion of the Game......Page 345
C. Data Adjustment with Sensitivites......Page 349
D. Estimation of Higher-Order Derivatives......Page 351
E. A Simple Example......Page 353
F. Extension to Parameter-Dependent Estimators......Page 356
G. Perturbation Estimation by Differential Games: The Taylor Series Approach......Page 357
III. Criticality Calculations......Page 359
l. First Method......Page 360
2. Second Method......Page 363
3. Third Method......Page 364
4. Fourth Method......Page 367
B. On the Convergence of the Source Iteration......Page 368
C. Practical Realizations......Page 370
D. Variance of the Estimated Multiplication Factor......Page 375
E. A One-Step Scheme: Acceleration of the Iteration......Page 377
F. Reactivity Change Due to Perturbations......Page 380
G. Parametric Derivatives of keff·......Page 389
IV. Estimation of Flux at a Certain Point......Page 390
A. The Next-Event Point Estimator......Page 391
B. Confidence Limits for Singular Estimators......Page 395
C. Point Estimators with First-Order Singularity......Page 399
D. Bounded-Variance Point Estimators......Page 404
E. Practical Modifications of the Basic Methods......Page 407
A. Optimum Combination of Sample Means......Page 412
B. Unbiased Estimation of Combined Variance from Small Sample Sets......Page 419
C. Estimation of a Common Mean from Rare Sets......Page 424
D. Estimation of the Combined Variance of Rare Sets......Page 429
E. Estimation of Ratio of Expectations......Page 432
F. On the Determination of Theoretical Variances......Page 439
Appendix 6A: Unbiased Estimation of Criticality Reaction Rates......Page 443
Appendix 6B: Accuracy of the Corrected Variance from Small Sample Sets......Page 444
Appendix 6C: Expectation of the Matrix ARA......Page 447
Appendix 6D: Empirical Third Moments......Page 448
References......Page 449
Chapter 7: Optimization of Efficiency-Increasing Techniques......Page 454
A. Optimum Splitting Schemes in the Straight-Ahead Model......Page 455
B. Optimization of Path Stretching in the Straight-Ahead Model......Page 462
C. Approximate Optimization of the Russian Roulette Parameter......Page 458
D. Optimization by Direct Statistical Approach......Page 471
II. Optimization of Geometrical Splitting......Page 475
A. Geometrical Splitting in Terms of Regional Importances......Page 476
B. A Simple Method......Page 477
C. Properties and Refinements of the Method......Page 481
D. The Continuous Splitting Model......Page 483
E. Optimization of the Continuous Splitting Scheme......Page 486
F. Practical Realizations......Page 493
G. The Weight-Window Technique......Page 499
III. Optimization of Path Stretching......Page 500
A. Zero-Variance Path-Stretching Schemes......Page 502
B. Discussion of the Schemes......Page 507
C. Practical Applications in Deep Penetration Calculations......Page 510
D. Special Problems Associated with the Method......Page 519
Appendix 7 A: Approximate Moments of the Number of Transmitted Particles Through Multilayer Slabs......Page 521
References......Page 522
Index......Page 526

Citation preview

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations Authors

Ivan Lux, Ph.D. Head Applied Reactor Physics Department Central Research Institute for Physics Budapest, Hungary and

Laszlo Koblinger, Ph.D. Senior Scientist Health Physics Department Central Research Institute for Physics Budapest, Hungary

CRC Press

Taylor & Francis Group Boca Raton London New York

CRC Press is an imprint of the Taylor & Francis Group, an informa business

First published 1991 by CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 Reissued 2018 by CRC Press © 1991 by CRC Press, Inc. CRC Press is an imprint of Taylor & Francis Group, an Informa business

No claim to original U.S. Government works This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright. com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Lux, I. Monte Carlo particle transport methods: neutron and photon calculations/authors, Iván Lux and László Koblinger. p. cm. Includes bibliographical references and index. ISBN 0-8493-6074-9 1. Neutron transport theory. 2. Photon transport theory. 3. Monte Carlo method. I. Koblinger, László. II. Title. QC793.5.N4628L88 1990 530.1’38—dc20

90-2108

A Library of Congress record exists under LC control number: 90002108 Publisher’s Note The publisher has gone to great lengths to ensure the quality of this reprint but points out that some imperfections in the original copies may be apparent. Disclaimer The publisher has made every effort to trace copyright holders and welcomes correspondence from those they have been unable to contact. ISBN 13: 978-1-315-89573-4 (hbk) ISBN 13: 978-1-351-07483-4 (ebk) Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

PREFACE With this book we try to reach several more-or-less unattainable goals, namely: • • • • • •

To comprise in a single book all the most important achievements of Monte Carlo calculations for solving neutron and photon transport problems; To present a book which discusses the same topics in the three levels known from the literature; To write a book which gives useful information for both beginners and experienced readers; To list both the well-established old techniques and the newest findings; To fulfill the functions of both a textbook and a handbook; and last but not least, To formulate everything in a manner that is understandable (and, perhaps, sometimes even enjoyable) for the interested readers.

These are the goals . . . The judgement whether we were at least partly successful in reaching them is for the readers. Ivan Lux Laszlo Koblinger

THE AUTHORS Ivan Lux, Ph.D., is Head of the Applied Reactor Physics Department of the Central Research Institute for Physics, Budapest, Hungary. Dr. Lux received his M.Sc. in Physics in 1968 and in Mathematics in 1973, both at the Roland Edtvos University, Budapest. He received his Ph.D. in Physics in 1974 and the Candidate of Physical Science degree from the Hungarian Academy of Sciences in 1981. He was guest researcher at the Nuclear Engineering Laboratory of the Technical Research Centre of Finland from 1977 to 1978 and 1981 to 1982. Dr. Lux is a member of the Roland EotvOs Physical Society and won the Janossy Prize in 1979 (together with Dr. Koblinger). He is the author of about 60 papers and has been an invited speaker at several conferences. Laszlo Koblinger, Ph.D., is a Senior Scientist at the Health Physics Department of the Central Research Institute for Physics, Budapest, Hungary. Dr. Koblinger obtained his training at the Technical University of Budapest, receiving his M.Sc. in 1969 and his Ph.D. in 1976. In 1978, he received the Candidate of Physical Science degree from the Hungarian Academy of Sciences. Dr. Koblinger is a member of the International Radiation Protection Association and the International Radiation Physics Society. From 1975 to 1984, he was the Secretary and for the period 1985 to 1990 he was the President of the Health Physics Section of the Roland Edtvos Physical Society. He is a member of the Editorial Board of Fizikai Szemle (Hungarian Physical Review). He won the Janossy Prize in 1979 (together with Dr. Lux). Dr. Koblinger is the author of more than 60 papers and has presented over 20 lectures at international conferences. He is co-author of a book on the history of radiation protection in Hungary.

ACKNOWLEDGMENTS The authors wish to express their most sincere thanks to Professor Ely M. Gelbard of the Argonne National Laboratory for his most thoughtful approach to their manuscript — concerning himself, as he did, with constructive criticism to the scientific content as well as to ensuring that the English should not lead to any misunderstanding. In addition, Dr. Herbert Rief, EEC Joint Research Centre, Ispra, Italy is thanked for his valuable comments. The support of Dr. Janos ValkO and Dr. Istvan Feher, leaders of the Reactor Physics and Health Physics Departments at the Central Research Institute for Physics, Budapest, is gratefully acknowledged. Mrs. M. Dobrocsi and Mrs. I. Nemeth are thanked for typing and draftsmanship, respectively.

TABLE OF CONTENTS Chapter I Scope and Structure of the Book References Chapter 2 Introduction I. Sampling Probability Distributions A. The Inverse Distribution Method B. The Probability Mixing Method The Rejection Techniques C. D. The Table Lookup Method E. Selection from Power Functions F. Sampling from the Normal Distribution G. Efficient Selections from the Exponential Distribution H. The Use of the First Derivative of the Probability Density Function I. Selecting Random Vectors J. Selecting Two- and Three-Dimensional Random Orientations II. Basic Physical Quantities The Phase Space A. B. The Particle Sources Flux-Type Quantities C. D. Elementary Interactions of Particles with Matter E. Free-Paths, Distances F. Collision Densities Quantities to be Determined: Reaction Rates, Responses, G. Scores Other Quantities H. References Chapter 3 Direct Simulation of the Physical Processes I. Analog Simulation of the Random Walk A. Selection of Source Parameters B. Path Length Selection Collisions — in General C. Interactions of Photons with Matter D. E. Interactions of Neutrons with Matter Direction Cosines of a Particle after Scattering F. Scoring G. II. Plausible Modifications of the Analog Game Replacement of Absorption and Leakage by A. Statistical Weight Reduction Replacement of Multiplication by Increase of the Weight B. Russian Roulette and Splitting C. D. Expected Values in Scoring E. Problems with Extremely Rare Events

1 2

5 6 8 9 9 12 13 14 16 16 19 20 22 23 23 24 25 27 27 29 30 30

33 34 34 39 41 43 47 54 54 55 56 58 58 59 62

III.

Statistical Considerations A. The Central Limit Theorem The Actual Computations B. The Efficiency C.

Appendix 3A: Energy Selection from the Klein-Nishina Formula A. Combination of the Direct Sampling and Rejection Techniques B. The Carlson Method Appendix 3B: Thermal Neutron Energy Selection Selection from the Maxwellian Distribution A. New Energy Selection from the Differential Thermal B. Neutron Cross-Section Appendix 3C: Fission Neutron Energy Selection Appendix 3D: Angle Selection for Anisotropic Scatterings Table Look up Method A. Sampling from Linear Anisotropic Angular Distribution B. C. Application of the Rejection Technique for the Legendre Expansion D. Selection of Discrete Angles from the Legendre Expansion References Chapter 4 Collision Density and Importance Equations and Their Solution by Monte Carlo Monte Carlo Calculation of Integrals I. A. Two Basic Ways for Solving One-Dimensional Integrals B. Generalization to Multi-Dimensional Cases C. Integration Domains of Complicated Shape Convergence of Numerical Integration Methods D. II. Elementary Variance Reducing Techniques A. Mean and Variance in Straightforward Sampling B. Importance Sampling C. Systematic Sampling Quota Sampling D. E. Use of Expected Values F. Correlated Sampling G. Further Methods III. Solution of Fredholm-Type Integral Equations A. Introduction B. Fredholm-Type Integral Equations, Functionals to be Determined C. Expansion into Neumann Series and Solution by Monte Carlo

62 63 63 65 65 66 68 69 69 70

71 73 75 75 77 77 79

81 81 81 83 83 85 86 86 87 89 90 91 92 93 93 93 94 94

Kernel Distortion, Importance Sampling D. Collision Density Equations A. Definition of the Collision Densities Definition of the Transition and Collision Kernels B. The Equations Connecting the Collision Densities C. D. The Theory of the Step-By-Step Solution of the Collision Density Equations E. Normalizations of the Transition and Collision Kernels F. Termination of the Monte Carlo Cycle V. Scoring A. General Formulation of the Reaction Rates B. Estimation of More than One Response Monte Carlo Estimation of the Responses C. Examples of Pay-Off Functions D. VI. Three Special Problems Path Stretching (Exponential Transformation) A. B. Perturbation Monte Carlo Criticality Studies C. VII. Adjoint Monte Carlo The Value Equations A. Solution of the Value Equations (Adjoint Monte Carlo) B. Sampling the Adjoint Source C. The Collision Kernel of the Value Equation D. Scoring in the Adjoint Monte Carlo E. Contributions of the Uncollided Particles F. VIII. Variances Variance Estimates by the Moment Equations A. The Value Used as Importance Function B. References IV.

Chapter 5 The Moment Equations I. Introductory Remarks Relation of the Expected Score to the Adjoint A. Collision Density Conditions of Existence and Uniqueness B. Analog and Nonanolog Simulation C. Definitions and Notations D. Heuristic Interpretation of the Moment Equations E. Moment Equations in Nonmultiplying Games IL Score Probability Equations A. Moment of a General Score Function B. Special Cases: Expectation and Second Moment of C. the Score An Analytical Example D. Extension to Multiplying Games III. Score Probability Equation A. Expectation and Second Moment B. An Equivalent Nonmultiplying Game C. Splitting: When a Nonmultiplying Game is Played as D. a Multiplying One Alternative Forms of the Collision Kernel E.

96 98 99 100 101 103 104 107 108 108 108 109 110 119 119 121 125 126 127 129 130 131 134 137 138 139 141 141

143 143 145 146 149 151 155 158 158 160 163 166 169 170 171 173 178 182

IV.

Further Generalizations A. Interruption and Restart of a Free Flight B. Geometrical Splitting C. Score Probabilities in a General Time-Independent Game D. Inclusion of Time Dependence V. Analysis of the First-Moment Equation A. Unbiased Estimators Weight Generation Rules B. A Nonanolog Game Without Statistical Weights: C. Importance Sampling D. Generalized Exponential Transformation E. Path Stretching F. Computing Time and Number of Events per History G. Feasibility of a Nonanolog Game H. Delta Scattering VI. Partially Unbiased Estimators A. Transformation Theorems B. Commonly Used Estimators C. Analysis of Variances in the Straight-Ahead Scattering Model VII. Approximate Solutions of the Moment Equations A. The Simplified Model B. The Separation Assumption On the Quality of the Approximation C. Effect of Surroundings D. VIII. Analysis of Second Moment Equations A. Zero-Variance Schemes B. On the Boundedness of the Variance Sufficient Conditions of Variance Reduction by C. Nonanolog Games D. Examples: Survival Biasing and ELP and MELP Methods Variance and Efficiency of the Equivalent Nonmultiplying E. Game F. Zero-Variance Partially Unbiased Estimators: The Minimum-Variance Composed Estimator Relative Merits of the Common Estimators G. H. The Self-Improving Estimator I. Variance Versus Efficiency in a Nonanalog Game J. Optimization of Source Distribution Miscellaneous Specific Moment Equations IX. Estimation of Bilinear Forms A. B. Correlation of Estimators Moment-Generating Equation C. Coupled Multiparticle Simulation D.

183 184 186 192 193 196 197 199 203 207 212 213 216 222 226 228 231 236 239 240 241 244 246 249 250 258 260 263 265 271 275 280 283 284 286 287 289 290 290

Appendix 5A: Solution of the Moment Equations in the Forward/Backward Model

294

Appendix 5B: Second Moments of Multiple Convolutions

297

Appendix 5C: Solution of the Moment Equations in the Straight-Ahead Scattering Model

300

References

301

Chapter 6 Special Games Correlated Monte Carlo: Perturbation Calculations I. Correlated Moment Equations A. Feasibility of a Correlated Game B. Correlated Difference Estimators C. Variance of the Correlated Score Difference D. Examples and Special Techniques E. Perturbation Source Method F. Parametric Perturbations: Integral Monte Carlo G. Differential Monte Carlo: Sensitivity Analysis II. Estimation of First-Order Derivatives A. Discussion of the Game B. Data Adjustment with Sensitivites C. Estimation of Higher-Order Derivatives D. A Simple Example E. Extension to Parameter-Dependent Estimators F. Perturbation Estimation by Differential Games: The Taylor G. Series Approach Criticality Calculations III. Principle of the Simulation: The Source Iteration A. 1. First Method 2. Second Method Third Method 3. 4. Fourth Method On the Convergence of the Source Iteration B. Practical Realizations C. Variance of the Estimated Multiplication Factor D. A One-Step Scheme: Acceleration of the Iteration E. Reactivity Change Due to Perturbations F. Parametric Derivatives of k, G. Estimation of Flux at a Certain Point IV. The Next-Event Point Estimator A. Confidence Limits for Singular Estimators B. Point Estimators with First-Order Singularity C. Bounded-Variance Point Estimators D. Practical Modifications of the Basic Methods E. Specific Problems in Statistical Evaluation V. Optimum Combination of Sample Means A. Unbiased Estimation of Combined Variance from B. Small Sample Sets Estimation of a Common Mean from Rare Sets C. Estimation of the Combined Variance of Rare Sets D. Estimation of Ratio of Expectations E. On the Determination of Theoretical Variances F.

305 305 307 311 314 315 321 324 327 328 329 332 336 338 340 343 344 346 347 347 350 351 354 355 357 362 364 367 376 377 378 382 386 391 394 399 399 406 411 416 419 426

Appendix 6A: Unbiased Estimation of Criticality Reaction Rates

430

Appendix 6B: Accuracy of the Corrected Variance from Small Sample Sets

431

Appendix 6C: Expectation of the Matrix ARA

434

Appendix 6D: Empirical Third Moments

435

References

436

Chapter 7 Optimization of Efficiency-Increasing Techniques Simple Examples of Optimization Methods I. Optimum Splitting Schemes in the Straight-Ahead Model A. Optimization of Path Stretching in the Straight-Ahead Model B. Approximate Optimization of the Russian Roulette Parameter C. Optimization by Direct Statistical Approach D. Optimization of Geometrical Splitting II. Geometrical Splitting in Terms of Regional Importances A. B. A Simple Method Properties and Refinements of the Method C. The Continuous Splitting Model D. Optimization of the Continuous Splitting Scheme E. F. Practical Realizations The Weight-Window Technique G. Optimization of Path Stretching III. Zero-Variance Path-Stretching Schemes A. Discussion of the Schemes B. Practical Applications in Deep Penetration Calculations C. D. Special Problems Associated with the Method

441 442 442 449 445 458 462 463 464 468 470 473 480 486 487 489 494 497 506

Appendix 7A: Approximate Moments of the Number of Transmitted Particles Through Multilayer Slabs

508

References

509

Index

513

1 Chapter 1

SCOPE AND STRUCTURE OF THE BOOK Monte Carlo methods are being efficiently used for solving widely varying types of physical problems. Although Monte Carlo is trivially a straightforward tool to stimulate random processes, it can also be used for solving problems that have no immediate probabilistic interpretation. The first inventions of the method go back very far in history,' however, extensive applications came along with the construction and use of modern digital computers, i.e., from the late 1940s. Historically, the Monte Carlo method has first been successfully used to solve particle transport problems and this is still one of the areas of most extensive use.' The general method was originally developed by Fermi, Ulam, and von Neumann,' the first comprehensive review was published by Kahn,' and the high quality of this early contribution cannot be better evaluated than by stating that people interested in the use of Monte Carlo are well advised even nowadays to start with Kahn's report. The first book written exclusively for photon and neutron random walk simulations was published by Cashwell and Everett,' a newer review' — also from the Los Alamos group — appeared in 1975. Neutron transport is discussed with higher mathematical apparatus by Spanier and Gelbard.9 There are many books describing the particle transport in general — in these books, Monte Carlo is studied as one of the tools, generally in a separate chapter.'"' On the other hand, many reviews dealing with diverse aspects of Monte Carlo contain chapters devoted to particle transport. 5 '7 '8 The novelty of this book — and therefore the justification of its edition — can be briefly summarized in two points. First, the latest textbook9 deals only with specific problems and since it was published more than 20 years ago, it obviously does not cover the developments of the last decade. The newest methods, the rigorous and unified theory of which is the product of the last 6 or 7 years, can only be found in journal articles. In our judgment, these new methods (variance reducing techniques, efficiency analyses, combined scoring schemes, etc.) have now reached the point where they can be discussed in a precise and comprehensive way, in such a manner that is necessary for a textbook. Secondly, the present situation in the relevant literature is the following: There exist several introductory books or chapters'." which describe the simplest direct simulation procedures. Other reviews1".' already discuss several more advanced methods, however, such complicated though generally the most efficient techniques are only briefly mentioned, and not all of them at all. For the present users of Monte Carlo there might seem to be a gap between the level of the existing books and the much more refined description of the newest methods in the journal papers. This book aims to constitute a bridge over the gaps of the different levels. We do hope that this treatise in a single book at successively deeper levels, will satisfy an existing demand. The structure of the book is planned to coincide with the above-mentioned purpose. After an Introduction (Chapter 2) describing the basic sampling processes and precisely defining the later used quantities, Chapter 3 deals with the heuristically obvious methods: the one-to-one numerical simulation of the original physical processes and those modifications which are easily understood without rigorous and tiring mathematical derivations. In Chapter 4 the integral form of the Boltzmann Equation is the starting point. Here,

2

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

by a little bit more mathematics, the introduction of several more advanced techniques (such as the adjoint treatment) becomes possible. In Chapter 5 the whole treatment is based on the moment equations. The investigation of the equations that govern various moments of the Monte Carlo estimates are extremely helpful for increasing the efficiency of the methods. Special games (correlated, perturbation and differential Monte Carlo; criticality and flux at a point calculations) are discussed in Chapter 6 — based on the mathematical treatment introduced in the previous part. The last Chapter is devoted to optimization of the techniques (splitting, path stretching, Russian roulette, etc.) widely used in deep-penetration Monte Carlo calculations. Our intention was to compile the book in such a way that readers whose interest extends just to the depth of the first or to the second level, are provided with a concise and easily intelligible treatise of all the necessary tools for preparing Monte Carlo programs and solving problems. Though the real novelties are necessarily subjects of the later Chapters, we do hope that the reader can find new approaches, or descriptions of well-known techniques from a new, and hopefully interesting, point of view even in the first Chapters. We also hope that the understanding of the first level treatment will trigger out the curiosity of many readers to start to study the second and even further, the third level. Readers of the last three Chapters become familiar with the complete set of the most sophisticated weapons of the Monte Carlo arsenal. An essential feature of the book is that the same nomenclature and a unified notation is used throughout the different Chapters, wherever possible. Cross references between the various levels (particularly from higher levels to lower ones) make it obvious for the reader that the different approaches represent different projections of the very same physical phenomena. References are given at the end of each Chapter, therefore, several basic sources are listed more than once in the book. From our point of view, such repetitions are not unnecessary, but rather make the reader's orientation easier. It is clear for us that the inclusion of all the knowledge accumulated during 4 decades by a lot of scientists is impossible. Apart from the limitation mentioned already in the title of the book, i.e., that we deal only with transport of neutrons and photons, the most serious stipulation is that with very few exceptions, the whole treatment is restricted to timeindependent, or steady-state problems. There are also many minor points not treated, for example we do not discuss the construction and tests of the basic random number generators. In these cases the reader is directed to the literature.

REFERENCES 1. Carter, L. L. and Cashwell, E. D., Particle-Transport Simulation with the Monte Carlo Method. ERDA Crit. Rev. Ser., National Technical Information Service, Springfield, MA, 1975. 2. Cashwell, E. D. and Everett, C. J., A Practical Manual on the Monte Carlo Method for Random Walk Problems, Pergamon Press, London, 1959. 3. Ermakov, S. M. and Mikhailov, G. A., Course of Statistical Modelling (in Russian), Nauka, Moscow, 1976. 4. Halton, J. H., A Retrospective and Prospective Survey of the Monte Carlo Method, SIAM Rev., 12, 1, 1970. 5. Hammersley, J. M. and Handscomb, D. C., Monte Carlo Methods, John Wiley & Sons, New York, 1964. 6. Kahn, H., Applications of Monte Carlo. AECU-3259 Report, Rand Corporation, Santa Monica, CA, 1954.

3 7. Shreider, Y. A., Ed., Method statisticheskih ispytani (Monte Carlo) (in Russian) Fizmatgiz, Moscow (1961) — German translation: Die Monte Carlo Methode and ihre Verwirklichung mit elektronischen Digitalrechnem. B . b. Teubner Verlags gesellschaft, Leipzig, 1964. — English translation: The Monte Carlo Method, Pergamon Press, New York, 1966. 8. Sobol, I. M., The Monte Carlo Calculational Method, (in Russian), Nauka, Moscow, 1973. 9. Spanier, J. and Gelbard, E. M., Monte Carlo Principles and Neutron Transport Problems, AddisonWesley, Reading, MA, 1969. 10. Stevens, P. M. and Trubey, D. K., Methods for Calculating Neutron and Gamma-ray Attenuation. Weapons Radiation Shielding Handbook, Report DNA-1892-3, Nuclear Defense Agency, Washington, D.C., 1972, Chap. 3. 11. Wood, J., Computational Methods in Reactor Shielding, Pergamon Press, Oxford, 1982.

5 Chapter 2

INTRODUCTION When we started to think about writing a book on Monte Carlo techniques for neutron and photon transport calculations it was clear that in the very first sentence a nice definition of the Monte Carlo method itself should be given. This task seemed to be very easy: just have a look on the earlier textbooks and copy the well-established definition! However, after reading more and more introductions it became more and more hopeless to find this sentence. Instead of exact definitions we have rather found illustrations and examples. In the book, which is the most sophisticated earlier description of the Monte Carlo applications on neutron transport," the authors, J. Spanier and E. M. Gelbard frankly confess that they found "it difficult to construct a definition which characterizes the Monte Carlo method accurately, completely and concisely". Their next sentence, however, already catches a basic feature of Monte Carlo, namely that "this method, in all its forms, involves some sort of random sampling process". And, really, random is the only word obligatorily contained in all definitions. Anyhow, after listing our excuses, we cannot avoid giving our definition, which may not be accurate, complete, and concise, but can help the reader begin to have a rough image about the method. In all applications of the Monte Carlo method a stochastic model is constructed in which the expected value of a certain random variable (or of a combination of several variables) is equivalent to the value of a physical quantity to be determined. This expected value is then estimated by the average of several independent samples representing the random variable introduced above. For the construction of the series of independent samples, random numbers following the distributions of the variable to be estimated are used. There are two requirements imbedded in this definition, viz.: First, a stochastic model adequate to the problem has to be constructed. Secondly, in the actual Monte Carlo calculations, the user has to be able to select random numbers with various distributions. There are basically two different ways to construct a stochastic model. In certain cases — as in particle transport, the topics of this book — the physical process is per se stochastic and thus the most straightforward Monte Carlo calculation is simply a numerical (or computer) simulation of the real physical events. Such direct simulations are called analog Monte Carlo games. When the computational process deviates more or less from the one-to-one simulation of the actual physical process, the game is called nonanalog. The distinction is not always clear. Several authors tolerate small deviations and still call simulations slightly differing from the straightforward one analog. (In our book at the beginning the strictness of this distinction has no importance, however, from Chapter 5 on, the term "analog" is used exclusively for the really analog simulations.) The other extreme case is when the stochastic model is constructed artificially, just for solving deterministic equations by Monte Carlo. In the simulation of a physically stochastic process, two expected values: that of the physical quantity and that of the average of random samples, must equal one another. Both expected values have their own variances, which may have no direct relationship. In the correct solution of a deterministic problem, the expected value of the random sample average equals the real value of the quantity in question which is not accompanied by any statistical uncertainty. Though the distinction between the two cases described above is clear, in transport calculations one can seldom — or rather never — find algorithms or computer codes based purely on one-to-one simulation of the physical processes, and — on the other hand — in

6

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

the procedures derived for solving the deterministic equations of the expected values (generally collision densities), one can still find many resemblances to a particle's random walk. Here we have again attained one of the aims of this book: we should like to illustrate with as many examples as possible that heuristically introduced plausible modifications of the simple simulations lead to techniques, the validity of which can be mathematically correctly proved by analysis of the deterministic transport equations. The opposite statement is not always true, there are special procedures which are hardly understandable heuristically. An immediate question arises: why are refined nonanalog methods worked out, if we know that an accurate, analog simulation of the real physical process does not necessarily serve us with correct results? The answer is very simple: to save computer time. In the physical experiments millions of particles are usually emitted from the source and only a small fraction of them is observed by a receptor (the word receptor is used hereafter in a most general sense, it may be, e.g., a physical detector, a cell in a reactor core, an organ in a human body). In the computation — even on the fastest machine — the simulations of all the interreactions of so many particles is impossible within reasonable running times. The use of less source particles may result in a very small number (none, in the extreme) of them reaching the receptor, thus causing very poor statistics, i.e., nonconfident results. This answer directly involves a precondition against the nonanalog techniques: they are worth application only if they decrease the computer time as compared to that of the analog simulation, assuming that the statistical uncertainties are the same in the two cases. Needless to say the first precondition is that the result, the expected value of the physical quantity to be determined in any accepted nonanalog technique, must be the same as in the physical reality or in the direct simulation of the process. The second requirement for building Monte Carlo games is the ability to select the proper random numbers. This is the topic of the next sections.

I. SAMPLING PROBABILITY DISTRIBUTIONS In most practical cases, sampling of any probability distribution is based on sampling one or more random number(s) uniformly distributed (or equidistributed) over the interval (0,1) (hereafter: random number) and on a transformation of it (them). The probability density function (PDF) of the random numbers is:

P(t)

=

o':

if 0 0 and continuous from the right at x = 0 and let: (i)

P(x) = 0, if x < 0

(ii)

p'(x) continuous, if x > 0

(iii)

g(y) =— [ap(y) + p1 (y)1(1 — ec`Y) % 0, if y %-0 a

(iv)

lira p(y)e" = 0

where a is again an arbitrary real number. The g(y) defined in this way is again a PDF and if 1 is a sample from g(y) then 1 = -- ln[l — (1 — e-ern)p] a is a sample from p(x). The proof is analogous to that of Theorem 2.7. and is thus not detailed here. Several illustrative applications of the above theorems are given in the original paper of Lux,' here we call the attention of the reader to one only which fits to our special field: If in Theorem 2.7 a is set to unity and A tends to infinity, then from Equation (2.11) g(Y) = P(Y) + Po'(Y) and if g(y) is non-negative then according to Equation (2.12) = —1np + 1 This selection procedure was first recommended by Mikhailov28 for sampling of the fission neutron spectrum and the Maxwell energy distribution. I. SELECTING RANDOM VECTORS Very often, a random vector of an n dimensional phase space (that is n coordinates of a random point) has to be selected. In the simplest cases the multidimensional probability distribution can be factorized into a product of one dimensional PDFs of mutually independent random variables. A simple example of it for n = 2 is the selection of points in a square: Here both and 1 are equidistributed on (0,1) and their representative values x and y can be set by the use of two successive random numbers: x = p, Y=

P2

20

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations '2 4 1

0

1

FIGURE 2.5. A random point in a square.

FIGURE 2.6. Selection of a random unit vector in two dimensions.

If the PDFs cannot be separated for the variables (i.e., the borders of the domain are complicated) generally only the rejection technique works. Let the n-dimensional domain — from which the samples are to be taken — be defined by the relation F( t ,kz,... and a;

fl)

0

b; for i = 1, 2 . . . ,n then select x;'s with x; = a; + (b; — ai)p;

and accept the point (a l , a2, . . . a.) if and only if

F(x„x2,...,xn)

0

J. SELECTING TWO- AND THREE-DIMENSIONAL RANDOM ORIENTATIONS There are practically no transport codes where there is no need for the generation of randomly oriented two- and/or three-dimensional unit vectors. In two dimensions, according to the notation in Figure 2.6, the connection between the Cartesian coordinates and the angle cp is = cosy

21 and = sinp and pp is equidistributed in (0,27r). Thus, for the random selection of the two coordinates a quite straightforward method is given by: --= 27rp and y =-- cosp x = situp

(2.13)

An alternate method was first suggested by von Neumann' where the circle is covered by a square. The procedure is as follows: 2p, — 1 P2 = 2P2 - 1

except if + 15;

(2.14)

1

i.e., the (042) point lies within the circle. A simple normalization will give: P,

x( = cos(p) =

v + ij; y(= sing)) —

P2

The square roots in the denominators can be eliminated by the use of the double-angle formulae of trigonometry and the final procedure is: = 2P1 — 1 P2 = 2p2 - 1

except if: 02 = Pi + Pig 1 then x=

—

22

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

and 2P102 02

(2.15)

The efficiency is trivially the ratio of the circle to the square areas, i.e., E=

7r

4

0.785

Though the straightforward procedure (2.13) seems to be much simpler than (2.15), in most computers both the generation of random numbers and the execution of the other elementary operations of (2.12) are less time consuming than the evaluation of the sine and cosine of an angle. If not a unit vector, but rather a point from the circle area has to be selected then Equation (2.13) should be supplemented by r = max(p„p2) (since: p(r) = 2r dr) and x and y should be multiplied by r. In the rejection method Equation (2.14) gives directly the necessary coordinates (there is no need for normalization), thus its preference to the direct method in Equation (2.12) is even more obvious. In three dimension the coordinates of a random unit vector are31 z = o.) x = V1 — co2 cosy y = V1 — co2 sing) where w is equidistributed on ( — 1,1) and cosy and sirup can be generated by one of the methods listed above. A point from inside of a sphere can again be selected by the rejection method, where the sphere is boxed in a cube. The efficiency here is the ratio of the sphere to the cube volumes: 7r

E= — 6 0.523

II. BASIC PHYSICAL QUANTITIES The reader is assumed to be familiar with the basic physical quantities that are characteristic to particle transport that will be used in the following Chapters of this book. There are, however, several quantities which are named differently in different papers (e.g., fluence, flux, flux density . . . ), or which have equivalent physical interpretations but different names in different contexts (macroscopic cross-section and linear attenuation coefficient). Even a larger variety is found in the use of symbols for a number of terms. Considering all the above arguments it seems appropriate to give a systematic review of the basic quantities, their notations, definitions, and symbols as well as the derivation of the basic relations between several quantities. The main source of the definitions given in the following sections is the most recent booklet of the International Commission on Radiation Units and Measurements (ICRU),13

23 though many quantities not listed there are taken from other sources and several notations are specifically introduced for later use in this book. The survey of the definitions is arranged in sections collecting quantities describing similar phases of the radiation transport. A. THE PHASE SPACE A migrating particle (neutron or photon) is represented by a set of coordinates that uniquely determine the state of the particle. The notations of the relevant coordinates are given below. The three spatial Cartesian coordinates x, y and z of the particle are often denoted by the single vectorial symbol r. The three-dimensional unit direction vector is denoted by w and, if necessary, its components parallel to the x, y, and z coordinates are denoted by cox, co, and cow, respectively. The symbol E represents the energy of a particle. Since in many cases the energy and the direction of a particle change simultaneously, sometimes the coordinates (w, E) are simply denoted by a single vector E. (If somebody does not like to see the "energy" described by a "vector", we would like to remind them that — at least for neutrons — the direction vector + the energy coordinates might have been replaced by the velocity of the particle, i.e., by a real vector quantity.) Further simplifying the notation, a set of the spatial, direction, and energy coordinates are united and described by a point in the six-dimensional phase-space: P. In the integrations fff . . . dx dy dz may be replaced by f . . dr, similarly ff . . . dw dE is often reduced to f . . . dE and the shortest way to denote an integration over the whole phase-space is f . . . dP. B. THE PARTICLE SOURCES The intensity of a neutron or photon source is denoted by Q and Q(P) means the number of particles emitted with coordinates in dP about P. Generally, for the Monte Carlo calculations the equations are established for one starting particle, i.e.,

IQ(P)dP = 1 and hence Q is called the source density. For radioactive sources the term activity is used which is the "quotient of dN by dt, where dN is the number of spontaneous nuclear transformations which occur . . . in the time interval dt". This quantity of the ICRU differs from our intensity in two respects: 1. Since we deal with stationary processes in most of this book, the differentiation with respect to time is not necessary for us, we shall consider all quantities (e.g., collision densities, reaction rates) as integrated over an arbitrary time interval (e.g., unity); 2. there are many isotopes where e.g., beta decay is the elementary "nuclear transformation" and gammas are emitted only in a fraction of decays thus the number of transformations (activity) is higher than the number of photons emitted (intensity). Neutron sources are often characterized by the yield which is the number of neutrons leaving the source. Thus in case of extended sources, the yield is decreased by self-absorption. In many cases, the sources are isotropic, monoenergetic, or point-like. In such cases the argument is simplified from (P) (r,w,E) to (r) or (r,E) or in any other way but the symbol Q is preserved even in these cases. Therefore, equations like

Q(r) = fQ(r,E)dE

(2.15a)

24

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

are written, and here the difference in the argument indicates the difference in the quantities and necessarily in their units. C. FLUX-TYPE QUANTITIES This is the point where perhaps the most loose use of words can be found in the literature. By the ICRU definition "the fluence, (I), of particles is the quotient of dN by da, where dN is the number of particles which enter a sphere of cross-sectional area da". The argument r is naturally joinable to the symbol 1 since it specifies the point around which the sphere is placed. The time derivative of the fluence (q) = F) is called by the ICRU as flux density or fluence rate, however, in most of the books and papers both fluence and flux density are simply called flux, and this word is used throughout our book too. Flux also can be considered as differential by energy and/or direction of flight, then it is denoted as (p(r ,E)

or

(p(r ,o3)

or

p(P)

In reactor physics, many times an alternative definition of flux is preferred. In such cases the particle density (n) is first introduced: n(P) is the number of particles per unit phase-space volume. Then the flux density is defined as cp(P) = vn(P) where v is the particle's speed (\/2E/m for neutrons and it equals to c, the speed of light, for photons). The speed of an individual particle is its track-length per unit time. Hence, the total flux can be conceived as the sum of track-lengths traced out by the particles per unit volume. A similar final result can be derived from the ICRU definition, if we take into account that for a convex body the mean length of randomly oriented chords (d) is": d

= 4V/A

(2.16)

where V is the volume and A is the surface area. In the ICRU fluence definition dN dN • El =—= da dV

(2.17)

since for spheres A = 4a, dA = 4da. The numerator in the R.H.S of Equation (2.17) is just the total chord length of the dN particles crossing the infinitesimal sphere. Flux is characteristic to the number of particles crossing an infinitesimal sphere — regardless of their orientation. If one is interested in the direction of the flow the current vector (J) can be introduced as J(P) = wt(P) Obviously, if a unit area normal to the unit direction vector n is placed into the phase space at about r, then the number of particles crossing it (.1„) is: Jn = J(r)n

25 D. ELEMENTARY INTERACTIONS OF PARTICLES WITH MATTER Neutrons and photons interact with matters in many ways. Interactions or collisions can lead to: • • •

Absorption, when the original particle entering the collision is absorbed and no particle of the same type is emitted, or Scattering, if the incoming particle continues its flight after the collision but possibly with altered direction and energy, or Multiplicative effects, where after certain nuclear transformations more than one of the same type particles leave the collision than that entered.

There are interactions when the type of the outcoming particle(s) is different from that of the colliding particle, e.g., reactions (n,y) or (n,ny). The outputs of these events give additional source terms in the joint neutron-photon transport calculations. The probabilities of these interactions both for neutrons and photons, depend on the colliding particle's energy and on the knocked element. The interaction probabilities are described by the cross-sections. The total microscopic cross-section is defined as the probability of an interaction in a mass element divided by the product of the number of nuclei and the fluence. Its unit is therefore cm2, and the generally used symbol is cr, however, we shall denote it by cr*. In the formulae of the transport processes another quantity is much more frequently used: 0-

= pN A — fm f*

(2.18)

where p is the density of the material, NA is the Avogadro constant, and M is the molar mass of the target element. The quantity defined by Equation (2.18) has the unit of 1/cm and in neutron physics it is called the macroscopic cross-section and denoted by /„ whereas in photon interactions the term linear attenuation coefficient and the symbolµ are preferred. Since in this book we deal with the transport of both particles the symbol if and the simple name cross-section is used for the quantity of Equation (2.18) and if the microscopic cross-section is referred, we distinguish it with the obligate attribute "microscopic" and the superscript asterisk. If the matter investigated is a compound of n elements then the resultant cross-section is the weighted sum of the elementary microscopic cross sections Cr i = pNA w; —

=i

(2.19)

where w, is the weight fraction, o-1` is the microscopic cross-section, and M, is the molar mass of the i-th component. In most cases different types of interactions may occur at a certain collision and thus the total cross section can be expressed as the sum of partial cross-sections. If crti denotes the partial microscopic cross-section of the j-th type of interaction on the i-th element, then the total cross-section is: Q = pNA

M;;=1

o-

(2.20)

where we assumed that altogether m types of interactions can occur in the case investigated.

26

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations By introducing the partial macroscopic cross-sections as w, = pNA 1‘4+ oEquation (2.20) reduces to n m

cr =

(2.21)

0-„

1= 1

The cross-section is a function of the incident particle's energy and in inhomogeneous media it also varies from site to site, thus generally has two arguments: cr(r,E) In non-absorption events, differential cross-sections can be introduced where cr(r,o3,E—>co',E')dco' dE' is proportional to the probability that the particle entering a collision of type j with direction w and energy E leaves it with new direction and energy coordinates in dw' about w' and between E' and E' + dE', respectively. Again only the argument will indicate whether differential or integrated quantities are mentioned, i.e., similarly to Equation (2.15a), formulae like crj(r,E) = f

dE'

can occur. Another quantity important for the study of scattering events is the expected number of the outcoming particles. One can assign such an expected number (v) to every type of interactions. This number will be trivially 0, for an absorption, 1, for simple scattering events, 2, for (n,2n) reactions, etc. For fission interactions of neutrons v is usually not an integer which indicates that different numbers of neutrons can be emitted even if the same type of nuclei are split by neutrons of the same incident energy. If a complete set of vu-s are assigned to possible interactions with Q j cross-sections, then the expected number of the outcoming particles is

EE v„ 0-, —

o-

which may be either less or larger than one. The probability of the occurrence of a certain reaction (type j) with a partial crosssection a-, is

Q and this quantity, from its definition could easily be called reaction rate (or more definitely

27 absorption, scattering, etc. rate if the type of reaction is definitely denominated). However, the term reaction rate shall be used in a much more general sense, as defined in Section G. of this Chapter. E. FREE-PATHS, DISTANCES In a homogeneous medium the mean free path between two collisions is

since the PDF of the path length (R) is p(R) = o-e —"R

(2.22)

and thus 1 X = = fR = — 0. Actual selected paths will also be denoted by R. In inhomogeneous media, where the cross-section changes during the flight between two collisions the PDF of Equation (2.19) is changed to: p(R) = o(R) exp( — foxo(R')dR') The quantity

T(R) =

exp( — o-(Ri)dRi )

(2.23)

is called the optical distance from the starting point to the next collision site at a geometrical distance of R. In most of the practical cases, the material does not change continuously, but there are different, clearly separated regions filled with different media and thus the integral in Equation (2.23) is replaced by a sum (see Figure 2.7):

T(R) =

E=1

F. COLLISION DENSITIES In Section D. the elementary probabilities of interactions were described. The expected number of collisions occurring in a phase-space element dP about P are characterized by two functions, two collision densities. X(P)dP is the number of particles leaving a collision with coordinates lying in dP about P and is sometimes briefly called the outgoing density, whereas ti,(P)dP

28

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations cross sections:

6n

62 n

R2

i+1)st collision site i-th collision 1-st material

2-nd material

1

2

n-th material n

FIGURE 2.7. A particle flight crossing boundaries of zones of different materials.

denotes the number of particles entering a collision with coordinates dP about P and its name is incoming density, or simply collision density. From the above definitions it is clear that none of these functions is a "density function" since generally the normalization conditions

f x(P)dP = 1 and

ILIJ(P)dP = 1 are not satisfied. The incoming density is closely related to the flux. Let us recall the definition of the flux (fluence, by the rigorous ICRU terminology): of,(r,E) dadE is the number of particles entering an infinitesimal sphere of radius dr, cross-sectional area of da = rr (dr)2 with direction and energy dE about E. The expected path length of these particles is (see Equation 2.14) 4 = — dp 3 Thus the expected number of particles entering collisions in the infinitesimal sphere (i.e. the collision density) is tKr,E) dr dE = o-(r,E) < cif > 4(r,E)dadE = o-(r,E) 4(r,E)drdE

29 therefore: tp(P) = cr(r„E) 4(P)

(2.24)

1 tif(P) cr(r„E)

(2.25)

The inverse relation (RP) =

is valid only if cr 0 0, which condition clearly reflects the very simple physical fact that in vacuum the flux is still a reasonable quantity whereas no collisions can happen if no material is present. G. QUANTITIES TO BE DETERMINED: REACTION RATES, RESPONSES, SCORES Generally, the aim of a Monte Carlo calculation is the estimation of the value of a physical quantity or values of several quantities. For the sake of simplicity we restrict our discussion to the determination of a single quantity. Extension of the considerations to parallel examination of several quantities is straightforward. These quantities can represent a great number of physically interpretable data varying from the number of collisions in a space element to leakage probabilities, detector responses or doses absorbed in certain regions of the core of a reactor or even in organs of an anthropomorphic phantom. Just to preserve generality of the discussion all these quantities will be called either as receptor responses or reaction rates. A common feature of these responses is that they can be formulated as weighted integrals (or functionals) of one of the collision densities. Thus a response (or reaction rate) is most generally formulated as: R = Ifx(P)x(P)dP

(2.26)

or R = 1f4,(P)14P) dP The fx and fq, weight function are derived from the physical connection between the collision density and the quantity to be determined. The integrations are extended to the region of interest, or, by other words the f weight functions have to vanish outside the range of interest. Since the subscript of f is trivial from the type of collision density used in the integrals of (2.26) it is generally omitted. Naturally if R is calculated not for a finite range but only for a point, e.g., for re,, then the weight function contains a Dirac-delta component: f(P) = fi(E)6(r — ro) Simply, if the number of collisions are to be computed for a phase space domain Po, then 11,

if P E Po

f(P) = 0,

otherwise

30

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

Another simple example can be derived from Equation (2.25), the formula for the flux integral in a phase-space domain Po is

= f cr `(r,E)1.15(P) dP

f+(P) = cr `(r,E)

here.

A Monte Carlo estimate of the reaction rate is called the (actual) score, and every simulated event has a — many times zero — contribution to the score. A reaction rate is usually estimated by several independent Monte Carlo histories and the final estimate will be the average of the individual scores. H. OTHER QUANTITIES There are many other quantities — which may or may not have direct physical meaning — introduced in the next chapters. They are, however, used only in connection with certain special examples or techniques and will be defined at the appropriate places. A general rule of our notation is that if a quantity is denoted by x in the analog simulation (in the numerical "copy" of the real physical process) then its counterpart used in the nonanalog simulations (e.g. , simulations deviating from the physical process) will be denoted by z. The statistical weights necessarily introduced for nonanalog simulations are denoted by W and the factors modifying it at a single step by w.

REFERENCES 1. Ahrens, J. H. and Dieter, U., Computer Methods for Sampling from the Exponential and Normal Distributions, Comm. ACM, 15, 873 (1972). 2. Barnett, U. D., The behaviour of pseudo-random sequences generated on computers by the multiplicative congruential method, Meth. Comp., 16, 63 (1962). 3. Carter, L. L. and Cashwell, E. D., Particle-Transport Simulation with the Monte Carlo Method. ERDA Critical Review Series, National Technical Information Service, Springfield, MA, 1975. 4. Coveyou, R. R., Serial correlation in the generation of pseudo-random numbers, J. Assoc. Comput. Mach., 7, 72 (1960). 5. Coveyou, R. R. and MacPherson, R. D., Fourier Analysis of Uniform Random Number Generators, J. Assoc. Comput. Mach., 14, 100 (1967). 6. Ermakov, S. M. and Mikhailov, G. A., Course of Statistical Modelling (in Russian), Nauka, Moscow (1976). 7. Everett, C. J. and Cashwell, E. D., A Monte Carlo Sampler. LA-5061-MS Report, Los Alamos Scientific Laboratory, Los Alamos (1972). 8. Everett, C. J. and Cashwell, E. D., A Second Monte Carlo Sampler. LA-5723-MS Report, Los Alamos Scientific Laboratory, Los Alamos, CA, (1974). 9. Everett, C. J. and Cashwell, E. D., A Third Monte Carlo Sampler. LA9721-MS Report, Los Alamos National Laboratory, Los Alamos, CA, (1983). 10. Forsythe, G. E., Generation and Testing of Random Digits at the National Bureau of Standards. Applied Mathematics Series, No. 12, U.S. Government Printing Office, Washington, D.C. (1951), 34. 11. Gruenberger, F., Tests of random digits, Math. Tables Aids Comput., 4, 244 (1956). 12. Halton, J. H., A retrospective and prospective survey of the Monte Carlo method, SIAM Rev., 12, 1 (1970). 13. Radiation Quantities and Units. Report 33, International Commission on Radiation Units and Measurements, Washington, D.C. (1980).

31 14. Irving, D. C., Freestone, R. M., Jr., and Kam, F. B. K., 05R, A general-purpose Monte Carlo neutron transport code. ORNL-3622 Report, Oak Ridge National Laboratory, Oak Ridge, TN (1965). 15. Kahn, H., Applications of Monte Carlo. AECU-3259 Report, Rand Corporation, Santa Monica, CA (1954.) 16. Kendall, M. G. and Babington-Smith, B., Randomness and random sampling numbers, J. R. Statist. Soc. A, 101, 147 (1938). 17. Kendall, M. G. and Moran, P. A. P., Geometrical Probability, Hafner Publishing Co., New York (1963). 18. Kinderman, A. J. and Ramage, J. G., Computer generation of normal random variables, J. Am. Statist. Assoc., 71, 893 (1976). 19. Lehner, D. H., Mathematical methods in large-scale computing units, in Proc. Symp. on Large-Scale Digital Calc. Mach., Harvard Univ. Press, Harvard, MA (1949), 141. 20. Lux, I., A special method to sample some probability density functions, Computing, 20, 183 (1978). 21. Lux, I., Generation of random numbers by iterative rejection technique (in Hungarian). Alkalmazott Matematikai Lapok, 1, 347 (1975). 22. Lux, I., HEXANN-EVALU - a Monte Carlo Program System for Pressure Vessel Neutron Irradiation Calculation. VTT Report 210, Technical Research Centre of Finland, Espoo (1983). 23. Marsaglia, G., Generating Exponential Random Variables, Ann. Math. Stat., 32, 899 (1961). 24. Marsaglia, G. and Bray, T. A., A convenient method for generating normal variables, SIAM Rev., 6, 260 (1964). 25. MacLaren, M. D., Marsaglia, G., and Bray, T., A fast procedure for generating exponential random variables, Comm. ACM, 7, 298 (1964). 26. MacLaren, M. D. and Marsaglia, G., Uniform random number generators, J. Assoc. Comput. Mach., 12, 83 (1965). 27. McGrath, E. J. and Irving, D. C., Random Number Generation for Selected Probability Distributions. ORNL-RSIC-38 Report, Techniques for Efficient Monte Carlo Simulation, Vol. II., Oak Ridge National Laboratory, Oak Ridge, TN (1975). 28. Mikhailov, G. A., On modelling random variables for one class of distribution laws, (in Russian), Teoriya Veroyatnostei i ee Primeneniya, 10, 749 (1965). 29. Neumann, J., Various Techniques Used in Connection with Random Digits. National Bureau of Standards Applied Mathematics Series, No. 12, U.S. Government, Printing Office, Washington, D.C. (1951), p. 36. 30. Rhoades, W. A. and Mynatt, F. R., The DOT III. Two-dimensional Discrete Ordinates Transport Code. ORNL-TM-4280 Report. Oak Ridge National Laboratory, Oak Ridge, TN (1973). 31. Shreider, Y. A., Ed., Metod statisticheskih ispytani Monte Carlo (in Russian) Fizmatgiz, Moscow (1961). - German translation: Die Monte Carlo Methode and ihre Verwirklichung mit elektronischen Digitalrechnern B.b. Teubner Verlagsgesellschaft, Leipzig (1964). - English translation: The Monte Carlo Method, Pergamon Press, New York (1966). 32. Sobol, I. M., The Monte Carlo Calculational Method, (in Russian), Nauka, Moscow (1973). 33. Spanier, J. and Gelbard, E. M., Monte Carlo Principles and Neutron Transport Problems, AddisonWesley, Reading (1969). 34. Taussky, 0. and Todd, J., Generation of Pseudo-random Numbers. Symp. on Monte Carlo Methods, University of Florida 1954, H.A. Meyer, Ed., John Wiley & Sons, New York (1956), 15. 35. Wood, J., Computational Methods in Reactor Shielding, Pergamon Press, Oxford (1982).

33 Chapter 3

DIRECT SIMULATION OF THE PHYSICAL PROCESSES "Life" of a neutron or photon, from its birth to its death is governed by nature via many random processes. Just at the very beginning: there is only a certain probability that a particle is "born" at all in the source in a given short time interval. The initial direction of flight of a particle is also a random variable and such is its energy (if the source is not mono-energetic) and its location (if the source is not so small in spatial extension that can be represented by a point). In addition, randomness remains with the particle throughout its further history. Neither the distance traversed up to its next collision site nor the type of the subsequent interaction can be determined in advance for an individual particle. Instead, probability distributions of them are known. Similarly, random variables are the energy and the direction of the scattered particles and even the number of the secondaries created in a multiplicative interaction. The same uncertainty characterizes the detection: only a certain fraction of the particles crossing the receptor region will interact in that volume. Again, only the interaction probabilities and not the reactions of any individual particles can be predicted even if all the physical parameters of both the particle and the receptor are known. As a consequence of these inherent stochastic processes all observed results will be accompanied by smaller or larger fluctuations, this is why, e.g. measured count rates are generally given together with their standard deviations. In measurements carried out under time-independent (steady-state) conditions, the easiest way to decrease the statistical uncertainties is the increase of the observation time. The alternative — but seldom realizable in practice — way is the increase of the source intensity. Anyhow, in most experiments the product of the source intensity and the observation time can be set to as large as 106 particles, or even higher by several orders of magnitude. If the reader compares the definition of Monte Carlo methods we gave at the beginning of Chapter 2 and the random nature of the neutron and photon migration outlined in the previous paragraphs, one cannot but wonder that many people got the idea to connect the two phenomena: to try to simulate the particles' random walk on computers. The word simulation here means the as accurate as possible realization of the coordinates of the particles — in a computer. At first sight one might think that the accuracy of the computer simulation depends on the answers to two questions: 1. How precisely do we know the probability distributions governing the physical processes, and 2. How correctly can we select random samples from these distributions? And really, lack of satisfactory knowledge of the distributions or application of incorrectly selected procedures may draw systematic errors into the computations. By using a terminology more familiar in Monte Carlo: the results will be biased. There is, however, another source of error. It is the statistical uncertainty which is — at least in the one-to-one simulation — completely analog — or even equivalent — to the random fluctuation observed at measurements. The origin of this uncertainty is theoretically the same for the physical processes and their computer simulations. Nevertheless, in practice they differ — unfavorably for the Monte Carlo technique — in their extent. The random walk simulations are very time-consuming and therefore an increase of the simulation number (the computational counterpart of the product of the source intensity and the measuring time) over about 105 is seldom realizable.

34

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

Thus, in most Monte Carlo programs, special techniques are introduced to decrease the statistical uncertainties. The introduction of such techniques leads to deviations from the one-to-one simulations, to so-called nonanalog games. Appropriate selection of the nonanalog procedures is of major importance and will be discussed many times, in different levels throughout this book. Now, however — in spite of all its disadvantages — let us turn back and investigate in detail the analog simulation, from which the idea of Monte Carlo application for particle transport originated in the 1940s. Still now, a deep understanding of the simplest analog procedures is the basis for understanding the more advanced techniques. In the same way as the life of a physical particle starts by its emission from some sort of source, in an analog Monte Carlo game first the initial coordinates have to be selected. The next step is the free flight of the neutron or photon up to its next collision, consequently, in the simulation, a path length has to be selected. From the starting point coordinates and the direction of flight, the site of the subsequent interaction is to be determined. At the collision site, a large variety of interactions with the different atoms constituting the material at that point can take place. Accordingly, in the numerical simulation, first the types of both the collided atom and the interaction have to be selected. If the actual collision does not lead to absorption the particle goes on its way with a new energy and direction — both of them are to be selected. In multiplicative events, or e.g. (n,),) reactions, new particles are also created, the parameters of which are generally immediately selected but temporarily stored and handled as coordinates of particles from secondary sources. The histories of these "secondaries" are followed after the termination of the "primary" particle. (For correctness it must be noted here that in, e.g. an (n,2n) reaction there is no physically correct distinction between the two outcoming neutrons as to which one is the primary and which is the secondary. The decision is arbitrary from the point of physics and is governed by practical consideration. ) After the simulation of a scattering event, the process is followed by a next path selection. The repetition of this two step (transition + collision) cycle is terminated by one of the following three events: • • •

An absorption takes place The particle leaves the system investigated in such a way that there is no possibility to return The energy of the particle falls out of the range of interest

If the event, whose frequency is just studied, occurs, the actual contribution is calculated either in the transition or in the collision phase. The sum of the contributions collected during the simulation of the history of a single primary source particle is called the score. And the average of an appropriately large number of scores is the Monte Carlo estimate of the physical quantity investigated. In the consecutive sections of this Chapter, the basic procedures used during these steps (source selection, transition and collision simulations, and scoring) are discussed, several specific procedures, frequently used in neutron and photon transport processes, are collected in the Appendices of this Chapter.

I. ANALOG SIMULATION OF THE RANDOM WALK A. SELECTION OF SOURCE PARAMETERS There are six fundamental parameters of a particle emitted from a source, viz: • • •

The three spatial coordinates: r = (x,y,z), in a Cartesian system Two coordinates of the direction of flight: w = (cox,o)y,wz), Ital = 1 and The energy (E) of the particle

35 The list presented above is not the only possible specification, one can use, e.g., spherical coordinates instead of Cartesian, or replace the energy of the photons by the wavelength, or prefer to describe the state of a neutron by the velocity vector instead of the energy and the unit length direction vector. However, in any representation, the number of the mutually independent parameters is six and simple transformation rules can help to change from one representation to another, if needed. Any set of the six parameters can be considered as the coordinates of a point in a sixdimensional phase-space. In nearly all practical cases the source density Q (r,to,E) can be factorized as: Q(r,w,E) = WO • Wu)) • QE(E) reflecting the physical fact that the spatial, directional, and energy distributions are mutually independent from each other. From the point of view of Monte Carlo selection, it means that one can select separately the r, w, and E coordinates. To simplify the description of the selection procedures, we assume — for this Section — that the source is normalized to unity, i.e.,:

JQ(P)dP = 1 moreover:

f Qr(r) dr = 1 f Q,,,(w)do) = 1 and 'WE) dE = 1 If these conditions are not fulfilled and

IQ(P)dP = Q. 1 then the only correction that has to be made is the multiplication of all results by Q. since the transport processes are linear with respect to the absolute source intensity. 1. Space Coordinates Sampling For point-like sources, the random selection is replaced by an assignment: r,

r0

for

i = 1,2, ..., n

i.e., all the n simulations start from the source point ro. If the source is uniformly distributed along a straight line (or rod of negligible radius)

36

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

FIGURE 3.1. Selection of a random point from a line source.

then let us put this line into the coordinate system as given in Figure 3.1, and define the axis t along the source line. Then Q(t) dt = p(t) dt = T dt if t = 0 at R, and R, = R, + T(R2 — RI ). Now the i-th source point is selected as: r1 = R, + p,T(R2 — R,) More complicated, one-dimensional sources (or more precisely sources that can be approximated by one-dimensional curves) seldom occur in practice. But if they do occur, the best approach is to describe the curve in a parametric form, then select the parameter t from p(t) and determine the Cartesian coordinates by transformation. After, the zero- and one-dimensional forms let us continue with sources that can be described by surfaces. If the surface is a region of a plane, let us fix a (,-q) coordinate system to the plane. Points from sources that have simple boundaries can be easily selected. Recipes to pick-up points from a square (or from a rectangle, after linear transformations) and a circle are given in Sections 2.1.1 and 2.I.J, respectively. For other surfaces with complicated boundaries, the application of the rejection technique is recommended — if the user cannot find a special, efficient tricky method for his problem. In the application of the rejection method the source region is to be covered by a rectangle (Figure 3.2.a) from which tentative coordinates are selected. The points lying out of the region of interest are to be rejected. If the area of the covering rectangle is much larger than that of the source, it is expedient to cover the source region by several smaller rectangles (Figure 3.2.b). In this case, first one of the rectangles is to be selected. The i-th one is chosen with a probability of =

t, T

37

a

FIGURE 3.2. Covering plane sources with rectangles — for sampling by the rejection method.

if t, is its area and T=

E t, =

then the coordinates are selected from the i-th rectangle — by the rejection method. The covering surface is not necessarily a rectangle. (The reader should remember that in the general description of the rejection technique in Section 2.I.0 we have also started with a majorant constant and subsequently proceeded to majorant functions.) Any geometrical figure which is easy to sample can be used. For example, in Figure 3.2.B the first covering rectangle should, generally, be replaced by a parallelogram, if the t,/t,* efficiency gain overcompensates for the time increase brought in by the more complicated sampling procedure needed. Surfaces extended into three dimensions seldom occur in practice. If the source is uniformly distributed on the surface of a sphere then the method given in Section 2.I.J for the selection of three dimensional unit vectors can be applied — with two additional transformations: first all the three Cartesian coordinates have to be multiplied by the actual radius, then the points have to be shifted by the coordinates of the actual center of the source. Most sources are extended into three dimensions. (Physically there are only threedimensional sources, the representations of them by points, curves, or surfaces are only approximations.) The ideas of random sampling in three dimensions are the same as in two dimensions. For simple regions (rectangular blocks, circular cylinders, spheres, etc.) direct procedures can be developed by establishing the appropriate PDFs and using the inverse distribution method (Section 2.I.A). Points from source zones limited by complex boundary surfaces can be selected by the rejection technique: now the zones (or separate parts of it) are to be imbedded in rectangular boxes — or sometimes into other volumes still easy to handle. 2. Sampling of Initial Directions In the great majority of problems the particles are emitted by isotropic distribution, in which case, the method for selecting the direction cosines of random vectors described in Section 2.I.J may be used. Sometimes a parallel beam of particles enters the system of interest. In such a case, the random sampling of the direction cosines is replaced by assignments of the actual values. A cosine distribution is achieved if a plane is placed into an isotropic field (i.e., into

38

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

the infinite space where isotropic sources are distributed uniformly). Let us fix the coordinate system's [x, y] plane parallel to the plane of interest. Then the outer normal to the plane has the following direction cosines:

and

The cosine distribution has the PDF:6 P((oz) = 2(0z,

coz

0

Thus, according to Equation 2.2 p = f ao,thoz and the

sampling rule can be used. (The reader should remember Theorem 2.6 and replace the execution of a square root by selecting the maximum of two random numbers.) Since the azimuth is uniformly distributed over (0,27r), in the formulae cox = V1 — (...q • cost and wy = V1 — (1) • since cost and sing) can be selected either directly (c = p,27r), or by the Neumann method given in Section 2.I.J, Equation (2.15). 3. Selection of the Initial Energies The simplest — and not too rare — case is if one has a monoenergetic source and can, therefore, replace the energy selection by an assignment. There are many gamma sources that emit photons of different discrete energies with different intensities (probabilities). The sampling from such line spectra can be carried out as described in Theorem 2.1. More skillful techniques are needed to pick-up energies from continuous spectra. A relatively easy, though not exact, method is to approximate the spectrum with either a step function or by a broken line. In both cases the inverse distribution method (Section 2.I.A) is applicable. If the continuous spectrum is described by an analytical formula, the task is to find the best of the methods discussed in Section 2.1, but the rejection technique is the only one that always works. Two source types have special importance in neutron transport calculations. Methods for selecting neutron energies from the Watt-fission spectrum and from the Maxwellian distribution describing the energy distribution of the thermal neutrons are given in Appendices 3B and 3C, respectively.

39 B. PATH LENGTH SELECTION As has been introduced in Section 2.11.E the path (R) traversed between two collisions has a PDF of: p(R) = o-(R)exp[ — j cr(10dR]

(3.1)

If the medium is homogeneous: o(R') -E cro then p(R) and an actual path length can easily be selected by the inverse distribution method (see Equation (2.2) and Equation (2.6) with the transformation x = cro R): R= — fnp o-o

(3.2)

Naturally, the fast methods listed in Section 2.I.G can replace the execution of the logarithm in Equation (3.2). The selection is more complicated if the medium is inhomogeneous. In practical problems, the cross-sections (or the compositions of the materials) are generally constant within extended zones.* Let us consider n regions with cross-sections 431 , r2 , • • • Crn• If the particle crosses the distances RI , R2 , . . . R0 in these regions (see Figure 2.7) then the PDF (3.1) becomes: JJ-1 p(R) = crjexp[ — (E criR) — cri (R — > R;)] , i= JE Ri R R, if =1 i=1 Thus, the path length selection can be carried out by determining j from the inequalities -

E

o-iRi < --enp

i=1

E

I

cr,R,

(3.3)

E

(3.4)

and then, calculating: R=

-I

1

E R, + — (—fnp

—

i=

The difficulty in solving Equations (3.3) and (3.4) lies not in the summations but in the

*

Though it is beyond the scope of this book it should be noted here that in the transport of charged particles the cross-section changes quasi-continuously even within a homogeneous medium, since the energy of the particle changes (decreases) quasi-continuously.

40

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

determination of the R, distances, i.e., in the determination of the coordinates, where the particle trajectories cross the boundaries separating the different homogeneous zones. This procedure may be extremely complicated in complex geometries thus another technique" is suggested instead of the "straightforward" method described above. The method is the following: 1. Define a majorant cross-section: r % cr(r), for all r-s along the path; 2. Select a distance R by Equation (3.2) and determine the tentative next collision site (r„'): rk =

, + Rco„

if rk _ , is the starting point and 6h_, is the direction of flight; 3. Play a rejection game: • •

with a probability of cr(r,')/o- accept this point as a real collision site (r, = r,'); with a probability of I — cr(r,')/o-0 do not accept r,' as a real collision site but select a new path starting from r,' with the unchanged direction wk _„ again taking the total cross section to cro (i.e. set rk _ = r,' and return to step 2).

A mathematically correct proof of this procedure (even for continuously varying crosssections) is given in Reference 10, but there is a very easy-to-follow method to understand its validity: Let us introduce an imaginary scattering event which changes neither the energy nor the direction of the particle. This definition implies that such imaginary scatterings are not physically observable, i.e. they can be introduced with any cross-section at any point. Now if we assume that the majorant cross-section (cro) is a sum of the real (cr) and imaginary (o-,) cross-sections, then in the procedure outlined above there is no acceptance and rejection, but in a fraction of: 1

cr(rk) Qo

o-,(11,) cro

the collisions an imaginary interaction is simulated. * The advantage of the method is clear: only the locations of the tentative collision sites have to be determined independently from the boundary crossings up to those points. The majorant cross-section method is trivially applicable for paths going through cavities (regions filled with vacuum). Tentative collision sites lying in vacuum are, per se, never accepted. (Those who like curious statements can say that there are solely imaginary scattering events in regions filled with no material.) The selection of the path length leads to termination of the simulation if the particle leaves the system and has no chance to return. Such situations never exist in physical reality. However, if the system investigated is surrounded by a low density material (most probably air) and even the nearest reflectors (e.g., walls of the room) are so far that the contribution of backscattering is negligible then placing the volume of interest into an infinite vacuum space is a reasonable approximation. * The imaginary or hypothetical collisions are called "delta-scatterings".' Delta scattering will be examined more closely in Chapter 51V. A proof of the unbiasedness of the selection procedure above will also be given there.

41 For the computer simulation outlined above, a trajectory going out of the system means that for certain p-s selected, the inequality:

E crR,

— enp

i=1

of Equation (3.3) cannot be fulfilled even with the largest j — the particle goes through all regions without collision. Such a finite extension of the non-zero, cross-section region has the mathematical consequence that the p(R) function given in Equation (3.1) is not a PDF. If Ro is the coordinate of the latest boundary crossing point in a certain trajectory (o-(R') = 0, if R' > R0), then P(00) = P(R0) < 1 The leakage of the particles will be discussed in this latter way, that is as a problem of having a not normalized probability density for the path length, in Sections 3.I.A and 4.IV.E. There is an alternative description of the isolated systems: a black absorber surface is placed on the outer surface. This approach of absorbing all particles crossing the system boundary is as acceptable as to let the particles fly away forever.* C. COLLISIONS — IN GENERAL There are many different ways in which an incident particle can interact with matter. The most general formulation of the interactions was given in Section 2.II.D. The quantity the value of which is proportional to the probability that an interaction will take place in an elementary volume is the cross-section. If there are n elements composing the material in which the particle flies, and m different types of interactions then the total cross-section is expressed by the sum: n m

cr(E) =

E E crii(E) i =1 J-1

(3.5)

where cr denotes the partial macroscopic cross-section of the j-th type interaction, if the ith type element is hit by a particle having an incident energy E. The cross-sections are assumed to be independent of the angle of incidence (w). This assumption is not fully true, however the influence of the orientation of the particle with respect to e.g., the axis of a molecule containing the element hit is always negligible in transport calculations. The probability that at the next interaction the i-th element will be hit and the actual interaction will be of the j-th type is: cr (E) Pi; = Q(E)

(3.6)

From Equations (3.5) and (3.6) it trivially follows that: n m

E E 13 =

i= I j= I

*

1

The author of this paragraph (L.K.) admits that, of these two physically fictional but very fruitful models he can more easily imagine the system immersed in an infinite sea of vacuum than covered by an impenetrable absorber.

42

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

i.e. the actual (i,j) values can be selected as described in Theorem 2.1. Since at the definition (3.5) we assumed that all interaction types are allowed for all elements, there might be (i,j) pairs for which 0 (For example, in neutron transport calculations, one of the possible interactions is fission, but the list of the fissionable elements is quite limited.) In practice generally one of the following two ways is followed: 1. First the type of interaction (j) is selected with a probability of

pi

-E

and then the element hit is chosen from the conditional probability:

POO

13;

The second selection is frequently omittable, for example, if the selected interaction is absorption it may be of no interest which type of element absorbed the particle. 2. One can first select the element i with a probability of: P, =

E 13, =

and follow with the selection of the type of interaction. The conditional probability for the second selection is now: P(.110

111 Pi

This method is advantageous if the types of interactions that have non-zero cross-sections (i.e., that can really happen) are different for the different elements involved in the same problem. (As an example, the reader can think of a reactor core containing water and uranium.) After the selection of the actual interaction the Monte Carlo treatment may be continued in three principally different ways: 1. If an absorption is selected then the history of the particle followed is terminated. The termination of the life of the followed particle, however, may lead to creation of new particles. Let us illustrate with two examples: • The neutron is absorbed in an (n,-y) reaction where a secondary photon is created which has to be followed if one executes a coupled neutron-gamma transport calculation. In practice, this may be done by storing the phase-space coordinates of the photons created, continuing the neutron simulations and start a gamma transport calculation phase afterwards. • At pair-production, the followed photon disappears and an electron-positron pair is

43 created. However, from the immediate annihilation of the positron, two new photons are created, by very good approximation just at the site of the pair-producing interaction. Since the scope of the book is restricted to neutron and photon transport problems, we assume that paths of charged particles are never followed. Thus, interactions like (n,p) or (n,a) are considered as absorptions. 2. In the simulation of scattering events, energy and direction coordinates of the scattered particles have to be selected according to the appropriate probability densities. 3. If a multiplicative event; such as a fission, or (n, 2n) reaction; is selected, then the energy and direction coordinates of the new particle(s) have to be selected. In case of events in which more than one secondary is created, the history of one is followed immediately, the initial coordinates of the other(s) are temporarily stored. Hereafter, we assume that the system is subcritical, that is the number of stored particles does not increase boundlessly. The simulation of the collisions is the only point in the analog Monte Carlo where the procedures applied in neutron and photon transport calculations basically differ from each other. Consequently, the types of interactions and their Monte Carlo treatments are described in separate Sections for neutrons and photons. The implementation of the techniques for neutrons are more difficult. There is a greater variety of the possible important neutron interactions and the simulation of several neutron interactions (in elastic scattering, first of all) need skillful techniques. Thus, purely for pedagogic reasons, interactions of photons with matter are discussed first. D. INTERACTIONS OF PHOTONS WITH MATTER The physics of photon interactions is nicely summarized in several textbooks. Readers interested in derivations of formulae and details of effects of even minor importance are advised to study the works of Heitler,22 Fano et al.,I5 Hubbell," or Hubbell et al." The papers of Hubbell, after a description of the physical background, present a large amount of cross-section data. There are several other well known large — and time-to-time reevaluated — cross-section libraries. The two best-known data compilations are edited at Brookhaven National Laboratory (the ENDF — Evaluated Nuclear Data File") and at the Lawrence Livermore Laboratory. 42 The three basic types of photon interactions that are exclusively included in most transport codes are described in the next three subsections, in the fourth one the other interactions are briefly mentioned. 1. The Photoelectric Effect In the photoelectric effect, the incident photon is absorbed and an electron of the collided atom is ejected. The ejected electron carries away the energy of the photon minus the binding energy of the electron. In the photon transport calculation the simulation of a photoelectric interaction means termination of the random walk. (Theoretically, there are two possible ways to create secondary photons, viz. by emission of Bremsstrahlung photons or fluorescent radiation, but generally none of them have significance.) The photoelectric cross-section (cr) depends on the energy of the incident photon and on the atomic number of the atom hit. It increases approximately by the 4.5-th power of the atomic number and decreases rapidly as the photon energy increases, except that it rises sharply when the photon energy becomes large enough to eject inner-shell electrons. Thus the effect is dominant at low energies and for materials containing high atomic number elements.

44

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

Cross-sections of individual elements and several frequently used compounds are available, e.g., in the compilations mentioned above. It is worth noting here that in the energy range which is typical in reactor physics calculations (-10 keV to 10 Mev), from among the three important interactions the uncertainties of the photoelectric cross-sections are the largest. 2. Compton Scattering In Compton scattering the incident photon loses a part of its energy and is deflected. The theory of this effect was developed by Klein and Nishina.22 In their derivation it is assumed that the electron is initially free and at rest. In this case the electron hit carries away the full energy lost by the incident photon. Under the free stationary electron assumption the Klein-Nishina differential microscopic cross-section is do-T(a„,15) 1 di2 —2

+ a0(1 — cos'4)] -2

x [ 1 + cost* +

a.2(1 — cosil-)2 1 ron2/electron l 1 + a0(1 — cosil-LI I_ steradian i

(3.7)

(The index 1 refers to the "one-electron" cross-section.) In Equation (3.7), the following notation is used: a. — is the incident photon's energy in electron rest energy units, i.e.,: a. = E0/€

where

e = mec2 == 0.511 MeV

'4 is the deflection angle of the photon; and re is the classical radius of the electron (re = e2/(mec2) = 2.818 x 10-'3 cm) From the conservation of momentum and energy, the energy loss of the photon and the angle of deflection are linked by the following relation: 1 x

a a.

1 1 + a.(1 — cos'4)

(3.8)

where a is the photon's energy after the collision, in electron rest energy units and 1/x is the energy loss ratio. Since the minimum and maximum values of cos i5 are — 1 and + 1, respectively, x can vary from 1 to 1 + 2a0. By substituting Equation (3.8) into Equation (3.7), and taking into account that dSt = — 27rd(cos19-) and from Equation 3.8 1 d(cosb-) = — — dx

45 one gets a new formulation of the differential microscopic cross-section: B C D do-T(a„,x) —KA+—+—+ dx x x2 x3

= f(a,„x)

(3.9)

where K = 7rr/oto A= B = 1 — 2(a0 + 1)/et,2, C = (1 + 2cc,,)/a,2, and D=1 The total cross-section is obtained by integration of either Equation (3.7) over all angles x 1 + 2a0. The result is or Equation 3.9 over 1 o-T(a„) = 2K

f 1 + oto F2 + 2a0 cc, I_ 1 + 2a„

fn(1 + 2a0) 2a0

fn(1 + 2a0)1 ao

1 + 3a0 (1 + 20.0)2

(3.10)

Equation (3.10) can theoretically be used for direct computation, however, it leads to numerical errors due to near-cancellation between the logarithmic and algebraic terms at low energies. For practical use the empirical formula of Hastings' is proposed: cil(oto) = Ka„

c + c21 + C3 r13 + di-q2 + dyri + d,

(3.11)

where -r = 1 +0.222037x„, c1 = 1.651035, c2 = 9.340220, c3 = — 8.325004, d, = 12.501332, d2 = — 14 . 200407 and d3 = 1.699075. The fit of Hastings is correct within 1.3% up to 100 MeV. Its use is recommended not only due to the lack of the numerical problems mentioned above, but also because the evaluation of the RHS of Equation (3.11) is much faster than that of Equation (3.10). The Klein-Nishina cross-section formulae describe the probabilities of scattering on a single electron. The cross-section of an element whose atomic number is z and thus contains z orbital electrons is: * = Z0'1

The macroscopic cross-sections, or rather the linear attenuation coefficients according to the terminology generally used in connection with photon interactions, are to be calculated as given in Equation (2.20) both for the differential and the total cross-sections. Since the angular (or energy change) distribution, in the Klein-Nishina approximation, is independent of the atomic number(s) of the element(s), the angle of scattering (or the energy of the scattered photon) can be selected by the same procedure for all elements and compounds.

46

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

In other words there is no need at all to select the type of element hit, thus method 1. of Section 3.I.0 is preferred. The PDF to be sampled is P(a,„x) =

na.,x) 0- T(cto)

where f and crT are defined by Equations (3.9) and (3.10), respectively, and values of the energy change x are to be picked up with given cto incident energies. Concrete recipes are given in Appendix 3A. In reality, the electrons of an element or compound are neither free nor at rest when the photons hit them. The binding corrections are usually taken into account by applying a multiplicative correction, the so-called incoherent scattering function S(q,z): ci-mc(a0,•0- ,z) = cr,(ctool) ' S(q,z) The first argument of S is the momentum transfer: q = (a,2, + a2 — 2actocos*)"2

(3.12)

the second is the atomic number. Tables of incoherent scattering factors are given e.g., by Hubbell et al.24 while Biggs et al.3 developed analytical approximations for them. Recently, Persliden' developed a sampling method, where first the Klein-Nishina density function is sampled and an additional accept or reject game is applied for the correction for electron binding. Data for the incoherent scattering factors and total cross-sections derived with them are included in most large cross-section libraries mentioned earlier. Trivially, if the electron binding effect is not neglected, the scattering angle (or energy change) distributions are not element independent anymore. The formula for incoherent scattering can be further refined: • • •

one can take into account that the electrons hit are not at rest but in motion the radiative correction reflects the emission and reabsorption of virtual photons there is a minor probability that double Compton effect occurs, in which an additional photon is emitted

All these corrections are described in details by Hubbell et al. 24 In practical transport calculations, however, most frequently the use of the Klein-Nishina approximation is satisfactory. 3. Pair-Production In this interaction the incident photon disappears in the field of either the nucleus or an electron, and an electron-photon pair is created. The threshold of the effect is Ince (the combined rest energy of the created particles). The cross-section rises monotonically from zero at the threshold. With the exception of the lowest atomic number elements, pair production in the field of electrons is much less probable than that in the field of the nucleus. The cross-section for pair-production in the field of the nucleus is proportional, roughly, to the square of the atomic number. The full phenomenon of pair-production is quite complicated, readers interested in details are advised to read the overview of Hubbell et al. 24 Cross-sections for particular elements can be found in the libraries already mentioned.

47 The life of the primary photon is trivially terminated at pair production. The two charged particles follow their paths, however the positron — generally in the closest vicinity of its birth — combines with an electron. In this annihilation, two photons are emitted. These "secondary" photons have twice mec2 initial energies, assuming that the annihilated electronpositron pair had virtually no kinetic energy. The two "secondary" photons start to fly in opposite directions, which are practically uncorrelated to the direction of the primary photon. Since the free paths of the positrons are negligible relative to those of the photons, in the Monte Carlo codes the annihilation games are started just from the site of the pair production event. 4. Other Interactions From among the other reactions Rayleigh (or coherent) scattering is a process which may be not negligible at low energies for materials containing high atomic number elements. During coherent scattering, by definition, there is no energy loss. The angle distribution is described as a product of the classical Thompson cross-section: r2 dcrT(b) — (1 + cos2*) 2 dfi and the so-called atomic form factor F(q, z), where q is again the momentum transfer. Since here ao = a, Equation (3.12) reduces to

q = 2aosin 2 In many photon transport calculations Rayleigh scattering is treated by combining it with the electron binding effect on the Compton event. Therefore, values of the atomic form factors are given in the same references as those of the incoherent scattering function. A scattering angle selection method is proposed by Persliden.' Photonuclear absorption — These processes, leading to emission of one or more neutrons, charged particles or photons, have small cross-sections, but there are broad peaks (giant resonances) in the 12 to 24 MeV region (at higher energies for the light elements). Since a large fraction of these reactions leads to the emission of neutrons, nuclear absorption may be interesting in high energy coupled photon-neutron calculations. Photonuclear cross-section compilations have been carried out at the NBS. Other photon interactions (e.g. Delbriick-scattering, or photomeson production) contribute less than 1% to the total cross-section and are always neglected. E. INTERACTIONS OF NEUTRONS WITH MATTER The ways in which a neutron can interact with matter show a much larger variation than is exhibited by photon interactions. A major physical difference is that photons interact mostly with the electrons of the atoms, whereas neutrons interact predominantly with the nucleus. Moreover, the crosssection vs. energy curves are not so smooth, for several important interactions no theoretical formulae are known, the user has to pick up both the total and differential cross-sections from compilations of measurements. There are resonances, where the cross-section jumps up and then back down in an extremely short energy interval and the measurement of the fine structure of such an abrupt cross-section change is very problematic. Thus problems with the unresolved resonances are discussed separately at the end of this Section.

48

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

In general, from the computational point of view, one can choose from two basically different approaches. In the first case the energy is allowed to change continuously. Then, the uncertainties in the shape of the near-resonance part of the cross-section curve may cause significant errors, especially since interpolation between different energies is a hard task. Similarly, problematic is the interpolation between different elements, and even various isotopes of the same element may have significantly different cross-sections at the same neutron energy. The other method is the use of the so-called multigroup treatment. Here, the energy range of interest is artificially divided into several groups and the cross-sections are averaged over them. Needless to say, the more groups are used the more accurate results may be expected. The fundamental problem with the multigroup treatment is that the cross-sections must be averaged by the flux-energy curve, which is not known in advance. Anyhow, the preparation of the multigroup cross-section libraries is a very difficult task and has its own large literature. In the following we shall discuss only the pointwise cross-sections. Even in the pointwise (continuous energy) treatment, neutrons that have slowed down by a sequence of scattering events to equilibrium with the thermal motion of atoms are handled in a separate group. These thermal neutrons either lose or gain energy during the scatterings, therefore, one can say that the "end" of slowing down is thermalization. Generally, neutrons whose energies are less than about 0.5 eV are called thermal neutrons. Classification and description of the neutron interactions are given in several textbooks, see e.g., References 20, 1, and 8. For the actual cross-sections, the reader is again advised to review the large compilations, such as the ENDF" and the Livermore' libraries. 1. Capture Using a somewhat loose terminology one could say that capture of neutrons is the counterpart of the absorption of photons. Really, in this process the history of the neutron is terminated — like that of the photon at a photoelectric interaction. Significant difference in the treatment arises only in coupled neutron-photon transport codes, since an important kind of capture events is radiative capture, where the emission of a photon follows the absorption of neutron. The cross-section for capture typically does not exceed a few percent of the scattering cross-section. Radiative capture is, however, a more important reaction for thermal neutrons. For nuclides which have a capture resonance near the thermal region capture represents the main contribution to the total cross-section. The thermal capture cross-section of cadmium for example, is higher by a factor of more than 300, than the scattering cross-section. 2. Elastic Scattering In an elastic scattering between two particles (the incident neutron and the target nucleus) the momentum and the energy are conserved. The basic assumptions involved in the above statement are that the target atoms are initially free and at rest. The change in neutron energy, from E0 to E, and the scattering angle (in the laboratory system) are linked by the following relation: t cos = — [(A + 1) 2

— (A —

(3.13)

where A = m,/m,,, the ratio of the mass of the target to that of the neutron or, with very good approximation, the mass number of the target nucleus. The new energy is determined

49 from Equation (3.13) as E=

E (A + 1)2

+ Vcos2* + A2 — 1)2

(3.14)

Substituting the two extreme values: cosil- = —1 and cos19 = 1, respectively, one gets the restriction E„a2 E E,, for the new energy, where a = (A — 1)/(A + 1). For a wide range of energies (especially for light target nuclei) the elastic scattering is isotropic in the center-of-mass system. Then its cosine can be simply selected as = 1 — 2p

(3.15)

and the cosine of the scattering angle in the laboratory system is cos. =

1 + A(cosil)„„ VI + A2 + 2A(cos'a)c„,

(3.16)

Hereafter the subscript cm indicates the variable measured in the center-of-mass system; angles in the laboratory system have no subscripts. The new energy is to be computed from Equation (3.14), or directly from E = 1 /2 E0R1 — a2)(cos10)e,„ + 1 +

(3.17)

where a = (A — 1)/(A + 1). The selection procedure can further be simplified if the target is hydrogen: A = 1. By preserving the isotropic scattering assumption in the center-of-mass system (which is valid up to about 10 MeV) and substituting Equation (3.15) into Equation (3.16) one gets cos* = The new energy is E = Eo(cos'0)2 = Eop When scatterings are simulated for such incident energies and target nuclei that scattering is not isotropic, theoretically the Equation P=

0'40 dP-,

( — 1 < p. < 1)

(3.18)

has to be solved for the actual selection, where p, denotes the cosine of the scattering angle — either in the laboratory or in the center-of-mass system. If the differential cross-section is given in the center-of-mass system, then there are two ways to proceed: 1. One can first transform the cross-section according to Cr( II) d

Crcm(P'cm) d P cm

(3.19)

50

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

2. Or select the center-of-mass scattering angle and compute the cosine of the scattering angle in the laboratory system from Equation (3.16). The differential cross-sections required to solve Equation 3.18 are found in the libraries in approximate forms, they are usually tabulated in terms of vi.„. If the elastic scattering is not assumed to be isotropic in the center-of-mass system, data specifying the differential cross section are required to solve Equation (3.18). Since anisotropy is much more frequently considered in simulating inelastic than elastic scattering events actual techniques are discussed in the next subsection and in Appendix 3D. 3. Inelastic Scattering In a scattering process, if the residual nucleus is left in an excited state, the scattering is called inelastic. The kinetic energy retained by the target nucleus is denoted by — Q. The value of Q is negative, since, by definition, Q is "the excess of kinetic energy of the product particles over that of the original particles".8 Again, from the conservation of energy and momentum, the scattering angle is 1 cos* = 2 — [(A + 1) — — (A — 1

QA VEE0]

(3.20)

and the new energy is 1 E — (A + 1)2 [cos*VE„ ± VE,,(cos2* + A2 — 1) + A(A + 1)Q12 (3.21) Naturally, Equations (3.20) and (3.21) reduce to Equations (3.13) and (3.14), respectively, if one substitutes Q = 0. If — Q is large, Equation (3.21) might lead to complex solutions, thus, there is a threshold energy (E„,) for a given cos*: =

A(A 6 + 1)Q cos2 + A' — 1

The minimum threshold RE0 ,)„,1 is given by cos* = 0: (E0.)„, —

A+1 A

There is no inelastic scattering if the energy of the incident neutron is less than — (A + 1)Q1 /A, where Q1 denotes the excitation energy of the first excitated state of the nucleus hit. One can see that two discrete values of E can be obtained from Equation (3.21) if VET, cos* > VE„(cos2* + A' — 1) + A(A + 1)Q Thus, in the region E, cos*) is also derivable from the free gas model." In several Monte Carlo codes the effects of thermal motion and chemical binding are simulated in great detail. From among the most important programs let us mention here the VIM (developed at Argonne National Laboratory) and the MCNP (Los Alamos National Laboratory).'9 In a simpler approximation, the new energy is frequently selected from Equation (3.25) and uniform scattering in the laboratory system is assumed. An effective method is to use few thermal groups (one, in the extremity) with preset thermal energies and cross-sections. If the thermal neutron group has an upper energy limit (or cut-off energy) E*, the density function has the following form: E TV 2

exp( — E/T),

if 0 E < E*

f(E) = P • 8(E — E*),

if E E*

where P=

2 f ir

exp( — E/T)

dE T

The truncation prevents the thermal neutrons from having higher energies than the thermal cut-off. A selection scheme for the truncated Maxwellian distribution is given by Lux. 35 In another paper of Lux36 a sampling scheme is presented to select the outcoming energy from the ingoing energy of a thermal neutron. The actual scheme is described in Appendix 3B. 5. Fission Many high atomic number nuclides decay after absorption of neutrons, with the simultaneous emission of high-energy secondary neutrons. The great majority of these neutrons is emitted within a few microseconds of the fission event. Delayed neutrons, both the yield and energy of which are lower, are generally of smaller importance. The average number of the secondary neutrons (v) is about 2.3 to 2.9 for most nuclides of interest and depends on the target as well as on the incident neutron energy. For 235U and thermal neutrons v ---- 2.43 is commonly accepted, a table for several elements and energies is given in Reference 28 and reprinted in Reference 8. In a Monte Carlo game the actual number of the secondaries can be selected in a

53 straightforward manner: select 3 neutrons with a probability of (v — 2) and 2 with a probability of (v — 3). The energy of the fission neutrons can also be approximated by the Maxwellian distribution, with appropriately selected temperatures. For 235U and thermal fission kT = 1.290 MeV gives the best fit.8 Watt45 published the following fission energy formula: f(E) = 0.484 exp( — E)sinhV2

(3.26)

where E = E/E,, and Eo = 1 MeV. A slightly modified fit is given by Cranberg.12 His expression for the energy distribution of the neutrons emerging from thermal fission of 235U is f(E) = 0.4527 exp( — E/0.965) • sinhV2.29E

(3.27)

A selection scheme for the distribution described by Equations (3.26) and (3.27) is given in Appendix 3C. It is usually assumed that fission neutrons are emitted isotopically in the laboratory system. High energy neutrons may cause fission events in which one or two of the emergent neutrons may have been scattered inelastically rather than emitted. Treatments for such secondary neutrons are suggested by Carter and Cashwell.' 6. (n,2n) and (n,3n) Reactions The cross-sections of the (n,2n) and (n,3n) reactions are very small or rather negligible in comparison to that of inelastic scattering. Furthermore, the thresholds for these reactions are at high energies for most materials of interest. If the reactions are not neglected, the correlation between the energies and the scattering angles of the secondaries is generally ignored. Thus, the same selection procedures can be applied as for the inelastic scattering. It should be, however, noted that this independent sampling yields unbiased results only on the average, at the individual event simulations the conservation laws may be violated. 7. Charged Particle Producing Reactions Charged particle producing reactions, such as (n,p), (n,a) or (n,np) may be important if the target is composed of light nuclei. For example the (n,a) cross-section of Be, N and O may exceed that of the inelastic scattering — for the same elements. However, these interactions are considered as absorption events, if only histories of neutral particles are followed. 8. Cross-sections in the Unresolved Resonance Ranges Probably the greatest weakness of modern Monte Carlo codes lies in the cross-section uncertainties, and especially in our inability to measure or use cross-sections in the unresolved resonance range.19 Let us illustrate the problem of extremely large storage requirements by an example given by Levitt.33 Over the energy range of 30 eV to 25 keV, approximately 33,000 resonances exist in the cross-sections of 239Pu. To describe each resonance adequately would require about 8 points per resonance, for each of the total, scattering, and fission crosssections; and the corresponding energy. This amounts to about one million words of computer storage! Thus, the use of point cross-section data is restricted to computers of extremely large capacity. The most commonly used method is the probability table method invented by Levitt.33

54

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

The cross-sections at a given energy are represented by a distribution function whose mean value is the infinite dilution smooth average of the energy dependent cross-section. The distribution functions are given in tabular form. These tables are called cross-section probability tables. Values of the probability tables are obtained from "ladders produced about a small energy range, sufficient to contain 50 to 100 resonances, insuring an adequate sampling of resonance interference and overlap effects while preventing significant variation in the energy-dependent average parameters."" In most of the practical applications a table size of 10 probabilities is enough. In the calculation, when a neutron enters the unresolved resonance region, the cross-sections are obtained by random selection from the appropriate probability table — following the general sampling rule given in Section 2.I.D. "The probability table method assumes that the resonances are so close together that the neutron enters a resonance randomly, and that the resonances are sufficiently narrow to ignore successive collisions in the same region."" F. DIRECTION COSINES OF A PARTICLE AFTER SCATTERING In the previous subsections different types of scatterings were briefly discussed. General laws governing the scattering of the particles at various interactions were described and several actual recipes for selecting the scattering angles are given in the Appendices of this Chapter. One task remains to be done: to compute the new direction cosines, co'x , co;, and wiz in wy2 the Cartesian system, if the pre-collision cosines: co„, wy, coz (o = 1) and the angle of scattering 0) as well as the azimuth (x) are assumed to be already known. This transformation needs only geometrical considerations, thus the derivation is omitted here. From the several formulations (naturally, differing from each other only in notation) that of Sreider et al.43 is given here: co; = wZcosiR + acosx Y

=

1 1—

(w (3 + w„asinx)

and wX =1

0).R — wyasinx) — w2 (

where a

= (1 — 01/2

13 = cos —

zo.);

and the azimuth x is assumed to be distributed uniformly over (0,27r). G. SCORING Since the whole treatment here is based on the completely analog simulation of the physical processes, scoring has to be done in the same manner: according to the physical process studied. Therefore, there are at least as many scoring possibilities as physical quantities to be determined. There are cases when more than one possible way exists to estimate the same quantity. Let us illustrate with the example of the flux integral, i.e., the flux integrated over a certain

55 space region. According to the two different definitions given in Section 2.II.0 the programmer can calculate the flux integral in two ways: 1. To store score contribution of 1/cr before the simulation of every collision taking place in the region of interest, or 2. To sum up the chord lengths of the paths in the region. (Needless to say the first method cannot be used if the flux integral is to be calculated in a region containing vacuum.) Reaction rates, net currents, etc. can be compiled in similar straightforward manner, by simply registering the occurrence of the events investigated. From the quantities used in dosimetry kerma (or kerma rate) is calculated most easily: the difference between the energies of a particle before and after collision is just the "kinetic energy released to matter" at the collision site. Absorbed dose can exactly be computed only by codes in which the histories of the charged particles are also followed. Let us briefly mention here another quantity which will be repeatedly discussed several times throughout this book. That is the flux-at-a-point. The importance of it needs no explanation. In measurements one can produce very small, "point-like" detectors, however, if the sensitive volume of the instrument is too small, large times are needed to achieve acceptable statistics. In the case of Monte Carlo simulations the same method is trivially usable: to surround the point of interest by a small but finite space element and to compute the flux integral in it. However, the difficulty with the statistics is aggravated in the numerical simulations since the required computer time may exceed even hours or days. Thus, other methods are preferred — and given later.

II. PLAUSIBLE MODIFICATIONS OF THE ANALOG GAME In the previous Chapter, the whole Monte Carlo game was based on an as precise as possible simulation of the physical processes. The application of such a method will necessarily lead to correct results — to the extent that the physical laws governing the random walk of the particles are well known and correctly built into the actual computer program. Reliable final results, however, can be reached only after averaging many individual scores obtained during the individual simulations. There might be many paths that end before they have made any contribution to the score. It is worth recalling here that the situation is the same again as in experiments: generally only a fraction of the emitted particles reaches the region of interest. In the physical measurements if the count rate in a detector is very low, long detection times, or many repetitions of the experiment are required to obtain good statistics. Similarly in the numerical experiments many particle histories have to be simulated in order to reach a reliable estimate of the quantity of interest. However, even in the fastest modern computers, the numerical simulation of a long series of collisions and transitions requires an incomparably longer time than the total flight time of a physical particle. A necessary — necessary from the point of good statistics — increase in the number of simulations might frequently lead to prohibitedly large computer times and thus the possibility of solving complex problems by analog Monte Carlo method would be out of the question. This problem motivated — from the very beginning of the Monte Carlo applications — efforts to find methods which modify the analog simulation process in such a way that: • •

More particle simulations have non-zero contributions to the score, than in the analog simulation, But these individual scores differ just so from the analog ones that the expected results of the analog and the modified simulations be identical.

56

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

Modified simulation processes are generally called nonanalog games and the scores leading to finally correct answers in the nonanalog games are called unbiased estimators. Nowadays a lot of different nonanalog game types and corresponding unbiased estimators are known and used. (It should be mentioned here, that unbiased estimators differing from the physical scores can be joined also to otherwise analog simulations, but this is rather a theoretical remark than a practical suggestion. Anyhow, examples illustrating this point will be given in Section D of this Chapter.) The introduction of the more sophisticated efficiency increasing nonanalog methods, or rather the proof of their unbiasedness requires complicated mathematical manipulations and thus will be discussed in the subsequent Chapters of this book. There are, however, several very simple, and at the same time very effective, methods that can be easily understood on a heuristic basis. Such plausible techniques will be described in the following. In all the modified games the numerical experiments follow pseudo-particles which have no analogs in nature, however, just for brevity, they will still be called particles. A. REPLACEMENT OF ABSORPTION AND LEAKAGE BY STATISTICAL WEIGHT REDUCTION In the description of the analog simulation processes we have seen that the random walk of a particle is terminated basically by three events: • • •

If the particle is absorbed If it leaves the region of interest and there is no possibility of its return If the energy of the particle decreases below preset limit

The termination by the first two criteria can be avoided by the introduction of so-called statistical weights (or simply weights). In the simplest case, the weight is unity at the start of the simulation (at the emission): Wo = 1 and is multiplied by the ratio of the nonabsorption and total cross sections at every collision, thus after the i-th interaction W, = W:

Q— o-a o-

(3.28)

where W: is the weight before the i-th collision. Absorption is not simulated, all particles survive all interactions but at the scoring the new, non-analog score (fna) is used: fn. = Wfa

(3.29)

where fa would be the score in the analog game and W is the actual weight. Otherwise, the new energy and direction of flight coordinates are selected in the same way as they were for scattering events in the analog simulation. Heuristically, one can imagine that a fraction of the particle was absorbed, and the other fraction — the less than unit weight represents how large this latter fraction is — continues its random walk. Similarly, if the geometrical region is finite and the possible path continues in vacuum in a certain direction, then the particle has a leakage probability (pi) of Ro

p, = exp( — I o-(R)d12)

(3.30)

57 where R0 is the distance to the region boundary. Instead of simulating the escape one can multiply the weight of the particle leaving the (i-1)-st collision by the non-escape probability (1 — p), thus the weight of the particle entering the i-th collision becomes (3.31)

=W_(1-

In such cases leakage of the particle is to be precluded and the next free path (R) has to be selected from the distorted, nonanalog density: 13(R) =

P(R) 1 — p,'

0 R Ro

where p(R) is the path length density given in Equation (3.1). Trivially, the modified score is still the same as in Equation (3.29), since from the point of simulation there is no difference whether in a particle's history the absorption or the leakage is replaced by weight reduction. In the previous case a fraction is considered as absorbed and the other fraction (the remainder) is scattered, now a fraction leaves the region and the other fraction is artificially kept in the region. In both cases the second fraction — with its decreased weight — continues its random walk. Now if both absorption and leakage are replaced by weight reductions the resultant weight modification for a particle's flight between leaving the (i-1)-st and leaving the i-th collisions follows from Equations (3.28) and (3.31) as:

=

1(1 — Pt)

CT

(Ta

cr

The simple multiplication of the two weight modifying factors indicates that the replacement of the absorption (during the collision) and the leakage (between two possible collisions), by weight reductions are mutually independent events. The demonstration of the variance decreasing effects of these two simple methods is the task of another Chapter of the book (namely, Chapter 5. VIII), it should only be mentioned here that in most computer codes absorption is replaced by weight reduction, whereas the non-escape modification is seldom used, mainly because it often increases the computing time to such an extent that the resulting efficiency is decreased. Computer time is increased by the greater number of collisions to play in a game without leakage and also due to the simple practical reason that the execution of the integral in Equation (3.30) for complex geometries may waste more computer time than earned by prevention of the escape of particles. Finally, we should like to comment here, why the third paragraph of this point was started with the words: "In the simplest case . . . " There is a possibility to start with a non-unit weight even at the emission. If the sampling from the source Q(r,E,w) is already difficult and one can find another density function Q(r,E,w) from which the selection is easier, Q can be used for selecting ro, Eo and wo and an initial weight of Q(r ,E ,w ) ° Q(ro,E0,w0 )

(3.32)

has to be assigned to the particle. Trivially, Q must have non-zero value, wherever Q is non-zero.* The question whether such a source biasing is finally efficient or not depends *

This non-zero condition is necessary not because of the hazard of singularity, but to ensure an adequate sampling (see Section 3.II.C).

58

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

on the balance between the time gain achieved by the introduction of the simpler density function and the increase of the statistical uncertainty caused by the fluctuation of the initial (and inherited to all successive) weights. B. REPLACEMENT OF MULTIPLICATION BY INCREASE OF THE WEIGHT In neutron transport in presence of fissioning materials v progeny neutrons with independently distributed energies and isotropic direction distributions can be released by fission. In the analog game first an actual integer value of n (see Section 3.I.E, point 5.) then n energies and directions are selected and all the n neutrons are followed individually. Instead, one can choose a single energy and direction and follow a single progeny with a weight of v — the expected value of n — times the weight the incident neutron that caused the fission (times the non-absorption probability, if absorption is replaced by weight reduction). Similarly, at pair-production interactions of photons, from the annihilation of the produced positron two photons with the same energy are emitted. Here, again it is more convenient to follow a single photon with a starting weight of twice the weight of the incident photon (times the nonabsorption factor — if it is applied). C. RUSSIAN ROULETTE AND SPLITTING The purpose in introducing the weight was twofold. In Section A loss of paths by absorption or leakage was prevented, in Section B unnecessary multiplications of simulations were avoided. There may, however, be cases when just the opposite is profitable for the user: unimportant paths should be stopped, or — in important regions — more independent histories are needed than are at disposal. By the absorption and/or leakage replacement method the weight of the particle monotonically decreases and may reach very small values: following it any further is more or less a waste of time, since it will surely have only small contributions to the quantity to be determined (see the scoring formula [3.29]). In such cases the Russian roulette method can help. Let us decide that if a particle's weight (W) falls below a preset minimum Wth, we either restore its starting (or any other preset) value, Wo, and continue its path, or we kill it. The process is trivially correct if the survival probability (p) is P= and the history is terminated with a probability 1-p. In an alternative way, one can fix the surviving probability po, and increase the weight of the survivors to

Wnew

Po

Since the importance of a particle depends not only on its weight but also, for example, on the region where (e.g., how near to the detector) it is, different minimum weights, or surviving probabilities can be set to different regions. On the other hand, in very important regions we can increase the number of histories by splitting the entering particle into several "fragments" which are afterwards followed independently. If the particle is split into n new ones, "n for one splitting", the new weight is trivially Wnew =

for each progeny.

59 By this technique the number of events and the number of score contributions is increased, which — assuming that splitting is applied really in important regions — increases the efficiency, even in spite of the increased number of simulations. It is worth noting that the above-described trivial ways are not the only possibilities. Splitting and Russian roulette are treated in full generality in Chapter 5.111. Russian roulette and splitting may — and frequently are — used together. In complex geometries, sometimes each region can be assigned an importance.25 Then, when a particle enters region n + 1 from region n, and the ratio I,/In > 1, the particle is to be split into v = 1 ,/In pieces, otherwise Russian roulette is to be played by a survival probability of p= If v, in the splitting, is not an integer, one can either choose the nearest integer, or to split into n = ent(v) with a probability of n + 1 — v and into n = ent(v) + 1 with a probability of v — n. In the previous paragraph we already assumed that importances can be assigned to certain regions of phase space. Such importances, however, can be exactly specified only if the problem is solved. Thus, in a first step one can use estimated values. Such estimates can be derived from non Monte Carlo (e.g., discrete ordinate) calculations carried out for simplified problems, or the user can set ad hoc values based on his earlier experiences with more-or-less similar problems. Sometimes the first estimates of the importances are successively corrected after a smaller number of simulations and the computation is followed by the better estimates. In the solution of real physical problems, with different materials in different zones; energy dependent cross-sections; region, energy and direction dependent importances; the proper selection of the survival and splitting criteria may strongly influence the efficiency of the calculation and is a hard task. Investigations on finding optimum or near optimum parameters will be given in Chapter 7.11. D. EXPECTED VALUES IN SCORING The replacement of absorption by the reduction of the artificially introduced weight parameter, from a certain point of view, is equivalent to the analytical calculation of the expected value of a random process. Specifically, if we assign a score of 0 to an absorption and 1 to a scattering event, and we know that the probabilities of their occurrences are o-a/o- and Qs/Q, respectively, then the weight reduction factor: cr — rya = Qs

1•

Qs

+ 0 • cra ± cra

is just the expected score of the absorption game. Expected value score contributions are frequently used to replace scoring at events that simulate the processes leading to real physical scores. In many physical problems only a small fraction of the emitted particles reaches the receptor, i.e. has non-zero score. However, there are many examples where many more contributions can be obtained by the use of the expected scores. Let us illustrate this with two simple examples. 1. Transmission Through a Slab In this example a homogeneous thick slab (infinite in two dimensions) is considered. Particles enter the slab at x = 0, parallel to the positive x axis. Let us investigate the very simple question: how many of the incident particles will cross the slab — and leave — at x = X? If the slab is thick, the fraction transmitted is very low. If it is in the range of, say, 10-5 then just to register 100 outcoming particles, i.e. get 100 scores, one has to start with about 102 entering particles, and the simulation of so many histories, each consisting of a great number of collisions, requires an enormous computing time.

60

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

FIGURE 3.3. A sketch of expected leakage scoring.

Better results can be achieved if we sum up the probabilities after every collision considering the entrance on the x = 0 surface as the 0-th collision, that the particle reaches the x = X surface without any more collision. The contribution to the score after the i-th collision is then = w; exp[

X

—x] cox;

if w, > 0

(3.33)

and f, = 0,

if co,

0

Here, all the positively oriented paths make their contribution to the transmission probability, however, due to the possibly large deviations in the exponents the score contributions may be very different for different histories, which results in large variance of the score. Thus for very deep penetration problems the application of this scoring method alone does not help. There are, however, thicknesses where the application of the expected leakage scoring technique increases the efficiency. 2. Point Flux Estimation Another very important problem arises when one wishes to calculate the flux-at-a-point. Let us briefly recall the definition of the flux: it is the number of particles crossing the surface of an infinitesimally small sphere divided by the cross-sectional area of that sphere. Clearly, there will be no actual paths crossing an infinitesimal sphere, however, flux contributions can be calculated via the sphere-crossing probabilities. Let us consider a particle entering the i-th collision at P with energy E„ weight w, and direction to. Let D be the detector point and da the cross-sectional area. The probability that the particle reaches the sphere is a product of two probabilities: 1. The scattering event turns the particle to the proper direction 2. There are no additional collisions along the PD path

61

R -1:0

FIGURE 3.4. Geometry for the point flux calculation.

Thus, if C(E, w --> Oda = C(E„-4,)d1I denotes the probability that the particle entering the collision at P will be scattered into the solid angle dS1 around and T(E,,,) is the optical distance between P and D, then the probability of crossing the sphere (dp,) is dp, = W,C(E, 01) di/ • expf - T(E, For the contribution of a single collision of one particle this probability is just the expected "number of particles" in the definition of the flux, therefore, the flux contribution ((F,) of the investigated collision is dPi a

(ID = —

di/ W ,C(E ,b)exp[ - T(E,, ,)] da '±

W, ± ,C(E,09)expI - T(E,, ,)] R2

(3.34)

i.e., we have got a non-zero contribution formula. The only problem with Equation (3.34) is the R2 in the denominator, which means that one can obtain infinite contributions if the detector is not surrounded by a vacuum. This divergence is a very serious problem and the question of point flux estimators will still be discussed several times in this book. 3. Comments on the Two Examples First, we should like to call to the attention of the reader that the two examples presented above represent, in a certain sense, two basically different cases: expected values can be derived either before or after the collisions. In the first case averaging is executed over the possible post-collision points, whereas in the second case the averaging is extended to the after collision directions and the post-collision sites. Methods where these two ways are combined will be presented in Chapter 6.IV. A second comment, which is absolutely trivial but perhaps worth noting, is that if one replaces the physically realistic scorings by probabilities then the real events must not be scored. In the very simple example of point 1 given above this simply means that if a particle really crosses the x = X surface it must not be scored, since just the surface-crossing probabilities have earlier been summarized. There may, however, be more complicated cases where the treatment of the events already estimated by probabilities is not so trivial. Finally, it should be mentioned that expected values of the scores can be used in otherwise analog games, as was already noted in the introduction to this Chapter. (Then the weights W, in Equations (3.33) and (3.34) are to be set to 1.)

62

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

a

0

vacuum

material 1

material

2

vacuum

FIGURE 3.5. An illustration of the problem of omission of rare events.

E. PROBLEMS WITH EXTREMELY RARE EVENTS It was already mentioned in Chapter 3.1 that total omission — in an actual series of simulations — of events with extremely low probabilities but high contributions may lead to false results. The same problem may arise in the modifications, too. Let us just extend example 1 of the previous point to a slab of two layers with different materials having different scattering characteristics. If the first layer is very thick, it has a high probability that all simulated collisions occur in it and thus the calculated angular distribution of the transmitted particles will trivially be false, since there will be no information taken from the scattering laws of the second material. There is a similar danger in the application of biasing. The biasing process may result just in the opposite type of error than the neglect of the physically rare events. For example, if the modified source density Q(r,E,w) is much smaller than the exact one Q(r,E,w) over a certain (r,E,w) domain, one may have no point at all selected — in an actual calculation — from that domain and thus there is no chance for the correction according to Equation (3.32), i.e., the result will be false. The word "false" was intentionally used here instead of "biased". This is not a classical example of a biased result, since if the number of samplings is extremely high (usually not realizable in practice) there will be no error exceeding the statistical uncertainties. In the actual computations both the very likely full omission of the rare events and the very improbable but accidentally possible selection of them in a small sample will result in incorrect results. (A nice numerical example is given in Chapter 2.5 of Reference 39.)

III. STATISTICAL CONSIDERATIONS As has already been discussed at the very beginning of this Chapter, randomness characterizes the migration of the particles. Therefore all results either by measurements or by Monte Carlo simulations have to be presented by two values: the average and the standard deviation. The latter characterizes the uncertainty of the average. In an analog numerical experiment with n simulations one may per se expect the same uncertainty as in a physical experiment in which the measuring time is selected such that n particles are emitted. (The "same uncertainty" does not mean exactly the same estimated value of the standard deviation, since due to the randomness even the empirical standard deviations fluctuate around the theoretical value.)

63 In the following Sections, an elementary treatment of the statistical uncertainties is given. Derivations of the formulae given as well as more thorough discussion of their validity can be found in many textbooks on probability theory and statistics. 11,16,29,40 A. THE CENTRAL LIMIT THEOREM Let us consider n independent random observations, LIL LL 2, • • • , tin of a random variable. Assuming that this random variable (p) is a function of t, with a PDF p(t), the expected value of p is defined by M(p) = JP(t)p(t)dt

(3.35)

The real meaning of tin our case is quite general, it symbolizes a variable by which all the possible random paths can be parametrized. The variance is defined as: D2(

= m ( cc> — 1N1 (

If one estimates the expected value by the average of the n samples, i.e., by

M(p)

=

E

n ;_1

then according to the law of large numbers the average 11 approaches the expected value M(y) with a probability that approaches 1 as the sample size increases (n —> 00). More precise information on the convergency of the estimation is given by the central limit theorem. Given the n observations described above liraP { a

tin — M(Y) Nrn b} = fe b dt D(y) V2'rr

(3.36)

where P{x} denotes the probability that x is true. Equation (3.36) means that the average of n independent observations of a random variable (with finite mean and variance) approaches a normal distribution. Substituting a = —1 and b = 1 into Equation (3.36); the probability that 111.

M(4))1 > D(qP)/ \/

is about 32%. The probability that, for example, the difference between the average and the expected value exceeds 3D/\/n is only 0.27%. In practice Equation (3.36) is not directly evaluable since the variance D2(p) is not known in advance. A method to estimate it is given in the next Section. B. THE ACTUAL COMPUTATIONS At the actual computations histories of n particles are followed resulting in scores p.„ 112, . . . ,Rn. The definition of the average is:

=n L ,=,

64

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

whereas the variance can be empirically estimated by the square of the standard deviation (s).

s=

/

1 " n — 1 iI =1

(3.37)

— 11)2

Equation (3.30) is an unbiased estimate of D(q:), i.e., M(s2) = D2

The standard deviation of the mean (sii) is an unbiased estimate of D/\/n: $,„ =

1 V1 =

E cw, —

n(n — 1) i--1

P2

(3.38)

This formula is hardly applicable in an actual calculation, since evaluation of the subtractions on the RHS assumes the storing of all µ,-s up to the end of the run. A more convenient formulation is achieved by elementary transformations:

E(pi

=E

— 11)2

— 211,11

n

=E4.- 2-7

+

P2 1 pi) 2 + n —

n

=Ew—1 '(

R,,) 2

This latest formulation drastically reduces the store requirement: one has to store only the sums of the scores and their squares. Now, the empirical standard deviation is computed as: n

1 = lin(n — 1)

nc /

\ 21

(3.39)

Frequently, the relative standard deviation, or coefficient of variation, is given. It is defined by the relation sr and hence can be computed as sr =-

I R;

n [ n — 1 La

R)2

n

During the simulation of the history of even a single particle, the score may be obtained

65 as a sum of many contributions. If the i-th particle undergoes m interactions, with f,„ contributions, each, then the score is

=E Naturally, for the statistical evaluation the score (p.) is to be handled as an independent estimate, its composition from the f',„ contributions is not of interest. C. THE EFFICIENCY From the computational point of view, it is not the variance itself but the computer time required to reach a given variance that should be reduced by the introduction of nonanalog games. A modification of the analog game will certainly change the average time required by the individual simulations. It follows from Equation (3.38) that if the standard deviation of the mean (the quantity which characterizes the statistical uncertainty) is limited to s., than the minimum number of simulations is n=

s' so

if s' is the variance of the method. If the average time per simulation is denoted by T, then a computer time of s' T = nT = 2T so

(3.40)

is consumed during the total game. If one compares techniques, the error limit (s.) is fixed, thus the time in Equation (3.40) is proportional to product of the variance square and the average time per simulation. The inverse of this product is called the efficiency: 1 E — s2 — T In the comparison of two techniques, that with higher efficiency is considered to be the better. A detailed analysis and comparison of the efficiencies of a wide range of techniques is the task of Chapter 7.

APPENDIX 3A: ENERGY SELECTION FROM THE KLEIN-NISHINA FORMULA As has been stated in Section 3.I.D the differential scattering cross-section of photons is generally approximated by the Klein-Nishina formula, the result of the derivation is exact for initially free and at rest target electrons.

66

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

From Equation (3.9), the PDF of the ratio (x) of the energies of the photon before and after collision (ao and a, respectively) is H

( B C D x x2 x3

if 1

x

1 +2a

p(x) = 0,

otherwise

(A.1)

where the reader should be reminded that x= A = 1/a,2, B = 1 — 2(ao + 1)/a,2, C = (1 + 20.,,)a,?„ D=1 H is K/o-*, the value K was given in Equation (3.9), the total cross-section cr* in Equation (3.10). However, the actual value of this normalization factor has no importance for the selection procedures. There are many procedures that may be used to select samples from the PDF (A.1). The five basically different types of sampling methods applied are 1. The rejection technique 2. Solution of the inverse cumulative distribution function (CDF) by numerical approximation 3. Creation of approximate formulae for the inverse CDF 4. Use of an approximate inverse CDF coupled with correction by a weight adjustment factor 5. Direct sampling Several procedures are cited in Reference 31 and also compared in Reference 4. In the following we describe two methods only. The first one (A). is a combination of two techniques and is found to be the fastest method containing no approximation.' The second method (B). is extremely simple but is proposed for uses in codes where the exactness of the new energy selection is not crucial since it is based on an approximate fit of the inverse CDF. A. COMBINATION OF THE DIRECT SAMPLING AND REJECTION TECHNIQUES The only selection procedure which uses neither approximation nor rejection is the direct sampling method proposed by Koblinger.31 This direct sampling is a combination of the probability mixing and the inverse cumulative distribution methods.

67 Let us express the PDF A.1 as a sum of four terms: p,(x) = HA B P2(x) = H — P3(x) = H

x2

and D P4(x) = H X s The probabilities for the selection of the term (see Section 2.1.6) are 2 P1 = P3 = H — a.

1 + 13) fn13

P, =

ot,2,

and P4

H

where 13 = 1 + 2a0 and 1 -y = 1 — — [32 The selection of the i-th term is carried out as given in Equation (2.1), however the whole probability mixing method is applicable only if all p,(x)-s are non-negative functions. This condition is automatically satisfied for p„ p, and p4 but, only if

P2(x) % 0

a, > 1 +

(A.2)

If the condition A.2 is satisfied and the i-th term is selected the inverse CDF method can be applied for each of the four terms. The actual recipes are x= 1 + 2otp

for i = 1

x = 13P

for i = 2

x—

13 1 + 2ap

for i = 3

and x = (1 — -yp)- "2 for i = 4

68

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

L.

Select p1 , p,, and p,

7

no

1 + 2a, ( P 9 + 2cco

yes

II

Compute

Compute x = 1 + 2132ao

x—

1 + 2a0 1 + 2p,a„

no (2p,

X ao ao

+ 1 +

yes

a = au x

FIGURE 3A.1. The flow chart of the rejection method for sampling the Klein-Nishina distribution.26

The energy of the photon after the collision is trivially a = ajx. The inequality (A.2) strongly restricts the applicability of the direct method. For low 1.4 MeV) another technique must be chosen. The rejection 1+ E energies (a method proposed very early by Kahn26 seems to be the best candidate, which is given in Figure 3A.1. Fortunately, the efficiency of the Kahn method is pretty good at low energies and worsens drastically only at high incident photon energies, where it can be replaced by the direct sampling method described above. Thus the combination of these two techniques leads to a fast procedure .4 B. THE CARLSON METHOD This very simple sampling method requires the selection of one random number per scattering simulation. The method is based on an accurate fit for the inverse CDF. For energies not exceeding a„ = 4 (E. 2 MeV) the following formula is to be used': a—

a. 1 + sp + (2a.„ — s)p3

(A.3)

69 where S=

oto

1 + 0.5625 ao

The applicability of the method can be extended' up to about a, = 10 (E 5 MeV) by adding the term 0.5(a0 — 4) • p2(1 — p)2 to the RHS of Equation (A.3).

APPENDIX 3B: THERMAL NEUTRON ENERGY SELECTION In most codes thermal neutrons are handled in one (or several) thermal group(s), thus there is no actual continuous energy change selection. Another approximation is the selection of the energy of the scattered neutron from the equilibrium spectrum — independently of its pre-collision energy. Selection schemes for this approach are given in Section A. and a method for choosing the outcoming energy from the ingoing one is cited in Section B. A. SELECTION FROM THE MAXWELLIAN DISTRIBUTION In the free gas model with no absorption, the thermal neutrons have a Maxwellian energy distribution of Equation (3.25). The probability density function can be written as P(R)

2

0 R
E,µ) = — 4,Tr Eo 27m€2 ex

mt

2kTe2

ip

K2 \

E 2mi)

(B.2)

where mt is the mass of the scattering nucleus, m is the mass of the neutron and E = 2m(E + E0 — 21IVEE0).

71 Following Erikson's results" the expression of Equation (B.2) can be transformed so that the resulting PDF (the normalized cross section) consists of the product of two independent probability variables: p and q. The new probability density function is (B.3)

h(p,q) = Cpexp( — q2) where C is a normalization constant, + Z2

p = V1 —

a q = — [Xp + (Z2 — 1)/p] 2

a= and X

m,

The new variables p and q have physical meanings only if their range consists of two sets36: (B.4)

R(p,q) = R,(p,q) + R2(p,q) where R,(p,q) = tp,q: — a < q a, °
r symbolizes the integration along the straight line from r' to r, r" = r + sw and 8(x) is the Dirac-delta function which selects the actual direction w leading from r' towards the point r just investigated. In Equation (4.31) the first factor (u) indicates that the probability of an interaction at r is proportional to the total cross-section there. The second factor is the exponential attenuation between r' and r, i.e., it is the probability that the particle will fly as far as r. The third factor expresses the fact that the flight may only reach points along the line from r' in the direction w. A more attractive form of Equation (4.31) can be derived if it is transformed to onedimensional form with the aid of Figure 4.2. If we replace the elementary relations given and illustrated in Figure 4.2, a one-dimensional form is finally obtained as: T(e-->rit.o,E) dr = cr(R,E)exp[ — T(R)]dR

(4.32)

where T(R) = f o-(R1 )dR' is the optical distance between r' and r (i.e., corresponding to 0 the geometrical distance R). The RHS of 4.32 will be briefly denoted by T(R)dR. *

A scalar product of two vectors is identified like ab, without a dot.

101

R —1 r — ril r = r' + Rc.) rn

r' +

"•-•

co

0••••-•

dFe= ds 2 dr = R dRd co FIGURE 4.2.

2. The Collision Kernel When a scattering takes place at a point r, the energy and the direction of motion of the particle will be changed. There are, however, collisions which lead to absorption of the particle, or to multiplication (e.g, fission). The collision kernel is to describe the total effect of all types of possible interactions. Let us denote the expected number of particles coming out from a collision at r with an energy between E and E + dE and a direction lying in the solid angle dw around w by C(to',E'—>co,Eir)dco dE assuming that E' and w' used to be the energy and direction before the collision. If there are i = 1,2, . . . ,n elements in the material considered and there are j = 1,2, . . . ,m possible types of interactions with v, expected numbers of outcoming particles then n m

=

E

E vu o-,,(r;o3',E'—>t.o,E) o(r,Er)

(4.33)

where r is the differential cross section for element i and interaction j — as defined in Chapter 2.11. The summations over i and j are interchangable, as it was discussed in Section 3.I.A. In the Monte Carlo solution either one selects first the element with which the particle collides and next the type of interaction, or vice versa. In gamma transport calculations at Compton scatterings, the energy and direction change independently of the element hit — as far as the Klein-Nishina formula is valid. C. THE EQUATIONS CONNECTING THE COLLISION DENSITIES There are two ways leading to the emergence of a particle at dr about r with an energy between E and dE and direction dw about w: either it is emitted from the source with these parameters or it comes out of a collision in this phase-space element. Thus according to the above definitions the outcoming collision density satisfies the following equation: x(r,E) = Q(r,E) + IdE' 41(r ,E') C(E'--->EI r)

(4.34)

102

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

Hereafter, the symbol E = (w,E) is used since it simplifies the formalae as the energy and the directions of the scattered particles change at the same events, and in many cases not independently of each other. (The same unification of the energy and direction coordinates was achievable by the use of the velocity vector v.) The second term in the RHS of Equation (4.34) summarizes the results of all possible collisions from all possible pre-collision coordinates (E' and w'). Particles emerging (either by emission or by collision) at r' can enter a next interaction in the neighborhood of r, if they are oriented in the proper way and reach that region, i.e., with the definition of the collision densities and the transition kernel the ingoing collision density is expressed as tli(r,E) = 1de x(e ,E)T(e-->r1E)

(4.35)

The pair of Equations (4.34) and (4.35) fully describes the collision densities. There are two points worth noting here. 1. Equations (4.34) and (4.35) are deterministic equations for expected values of stochastic variables, and thus — at least theoretically — they can be solved by deterministic methods. Thus when discussing Monte Carlo solutions of these equations we make — in a certain sense — a double maneuver: first deterministic equations are established for a stochastic process and then a stochastic model is built up to solve the deterministic equations. In the simplest cases the stochastic model leads directly back to the simulation of the basic stochastic process, whereas in more refined techniques, the physical analogue does not exist anymore, the original physical and final artificial stochastic models are connected by mathematical manipulations only. 2. By substituting Equation (4.35) into Equation (4.34) an integral equation solely for X(P) is obtained: x(r,E) = Q(r,E) + JJdr'dE'X(r' ,

(4.36)

Similarly, if one substitutes Equation (4.34) into Equation (4.36) the following equation for the ingoing density is obtained: = Qe(r,E) +

dE' dr' Ve ,E')C(E r—>Ele)T(e—>rIE)

(4.37)

where the first (or first-flight) collision source Q is defined by: Qc(r,E) = fdr' Q(r' ,E)T(e—n.lE)

(4.38)

Qe(r,E) is a physically correct description of the density of the particles entering their first collision at (r,E). The whole Monte Carlo treatment of particle transport can be carried out in terms of only one of the densities LI; or X. It is reasonable to express the equations in terms of the ingoing collision density lift which is directly connected to the particle flux, a fundamental quantity in particle transport theory. In many Monte Carlo descriptions indeed this single quantity is used, (and is referred to briefly as collision density) nevertheless for the moment we keep both densities in the derivation since parallel use of x and offers more opportunity

103 for simple plausible explanations — especially in the derivation of the value of equations in Chapter 4.VI. Equation (4.37) can be further simplified by using the general phase-space coordinate symbol P and by introducing the combined transport kernel (K) tp(P) = Qc(P) + fdP'

K(P',P)

where K(P',P) = K(r',E'; r,E) D. THE THEORY OF THE STEP-BY-STEP SOLUTION OF THE COLLISION DENSITY EQUATIONS Here, the Monte Carlo solution of the collision density equations introduced in the previous Section is outlined on the basis of the Neumann series expansion of the collision density functions. The separate terms of the Neumann series expansions can be recursively generated with the aid of Equations (4.34) and (4.35). The method is basically the same as described in connection with the solution of a general Fredholm-type integral equation, however here a coupled integral equation system is considered. The corresponding recurrence is given by the following equations: )ar,E) = Q(r,E)

(4.39)

tilo(r,E) = idr' xo(r' ,E)T(e-4r1E)

(4.40)

x,(r,E) = JdE'~(r,E')C(E'~E~r)

(4.41)

+1(1",E) = fdri x, ÷i (r',E)T(r'—>rIE)

(4.42)

for the zeroth terms and

111

for i + 1 = 1,2, . . . ,00. Equations (4.39) through (4.42) have an obvious interpretation. xo in Equation (4.33) is the density of source particles at (r,E), i.e., the density of the particles coming out of their "zeroth collision". Similarly, tllo in Equation (4.30) defines the number of particles entering their first collision in the neighborhood of (r,E). x„ in Equation (4.41) is expressed as the density of the particles leaving their (i + 1)-st interaction at r and leaving the collision with a direction + energy coordinate about E. Thus x,± , is the density of particles leaving their (i + 1)-st collision and tli, is the density of the particles entering this collision. Hereafter, we assume (as in Chapter 3) that the system is subcritical, which means that

f

f dr dE x(r,E) = fidr dE x„(r ,E)

and q4, if dr dE 41,(r,E) = if dr dE qi„(r,E)

104

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

where q, < 1, qq, < 1. (The possiblity of the use of the same q values for all 1-s is a consequence of the physical fact that the process is Markovian, i.e., the laws of transport are independent of the number of previous collisions.) Subcriticality means also that the Neumann series' for the collision densities are convergent. The Monte Carlo realization of the recurrence in Equations (4.39) through (4.42) is given in the following steps: 1. Set i = 0. Select initial coordinates (ro,E.) from Q(r,E). This gives a sample from X.(r.,E,„) in Equation (4.40). 2. Select the next collision site r,,, from T(r,—>r1E,)dr This will define a sample from 11),(r,,,,E) in Equations (4.40) and (4.42). 3. Select the post-collision coordinates in the (i + 1)-st collision from C(E,-->E1r,,)dE which gives a sample from x,+i(r,+,,E,+ i) in Equation (4.41). 4. Set 1 = i + 1 and return to Step 2. There are, however, two problems with the procedure given above: •

•

Selection of new coordinates can be made only from probability density functions, i.e. from functions whose integrals over their whole range is unity, however this condition is not a priori fullfilled. The cycle of Steps 2 through 4 above is infinite i.e., for practical applications a terminating criterion must be found.

As it has already been seen at the heuristical level and will be demonstrated in Section F. the two problems are not independent of each other. The next Sections are devoted to a discussion of these questions. But before turning to this a last — and very important for the future (Chapter 5) — remark is to be made here. If one compares Equations (4.39), (4.40), and (4.38) then the equality Qc(r,E) = tlio(r,E)

(4.43)

is obtained, i.e., the first-flight collision source is just the zeroth term in the Neumann series of the ingoing density function. Hence, inserting Equation (4.43) into Equation (4.37) the resulting equation reads 115(P) = +.(P) + f dP' tp(P')K(P' ,P)

(4.44)

and this form will be the starting equation in Chapter S.I. E. NORMALIZATIONS OF THE TRANSITION AND COLLISION KERNELS If one assumes that both the collision and the transition kernels are normalized to unity then the selection procedure mentioned in the previous Section needs no more modification. However, this is seldom the case. Let us now analyze the two kernels separately, since the physical phenomena described by them are substantially different — though the mathematical problem is the same.

105 1. The Transition Kernel If the whole geometric) space if filled with non-zero total cross-section material then the integral form of the transition kernel in Equation (4.32) becomes T(r'—>rIE,o3) = f dRT(R) = dR o-(R,E)exp[ — o-(R',E)dR'l 0 0 0 -dR dR — d exp [ — f Rcr(R',E)dR'] 0

f0

= 1 — exp [ — cr(R',E)dR'] or by introducing the optical distance: fdRT(R) I = 1 — exp[ — T(00)]

(4.45)

Now, since T(00) = 00 for a non-zero cross section extending to infinity, we find that the integral of T(R) is unity. However, if in any direction there exists a distance value Ro such that the cross-section of the medium R' > Ro is zero, then T(00) = T(Ro) < 00 and in this direction the integral of T(R) is less than unity. This problem can be eliminated by the introduction of a hypothetical black absorber around the real system — just as we discussed at the level of heuristical interpretation (see also Section 5.I.C.). The other approach is the normalization of the transition kernel. The normalization factor is trivially T* = 1 — exp[ — T(00)]

(4.46)

Equation (4.35) — and its successors: Equations (4.40) and (4.42) — can be reformulated as: tli(r,E) = fdr' x(r',E)T*

T(r'—>rIE) T*

(4.47)

where T/T* is now the probability density function that can be used for selecting a new R Ro. collision point within the region: 0 The factor T* in Equation (4.47) is to be handled as a weight — like the weight introduced in Section M.A., this Chapter — and now the actually selected points will represent the 4i(r,E) distribution only if all points are considered with their "scaled down" contributions, or, in other words, not only the r,, E coordinates but also the weight "coordinates" are essential characteristics of the collision density. When, during the successive selections in Steps 2 of Section D., this Chapter, such weight corrections are applied many times, then at every step a new net weight correcting factor has to be applied. If T'kl" denotes the normalization factor in the k-th flight of a particle then the weight factor in the flight selection is: Wk,T

=

106

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

and the resultant weight after K selections is: K

W =

n W

k=I

(4.48)

k.T

There may be geometries where the transition kernel is normalized in one direction and is not in another one. In such cases several wk, factors in Equation (4.48) are unity. This weight treatment corresponds to the method introduced on a heuristical basis in Section 3.II.A. One has to realize that normalization of the transition kernel and introduction of the weight factor w,„ imply that the particles are artificially kept inside that part of the system where the cross-section is non-zero. If one does not like to use such weights, one can play a Russian roulette after every transition. If the weight of a survivor is fixed to W. = 1 which trivially means that weights need not to be used at all, then the simulation is terminated with probability of (cf. Equations (3.18) and (4.46): T* p=1—w — = exp[ T(00)1 which is just the probability that the particle can leave the region of interest, i.e., it is the leakage probability! It is worth recalling here that in the heuristic interpretation of the Monte Carlo technique the simulation of the leakage was plausible and its replacement by weight reduction seemed to be artificial. Now the case is just the opposite. If one approaches the problem from the points of view of solution of deterministic integral equations then the application of weights is trivial and the simulation of the leakage has to be introduced artifically, by repeated applications of Russian roulette. 2. The Collision Kernel In the numerator of the RHS of Equation (4.33) the expected number of particles coming out is summed over all possible interactions, whereas the denominator is the total crosssection. If there are only scatterings, i.e., interactions with one outcoming particle, C is trivially normalized to unity and therefore can be used as a probability density function for selecting the post-collision E and w coordinates: if v, = 1 for all i-s and j-s, then —

if do) dE C(u.) I

o-(r,E')

nm

1 u(r,E')

IdolclEcrjr;6.)',E'—>63,E)

= "

In the real physical processes widely varying v-s occur, from v = 0 (for absorption) to v = 2 (for (n,2n) reactions, for the positron annihilation following a pair production event in photon transport), or even to v > 2 (for fission events). Under these realistic conditions the collision kernel is not normalized to unity. The method described for the transition kernel is to be applied here again. The normalizing weight function is now trivially n m Wk,c = C*

=EEidEtho vijo-Jr; to' ,E'—>w,E) 1=1 j=1

107 and now Equation (4.34) — and its successor Equation (4.41) — can be reformulated as X(r,E) = Q(r ,E) + f dE' C*

EIr) C*

IP(r,E')

In non-multiplying media Wk,c = C* is by definition the non-absorption probability and its use can be avoided by the application of a Russian roulette game which leads to the simulation of absorption just in the same way as we saw in connection with the leakage simulation in the previous subsection. In multiplying media if C* > 1, the particle's weight increases at the interaction, thus, if one wishes to fix the weight at W = 1, one has to "split" the particle into C* fragments, or if C* is not an integer into ent (C*) or ent (C*) + 1 fragments with ent (C*) + 1 — C* and C* — ent (C*) probabilities, respectively. (Splitting and Russian roulette will be treated in more detail in Chapter 5.) The relation between the former heuristic and the present interpretations is again similar to that which we saw in the case of weight reduction: in the simulation of the physical process the selection of more than one outcoming particles was trivial and its replacement by weight increase was artificial. Here, in the solution of the integral equation the introduction of weights was the straightforward method and artificial introduction of splitting leads us back to the analogue treatment of the physical events. When the particle's weight is modified at K successive collisions, the resultant weight is: K

W = II Wk,C

k= 1

If weight corrections are applied — or, in this presentation it is more consistent to say that neither the transition nor the collision normalizing factors are renormalized by Russian roulette or splitting — then after K transitions and collisions, the resultant weight is: W=

k=1

Wk,C Wk,T

since the transition and collision simulations are independent of each other. F. TERMINATION OF THE MONTE CARLO CYCLE The first problem mentioned in Section D of this Chapter — the normalization of the kernels — is essentially solved now. The introduction of the weights or their replacement by Russian roulette or splitting automatically revealed two possiblities of termination: • •

The particle leaves the region with no return possiblity, or is absorbed, if no weights are used; Particles can be killed by Russian roulette if their weight becomes too low.

These two termination criteria are identical with those found in the heuristical interpretation Chapter 3, Sections I.B. and I.C. There is a third criterion which is also the same as the criterion which we earlier derived heuristically, viz. points representing terms of the Neumann series which make no contribution because the energy coordinate fell outside the region of interest need not be computed. Feasibility of Monte Carlo games, i.e., conditions under which games do terminate with a probability one will be discussed in a rigorous way in Chapter 5.

108

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

V. SCORING A. GENERAL FORMULATION OF THE REACTION RATES In Chapter 4.IV., methods for finding the r, E and w coordinates representing either the ingoing or the outcoming densities were described. For practical purposes generally not the collision densities themselves but a functional of one of them, a reaction rate or receptor response is to be determined. The general formulation of such functionals was given in Equation (4.27) and the recipe for a Monte Carlo estimate was described in Theorem 4.8. Theoretically, most reaction rates or responses can be derived either from the ingoing or the outcoming densities and it is up to the user to choose the most suitable approach. Accordingly, there are two basic definitions of any reaction rate R: R = fdPfx(P)x(P)

(4.49)

R = fdP fq,(P) tir(P)

(4.50)

and

where 1', and f1, are the pay-off functions related to the respective collision densities. The actual forms of fx and fo, are determined by the physical quantity to be estimated. However, from the elementary requirement that the result R must be the same, whether it is derived from Equation (4.49) or Equation (4.50) a relation between the two pay-off functions can be derived in the following way: Let us substitute Equation (4.35) into Equation (4.50): R = Ifdr dEfq,(r,E)fdr' T(r'—>r1E) x(r' ,E)

(4.51)

If now we interchange the order of integration over r and r' and then the symbols r and r' themselves Equation (4.51) becomes: R = fidr dE x(r,E)Ide T(r—>elE) fq,(r' ,E)

(4.52)

From comparison of Equations (4.52) and (4.49) the relation fx(r,E) = fdriT(r—>elE)4(r',E)

(4.53)

is obtained. B. ESTIMATION OF MORE THAN ONE RESPONSE The same set of random walk simulations can be used for the estimation of many different responses or reactor rates. One may be interested in the flux integrals in two or more different regions, or in calculating differential quantities; e.g., in determining the fluxes for different energy intervals. In such cases a set of responses is to be determined by the use of a set of pay-off functions: R' = f dP f:1,(P)41(P)

109 or R1 = dPf;,(P)x(P) The crucial point here is that all estimations are based on the same representation of the collision densities and therefore the results W'' will not be statistically independent of each other. This technique, therefore, can be regarded as a use of the correlated sampling outlined in Section 4.II.F. When talking about the "result" of a Monte Carlo game it is to be stressed that the simulation process and the estimation procedure are essentially independent processes. This is obvious from the fact that several different games may produce the same estimate (response). The same follows also from the derivation above, i.e., that a given reaction rate can be estimated with the aid of different pay-off functions. The "separation" of the random walk simulation from the evaluation process is clearly demonstrated e.g., in the 05R program' where the output is a magnetic tape containing all data obtained from all collisions and the user provides a supplementary code to derive the quantity which interests him from the collision data. C. MONTE CARLO ESTIMATION OF THE RESPONSES The Monte Carlo estimation of a functional of the function go (x) satisfying the Fredholmtype integral Equation (4.26) was described in general in Section 4.III.B. Now, after the expansion of the collision densities into Neumann series, an analog expansion of the response or reaction rates is straightforward: R=

E Ri.X =

R=

E

=

0

fdPfx(P)X,(P)

E fdpf+(p)tp,(p)

,=.

Points with coordinates r„ E,, W c represent the x, outcoming density, and points r1,„ E„ W,,, represent the ingoing density, if the weights here are defined (after Equation [4.48]) as

Wi,c = 11 W k,C • 11 Wk,T k=0

k=0

and

Wi,T =1 I W k,C Wk,T k=0

Thus the contribution from the i-th collision is either r;., = Wi.0 fx(r„E;) or = W, r f,(r,±

110

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations The score from an entire history is the sum of the contributions:

E

(4.59)

=E

(4.55)

P-x =

i=o

or

i=0

If the fx and f, pay-off functions are correctly chosen then the expected values of the (4.54) and (4.55) estimates must be the same: (P,x) = (1-1.,p) = R After following j = 1,2, . . . ,n histories each scoring fix., and/or 114,,, the average of the scores is an unbiased estimator of R: 1" R = (— E= n J =1

1"

E n ;-,

D. EXAMPLES OF PAY-OFF FUNCTIONS An extremely wide variety of physical quantities may be determined by Monte Carlo. In the following subsections several representative examples are shown and discussed. 1. Collision Density and Reaction Rate Integrals The simplest question to raise in a Monte Carlo game is: how many collisions occured, i.e. how many particles entered collision, in a certain volume element (or domain) F? The answer is trivially: R

dr .idE tir(r,E) = I f dr dE11,-(r) tli(r,E)

= fr

11, where

if r

E

(4.56)

F

hr(r) = 0,

otherwise

Thus from a comparison of Equations (4.50) and (4.56) the pay-off function is: f 4,(r,E) = hr(r) If only a part of the spectrum is of interest, e.g., one wishes to count the number of collisions which the particles enter with energy between E, and E2, then the corresponding pay-off function becomes f4,(r,E) = hr(r)hE(E) 1,

if E, < E < E,

0,

otherwise

where hE

111 Similarly, solid angles of interest can be "cut out" by a window function h0,. The use of the counterpart pay-off function f, can inform us about the number of particles leaving collisions in a certain phase-space domain. The rate of a certain kind of interactions is also easily determinable. The number of interactions type i (having a cross section o,) is trivially: = If dr dE

cr(r,E) (r ,E)

,E)

The corresponding pay-off function is obvious from this formula. The flux integral can also be handled as a reaction rate. From the elementary connection between the ingoing density and the flux, the flux integrated over a domain I' is expressed as R = J dr idE (p(r ,E) = frdr idE l(r,E)115(r ,E) = f fdr dE hr(r) cr '(r,E) tli(r,E)

(4.57)

i.e., the pay-off function is now Icr i(r,E),

if ref

t,(r,E) = 0,

otherwise

In an actual simulation of N histories that result in altogether i= 1, 2, . . . ,n collisions with (r„E„w,) pre-collision coordinates representing 111(r,E), the flux integral is estimated by:

E

R=1 w,hr(r1 )cr -1(ri,E) n i=1

(4.58)

Equations (4.57) and (4.58) are meaningful only if cr(r,E) 0, i.e., only in domains which are filled with material having non-zero cross section. However, as was already discussed in Section 2.II.F., the flux is a reasonable and therefore computable quantity even in vacuum where there are no interactions and thus no collision densities. In such cases the other definition of the flux, which was discussed in Section 2.II.C., and which is based on the sum of track lengths, can be used. In order to illustrate the flexibility of the integral equation systems applied throughout this Chapter, we derive the track-length type estimation of the flux integral by starting from Equation (4.57) — without considering the alternative definition mentioned above. Let us substitute Equation (4.35) into (4.57): R = Jdr f dE cr -1(r,E) f dr' x(r' ,E) T(r'—*rIE)

112

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

R

0

r'

FIGURE 4.3.

After changing the order of integrations with respect to symbols themselves, one gets:

r

and

r',

R = f fdr dE x(r ,E) Irdr` o- i(r ,E)T(r—>r '1E)

and then these two

(4.59)

Thus the pay-off function of the outcoming particles is fx(r,E) = frde cr -1(r,E)T(r-->e 1E)

(4.60)

Let us further analyse this pay-off. If the transport kernel is rewritten with the same transformation as applied in the derivation of Equation (4.32), and the points where the path crosses the F domain are at R, and R, distances from r' (see Figure 4.3), then after substituting Equation (4.32) into Equation (4.59) one gets: 122

fx(r,E) =

Ri

dRexp[ — cr(R')dR']

On the other hand let us derive directly the expected path length in F of a particle starting a flight from r' with a direction of 63'. Three different events may happen to this particle: 1. It does not reach R,, thus the path length in F is zero; 2. It collides in dt about inside F, then the track length is trivially C and the probability of such an event is: Ri“

Pb de = 0.(R,

f) exp[

0

cr(Ri )dR'] df

(4.61)

3. It flies through F without interaction, and thus the track length is max = R2 — and the probability of this passing through is R2 =

exp[ — I cr(R')dR]

RI

(4.62)

113 The expected track length is a sum of the contributions of the three (practically two, since the first one is zero) events: err., (f) = (10 df f

+ emax pc

(4.63)

Substituting Equations (4.61) and (4.62) into Equation (4.63) and integrating by parts one finally obtains the expected track length as: R2 (e) = f dR exp [ — o(R')dR1 ]

R,

(4.64)

Since the right hand sides of Equations (4.60) and (4.64) are equal to each other, fx(r,E) = (f) i.e., the pay-off function in Equation (4.59) is simply the formula describing the expected path length of a particle in the domain F if it enters a collision at r with direction and energy (w,E). (The derivation presented above is easily generalizable to paths crossing the F domain more than once.) As in many cases already discussed these expected paths can be replaced by actual selected path lengths as if an "internal Monte Carlo" were played to estimate the expected path — by a single experiment. In an actual game the flux integral estimate is now 1 R = — 2, w f" N

(4.65)

where f, is either the expected or the actual track length of the i-th flight, if there are altogether n flights of N history simulations. Naturally, the expected track length is computable just after the simulation of the collision investigated, whereas for the calculation of the actual length the knowledge of the next collision site is a precondition for the determination of the pay-off contribution. The expressions of the Equations (4.57) and (4.59), and their Monte Carlo estimates given in Equations (4.58) and (4.65), respectively, are based on different collision densities. Nevertheless the estimates can be evaluated during a single run, from data observed from the same set of simulations. Therefore it is straightforward to assume that a linear combination of the two unbiased estimators will also give unbiased results. The combination of the different estimators will be discussed in Chapter 5. 2. Transmission Through Slabs Let us re-examine the problem of Subsection 3.11.D.1. particles are entering a slab and the task is the calculation of the number of particles transmitted (see Figure 3.2). If the flux is denoted by cp (r,w,E) then the total flux of the particles crossing the plane x = X is: R = ffdrdEcp(r,E)8(x — X)

(4.66)

where the 8 (x — X) factor selects out those paths that really cross the x = X surface. By the use of the elementary Equation (2.22) connecting the flux and the ingoing collision

114

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

density the flux at X is rewritten as R = Ifidr dEo-- V,E)tli(r,co,E)8(x — X)

(4.67)

Now the density of the ingoing particles can be replaced by that of the outcoming ones from Equation (4.35) and Equation (4.67) becomes: R = IfidedEx(r',E)fdrcr'(r,E)T(e-->rIE)8(x — X)

(4.68)

After substituting Equation (4.31) for the transition kernel and transforming it into its onedimensional form (like in the derivation of Equation (4.32), Equation (4.68) becomes: R

[ icx-x')/.‘ fildr'clEx(r',E)I dxexp — 0 o-(r' + R'co,E)dR'] 8(x — X) = fifth' dE x(r ' ,E) exp [ —

J

oc-x,y.„ cr(r' + R'w,E)dR']

Changing r' to r (and consequently x' to x), the new form for the transmitted flux becomes: R= iffdr cico x(r,E)exp [ — cr(E)

cox

i.e., the pay-off function is: fx(r,E) =- exp[ — Q(E)

X — xl Wx

It is worth noting here that "collision density estimation": was first proposed by Berger2 just for the solution of these types of problems. If w„ E„ cox denote the coordinates of a particle leaving a collision then the contribution from this collision is: f, = w,exp[ — cr(E,)

X—

xi l

(4.69)

Equation (4.69) is identical with Equation (3.33) i.e., again the estimation via the expected value — a little bit artificially introduced in Section 3.II.D. — is the straightforward way here. Counting of the actual crossing events can be derived artificially, by defining an internal Monte Carlo game for a 0 or 1 estimation of the probability given by Equation (4.69). The warnings about the estimation of extremely rare events (discussed in Section 3.II.E.) are per se valid here, too. 3. Flux-at-a-Point As a last example of pay-off functions we consider a delicate problem of transport Monte Carlo calculations: estimation of the flux-at-a given spatial point. A systematic investigation of this problem will be given in Chapter 6.

115

In regions imbedded in non-zero cross-section materials the flux 41) can be determined by Equation (2.22); i.e., the flux at r is (4.70)

ckr„) = f dElI)(ro,E) = idEo-- V0,E)tlr(ro,E) By the general formulation of Equation (4.50) R = 4,(ro) = fdr IdE u(r E) tIr(r,E)8(r - ro) i.e., the pay-off function is: =

1 s cr(r,E) -(r

(4.71)

1.0)

The use of Equation (4.71) in practice is not possible since the probability of having a collision exactly at 1.0 is zero, moreover, in points surrounded by vacuum the division by ocauses divergence. To achieve a nicer pay-off let us turn to the outcoming density by substituting Equation (4.35) into Equation (4.70): R = 41(ro) = f dEo- '(ro,E) f dr' T(r'-->rIE) x(r' ,E) After replacing r' by r the equality becomes (4.72)

R = f fdrdEo-- l(ro,E)T(r—>rolE)x(r,E) i.e., the pay-off function related to the outcoming density is

(4.73)

fx(r,E) = o-- `(ro,E)T(r-->rolE) After substituting Equation (4.31) into Equation (4.73) the f, pay-off becomes: 8(0)

fx(r,E) = exp( - f rocr(r",E)ds)

Iro - 1.1 Ir„ - rl

1)

where r" = r + so)

(4.74)

Now, the problem with the possible divergence of the factor o ' is overcome, however the presence of the Dirac-delta function prevents the practical application of the pay-off function in Equation (4.74), since only those paths should give non-zero contributions that pass exactly through the r„ point — and the probability of the random occurence of such predetermined directions is again zero just like the probability of having collisions at a fixed point. Thus, a further transformation is needed.

116

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations Let us now substitute Equation (4.34) into Equation (4.72), the reaction rate then reads: R= fidr dE cr - Vo,E) T(r-->ro jE) [Q(r,E) + f dE1 C(E'—>E1r) tlf(r,E')] (4.75)

The total flux in Equation (4.75) is a sum of two terms (R = R. + R'). The first one is the direct contribution from the uncollided source particles: R. = fidr dE

(ro,E) T(r—>ro jE) Q(r,E)

(4.76)

The second term describes the contribution of the collided particles: R' = fidr dE o- Vo,E) T(r-->ro jE) f dE' C(E'—>Ejr) ‘11(r,E')

(4.77)

By interchanging the order of integrations over E' and E, and then the two symbols themselves, Equation (4.77) becomes: R'

ifdr dEtp(r,E)IdE1 cr - '(ro,E')T(r-->ro jE') C(E—>E'jr)

(4.78)

In the majority of practical cases (in every case in photon transport) the post-scattering energy and the scattering angle are uniquely related. If so, the collision kernel can be factorized as follows C(E-->E'lr) = C(o)-->co'lE,r)8[E' — G(ww' ,E)]

(4.79)

where the first factor describes the change in the direction of flight, whereas the relation E' = G(6363',E) gives the new energy of a particle entering a collision with energy E and being scattered by an angle *, cos* = cow. Now by substituting the actual forms of the transition and collision kernels as given by Equations (4.31) and (4.79), respectively into (4.78), the contribution of the collided particles to the flux is R' = fidr dE tp(r,E) f idco' dE' exp [ —J cr(r' ,E)ds] (03,

ro

—r

1 1.0

rl

Iror 2

1 ) C(63-->wlE,r)6[E' — G(tow',E)]

The first Dirac-delta selects out from among all directions the only direction co* = (r0 — r)/Iro — rj oriented towards the receptor point at r0 , the second one selects out the energy after scattering: E* = G(oo*,E)

117 Thus, —->w*IE,r) R' = fidrdE tli(r,E)exp[ — T(E*)] C(o.) Ir. — rI2 where the optical distance T is introduced (see Section 2.II.E.). In the following the distance Iro — rl is denoted by R and we assume that the value of the collision kernel is independent of the direction of incidence but is determined by the incident energy and the angle of scattering. C(w-->w*IE,r) = C(E,*,r) Then the pay-off is: f4,(r,E) = exp[ — T(E*)]

C(E,*,r) R2

(4.80)

If at the i-th collision of an actual game the pre-collision energy is E„ the energy of the outcoming particle directed towards the receptor location is denoted by E,± „ and the game is played with the use of weights, the use of the pay-off function given in Equation (4.80) leads to the contribution formula derived via heuristic interpretation and given in Equation (3.34) Accordingly, all the problems caused by the possible divergence due to the 12 -2 factor, and listed in Section 3.II.D.2. are still valid here. After this relatively long derivation of the pay-off function to be applied to the collided particles it is worth turning back to the source contribution. Since Q(P) = X. (P) from the source contribution described by Equation (4.76) and the definitions given by Equations (4.49) and (4.54) we see that the pay-off function for the uncollided particles is: fx.(r,E) = o- -(ro,E)T(r-->rolE) If we assume that the source is isotropic and uniformly distributed in a spatial region v, then 1 q(E), v,47

if

0,

otherwise

r

E V,

Q(r,E) =

Further, assume that q(E) is a PDF that is

f

q(E) dE = 1.

By substituting Equations (4.81) and (4.31) into Equation (4.76):

(4.81)

118

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

R

= ro —

2 dr = —R dR dko, r' = ro _ Wt..)

FIGURE 4.4.

Ro =

dr idEexp [

Jr ro

cr(r' ,E)ds]

1 dr =— — idE q(E) exp [ — 47 v, v,

8 ( ro — r —

)

1ro — r12

cr(r' ,E) ds] r I ro

1

1 q(E) v,47r

r12 q(E)

(4.82)

Thus if i = 1, 2, . . . ,n points with initial coordinates ro, and energy E1,o are selected then Ro is estimated by

—

1 v n n • 47r

exp—J [

o-(r' E, )ds]

Iro —

A much more exciting result can be achieved by several elementary modifications of Equation (4.82). Let us first note that by the definition in Equation (4.31) the relation T(r—>rolE,w) o(ro,E)

T(ro—>rIE, — to) o(r,E)

(4.83)

holds. Thus, Equation (4.76) can be reformulated as Ro = f fdr dE V,E) T(ro—>r1E, — to) Q(r,E) 8

=

(to r — ro

111.0 dr f dE exp [J o-(r' ,E)ds] IIr v47r 1..12

Since the source is isotropic and uniformly distributed over the region v„ let us transform the transition kernel into its one dimensional form but now define the w vector with an orientation opposite from that used earlier, (see Figure 4.4). With this notation the flux at

119 ro reads R =

f RI vs4'rr

R2

— dRidE exp [ — f °o-(R ') dR'] q(E)

(4.84)

where R2 and R, are the coordinates of the points where the line drawn from r to r0 enters and leaves the source region, respectively. By elementary transformations and introducing R" = R — R', the final form of Equation (4.84), i.e., the uncollided contribution is R2 1 ftho Ro -= — — 4m. IdE q(E) f dR exp [ — oo-(R") dR"] vs R,

where the integrations from R, and R2 and 0 to R follow the orientation of a line drawn from ro to r. Thus, if i = 1, 2, . . . ,n points at ro are selected with direction w and energy E, then Ro is estimated by: 1

R

'.2

Ro = — dR exp [ n vs R.

0

o-(R") dR"]

(4.85)

Comparing Equations (4.85) and (4.64) one observes the peculiar result that the contribution of the source particles to the flux-at-a-point can be estimated by summing up the expected length of paths in the source region. In this way the difficulties originating from the R -2 factor are eliminated. Such an interchange of the source and target positions for the total reaction rate will be achieved by the introduction of the adjoint Monte Carlo method — described in Chapter 4.VII. A systematic treatment flux-at-a-point estimation, based on the moment equations, will be given in Chapter 6.IV.

VI. THREE SPECIAL PROBLEMS In the previous Chapters of this Part the Monte Carlo treatment of particle transport was approached via the solution of integral equations. This approach is more elegant — or, at least, uses more complicated mathematics — than the heuristic interpretation, however most of the final results, the recipes given for actual computations remained quite the same as those that we were able to derive in Chapter 3. With the next three examples we should like to demonstrate the power of the tools we gained by the introduction of the integral equation formalism. These three special problems will be investigated more rigorously in the following Chapters, however, we hope that the presentation given here outlines the basic principles and, moreover, gives practical guidance for the users. A. PATH STRETCHING (EXPONENTIAL TRANSFORMATION) Calculation of deep penetration is one of the most challenging tasks of Monte Carlo. The problem arises from the physical fact that, the thicker the region is, the lower the probability becomes that a particle traverses it. In an analog game only a very small fraction of incident particles will fly across a thick layer. Thus the number of simulations required to obtain statistically correct results may be so high as to preclude the use of the method. This problem was first met in this book in Chapter 3. In Section 3.II.D. counting of

120

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

the real transmission events was replaced by summing up the transmission probabilities, however, it was mentioned already there that this scoring technique helps only in a limited region and constitutes no solution for really deep penetrations. A more promising method is path stretching (or exponential biasing, or track lengths biasing). The basic idea is the modification of the transport kernel in such a way as to move the particles in the preferred direction by artificially reducing the cross-section in the preferred direction and increasing it in the opposite direction. It was mentioned in Section 4.III.D. that the distortion of the kernel(s) is one of the possible variance reducing methods. The transition kernel was modified in Section 4.IV.E. where the introduction of the new T/T* kernel was a necessity for normalization. Here, a new kernel T is introduced for the purpose of variance reduction. It follows from Theorem 4.10 that the estimates based on the game with the modified kernel are unbiased if the statistical weights of the particles are multiplied by T/T after every selection. (Since we do not modify the collision kernel, K/K = T/T.) In the following derivation the one-dimensional form of T given in Equation (4.32) as

T(R) = cr(R ,E) exp [ — I o-(R') dR'

(4.86)

is used. Let us now introduce the modified cross-section as 6- as &(R,E) = (1 — pp)o-(R,E)

(4.87)

where p, is the cosine of the angle between the preferred direction and the direction of flight of the particle, and p is the biasing parameter. The principle of path stretching is clearly reflected in Equation (4.87). If p 0 the cross-section & has the minimum value of = (1 — p)o-

if p, = 1

= (1 + p)cr

if µ= — 1

and the maximum

To keep the cross-section positive, we restrict the biasing parameter to p < 1. Let us now define the new transition kernel, through use of the modified cross-section as =

Csr(R,E)exp[ — 6-(R',E)dR]

(4.88)

The path length R is now to be selected from T which has the same form as T, except that the cross-section is changed. From Equations (4.86) — (4.88) the statistical weight factor is

T

exp -[

cr(R',E)dR'] 1 — pp,

(4.89)

121 If cr is constant in the region, then the weight factor becomes: T —

exp( — lap crR) 1 —pp.

In cases where the new transition kernel given in Equation (4.88) is not normalized the same approach is followed as in the case of the analog sampling. One can normalize the kernel and apply another weight factor, or simulate leakage. The application of weight factors described in Equation (4.89) can lead to large dispersion in the statistical weight, the factor may become very large if the particles fly opposite to the preferred direction, i.e., if (1 — pp.) is very small. This weight fluctuation limits the efficient use of path stretching. The proper selection of the biasing parameter p is of crucial importance. Thompson et al. 26 warns that "it is generally chosen too high — especially by novice users". Chapter 7.111. of this book deals with optimization of path stretching. We have seen already in Section 4.III.D. that importance sampling also leads to distortion of the kernel(s). It can be proved that special importance sampling called exponential transformation leads to the same cross-section change as path stretching. z< In a one dimensional interpretation, if a thick semi-infinite slab extends from 0 T and the source is near z = 0, then the flux of particles falls approximately exponentionally in the positive z direction, thus the importance function V(z) = eaz

(4.90)

where a is a constant, is a good selection. It can be shown after some mathematical manipulations of the one-dimensional form of the collision density equation that the introduction of this importance function (i.e., the use of exponential transformation) leads to the same distortion of the game as path stretching, if a = po-. A sophisticated study of the relation between exponential transformation and path stretching is presented in Section 5.V.D. and E. Here, we should like to call the attention of the readers to the fact that in many papers the application of path stretching is mentioned as an implementation of exponential transformations. Finally, it is worth noting that exponential transformation was the first elaborated importance sampling device. The idea was suggested by Kahn," the actual one dimensional procedures were developed by Leimclorfer." B. PERTURBATION MONTE CARLO Small changes in a system are called perturbations. The question of how large a change in a reaction rate is caused by a perturbation of a system parameter, is raised mainly for two reasons, viz. if one wishes to know the uncertainty of a result vs. the uncertainty of an input parameter, or if one studies the effects of changes made during operating maneuvers. In a straightforward perturbation computation two independent runs are made and the difference of the two results calculated. This type of solution is, however, very time consuming since, for a reasonably small uncertainty in a difference, an extremely low statistical error must be achieved in the two independent runs carried out for the unperturbed and perturbed systems, respectively. Two more efficient methods are outlined below, viz. correlated sampling and differential Monte Carlo. The interpretations presented here are based on a paper of Rief.' The two techniques are discussed in general in Chapters 6.1. and 6.11., respectively.

122

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

1. Correlated Sampling As it was already seen in Section 4.I.C. for integrals, correlated sampling can efficiently reduce the error in differences. In particle transport the technique is applied in such a manner that the same set of particle tracks is used to analyse two problems simultaneously. Let us denote the kernel of the unperturbed problem by K(P',P) and that of the perturbed problem by K(P',P). If the unperturbed problem is solved by an analog game, then from the point of view of the perturbed problem its kernel K is replaced by the "distorted" kernel K and thus a weight factor of

w=

k(P1-1,13)

(4.91)

K(P; -1, P1)

is to be applied whenever the coordinate P changes from P,_, to P. (Equation (4.91) follows from Theorem 4.10.) If the pay-off function is f(P) then the contributions of the i-th event are r, = f(139 and = f(131 )

n we e =1

for the unperturbed and perturbed games, respectively. The difference in the scores for a total history reads:

= 13,

- µ = i=i E e=i w, -

1) f(P)

After following j = 1,2, . . . ,n histories each scoring (At.L),, the average

-n;=1 E (AV); is an unbiased estimate of the reaction rate difference. Let us illustrate the derivation of the w, factors defined in Equation (4.91) in a simple example. The task is the calculation of the effect of a small change in the density of a medium. The density p is changed to p + Ap. As was seen in Chapter 2.11., all macroscopic cross-sections are proportional to the density. Thus the collision kernel, being a ratio of scattering to total cross-sections (see Equation [4.331) does not change with the density. Consequently the transport kernel ratio _

K CT — T reduces to the ratio of the transition kernels. Using the one-dimensional form given in Equation (4.32) and assuming a homogeneous medium: T(ri „r,) = o- exp( crR,)

123 and t(r,_,,r,) = (1 + ) crexpE — (1 + if R1

o-R

P

1r1

The weight factor, i.e., the ratio of the two kernels is T — 1 + 12exp — wi = — A

)

If zp E1r) where the fission collision kernel

Cf

is defined as

E vio-,f(r;E'—>E) =f

o-(r,E') Here, v, denotes the expected number of outcoming neutrons per fission and differential fission cross-section for element i. After integrating both sides of Equation (4.95) one gets

is the

JdP fdP' xf(P')Kf(P' ,P) (4.96)

keff = JdP xf(P)

The eigen value Equation (4.95) is typically solved by iteration. A guess function xf(P) is selected as the neutron source and the resulting progeny construct the source for the next generation. If the source distribution converges after a sufficient number of generations (we have the eigenfunction), k, can be calculated from Equation (4.96). A good guess function can be found, for example, from approximate calculations. Less computer time is required by replacing the Monte Carlo calculation with a fission matrix iteration.4 The volume is divided into a number of contiguous cells and the matrix element a, equals the number of first generation fission neutrons produced in cell i from one fission neutron starting in cell j. The numerical iteration using the matrix with an initial source

126

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

vector is very fast. Here, the a,, matrix elements can be computed in a one generation Monte Carlo calculation. Let us now investigate the case where a small perturbation changes the fission neutron collision density to xT, the fission kernel to KT(P',P) and the effective multiplication factor to lc*e,24.

k 4:ff —

f dPidPixT(P')10(P' ,P) f

(13)

dP

If the differences are small we may approximate x'; as XT(P) = Xf(P) + 8Xf113) and

1

1

f

1

f dP8x,(P)1 f dP xf(P)

dP xT(P) f dPx,(P)

By these approximation the difference between the two multiplication factors is: Ek„ =

1

— k,, — f

ifdP dP' x,(P)[KT(P' ,P) — Kf(P' ,P)]

dP VP)

+ f f dP dP ' 8x,(P)K,(P' ,P)

f

dP 8x1(P)

f

fidP dPx,(13)

b(f(P) + 8Xf(P)] (P' ,P)} (4.97)

If the second and third terms of Equation (4.97) may be neglected (8x < x) then the equation reduced to

Ske,

Jf dPdP' xf(P' ,P) kf(P' ,P) f dP xf(P)

(4.98)

where IZ,(P',P) = KT(P',P) — K(P',P). In this approximation Ske„ can be obtained in parallel with k, in a correlated Monte Carlo, since only the transport kernels of Equation (4.96) and (4.98) differ from each other. Methods to evaluate Equation (4.97) with all terms are reviewed by Bemnat. 3 Criticality is discussed in detail in Chapter 6.111.

VII. ADJOINT MONTE CARLO Investigation of the adjoint transport equations plays an important role in modem Monte Carlo research. Equations adjoint to the transport (collision density) equations can be intro-

127 duced and discussed in various ways. Basically, one may consider the adjoint to either the integral or the integro-differential transport equation. Some confusion may arise if one does not take into account that conversion from integro-differential to integral forms and conversion to adjoint do not commutate." The adjoint to an equation can be found purely mathematically. Instead, we approach the problem from a rather physical point of view. First, the value of the particle is defined and then equations describing this value are set up. In this approach it is quite an accidental discovery that the value equation is adjoint to the collision density equation. This presentation is based on the paper of Irving." A most comprehensive treatise of the adjoint equations is given by Lewins.18 The name value already indicates that this quantity characterizes the value — or importance of the particles. Thus — as we shall demonstrate — the value is a good choice for importance function in the Monte Carlo solution of the collision density equations. Beside its role in the efficient solution of the direct transport equations, the adjoint formalism has its own merits. The Monte Carlo solution of the adjoint equations itself provides a new way to estimate reaction rate type physical quantities in radiation transport. In this alternative method the roles of the source and the receptor are interchanged. Consequently, the solution of the adjoint equation is preferred if: • •

The same reaction rate in the same geometry is to be calculated with various sources, and if; The phase-space volume of the receptor is small, i.e., the efficiency of estimations based on the direct (collision density) simulation is poor.

To illustrate the importance of the second problem let us remind the readers of the problems of estimating the flux-at-a-point (Section 4.V.D., third example). A. THE VALUE EQUATIONS Similarly to the duality of collision densities defined in Section 4.IV.A. let us now introduce a pair of values. (The name "importance" is alternatively used in several papers, however "importance function" was defined differently in Section 4.III.D. of this book.) Let ij*(r,E) be the value of a particle when it enters a collision at point r with energy and direction E. This value is the sum of two terms, the immediate pay-off f41(r,E) and the pay-off that may be expected to result from all future collisions. In the latter term the products of the values at other points of the phase-space and the probabilities of having collisions there are integrated over the whole phase-space. Thus, with the kernels defined in Section 4.IV.B., the value of the ingoing particles can be written as: 41* (r,E) = fo,(r,E) + f f dE' dr' C(E—>E' 1r)T(r—*OE')41*(r1 ,E')

(4.99)

The counterpart of tli*(r,E), the value of a particle leaving a collision at r with energy and direction E can be expressed by the equation: X*(r,E) = fx(r ,E) + 1de dE' T(r—>e1E)

le)x*(r',E')

(4.100)

Equations (4.99) and (4.100) are similar to Equations (4.36) and (4.37), respectively. One of the differences is that the physical source terms of the collision density equations are replaced by the pay-off functions in the value equations. The second difference is that the kernels in Equation (4.99) and (4.100) are transpositions of the kernels in Equations

128

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

(4.36) and (4.37), respectively. Consequently, Equations (4.99) and (4.100) are adjoints of Equations (4.36) and (4.37). For this reason, Monte Carlo games played according to Equation (4.99) and (4.100) are generally called "adjoint calculations". (Later in this Section we shall modify the value equations, however, the games built on the modified equations still retain the name of "adjoint calculations".) Based on the similarity between the collision density and value equations one can imagine that the latter equations also describe collision densities — for some kind of imaginary particles: pseudo-particles. These curious particles start their histories from the receptor, since the source terms in Equations (4.99) and (4.100) are the pay-off functions of the original physical problem. Moreover, the change from E' --> E to E —> E' in the collision kernel means that the energies of the pseudo-particles increase at every collision (or rather pseudo-collision). The replacement of r' —> r in the transport kernel of the collision equations to r —> r' in the value equations can be interpreted to mean that the pseudo-particles fly in the direction opposite to their direction vector. (An analysis of the definitions of the collision and transport equations given in Equations (4.31) and (4.33), respectively, proves the validity of the above statements.) Referring to the adjoint equations, the pseudo-particles are called adjunctions in several papers. The connection between the two value functions is clear if we rearrange Equation (4.99) and (4.100) as*: 41* (r,E) = fq,(r,E) +f

C(E—>E'lr)x*(r,E')

(4.101)

and X*(r,E) = ide T(r—>r' 1E)

(4.102)

Nothing, in principle, prevents us from a solution of the value equations by a similar step-by-step solution as described in Chapter 4.II.D. for the collision densities. However, a more attractive Monte Carlo treatment can be constructed if we modify the value functions via the following transformations: kji(r,E) = cr(r,E)ili*(r, —E)

(4.103)

Vr,E) = cr(r,E)x*(r,

(4.104)

and

where —E is the shorthand notation for (— After elementary manipulations and taking into account Equation (4.83), it can be proved that the modified value functions satisfy the following pair of equations:

LT, (r,E) = 14,(r,E) +

C(E—>E'lr)Vr,E')

.=

fx + fdr'Tlt* and "kV* = There is another mathematically possible expansion leading to x However, such a solution implies that f,i, = SdE'Cfx, and the use of such pay-offs would result in reaction rates differing from each other if in the scoring Equation (4.49) and (4.50) were used, respectively. The expansion given in Equations (4.101) and (4.102) implies the relation given in Equation (4.53), i.e., the condition required on physical grounds is fulfilled.

129 and j((r,E) = f dr' T(e--->rIE)41(r' ,E)

(4.105)

with the following further notation: f 4,(r,E) = o-(r,E)f y(r,—E)

(4.106)

and C(E—>E1r) = C(E--->E1r)

o-(r,E) o-(r,E')

(4.107)

The transition kernel in the modified value Equation (4.105) is now identical with the "original" one in the collision density Equation (4.35). Taking into account the definition of the collision kernel given in Equation (4.33), the new kernel in Equation (4.107) can be expressed in terms of the cross sections as n m

EE

C(E—>E'lr) — 1 =1 ' 1

vij o-ii(r;E—>E') r(r,E')

(4.108)

B. SOLUTION OF THE VALUE EQUATIONS (ADJOINT MONTE CARLO) The comment in the previous section that the value equations can be solved by Monte Carlo in the same way as the collision equations is per se valid for the modified value equations, too. The modified values can be expanded into Neumann series, and the corresponding recurrence is given by the following equations: d0(r,E) =

E)

jZo(r ,E) = fdr' T(r'—*rIE) LIJo(r' ,E)

(4.109)

for the zeroth terms and tfl,,(r,E) =

C(E—>E' Ir) jiSr,E')

f dr' T(r'—>rIE)41(r' ,E) for i + 1 = 1, 2, . . . 'GC. The Monte Carlo realization is similar to that given for the collision densities in Section 4.IV.D.: 1. Set i = 0. Select initial coordinates (r„,E,,) from 1.4,(r,E) fidr dE t4,(r,E)

130

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations and set initial weight to = if dr dE

2. Select next collision site r;+ , from T(r,—>r1E.)

J Jdr T(r,--*r1E,) and multiply the initial weight by wi,T = f fdr T(r,—>rIE;) 3. Select the next energy E1 ,1 and direction from

+ 1)

idE and multiply the weight by C(Ei+1—*Eilr,+ C(Ei+,—>K1r,+

i)

f dE 'e(E—>Eilr,+1)

4. Set i = i + 1 and return to Step 2. Note that here in the adjoint game the value d is represented by the "source" coordinates thus (I( is the counterpart of x and X is the counterpart of tfr. In other words, the value of a particle leaving a collision is the density of pseudo-particles entering a pseudo-collision, and vice versa. Two steps of this cycle are further discussed in the following. First, we analyze the new source term, the pay-off function. In Section D. an explanation is given as to why a distorted kernel (0 in Step 3 replaced the adjoint collision kernel C. C. SAMPLING THE ADJOINT SOURCE As has already been seen in Section A. of this Chapter, in the adjoint game the payoff function plays the role of the source. On the other hand the pay-off describes the quantity to be determined, or simply the characteristics of the receptor, if we use this word in a general sense. Thus, with a slightly loose terminology one can say that the physical receptor is the adjoint source. Several examples of pay-off functions are given in Section 4.V.C. Let us first remind the reader of the problems of estimation of the flux-at-a point. The pay-off function given in Equation (4.71) could not be used as a score in the direct simulation, however, is an ideal function for selecting the initial coordinate in the adjoint game. The relation i4,(r,E) = 8(r — ro) means that all particles start from r = ro, that is the adjoint source is a point source.

131 The flux-at-a point pay-off given in Equation (4.80) was obtained via a relatively long derivation and could be used in the direct simulation — at least if the point of interest was not imbedded in media having non-zero cross sections. At the same time this pay-off formula is useless for the adjoint simulation, since its sampling is complicated. There may be problems where it is easier to determine the pay-off of the outgoing particles. If the form of fx (r,E) is more convenient for adjoint source sampling one can start the simulation cycle from it. Taking into account Equation (4.53) and the algorithm given in Section B. such a simulation leads to the omission of the very first step. This means that there will be no points representing the adjoint source particles, but the full sequence of all further movements of the pseudo-particles is numerically simulated. A possible substitute for the missing adjoint source term is given in Section F. D. THE COLLISION KERNEL OF THE VALUE EQUATION In the definition of the collision kernel given in Equation (4.33) both the numerator and the denominator have their physical meaning. If we integrate C(E'—>Elr) over all possible energies after the collision: C* = idE C(E'—>E1r) the result C* is the expected number of particles leaving a collision at the site r. If there is no multiplicative event, 0* is the non-absorption probability, i.e., less than unity. For non-multiplicative, nonabsorbing materials C* = 1. If multiplicative interactions and events leading to absorption of the incident particle without emission of secondaries of the same type can take place in the same medium, C* may be either smaller or greater than unity. However, it follows from the physics of the interactions that C* cannot be substantially greater than two. Thus, the statistical weight corrections applied in parallel with the normalization of the collision kernel (see Section 4.IV.E.2.) do not cause extremely large weight fluctuations. This is not the case with the adjoint collision kernel. First of all, it should be emphasized that the integral C* = idEC(E—>E'lr)

(4.110)

that should be used for the normalization has no physical meaning. It follows from Equation (4.108) that the evaluation of the integral given in Equation (4.110) means the calculation of terms of the form fdE o-(r;E—>E')

(4.111)

and such integrals may have extremely high values. Consequently, very large fluctuations may occur in the statistical weight and thus in the score contributions. There are, in fact, cases when the integrals given by Equation (4.111) are even divergent. Let us show this in two examples. First, consider the Klein-Nishina formula describing the Compton-scattering of photons.

132

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations The differential cross section given in Equation (3.9) can be rewritten in terms of energy

as o-(E—>E') = — Ek2 [— EE' + — EE, + (1 + —

— 1]

(4.112)

where E is the electron rest mass energy (mec2 — 0.511 MeV), the value of the constant k is not relevant at the moment. From the energy ratio relation given in Equation (3.8), the limiting values of E (the energy of a real photon before a real collision which is the energy of a pseudo-photon after a pseudo collision) are E'

E' E

E
E')dE' relation and Equation (4.113) the differential cross-section of the isotropic elastic scattering becomes: 6(E—>E') = k

1 E(1 — a2)

where k is again a constant of no importance from the point of energy selection. The limits of integration in the calculation of Equation (4.111) are derived now from Equation (3.15) and the result of the integration itself is:

JE E(1

a2) dE =

k 1—

a2

2 1 k 1 log — dE = E 1—a a2

(4.114)

133 If the scatterer atom is hydrogen, then A = 1, a = 0, i.e., the integral given in Equation (4.114) is again logarithmically divergent. These two examples clearly demonstrated that the factor C* introduced by Equation (4.110) for the normalization of the adjoint collision kernel may become infinite both in photon and in neutron transport calculations. The methods of solution are basically the same for the two types of particles. The first method is based on the fact that there is always an upper limit for energies of interest. There is no use in following the history of any pseudo-particle whose energy exceeds that of the source (the real, physical source). Let us denote the maximum source energy by EM, then the new energy E can be selected from the distorted kernel a(E—>E'),

if

E --.. Em

0,

if

E > EM

C(E—*E') =

Now, for the normalization the integral becomes: EM j.dE a(E—>E')

=

dEC(E—E')

and this integral is convergent for all finite Em's. A disadvantage of this method is that the form of the new collision kernel is problem dependent. A change of the source leads to change of the kernel. A better approach is the distortion of the collision kernel in the whole energy range — independently of the highest source energy. If the distored kernels are defined as a(E—>E') = E C(E—E') then the normalization factor of C* = JdEC(E—E') is bounded. Proofs are given in the literature both for photons' and for neutrons.' It follows from Theorem 4.10 that if the E'/E distortion is applied, the weight has to be multiplied by E/E' (E > E') after every pseudo-collision. At first sight it may seem that now the weight factor may become unbounded. This problem is solved again by the practical limitation that no pseudo particles collided to above EM are followed further. Several techniques developed for sampling the adjoint collision kernels are described and compared in two papers of DeMatteis and Simonini.6'7 for neutron and photon transport, respectively. Finally, we should like to call the attention of the reader to the fact that the values of integrals like that given in Equation (4.110) may depend on the choice of units selected. (This statement is not a physical nonsense, since the quantity C* has no physical meaning.) Kalos" has published a paper on adjoint photon transport where the Compton wavelength (X) is used rather than the energy. The relation between the energy and the wavelength is (4.115)

134

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations The collision kernel can be expressed in terms of wavelengths from the condition C(X—>X')dX' = C(E—>E')dE'

(4 . 116)

since these two quantities describe the same physical probabilities. From Equations (4.115) and (4.116) the collision kernel is C(X—>X') = —

(E )2

C(E—>E')

and now the integral

IdX

(X—>x ')

is convergent. It can be proved'6 that this transformation leads to the same sampling as the application of a distortion factor (E'/E)2 in the energy representation. Hoogenboom9 described the energy of the neutrons in lethargy (u) units: E0 u = log — E where E0 is a reference energy. In this presentation the relation dE du = — "automatically" introduces the

' distortion and thus

Jdu C(u—*u') is convergent. E. SCORING IN THE ADJOINT MONTE CARLO In the way described in the previous sections sequences of points in phase-space are generated and the densities of these points represent the values of particles entering or leaving collisions at these points. The knowledge of these densities is the aim of the adjoint treatment if one wishes to use the value as importance function in the collision density simulation. However, if one uses the adjoint simulation solely, then relation between the value functions and the physical quantity (reaction rate) to be calculated must be found. 1. Reaction Rates Theorem 4.11 — Let Equation (4.36) describe the outcoming collision density of certain particles and Equation (4.100) describe the value of particles leaving collisions. Then the reaction rate defined by Equation (4.49) can be expressed in terms of the value of the particles leaving the collision as R = f fdr dEx*(r,E)Q(r,E)

(4.117)

135 Proof. Multiply Equation (4.36) by X*(r,E) and Equation (4.100) by x(r,E) and integrate all terms over r and E. From comparison of the two equations one concludes: ff dr dE x(r ,E) Px(r,E) = f fdr dE x*(r ,E) Q(r ,E) The left hand side integral is the definition of the reaction rate, as given in Equation (4.49)

The new expression of the reaction rate is extremely plausible. If one integrates the values (i.e., the sums of the present and all future pay-offs) for all source particles, the result is the total pay-off to the reaction rate. Theorem 4.12 — Let Equations (4.37) and (4.99) describe the density and the values of particles entering collisions. Then the reaction rate defined by Equation (4.50) can be expressed as: R = ifdrdEti,*(r,E)Qc(r,E)

(4.118)

The proof is analog to that of Theorem 4.11. The equality of the two integral formulae given in Equations (4.117) and (4.118), respectively, follows from Equations (4.38) and (4.102). It follows from definintions of Equations (4.103) and (4.104) that the reaction rates expressed in terms of the modified value functions have the forms of R = if dr dE X(r,E)

Q(r, —E) o(r,E)

(4.119)

R = f f dr dE tir(r,E)

Q(r—E) cr(r,E)

(4.120)

and

2. Estimation of Responses For More Than One Source The possibility of estimating more than one response from the same direct random walk simulation was outlined in Chapter 4.V.B. In the adjoint game the roles of the source and the receptor are interchanged. Thus, if the quantity of interest is to be determined with several varying source configurations, a single set of random walk simulations can be used to compute the series of responses IV') = f fdr dEX(r,E)

Q(`)(r, —E) 6(r,E)

or

= fidr dE tkr,E)

6(r,E)

if Q(') and Qc(') denote the set of sources and first-collision sources, respectively. The values R(') are not statistically independent of each other, since all the estimates are based on the same representation of the values.

136

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

3. Monte Carlo Estimation of the Responses In a fully analog way as described in Section 4.V.C., the responses can be expanded into Neumann series as R=

i=0

12; =

=0

fidrdE

Q(r,—E) jar,E) o-(r,E)

if

Qe(r,— E) f4r,E) ff(r,E)

and R=

= i=0

dr dE

(4.121)

The contributions of the pseudo-collisions can be expressed in the following way: If points (r„E„W,,,) represent the 4, modified value distribution, then r-=W

Q (r ,—E) er(r„E)

(4.122)

If points (r1 ,„Ei,W,,,) represent the ki modified value distribution then r1 = W

'

Q(r 1 ,—Ei) ' o-(ri,E;)

(4.123)

The resultant weights are defined as in the direct simulation: Wi.c =

k=0

wk4

k=O

wk,T

and WI,T

= 1I Wk,074,T k =0

The score from a total history is again the sum of the contributions, analogs of Equations (4.54) and (4.55) are to be used. The division by the cross section in Equations (4.122) and (4.123) is not a source of possible divergencies here if there is a void in the system investigated since no collision takes place in vacuum. Analognus to the problems of the flux-at-a-point estimation in the direct simulation, the adjoint contribution cannot be easily calculated if the source is point-like. Thus, as a rule of thumb, we can state that direct simulation is effective for geometries with detectors extended in phase space and adjoint simulation can work well if the source region is large. Neither of the two types of games is promising if both the source and the receptor are point-like. In the example of the flux integral Chapter 4.V.D. we have demonstrated that the use of the two collision densities may lead to two types of estimators: one based on the event densities (see Equation [4.58]), the other based on the expected path lengths. It follows that, according to Equation (4.65) the sum of either the expected or actual track lengths in the recpetor volume is an unbiased estimator of the flux integral.

137 Let us assume now that the source is uniformly distributed over a certain spatial region v, (see Equation [4.811) 1 q(E), vs47r

if re

0,

otherwise

v,

Q(r, — E) =

Then the reaction rate given in Equation (4.118) becomes: R=

vs4IT

f dE q(E) dr j((r,E) vs

'(r,E)

(4.124)

Substituting Equation (4.105) into Equation (4.124) one gets: R=

1 IdEq(E) dr I(r,E) = ide tp(r',E)T(r'—>r1E) v, vs47r

Now by interchanging the integrations over r and r' and then the two symbols themselves the reaction rate becomes: R=

1 if dr dE kffr(r,E) q(E) f dr' o- '(r' ,E) T(r—>r '1E) vs4ir

(4.125)

The last integral in Equation (4.125) is the expected path length of a particle starting to fly from r with a direction and energy E. (A proof can be constructed via transformation of the transition integral into one dimensional form, in the same way as for the derivation of Equation [4.851.) Thus, if points representing the 11J modified value are obtained during the simulation, the contributions to the score are products of the source intensity (on the energy selected) and the expected path length of the pseudo-particles traversed in the source region. A similar result was derived for the flux-at-a-point of the uncollided real particles — at the very end of Chapter 4.V. In mathematical terms, if i = 1, 2, . . . ,n points (r„E,W,) represent the modified value 4 then the reaction rate can be estimated by R—

1 q(E,)(f,) nv,47r i = 1

E

where is the expected path in the source region: R2 (C) =

dr' r '(r1 ,E,) T(r,—>r'1E,) = vs

if R, and

R2

R,

dR exp [ — u(R') dR'

are the points where the pseudo-particle enters and leaves the source region.

F. CONTRIBUTIONS OF THE UNCOLLIDED PARTICLES A separate estimate of the contribution of uncollided particles (i.e., particles which have their first interaction in the detector volume) may be important, for example, in gamma

138

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

spectrometry if one wishes to compare the calculated results with measured full-peak efficiencies. In the Monte Carlo treatment, we gave estimation formulae separately for the uncollided and already scattered particles in the flux-at-a-point example (Section 4. V.D. , third example). Later, in the discussion of the sampling of the adjoint source the possibility of playing an adjoint game without estimation of the uncollided pseudo-particle contribution was mentioned. In this Section, we shall prove, that though the collision density simulation and the adjoint game lead to basically different sets of collision (and pseudo-collision) densities, the contributions of the uncollided particles in the direct game and those of the uncollided pseudo-particles in the adjoint game are equal to each other. Theorem 4.13 — Let us define the contribution of the uncollided particles to the rate — as a consequence of Equations (4.39) and (4.49) — by Ro = fidrdEQ(r,E)fx(r,E) and the contribution of the uncollided pseudo-particles — as follows from Equations (4.109) and (4.121) — by offdr

dE f q,(r,E) Qc(r' cr(r,E)

(4.126)

Then the two contributions are equal to each other, that is: Ro = Ro Proof. Let us substitute Equation (4.106) into Equation (4.126) and replace — E by E: Ro = ffdr dE fq,(r,E)Qc(r,E) After substituting the defining Equation (4.38) of the first collision source, then changing the order of integrations over r' and r and finally changing the symbols r' and r, one gets: Ro = fidr dE Q(r,E) fdr' T(r—>elE)4(r',E)

(4.127)

From the relation of the two pay-offs given in Equation (4.53), Equation (4.127) reduces to: Ro = f fdr dEQ(r,E)fx(r,E)

A consequence of this theorem is that if the adjoint game is played without the adjoint source simulation, source contribution computed by direct calculation can be added to the contribution from the scattered pseudo-particles.

VIII. VARIANCES The formula given by Equation (3.38) is of general validity for computation of the

139 empirical variance. It can be used in any Monte Carlo game resulting in I.L„ µ2, . . . µn individual scores from n simulations. Thus, the introduction of the integral equation formalism does not influence the simple straightforward a posteriori estimation of the statistical uncertainty. There is, however, an important benefit of the use of integral equations in the field of variance analysis. Equations can be derived, by the use of which efficiencies of techniques can be estimated a priori. Though the solution of these equations is at least as complicate as that of the collision density equations, even an approximate estimation of the variances may clearly indicate whether a certain nonanalog procedure increases the efficiency of the game, or not. Variance analysis is based on the moment equations. Since these equations are investigated in detail in Chapter 5, only a short introduction is given below. A. VARIANCE ESTIMATES BY THE MOMENT EQUATIONS The variance of a certain random variable is defined as the difference between the expected value of the square of the random sample and the square of the expected value. If the random variable in our Monte Carlo game is the score µ, then the variance is:

1)2(1) =

- (02

In particle transport, if Q(r,E)dr dE particles start from the phase-space element dr dE about (r,E) and the score depending on the starting point is denoted by µ(r,E) then the total score, i.e., the reaction rate, is (µ) = R = fidrdEQ(r,E)(µ(r,E)) where parentheses denote the expectation over all possible histories started from (r,E). If M, (r,E) denotes the first moment of the score, i.e., the expected score due to a particle started from P then M,(r,E) = (µ(r,E)) and the reaction rate reads: R = fidrdEQ(r,E)M,(r,E)

(4.128)

It is obvious from the comparison of either the definitions, or, formally, Equations (4.128) and (4.100) that the first moment is equivalent to the value of the particles leaving collisions: Mi(r,E) X*(r,E) Consequently, the integral equation of the first moment is (see Equation 14.1001): Mi(r ,E) = fx(r,E)11 dr' dE' T(r—>r '1E) C(E—>E' ) M (r ' ,E')

(4.129)

The reader should be reminded, here, of our comment at the end of Chapter 4.III.C. that in most papers the collision density of particles entering a collision is used solely. Similarly, in most treatments the value of the ingoing particles (40) is the only adjoint

140

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

density introduced. Consequently, the first moment is related to the ingoing value as: M,(r ,E) = fdr' T(r—>elE)1)*(e ,E) Only this treatment is followed in the remainder of this book, too. In consequence the pay-off f,4, is defined uniquely and the "source term" of Equation (4.129) is replaced (see e.g., References 1 and 5) by

f

dr' T(r—>r'lE)

,E)

according to Equation (4.53). Thus, the first moment in the ingoing collision reads: M,(r,E) = f Ide T(r-->elE) f4,(e ,E) + f ide dE' T(r—>r'1E)C(E--->E' le M, (e ,E')

(4.130)

The expected value of the second moment is defined by the relation: (112) = f dr dE Q(r,E)M2(r,E)

(4.131)

In the RHS of Equation (4.129) the first term describes the immediate pay-off, the second one is the contribution of the future collisions. Heuristically, the square of this sum is the sum of the squares of the two terms plus the twice of their product. Consequently, the second moment M2(r,E) in Equation (4.131) is the solution of the following integral equation: M2(r,E) = f)2,(r,E) + 2fx(r,E) f Ide dE' T(r—>elE)C(E—>E' le) M,(r' ,E') + fidr dE' T(r—>r '1E) C(E—>E' le) M2(e ,E')

(4.132)

A score is received when a particle enters a collision rather than it leaves, and the immediate pay-off is 4. The two first moments given in Equation (4.129) and (4.130) are identical to each other which means that neither of the two estimators gives biased results. The second moments are, however, different from each other. With the same philosophy as applied for the introduction of Equation (4.132), the second moment of the estimator based on the ingoing density M42' reads: fs/q(r ,E) = f dr' T(r—>elE)

,E)

+ 211 dr dE' T(r-4e1E)4(e ,E)C(E—>E1 1e)M,(e ,E') + f f dr' dE' T(r—elE) C(E—>E' le) Mt.`(e ,E1) This formula is given in Chapter 5.

(4.133)

141 The difference between Equations (4.132) and (4.133) can be explained in the following way: if scoring is undertaken when a particle leaves a collision, then the contribution of the free-flight up to the next collision site is calculated from the expected path, whereas if scoring is executed when the particle enters the collision we already have an actual (selected) path and its contribution. Actual forms of M, and M2 will be given for many special examples in subsequent Chapters. B. THE VALUE USED AS IMPORTANCE FUNCTION Importance sampling was first mentioned in this book in Section 4.III.D. It was stated there that the importance function V(P) should measure the importance of an event at P. Since the value functions ill* and x* are defined specifically to describe the value (or importance), their use for importance function is trivial. It can be shown,' "that an optimal importance function may resemble the value function, but is, in most cases, somewhat different from it". The - generally slight - difference between the optimum importance function and the value function tli*(P) has no real practical influence, since calculation of the value function is as complicated as that of the collision densities. Thus, only approximate values are known, if any. Coveyou et al.' proved that if a particle contributes to the score only if it is absorbed (i.e., the so-called last event estimator is used), then the use of the value function ill* as importance leads to zero variance. Other zero variance solutions are presented by Noack.21 •22

REFERENCES 1. Amster, H. J. and Djomehri, M. J., Prediction of statistical error in Monte Carlo transport calculations, Nucl. Sci Eng., 60, 131 (1976). 2. Berger, M. J., Reflection and transmission of gamma radiation by barriers. 1: Monte Carlo calculation by a collision-density method, J. Res. Nat. Bur., Stand., 55, 343 (1955). 3. Bernnat, W., Abschatzung der Reaktivitat beweglicher Reflektorelemente kompakter Reaktoren mit der Monte Carlo Methode, Ph.D. thesis, University of Stuttgart, Germany (1972). 4. Carter, L. L. and Cashwell, E. D., Particle-Transport Simulation with the Monte Carlo Method. ERDA Critical Review Series, National Technical Information Service, Springfield (1975). 5. Coveyou, R. R., Cain, V. R., and Yost, K. J., Adjoint and importance in Monte Carlo application, Nucl. Sci. Eng., 27, 219 (1967). 6. DeMatteis, A. and Simonini, R., A new Monte Carlo approach to the adjoint Boltzmann equation, Nucl. Sci. Eng., 65, 93 (1978). 7. DeMatteis, A. and Simonini, R., A Monte Carlo biasing scheme for adjoint photon transport, Nucl. Sci. Eng., 67, 309 (1978). 8. Hall, M. C. G., Cross-section adjustment with Monte Carlo sensitivities: application to the winfrith iron benchmark, Nucl. Sci. Eng., 81, 423 (1982). 9. Hoogenboom, J. E., Focus, A Non-Multigroup Adjoint Monte Carlo Code with Improved Variance Reduction. Proc. NEACRP meeting of a Monte Carlo Study Group. ANL-75-2 report. Argonne National Laboratory, Argonne, (1975). 10. Irving, D. C., Freestone, R. M., and Kam, F. B. K., 05R, A General-Purpose Monte Carlo Neutron Transport Code. ORNL-3622 report, Oak Ridge National Laboratory, Oak Ridge (1965). 11. Irving, D. C., The adjoint Boltzmann equation and its simulation by Monte Carlo, Nucl. Eng. Des., 15, 273 (1971). 12. James, F., Monte Carlo Phase Space. CERN 68-15 report, European Organization for Nuclear Research, Geneva (1968). 13. Kahn, H., Modification of the Monte Carlo Method. P-132 report, Rand Corporation, Santa Monica (1949). 14. Kahn, H., Applications of Monte Carlo. AECU-3259 Report, Rand Corporation, Santa Monica (1954). IS. Kalos, M. H., Monte Carlo integration of the adjoint gamma-ray transport equation, Nucl. Sci. Eng., 33, 284 (1968). 16. Koblinger, L., A New Energy Sampling Method for Monte Carlo Simulation of the Adjoint Photon Transport Equation. KFKI-76-57 report. Central Research Institute for Physics, Budapest, (1976).

142

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

17. Leimckirfer, M., A Monte Carlo method for the analysis of gamma radiation transport from distributed sources in ionizated shields, Nukleonik, 6, 58 (1964). 18. Lewins, J., Importance, The Adjoint Function, Pergamon Press, Oxford (1965). 19. Matthes, W., Calculation of reactivity perturbations with the Monte Carlo method, Nucl. Sci. Eng., 47, 234 (1972). 20. McGrath, E. J., and Irving, D. C., Variance Reduction. ORNL-RSIC-38 Report, Techniques for Efficient Monte Carlo Simulation, Vol. III., Oak Ridge National Laboratory, Oak Ridge (1975). 21. Noack, K., On the relation between adjoint and importance in Monte Carlo solutions of linear particle transport problems, Part I, Kernenergie, 22, 346 (1979). 22. Noack, K., On the relation between adjoint and importance in Monte Carlo solutions of linear particle transport problems, Part II, Kernenergie, 23, 372 (1980). 23. Rief, H., Generalized Monte Carlo perturbation algorithms for correlated sampling and a second-order Taylor series approach, Ann. Nucl. Energy, 11, 455 (1984). 24. Schmidt, F. A. R., Status of Monte Carlo development. IKE Bericht No.4-3. Institut filr Kernenergetik, Universitat Stuttgart (1972). (Reprint from "Numerical Reactor Calculations". International Atomic Energy Agency, Vienna, 1972.) 25. Sobol, I. M., The Monte Carlo Calculational Method (in Russian), Nauka, Moscow (1973). 26. Thompson, W. L., Duetsch, 0. L., and Booth, T. E., Deep-Penetration Calculations, in A Review of the Theory and Application of Monte Carlo Methods, ORNL/RSIC-44 report, Oak Ridge National Laboratory, Oak Ridge (1980).

143 Chapter 5

THE MOMENT EQUATIONS We have so far reviewed the main idea behind the transport Monte Carlo methods and the basic techniques of performing a Monte Carlo game. A direct application of these techniques may be sufficient when one intends to build up a simple Monte Carlo procedure for the occasional solution of a simple problem. However, transport Monte Carlo is often used for complicated problems, the solutions of which necessitate extensive programing work, require a large amount of computing time, and usually are to be solved several times with different parameters. For the solution of large-scale Monte Carlo problems, a great number of advanced general- and special-purpose Monte Carlo computer codes have already been developed and, in all probability, will also be elaborated upon in the future. In such cases, when a considerable amount of programing and long running are involved, it is far from immaterial what the efficiency of a program is. In Monte Carlo practice, efficiency is defined as the inverse of the product of the variance and the computing time necessary for estimating the required quantity with the given variance. If k is a Monte Carlo estimate of some quantity X in a run of time t, then the efficiency is E

{D2[t] • t}-'

(5.1)

[In most practical cases, the efficiency in Equation (5.1) is independent of the time — at least for sufficiently large values of t.] Obviously, the empirical variance s can be estimated in the same run which determines t and the computing time can be measured; thus, the estimation of the efficiency a posteriori is straightforward. This estimate may be of interest in comparing the capabilities of existing codes. It is more important, however, to have information on the efficiency of a given method a priori, so that the program designer can chose between several possible methods of solution to a given problem. Obviously, the a priori determination of the variance of some estimate to be calculated will, in general, not be any easier than the determination of the estimate itself (i.e., the solution of the problem at hand) before the simulation. Since Monte Carlo methods are usually applied in cases when all other (analytic and deterministic-numeric) methods fail to work, it is hopeless to seek methods which would give exact variance values a priori). On the other hand, rough but nevertheless reliable estimates of the relative variance for several designs may be helpful in the program design and efficiency analysis. This is why so much effort has recently been invested in the investigation of equations that govern various moments of the Monte Carlo estimates in specific games. These equations are called the moment equations and their derivations and discussion in more and more general forms is the main subject of this Chapter. The equations for the moments, in particular for the expected value and variance, have recently been derived and generalized to cover most variance-reducing schemes. They have the advantage of directly demonstrating that these schemes are unbiased by reproducing the same expected value, while at the same time yielding different second moments for use in comparing variances.

I. INTRODUCTORY REMARKS Let us assume again that the quantity to be estimated by the Monte Carlo method is some weighted integral, R, of the particles' collision density, i.e., R = fdPtp(P)f(P)

(5.2)

144

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

where, as before, ii(P) is the collision density of particles at the phase-space point P and f(P) is some given weight function. R is called a reaction rate since if o-r(P) is the cross section of the reaction r and f(P) = u,-(13)/cr(P)Xv(P) )(JP) being the characteristic function of some phase-space domain V, then R = fdP tp(P) o-,.(P)/cr(P) = fdP cp(P)cr,.(P) is the rate of the reaction r in the domain V. Note that the integrations, unless explicitly noted otherwise, are always assumed to extend over the entire phase space. For the integrals of functions that are defined only over a part of the phase space, the definition of the functions is tacitly extended to the entire space, with the function taken as zero outside the original domain of definition. We have seen in Chapter 4 that the integral, Equation (5.2), can be estimated in a Monte Carlo game by simulating the particles' collision density in a number, N, of histories according to the transport equation and, at the i-th collision point of the a-th history, 13„„ the quantity f(P,,) is added to the estimate. Then the estimate 1 N

,

f(P.,' ,)

is unbiased with respect to R in Equation (5.2), i.e.,

f

dP klfi(P)f(P) = (— L 2_, f(Pc, ,)) N

Now, denoting

= E f( ,„„) 11(P“.0) is the score due to the history that was started from the point P„,,. If M I (P) denotes the expectation of µ(P), then µ(13,,,0) is an unbiased estimate of M,(P) and, if the starting point 13„„ is selected from the source density Q(P„„), the score 1 ,,,„) is an unbiased estimate of the integral, i.e., N R = (— E µ(P4.0)) = fdpQ(p)mi(p)

(5.3)

It is heuristically obvious that an equation that describes the pointwise expected score, (P), is more naturally related to the actual simulation procedure than the particle-transport TN/11 equation, which does not account for the weighting function, f(P) — a quantity very characteristic of the goal and technique of the estimation. It will be seen later in this chapter that the equations governing the various moments of the score are, indeed, very useful in the analysis of the quality of various Monte Carlo techniques. In the next section, we show how the equation that describes the expected score, M,(P), is related to the equation adjoint to the integral transport equation.

145 A. RELATION OF THE EXPECTED SCORE TO THE ADJOINT COLLISION DENSITY As was seen in Chapter 4.IV, the collision density, tifr(P), satisfies the integral transport equation = kii,„(P) + fdP"VP")K(P",P)

(5.4)

where tlio(P) is the first-flight collision density tlr„(P) = fdP' Q(P')T(P',P)

(5.5)

and Q(P) is the probability density of the source particles. Recall that the transport kernel K(P",P) is related to the transition kernel T(P',P) and collision kernel C(P",P') as K(P",P) = f dP' C(P",P')T(P' ,P)

(5.6)

T(P',P) = T(r'—>rIE)8(E' — E)

(5.7)

C(P",P') = C(E"—>E'lr")8(r" — r')

(5.8)

where

and

while T(r'-->rIE) and C(E"-->E'lr") are given in Chapter 4.IV. In Chapter 4.VII, it was proven that an equation adjoint to the integral transport Equation (5.4) has the form tifr*(P) = f(P) + f dP"K(P,P")*(P")

(5.9)

Between the solutions and source terms of mutually adjoint integral equations, the following relation holds:"

f

dPiii(P)f(P) = Id13 4,*(P)tii,,(P)

(5.10)

Now, if the source term, f(P), in the adjoint Equation (5.9) is just the weighting function in the reaction rate, Equation (5.2), then the LHS of Equation (5.9) is the reaction rate itself. Inserting into the RHS of Equation (5.10), the expression of the first-flight collision density, ii)„(P) in Equation (5.5), the equality in Equation (5.10) takes on the form R = fdP' Q(P')I dPT(P' ,P)iii*(P)

(5.11)

M,(P) = idP' T(P,P')t1;*(P')

(5.12)

Finally, denoting

146

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

Equation (5.11) becomes R = f dPQ(P)MI(P) This is the same equality as in Equation (5.3) and it is concluded that the quantity defined by Equations (5.12) and (5.9) is the expected score due to a starter from P. The meanings of the functions 115*(P) M,(P) are heuristically clear. They represent the expected future contribution (or "importance") of a particle entering a collision at P or leaving a collision at (starting from) P, respectively. An equation that describes M,(P) follows easily from Equation (5.9) by multiplying it by T(P,P') (after changing P to P') and integrating with respect to P': M,(P) = idr T(P,P')f(P') + idP"L(P,P")MI(P")

(5.13)

where, with Equation (5.6), L(P,P") = fdP' T(P,P')C(P' ,P")

(5.14)

It will be seen in the following Sections that higher moments (and in certain cases, moments of other functions) of the particle's score obey similar equations. (Higher-order moment equations will be seen to be expressed in terms of successively determined lower-order moments.) Equations of the type represented by Equation (5.13) will be called the moment equations. Although normally we do not attempt to solve these equations, the existence and uniqueness of the solutions will often be exploited. In the next section, conditions are given under which the solution of an equation of the form of Equation (5.13) uniquely exists, and some consequences will be discussed. B. CONDITIONS OF EXISTENCE AND UNIQUENESS The majority of the results quoted here are proved in the basic textbook" by Spanier and Gelbard. The difference between the derivations there and here is that in Spaniers' book the conditions of existence and uniqueness of the solution to the transport equation (5.4) are established, while we are interested in conditions that grant a unique solution to the moment equation of the Equation (5.13) type. It is, however, easy to see that by putting K(P",P) —> L(P,P") in the original derivations in Reference 42, the assertions below remain valid. Let us write Equation (5.13) in the shortened form M(P) = I(P) + f dP"L(P,P")M(P")

(5.15)

where I(P) is a known function and L(P,P") is defined in Equation (5.14). Note that if the Monte Carlo simulation is performed according to the kernels T(P,P') and C(P',P") (i.e., if the site of the next collision and the postcollision coordinates are selected from T and C, respectively) then Equation (5.15) has an evident Monte Carlo interpretation. The expected future score of a particle departing from P consists of two parts.

147 The source term of the equation, I(P), represents the expected direct score due to the flight started from P (first-flight contribution), while the integral term is interpreted as the expected score due to the rest of the history, since L(P,P")dP" is the probability that a particle that departs from P will emerge from its next collision in the phase-space element dP" about P" and M(P") is the expected score of the emerging particle. As for the solution of Equation (5.15), the following theorem holds: Theorem 5.1 — Equation (5.15) has a unique, bounded solution if the following conditions are met: 1. There exists a constant B < 00 such that supf dP"L(P,P")

B

(5.16)

2. There exists a constant b < 1 and an integer N such that for n N supf dP"Ln(P,P")

b wm(P")

and

zo(P",W") =

1 — W"/w,p(P")

if

W" < wth(P")

0

if

W" > wth(P")

The second-moment equation will be derived in the special case when the splitting probabilities do not depend on the particles' weights, i.e., if zm(P",W") = Z m(P") Then Equation (5.55) applies and if again M2(P) denotes the second moment of the score due to a unit weight starter from P, then from Equation (5.95) and by making use of Equation (B.9) in Appendix 5B, the second-moment equation reads W2K12(P) = fdr t(P,P')Ca(P')[Wq(P,P') + Waia(P')12 + fdP' t(P,P')f dP" C(P',P"){[Wi(P,P') + Wt(P',P")12. + 2[W'f(P,P') + Wis(131 ,PNW"Mi(P") • W7Kli(P")

W4K12(P")

M;(13"))}

where E r' w'('„,. = =1

j

the total weight of the fragments in an m for one splitting and 1,V7 = E zm(P")(wD2

W'; = ,„=, E z„,(P") E ( AP,„„,)2 J

(5.98)

182

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

and we have made use of Equation (5.97). A little algebra yields an alternative form of the equation: W2K12(P) = fdP't(P,P'){Ca(P')[Wi(P,P')

• (2r ) fdP"C(P',P")[Wi(P,P')

W'ia(P')]2 W"is(P',P")]2-r(W")`Mr(P")}

- — (W")1 1142(P") ' + fdP't(P,131)fdP"(P',P"){[W

+ ox77 - wD*(P")}

(5.99)

Comparing this equation to Equation (5.58) that describes the second moment of the score in the original game with no splitting, it is apparent that the effect of the splitting on the variance basically is determined by the last term in Equation (5.99). E. ALTERNATIVE FORMS OF THE COLLISION KERNEL The collision kernel in Equation (5.74) accounts for the possible physical processes: absorption, scattering, or multiplication.* Notice that multiplication may also yield a single progeny (with a probability q,), and in this case multiplication and scattering are not different from a simulational point of view. The distinction between the two processes is justified when their contributions to the score, is and f1 , respectively, are different. On the other hand, if is = f1 , the collision kernel can be used in a simpler form as C(P',P") = Ea(P')8(P" — P) + [1 — ca(P')]

E

rfcin(P') en(P',P")

(5.100)

where, by putting C1(13',P") = 1,(131 )as(P',P")

‘ f(13')41(13' )1(P',P")/[6s(13')

'6f(r)ell(13')]

and Chan = W13') + f(P') 41(P')]/[e(P') + e‘f(P')] while Cn(PI ,P") = '&(P',P")

n = 2,3,...

and 40(P1 ) = 6f(Pi )gn(P1 )*JP')

OfOni

the kernel will be identical with that in Equation (5.74). Inclusion of this form of the collision kernel will make the moment equations somewhat simpler as no scattering kernel appears explicitly. *

These processes represent the possible reactions for particles of a given type. In case of the possible presence of different particles, there are other kinds of reactions le.g., (n,y) or (n,rry) collisions]. Such processes will be briefly discussed in Section 5.IX.D.

183 In certain applications, the kernel of multiplying processes is given in the form C‘,(P',P") = VP') C„(P',P") instead of the summed form of Equation (5.39). This is the case when only the mean number of secondaries per multiplication, vf(P'), and a common postcollision density, C„,(P',P"), are known, and also when the postcollision density Cr,(P',P") in Equation (5.39) does not depend on the number of progenies: en(P',P") = C,(P',P"),

n = 1,2,...

In general, v, is not an integer and the simulation of the multiplication goes as follows. Let k denote the integer part of v, k = ent[vf(P')] Then k progenies will leave the collision at P' with a probability Elk(131 ) = k + 1 — vf(P') and k + 1 secondaries are born with a probability

g„,,,f(P') = vf(P') — k The postcollision coordinates of all the progenies are selected from C„(P',P"). Obviously, the expected number of progenies in a multiplication at P' is kg, + (k + 1)g,, = vf(P') Setting all the other probabilities .4n equal to zero, the kernel of the above procedure reads k+1

(P' ,P") = E ng„(P') C„(P' ,P")

f

n=k

If the multiplication event contributes to the final score a value iv(P',P"), the formulas derived in this chapter remain valid by putting in(13' >13") =

,P"),

n = k,k+ 1

IV. FURTHER GENERALIZATIONS Although the majority of the practically applied simulation procedures are covered by the cases discussed in the previous Chapters, certain common Monte Carlo methods cannot be fully described by the results above. One of them is the geometrical splitting procedure mentioned in the previous Chapter. This procedure implies branching of histories during the particles' free flight. Time dependence is also beyond the scope of the moment equations so far derived. Geometrical splitting is examined in Section B. The time-independent description of a very general game is given in Section C and extension of the formulas to timedependent problems are briefly outlined in Section D. Other specific games which do not conform entirely with the formulation of this chapter will be considered in Chapter 6.

184

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

Before turning to more general games, a specific feature of the transition kernel is discussed. A. INTERRUPTION AND RESTART OF A FREE FLIGHT According to Equation (5.32), the transition kernel is expressed as D'

6-(P')expt 1 for P' = (r + D'w,E),

D' ---- 0

T(P,P')dP' = (5.101)

0 otherwise

It follows from this specific form of the kernel that it satisfies the following relation: i(13,13' ) = [ - j.

dQT(P,Q)It(P,,P')

for P, = (r + D,o),E),

0

D,

D'

(5.102) (5.102)

where .1:dQ denotes that the integration is performed along the direction w from P = (r,w,E) to P, = (r + Dico,w,E). Equation (5.102) has a definite physical meaning. It states that the probability of a collision after a free flight of length D' is equal to the probability that the particle reaches a point at a distance D, from the starting point without suffering a collision and then continues its flight as if it were started from that point. This property of the transition kernel is due to its exponential form, as has been discussed in Chapter 4.IV. Since Equation (5.102) holds for any point P, between P and P', it implies a generalized selection procedure of the next collision point. Let t(P,P,) be some given probability density function defined along the half-line P, = (r + D,co,E), D, > 0, and let us define the following selection procedure: 1. 2. 3.

Select a point Q from t(P,Q); Q = (r + Dw,E). Select a point P, from t(P,P,); P, = (r + D,co,E). If D, > D, then let the next collision point be P' = Q

4.

If D < D,, then let P = P, and return to step 1.

The interpretation of the procedure is obvious: a flight from P to P' is started with a probability T(P,P')dP', but it is interrupted at P, with a probability t(P,131)dP, and restarted at P, by selecting a new distance from T(P,,P). The unbiasedness of this altered selection procedure is proven in the following theorem. Theorem 5.5 — The procedure (steps 1 through 4) above is equivalent to the selection of the next collision point from the density function T(P,P')

185 Proof. Let Y(P,P') be the probability density of the point P' selected in the procedure above and let

x(PI,Q) =

1,

if D, -?-- D

0, 1

if D, < D

It is to be shown that 9-(P,P') = T(P,P'). According to steps 1 through 4, the density of P' satisfies the equation 3-(P,131) = idQfdP,t(P,Q)t(P PI)[X(P1,Q)8(13' = f dQ[f dPit(P,Pi)]i(P,Q)8(13'

Q)

f

VQ,130(P1,13')] r r. A dQT(P,Q)it(P,P1)-9-(P1,r)

or, since the densities are normalized to unity P'

.7(P,P1 ) --= [1 — f dP,t(P,P4t(P,P') fP1 fdP,

t(P,Pii 1 —

dQT(P,Q)] 5-(P„Pi)

(5.103)

First we show that T(P,P') satisfies Equation (5.103). Inserting T into the RHS of the equation and taking into account that T(P„P') = 0 for such P' that do not belong to the half-line from P, along w„ the second term on the RHS of Equation (5.103) becomes dP, t(P,P,) T(P,P') where we have made use of Equation (5.102). The sum of the two terms gives Ff(P,P') = T(P,P') i.e., T(P,P') does indeed satisfy Equation (5.103). To conclude the proof, it remains to show that Equation (5.103) has a unique solution. This, however, follows from the fact that the integral kernel P, t(P,131 )[ 1 —

jP dQT(P,Q)]

in Equation (5.103) has a norm definitely less than unity [unless t(P,P,) = 8(P — P,), the Dirac delta function], since both t and T are everywhere positive and are normalized to unity. Thus, the uniqueness of the solution is ensured by Theorem 5.1.

We have thus established the identity P'

T(P,P') = J dP,t(P,P,) T(P,P') + P'

A

A

dP,t(P,P,)1 dQT(P,Q)T(PI,P P1

186

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

or, what is the same: i(13 ,13' )1

dP, t(P,P1 ) =

P

dP, t(P,P,)

P1

dQT(P,Q) T(1),,P')

(5.104)

The selection procedure defined by steps 1 through 4 might seem unnecessarily complicated; however, it is just this procedure that makes geometrical splitting feasible. In the simplest geometrical splitting procedure, the particles are to be split into a number of fragments whenever they cross some given geometrical surfaces (this is why the procedure is also called surface splitting). In this simplest case, t(P,P,) is a Dirac delta function at the crossing point of the flight with the nearest splitting surface, and the above restart option makes it possible to start the split fragments independently from the previous collision point and from each other. Moment equations describing score moments in a game with surface splitting were first derived by Juzaitis'6 and were used to optimize the location of the splitting surface and the number of fragments in monoenergetic isotropic transmission through infinite slabs. In the following section, a rigorous derivation of a general unbiased geometrical splitting procedure is given with no restriction on the form of the density function t(P,P,). This density will be called the splitting probability as it represents the probability that, in a flight started from P, splitting occurs in dP, about P,. B. GEOMETRICAL SPLITTING The main difference between the collisionwise and geometrical splitting procedures is that while in the former it is played at collision sites only, the latter may occur at any point of the free flight and the probability of its occurrence is given by the density t(P,P,), a quantity having no direct connection to the original simulation. Let us assume that if a splitting procedure is played at PI , the probability that a particle of weight W suffers a kfor-one splitting is gi,(P„W) with

E

k =0

gk(Pi,w) =

(5.105)

Also assume that splitting itself has no contribution to the score. However, we shall allow the contribution probability assigned to a free flight between the point of departure, P, and the site of splitting, P„ to be different from that assigned to an ordinary intercollision free flight. Let the probability that a flight from P to a splitting site P, results in a score in ds about s be pg(P,P„W,$)ds for a particle of weight W. In the derivation below, deterministic scores will again be assumed, i.e., Pg(P,P„W,$) = ais — ig(P,PI)Wi There is one more point to be discussed in greater detail in connection with geometrical splitting. We have seen in Section 5.I.A that whenever the transition kernel used in the simulation is different from the physical (analog) kernel, the statistical weight of the particle is to be altered according to the values of the analog and nonanalog transition kernels at the arguments determined by the actual flight. In a game with no geometrical splitting, it is irrelevant whether the weight is changed at the beginning or at the end of the flight. In the case of geometrical splitting, however, the weight of the particle arriving at the site of splitting is to be fixed. Let us assume that the particle's weight is altered only when it reaches a collision point, i.e., that it enters the splitting procedure with its original weight W. Naturally, this means that the weights of the split fragments also must eventually account for the nonanalog transition of the original particle. Let W(I)k denote the weight of the i-th fragment from a k-for-one splitting at the point P,.

187 As the splitting probabilities g, depend on the statistical weight of the particle to be split, relation (5.55) does not necessarily hold, i.e., it is not granted that a starter with a weight W will yield an expected score equal to W times the expected score due to a unit weight starter. Therefore, in establishing a score probability equation and also the conditions of an unbiased splitting procedure, we have to proceed with a certain caution. Let us first consider the case when splitting may only be performed once during the first flight of the starter. Let 'ir(P,W,$)ds denote the probability that the starter form P with a weight W yields a final score in ds about s if splitting is allowed in the first flight. Again let Tr(P,W,$)ds denote the same probability in a game without splitting, and let *(P',W',$)ds be the probability that a particle entering a collision at P' with a weight W' will contribute to the final score in ds about s. Then Tr satisfies the equation p, ir(P,W,$) = j-dP' f(P,P') [1 — J dP, t(P,P1) P(P,P',W',$) *

f

,W' ,S)

dP, t(P,P,)[1 — f P dQT(P,Q)] Pg(P,P1,W,$)

E

* { g0031,W)8(S)

k= 1

gk(13,,w) il*Tr(13,,w(ok,$)} 1=1

(5.106)

where, as before, the integration with no definite boundaries is extended from P to infinity along the direction w. The first term on the RHS of Equation (5.106) represents the score probability due to a free flight from P to P' without splitting followed by a collision at P', and the second term describes the probability of the score when splitting is played before the next collision. Let IV1,(P,W) = f ds the expected score due to the starter to split. Furthermore, let N1 (P') = f oodss*(P',1,$) the first moment of the score due to a particle of unit weight which is about to collide at P'. Then the expected score due to a particle starting from P with a weight W follows from Equation (5.106) and from the expressions for the deterministic scores in Equation (5.46) and (5.48) as M I (P,W

= i'dP ' t(P,I1 1

fdP,

+E

k=1

—

t(P,P,) [1 —

gk(Pi,w)

E

1=1

dPit(P,P1)] W'[f(P,P') + &,(131)]

dQi(P,Q)][

k=0

gk(131,W) Wig(P,P0 (5.107)

wwkKii(Poi

If we denote by W

=E

k=1

k

gk(PI,W)

E

i=1

W(ok

(5.108)

188

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

the expected total weight of the fragments coming out of the splitting, the first-moment equation reduces to P'

KI,(P,W) =

dP' T(P,P41 — j dP, t(P,P,)] W'[f(P,P') + N,(P') f dP, t(P,P,)1 dQT(P,Q)iWig(P,Pi) + WKII(Pi)]

+

(5.109)

where we have made use of the normalization in Equation (5.105). Any game with splitting may only be useful if it results in the same expected score as the corresponding game without splitting. According to Equation (5.75), the expected score in the game with no splitting satisfies the equation WKI,(P) = j dP' t(P,P')WV(P,P') + &,(P')]

(5.110)

Comparison of Equations (5.109) and (5.110) shows that the two expectations are equal if

rdP, t(P,1301-dQ i(P,Q)Mig(P,P,) + WKI,(P,)]

JP

Pi

r

dP' t(P,P1 )1P dP, t(P,P,) WV(P,P ') + &,(P')] = 0

(5.111)

Now let us consider a particle that starts a flight from P, with the weight W. Let W' denote its weight after a free flight from P, to P'. The expected score due to this particle satisfies Equation (5.110) in the form WKI(P,) =f

[4131,P1 ) + r`11(13')]

Before substituting this equation into Equation (5.111), a number of identities are established. Notice that by interchanging the order of integrations with respect to P, and P' and making use of Equation (5.104), the following relations hold: P'

J dPit (P,P,)1 dQ

[..

= J dP' T(P,P').1 dP, t(P,131 ) [...]

(5.112)

and fdP, t(P,Pdf dgt(P,Q).1 Pi

fP

dP'

P

dP, t(P,P,) -,z1Q

,P') [...1 T(PI ,P') [..

P' = fdP't(P,P1 )f

dP,t(P,P1 )[.• .]

j dP,t(P,131 )dPT(P,P')[...] Pi

(5.113)

189 Inserting the equations above into Equation (5.111), it is seen that the expected scores in the two games (with and without splitting) are equal if dP, t(P,P,)f dP"i(P,P')EWig(P,P,) + Wq(P„P') — Wi(P,P1 ) Pi

+

- AA,P)&i( r)]

=0

for the arbitrary density function t(P,P,). This equation ensures an unbiased estimation of the expected score M,(P) in a game where, at most, one splitting is played in the first flight. Let us now consider a game where two splittings are allowed in the first flight. Then according to the arguments above, the particles that emerge from the first splitting will yield the same expected score independently of whether the second splitting is played if the equality above holds; that is, under this condition, the games with one and two splittings are equivalent from the point of view of the expected score and since the game with a single splitting is equivalent to the game without splitting, so is the game with two splittings. Recursive application of the arguments above shows that introduction of an arbitrary number of splitting procedures into the first flight leaves the expected score unchanged. Furthermore, since the split fragments are simulated independently and splitting has an effect on the future contributions only, any flight where a particle is split can be considered the "first flight". The conclusions are summarized in the following theorem. Theorem 5.6 — A game with geometrical splitting results in the same expected score as the game which is played with identical kernels and contribution functions, but without splitting, if the following conditions hold: 1. The weights of the split fragments are such that W' = W' where W' is the weight of a particle at P' that starts a flight at P with a weight W and enters its next collision at P' without suffering a splitting.

2. The contribution assigned to the flight from the starting point P to the splitting site P, is Wig(P,P,) =T(P,P') W' Pi

I p,

dP'

— i(P„P')]/I'dP'T(P,P') Pi

—

,P')]

(5.114)

Condition 1 formally means that if a particle starts from P and is split into a number of fragments and the fragments all collide at P', then the total expected weight of the fragments at P' must be equal to the weight of the original particle when it enters a collision at P' without splitting. In order to make condition 1 less abstract, let us recall the introduction of statistical weights in Chapter 5.1.D. It was stated that in the majority of practical cases, the bias due to the selection of a free flight from a nonanalog transition kernel is compensated for by multiplying the statistical weight of the particle by some weight factor w(P,P') in a

190

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

flight from P to P'. [In fact, it was mentioned in Chapter 3 and also will be seen in Section 5.V.B. that the simplest choice of w(P,P') is the ratio of the nonanalog and analog transition kernels at the argument (P,P').] Making use of this rule, the weight in the game without splitting is changed as W' = W • w(P,P') Similarly, in the game with splitting W' = W • w(P„P') Thus, in view of Equation (5.108), condition 1 reads 1. w(P,P')W = w(P1, P')

E

E

gk(P„W)

k=1

W(ok

i=1

Condition 2 amounts to saying that the expected contribution due to an intercollision flight must not be influenced by a possible splitting. Note that the theorem has an important practical consequence. Recall that the statistical weight of a particle is changed in a free flight because the transition kernel T, which is used to select the length of the free flight, is different from the analog kernel. Let us realize that condition 1 leaves some freedom in choosing the postsplitting weights since the condition concerns the total weight of the split fragments at the next collision, while this weight is determined by the weights of the fragments after the splitting and by their change due to the reselection of the free flight at the site of splitting. In practical realizations of the procedure, this uncertainty must be excluded. Now it is physically reasonable to fix the rules according to the following scheme. 1. The starter is at P and its weight is W. Let Po = P,

Wo = W

2. Select a point Q from t(Po,Q): Q = (ro + Dw,E). 3. Select a point P, from t(Po,P,): P1 = (ro + D,o),E). 4. If D, D, then let the next collision point of the particle be P' = Q and let its weight and contribution be determined as in a game without splitting for a free flight from P to P', i.e., they will be W'o and Woi(P,P'), respectively. 5. If D, G D, then split the particle into k fragments with a probability gk(P,,Wo). 6. If k = 0, let the contribution be Woig(P,P,) as defined by Equation (5.114) and start to process a new particle. 7. If k > 0, let the weight of the i-th fragment be W(ok such that

E

k=1

go1,w0)

E w1ok = W

i=1

o

Take the fragments one after another. For the i-th fragment, set Po = P1 and Wo = W(ok and repeat steps 2 through 6 until every fragment reaches a collision point or is eliminated from the system.

191 It is easy to see that the procedure above grants the fulfilment of the conditions of Theorem 5.6. Geometrical splitting, like most of the tricks in Monte Carlo methods, is introduced for reducing the variance of the score. This can be investigated on the basis of the second moment equation. However, it is not easy to establish, in general, a second-moment equation similar in form to that concerning a game without splitting because the simulation may depend on the weight of the particles. By arguments like those leading to Equation (5.106), it is easy to show that the score probability ir(P,W,$) for a game with geometrical splitting satisfies the equation ir(P,W,$) = i'dP"i'(P,P41 — J dP,t(P,PI)iP(P,P1 ,W1 ,$) * *(P',W',$)

f

dQT(P,Q)] Pg(P,P„W,$)

dP, t(P,Pi) [ 1 —

* tgo(P,,W)*(s) +

E

g,(P,,W) i rli *Tr(P„Ww„,$)}

(5.115)

Let us consider a game in which the splitting probabilities do not depend on the particle's weight, i.e., gk(P,,w) = gk(PI) Inserting these probabilities into Equation (5.115), multiplying the equation by s2, and integrating over s, the equation for the unit weight second moment M2(P) follows as below (details of the derivation are given in Appendix 5B): P' W2M2(P) = j-dP' t(P,P') [1 —

dP, t(P,P,)] (W')2 (i2(P,P')

+ 2i(P,P')&i(r) + 1`12(r)] + f dP,t(P,P41 — i dQT(13,Q)]1W2i2,(P,P1) JP

(5.116)

+ W2[M2(P1) — Mi(P1 )11

+ 2WWig(P,P1 ) Mgt) +

where W is the expected total weight of the split fragments as defined in Equation (5.108), while k WI =

E g,(PocE i=1 k=1

\ W(i)k) 2

and k

WI

=E gk(PI) 1=1 E k=1

We recall that in the procedure discussed in this section, the functions t(P,P,) and gk(P,W) and the split weights Wwk are independent of both the kernels and the contribution functions of the simulation. This means that even if the nonanalog kernels and contributions are fixed, for some reason the functions and weights above can be chosen almost arbitrarily and their

192

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

proper choice may help the optimization of the game. On the other hand, an erroneous choice not only decreases the efficiency, but, in extreme cases, can also make the simulation unfeasible. To conclude this section, let us consider the special case of surface splitting. Let t(P,P1 ) = &(P1 — 1)*) where the splitting surface nearest to P along w is situated at P* = (r + D*co,E). Then Equation (5.115) reduces to W2M2(P) = .1* dP' T(P,P')(W')21i2(P,P1) + 2.i(P,P')N,(P') + l'C1' 2(1n1 dP' T(P,P')] {W21(P,P*) + 2WWig(P,P*)KI,(P*)

+ [1 —

W k.(P*) +

EKUP*)

MCP*)]}

(5.117)

Equation (5.117) reflects the interesting fact that the second moment of the score is not necessarily a continuous function of its coordinates. Indeed, letting P tend to P*, from Equation (5.117) we have lim W2M2(P') = W;*(P*) + W2[M2(P*) - *(P*)]

P-,P"

+ lim [W2i2g(P,P*) + 2WW - k(P,P* ) M1 (P*)]

(5.118)

Obviously, the discontinuity is due to the stepwise change of the number of particles at the surface. Finally, we note that in most practical cases, geometrical splitting is played in one of the three ways discussed in connection with collisionwise splitting in Section 5.III.D. either a threshold weight is given above which the particle is split into a number of fragments, each leaving with this weight (weight-dependent simulation), or the expected number of split fragments is fixed irrespective of the weight to be split. Obviously,the second-moment Equations (5.116) and 5.117) concern only the latter procedure. C. SCORE PROBABILITY IN A GENERAL TIME-INDEPENDENT GAME Combining the results of the present and previous Sections, the score probability equation of a very general time-independent Monte Carlo simulation is obtained. We consider a nonanalog multiplying game where both collisionwise and geometrical splitting are applied. Again let 7r(P,W,$) be the score probability density assigned to a particle that starts its flight at P with a weight W. Let b-(P',W',$) and i1(P",W",$) be the score probability densities due to a particle entering a collision at P' with a weight W' and leaving a collision at P" with W", respectively. We have seen in the previous section that 'r is expressed by' and by itself according to Equation (5.115): 7r(P,W,$) = f dP' T(P,P1 )1 dP, t(P,P,) p(P,P',W',$) * *(P',W',$) P'

di) t(P,P, )1 dQ T(P ,Q) pg(P,P„W ,$)

.1

Pi

gk(P1 ,W) * kE =0

k

11 * *PI ,Wwk,$) =0

(5.119)

193 where we use the notational convention 7r(P,W1„),,$) = S(s)

(5.120)

The connection between 11- and -q is due to the collision process and it follows from Equation (5.76) in Section 5.III. A as ,W' ,$) = Ca(P') pa(P' ,Wa,$) + es(P')IdP" ,(P ' ,P") ps(P' ,P",W",$) * Ti(P",N",S) + 4P')

E

4n(P ')

n=1

* II f dro Cn(P',13%)p„(P',13%,W%,$) =1

(5.121)

(F('0, W('),$)

Finally, -q and -8 are interdependent due to collisionwise splitting according to Equation (5.95) in Section 5.III.D as oo

-q(P",W",$) =

E zra(P",W")Tl,=0*Tr(P",W(,,,a„s) 0,=0 m

(5.122)

where again the convention of Equation (5.120) is valid. Equations (5.119) through (5.122) describe the score probability in a general Monte Carlo game. Their moments can be obtained according to the procedures followed in the previous chapters. Here we leave the description of the general game since all its relevant instances have already been discussed. On the other hand, the system [(5.119) through (5.122)] as a whole is mainly of theoretical interest since, in practical cases, one or another specific trick is to be investigated at a time and very seldom a general procedure. D. INCLUSION OF TIME DEPENDENCE We have so far limited our discussion to problems where time dependence plays no role either because the quantity of interest is the cumulative effect of a certain amount of particles or because the distribution of particles is assumed to be constant in time and the estimated quantity is some reaction rate per unit time. In the sequel, we refer to such problems as stationary. On the other hand, simulation of particle transport inherently involves modeling of events at successive times and therefore estimation of time-dependent quantities is not in contrast to the very nature of Monte Carlo methods. Time-dependent problems may be classified into two main types. In the first type of calculation, evolution of a certain quantity, e.g., reaction rate in or escape rate from a given region, is investigated as a function of the time elapsed since the start of the particle's batch. The second type of time-dependent estimation concerns the investigation of the variation of some quantity due to the change in time of some characteristic function [e.g., the weighting function in the RHS of Equation (5.2) or the kernels that govern the transport]. Monte Carlo simulation of the second type of time dependence is only occasional (e.g., for investigation of reactivity change due to the moving components of a nuclear reactor) and will not be discussed here. Moment equation accounting for evolutional-type time dependence were first derived by Booth and Cashwell4 and by Booth' for multiplying nonanalog games with geometrical splitting. To illustrate the method to be used when deriving time-dependent moment equations, we consider a nonanalog nonmultiplying game. Extension to more complicated cases (with multiplication or splitting) goes along the lines of the previous Chapters.

194

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

Let 'rr(P,W,s,t)ds be the probability that a particle at P = (r,E) with a weight W will contribute to the final score in ds about s during a time t. Let Pt = (r+vt,E) where v = v • to, v being the speed of the particle. In contrast to the stationary case, the particle will not necessarily enter a collision during the time t and therefore three possible events may follow the start of a free flight in a nonmultiplying game: 1. The particle is flying without any collision all the time t. The probability of this event is jp, dP' T(P,W) 2. The particle enters a collision at a time t' < t and it is there absorbed. The probability of this is

oa(PD i(P,PD with P; = (r + vt',E). 3. The particle suffers a scattering at a time t' < t and emerges from the scattering in dP" about P". The corresponding probability is T(P,P;) dP; C(P; ,P")dP" where P"

(r + vt',E')

Accordingly, the score probability equation reads Tr(P,W,s,t) =dP' T(P,P')] p,(P,P„W,$) P,

dP; T(P,P;)Ca(P;)p(P,P;,W',$) * pa(P;,Wa,$) P,

dP; T(P,P;) dP"C.(P;,P")p(P,P;,W' ,$) * ps(P;,P",W",$) * Tr(P",W",s,t — t')

(5.123)

where p,(P,P„W,$)ds is the probability that a free flight from P tot), by a particle of a weight W at P results in a score in ds about s. The RHS of Equation (5.123) is the accumulated probability that some score is recorded in the first flight and collision during a time t', and the rest of the score up to in ds about s is the result of the subsequent history during the time t t'. If Mr(P,t) denotes the r-th moment of the score contributed by a unit weight starter at P during a time t: Kir(P,t) = f oods sr7r(P,1,s,t)

195 and assuming again deterministic scores [cf. Equations (5.46) through (5.48) and (5.51) through (5.53)1, the first-moment satisfies the equation WKI,(P,t) = [I'dP' f(P,P')] Wit(P,13,) P,

f(P,P:)[Wi(P,PD + a(P')Waia(PD + fdP"(P:,P")W"is(P,P")] P,

+

dP'T(P,P:)fdP"(P:,P")W"i/1,(P",t— t')

J

(5.124)

The interested reader may repeat the derivation for multiplying games with splitting and will find that the resulting equations are similar in form to those derived for stationary games. The main difference between them is that in the time-dependent case, integrals from P to P, replace the integrals from P to infinity in the stationary equations. This difference, however, is in a sense essential; namely, a stationary equation has a unique bounded solution only if the kernels satisfy the conditions of Theorem 5.1 in Section 5.I.B. On the other hand, a time-dependent equation may have a bounded solution for every finite t even if the conditions of Theorem 5.1 are violated. The reason for this difference is obvious. The stationary solution obeys an integral equation in which the integration is extended over the entire phase space and therefore the norm of the integral kernel is also connected to all possible values of the kernel. On the other hand, time-dependent integral equations concern only the region of the phase space that can be reached by a particle during a finite time t and therefore the existence of the solution depends only on the value of the integral kernel inside this region. In mathematical terms, this difference is analogous to the difference between the Fredholm and Volterra-type integral equations of the second kind. In terms of reactor physics, this means that stationary moment equations concern only subcritical systems with sources; the time-dependent description, on the other hand, may also follow the evolution in time of supercritical systems. (As will be seen in Chapter 6.111, critical systems are also treated as time independent, but a method different from the stationary solution is used.) We will not discuss time dependence more thoroughly. The techniques to be used in specific problems are analogous to those used in the previous derivations. To illustrate the difference between the stationary and time-dependent treatments, let us consider a monoenergetic analog game played in an infinite homogeneous medium. Let the quantity to be estimated be the number of particles absorbed during a time t. In such a game, the expected score depends on t, but is obviously independent of the starting point P of the particles. If the total cross section of the medium and the speed of the particles are taken as unity, then the transition kernel in Equation (5.32) has a simple exponential form with the time in its argument, and the first-moment equation (5.124) becomes Mi(t)

fo dt'e-e ca +

where c = fdP"C(P',P")

tit' e-e cMi(t — t')

196

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

the mean number of secondaries per collision. The solution of this equation is easily obtained as call — e -"

— c)

if c < 1

M,(t) =

(5.125) cat

if c = 1

The stationary expected absorption rate follows from the solution Equation (5.125) by letting t tend to infinity, and it is seen that the solution tends to a finite value with increasing time if c < 1, but it is unbounded at c = 1. This is in accordance with Theorem 5.1 since the conditions of the Theorem are met only if c < 1. Finally, we note that in view of the results in Section 5.III.C, the solution for c < 1 in Equation (5.125) also defines the expected absorption rate for c > 1.

V. ANALYSIS OF THE FIRST-MOMENT EQUATION We have established the moment equations that describe the expectation of various powers of the total score. There remains, however, a number of open questions. First, it is to be clarified what kind of contribution functions result in estimates of the required reaction rate, Equation (5.2). Second, it is not yet clear how to choose the statistical weight of a particle in order to keep a nonanalog game unbiased, i.e., in order to ensure that a nonanalog game does result in the same expected score as the analog game it corresponds to. These questions will be answered in Sections A and B of this chapter. We consider here the firstmoment equation of such multiplying analog and nonanalog games in which the secondaries of a multiplying event are indistinguishable. The considerations below can be easily generalized to the case of distinguishable secondaries. We remind the reader that the statistical weights of the split fragments in a game with splitting are essentially arbitrary (except for the conservation rules in Theorems 5.4 and 5.6) and they do not follow from the requirement of unbiasedness. Selection of the split weights will be investigated in Section 5.VIII.I. For easy reference, let us recall the first-moment equation for a multiplying game. The equation concerning an analog game is given in Equation (5.80) as M,(P) = I,(P) + fdP' T(P,P') fdP"C(P' ,P")M,(P")

(5.126)

where I,(P) = f dP' T(P,P')[f(P,P') + ca(P')fa(P') + cs(P')fdP" C,(P' ,P") f,(P' ,P") + cf(P')

E

nqn(P')fdP"Ca(P' ,P")

,P")]

(5.127)

nqn(P')Cn(P',P")

(5.128)

n=1

and CO

C(P',P") = cs(P') CAP' ,P") + cf(P')

E

n=1

197 [Note that we have omitted the absorption term ca(P')8(P' - P), which leads out of the simulation and plays no role in Equation (5.127)1. The nonanalog first-moment equation, from Equation (5.79), is M,(P) = 11(P) + fdP'i(P,P1 )IdP"C(P',P")M,(P")

(5.129)

with I(P) =

+ -W w : ia(P')

dP'

± w" is(P' ,P") + C,(P')1dP" as(131 ,P") 11

+

E

W

„

f (P',P")]

(5.130)

nein(P') an(P' ,P") 1117:-

(5.131)

nein(r)fdP"„(P',P")

n=1

and C(P'P") = Cs(P'),(P',P") W --: + Cf(P') n=1

A. UNBIASED ESTIMATORS It has been seen in Chapter 5.1 that starting the particles from the real physical source density in an analog game, the expected score due to a particle that departs from point P satisfies the equation M1 (P) = 1(P) + fdr T(P,P')I dP"C(P' ,P")MI(P")

(5.132)

I(P) = fdP'T(P,P')f(P')

(5.133)

with

where f(P) is the weighting function in the reaction rate integral R = idPIKP)f(P) to be estimated. It was also seen in Section 5.I.A that Equation (5.132) determines the expected score in such a game, where a history consisting of the collision points P;, 13;,...,P„' yields a total score of I-L(P) =

E=i f(P:)

In Section 5.I.D and subsequent chapters, we have constructed a general Monte Carlo simulation in which every event that happens to a particle may contribute to the total score. The first moment of the score in a general analog game is given in Equations (5.126) and (5.127). Comparing these equations to Equations (5.132) and (5.133), it is apparent that in the simplest game, the contribution functions (or estimators) are defined as f( 3,13 ') = f(P'); fJP') = fs(P',P") = fn(P',P") = 0

198

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

or, alternatively, f(P,P') = 0; fa(P') = f,(P',P") = fn(P',P") = f(P') It is heuristically obvious that there may exist several other estimators that result in the same expected score in a given game. A wide class of such estimators is defined in the following theorem. Theorem 5.7 — All contribution functions (estimators) that satisfy the relation I,(P) = I(P)

(5.134)

yield an unbiased estimation of the expected score, where I i(P) and I(P) are given in Equations (5.127) and (5.133), respectively. Proof. It follows from the uniqueness of the solution of Equation (5.132) (cf. Theorem 5.1) that if the source terms of Equations (5.126) and (5.132) are equal, then so are the solutions to them.

The sets of estimators {f,fa ,f„f„...,fn,...} that satisfy Equation (5.134) are called partially unbiased estimators". The name follows from the fact that the condition in Equation (5.134) ensures an unbiased estimation of not only the total score, but also the score due to any event pair consisting of a free flight and a collision, i.e., partial expected scores of a history are also preserved by these estimators. Indeed, 11 (P) is just the expected partial score due to a free flight and a collision in a general game and I(P) is the same in the simplest simulation. The theory of partially unbiased estimators will be developed in Chapter 5.VI. Scoring in a general analog game goes as follows. Every time a free flight is simulated from a point P, to Pk, the score of a history started from Po is increased as 11(P.) = 11(P.) + f(1)01)0

(5.135)

If the flight is followed by an absorption, scattering, or multiplication, the score is changed to 1-1 (P0) = 1-(P.) + fa(Pk)

(5.136)

1-1 (P.) = N(P.) + fs(PioPk i)

(5.137)

or

1-L(130) := 1-1(P.)

E f.(13:0Pk +1(i)

(5.138)

respectively. The total score due to a starter is the sum of the contributions from all the progenies in the simulation. Similarly, scoring in a nonanalog game is performed by accumulating the nonanalog contributions weighted by the actual statistical weights of the particles [cf. Equation (5.51)]. Thus, f in Equation (5.135) is replaced by and Waia, W"i„ and W:;(,)in stand for fa, f„ and fn, respectively, in Equations (5.136) through (5.138). Proper determination of the weights and nonanalog contribution functions is discussed in the next Section.

199 The contribution functions f(P,P') are usually called track length-type estimators as the most commonly used such estimator is proportional to the optical track length of the particle between two successive collision points (cf. Chapter 4.V). fs(P',P"), and f„(P',P") are called collision-type estimators, for obvious reasons (cf. Chapter 4.V). In the majority of practical applications, they do not depend on the postcollision coordinates P". fa(P') is referred to as the absorption or last-event estimator which scores at the collisions that terminate a history. If the contribution function assigned to a free flight depends only on the starting point P, i.e., f(P,P') = f(P) then it is called an expectation-type or next-event estimator because the contribution from the flight does not depend on the actual length of the free flight and hence it is equal to its expectation over the next collision points (cf. Chapter 3.11). Obviously, the simplest expectation-type partially unbiased estimator is the source term I(P) of Equation (5.133) itself. Equation (5.134) becomes especially simple in special cases. If, for example, the only nonvanishing estimator is the track length-type estimator, then it is partially unbiased if

f

dP' T(P,P') f(P,P') = fdr T(P,P') f(P')

On the other hand, if collisions also contribute to the score but their contributions are independent of the postcollision coordinates and of the number of secondaries emerging from the collision, i.e., if fs(P',P") = f.(13',P") = fc(13') then Equation (5.134) reduces to

f

dP' T(P,P')[f(P,P') + ca(P')fa(P') + c(P')fc(P')] = f dP' T(P,P1 )f(P')

where c(P') is the mean number of secondaries per collision at P' as defined in Equation (5.19) In the derivations of this chapter, we have assumed that the contributions to the score from various events in a history depend only on the coordinates characteristic to the events, and are independent of the sequence number of the collision point in the history to which they are related. In other words, we assume that identical events give identical contributions irrespective of the stage of the simulation at which they occur. It is this assumption that makes it possible to construct integral equations for the score moments, and only such estimators belong to the class of the partially unbiased estimators. Although in the majority of the practical cases partially unbiased estimators are applied, it is obvious that they do not exhaust all the possible unbiased estimators. In fact, it is possible to define a wide class of unbiased estimators, the forms of which explicitly depend on the sequence number of the actual collision point in the history. Such estimators are treated in full generality by Khisamutdinov19 and also by Mikhailov.37 Certain special forms of such estimators are also treated in Reference 40. As these special estimators have limited application in usual Monte Carlo problems, they will not be detailed here. A special application of such estimators is demonstrated in Section 5.IX.A. B. WEIGHT GENERATION RULES The second important open question of a general Monte Carlo simulation is how to

200

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

choose the statistical weights of a particle in a nonanalog game in order to obtain an unbiased estimate of the quantity of interest. In other words, we seek the weight values W', W', W", and W'n' appearing in the nonanalog first-moment equations [(5.119) through (5.121)] that ensure the equality of the nonanalog and analog expected scores Mi(P) ' M1(P)

(5.139)

Now, it follows from the uniqueness of the solution of Equation (5.126) (cf. Theorem 5.1) that Equation (5.139) holds if the integral kernels and the source terms of the analog and nonanalog first-moment equations are identical, i.e., if, with the notation in Equations (5.130) and (5.131), T(P,P')C(P',P") = T(P,P')C(P',P")

(5.140)

11 (P) = I,(P)

(5.141)

and

Theorem 5.8 — Equalities (5.140) and (5.141) hold and, thus, a nonanalog history started from P with a weight W results in the same expected score as the corresponding analog history with the same starter 1. if the statistical weights of the particle are changed in the various events according to the following rules: W' = w(P,P')W

(5.142)

w(P,P') = T(P,P')/T(P,P')

(5.143)

in a free flight from P to P' when the weight at P is W. Wa = wa(P') W'

(5.144)

wa(P') = c,(P')/6a(P')

(5.145)

in an absorption at P'. W" = ws(P',P")W'

(5.146)

ws(P',P") = [cs(P')/Ca(P')][Cs(P',P"))/Cs(P' ,P")] ws(P') wc(P' ,P")

(5.147)

in a scattering from P' to P" where Ws(P') = s(P')/Cs(P')Wc(P',P") = cs(P',P")/es(P',P") = wn(131 ,P")W'

(5.148)

wn(Pr ,P") = [cf(P')qn(P')/Cf(P')4,,(P')][Cn(P',P")/an(P',P")] wfn(P') wc„(P' ,P")

(5.149)

201 in an n-fold multiplication from P' to P" where the weight factor in Equation (5.149) was split up as w,(P') = cf(P') qn(P')/ef(13') 4n(P') wcn(P',P") = Cn(P',P")/ n(P',P") and 2. if the nonanalog and analog contribution functions (estimators) are related as

f

dP' T(P,P'){i(P,P') — f(P,P') + ca(P')[ia(P') — f.,(P')]

+ c,(P')I dP"C,(P',P")(Es(P',P") — fs(P',P")] + cf(P')

E

nq,(P')IdP" Cn(P' ,P") [in(P' ,P") — fn(P' ,P")]} = 0

n=1

Proof. Simple substitution of Equations (5.142) through (5.149) into Equations (5.127) through (5.131) shows that the equality (5.140) is satisfied, while condition 2 is just the detailed form of Equation (5.141) after substitution of the weights. Thus, the conditions above imply Equations (5.140) and (5.141) as was to be shown.

Equations (5.142) through (5.149) will be called the weight generation rules of a nonanalog game. The rules have an obvious interpretation. In a nonanalog game, the coordinates of an event are selected from a probability density different from the analog (physical) one. The weight of the particle participating in the event, however, is multiplied by the ratio of the analog and nonanalog probability densities, i.e., if the probability of an event in the analog game is lower than in the nonanalog, only a "fraction" of an analog particle takes part in the event of increased probability and vice versa. Equation (5.140) expresses the fact that the effective number of particles undergoing various events in a collision that follows a free flight is the same in both games. It is to be emphasized that the weight generation rules of the theorem represent sufficient but not necessary conditions of the fulfilment of Equation (5.140). In some applications, e.g., the multiplying part of the collision kernels in the nonanalog and analog games cannot be related as in Equations (5.148) and (5.149) because the possible number of secondaries are different in the two games. In such cases, the weight generation rules in Equations (5.148) and (5.149) can be replaced by the implicit relation o,(P')

w" ,P") W, + CP')

E

nifjn(P')

,P")

W"

n=1

= c,(131)c(P',P") + c(P')

E

nq„(P')Cn(P',P")

(5.150)

n=1

Equation (5.150) follows from Equation (5.140) by making use of conditions (5.142) and (5.143) to obtain W

C(P',P") = c(p' Jr)

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Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

Substituting the explicit forms of the kernels given in Equations (5.131) and (5.128), we obtain Equation (5.150). Similarly, in the case of the equivalent nonmultiplying game defined in Section 5.III.C, the rules above do not apply; nevertheless, Equation (5.140) holds under the alternative conditions of Equations (5.84) and (5.87). Condition 2 of the theorem restricts the class of functions that can be used as nonanalog contribution functions (estimators). The simplest way of satisfying this condition is to choose the nonanalog estimators identical to the analog ones, i.e., to put

This, however, is not the only possibility. In fact, the condition states that nonanalog contribution functions should also be partially unbiased. Indeed, in view of Theorem 5.7 and Equations (5.134) and (5.141), condition 2 is equivalent to the condition I1 (P) = 1(P) where I(P) is the expected score due to a flight from P followed by a collision in an analog game with the simplest contribution f(P), as given in Equation (5.133). Correspondingly, the analog and nonanalog estimators are interchangeable and they all belong to the class of partially unbiased estimators. Therefore, in nonanalog games that satisfy the conditions of Theorem 5.8, there is no need to distinguish the estimators from the analog ones. Such games will be called partially unbiased nonanalog games. Theorem 5.8 establishes the conditions under which the expected scores due to a unit weight starter in the analog and nonanalog games are equal. These conditions, however, ensure an unbiased nonanalog estimation only if the source densities (from which the particles start) are identical. In this case, the weights of the starters may be chosen equal and the equality of the expected scores per history(ies) calls forth the equality of the final scores in the two games. In other words, the theorem gives the generation rules of the weights during the simulation, but it does not fix the weight of a nonanalog starter. If the nonanalog source density differs from the analog source, the statistical weight of a starter should depend on the difference of the actual source density from the analog one. The generation rule of the starting weight is established in the following. Theorem 5.9 — A nonanalog game will yield the same final expected score as the analog game if the weight generation rules of Theorem 5.8 are satisfied and the weight of a starter at P in the nonanalog game is chosen as W = wg(P) = Q(P)/Q(P) where Q(P) and '0(P) are the source densities in the analog and nonanalog games, respectively. Proof. The final expected score in a nonanalog game is 1

= f dP 0(P)WK4 ,(P)

If W is chosen according to the theorem and the conditions of Theorem 5.8 are met, then Kii(P) = M1(P)

203 and making use of the results of Section 5.I.A,

i

d131:5(P)1Q(P)/0(P)M4,(P) = fdPQ(P)MI(P) = R

as stated. o C. A NONANALOG GAME WITHOUT STATISTICAL WEIGHTS: IMPORTANCE SAMPLING In the previous section, we proved that the statistical weights of any nonanalog game may be chosen such that the simulated particle field is identical to the analog (physical) field, provided one interprets the nonanalog particles as fractions (or multiples) of physical particles, their amount being characterized by their weights. Although this is a very reasonable method of simulation, there is no reason to exclude the possibility of games that do not reproduce the physical particle field and nevertheless result in unbiased estimates of the required quantity. Indeed, such games exist and a simple but sufficiently general construction of such games is given below. Suppose that the nonanalog and analog kernels are related as T(P,P') = T(P,P')U(P')/V(P)

(5.151)

e(P' ,P") = C(P' ,P") V(P")/U(P')

(5.152)

and

where V(P) is some arbitrary known function and U(P) is such that

f

dP't(P,P') = 1

V(P) = f dP' T(P,P') U(P')

(5.153)

In this case, the transport kernels [cf. Equation (5.14)] in the nonanalog and analog games are related as L(P,P") = L(P,P")V(P")/V(P)

(5.154)

Note that transformation of the analog collision kernel according to Equation (5.152) may result in a nonanalog kernel qualitatively different from the analog. Thus, if the analog game is nonmultiplying, i.e., if c(P') = fdP"C(P',P") = cs(P') - -c. 1 then the mean number of secondaries in the game with the transformed collision kernel is c(P') = f dP"C(P',P") V(P")/U(P')

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Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

which may be greater than unity, thus resulting in a multiplying game. Similarly, in case of a general analog kernel, the transformation may result in a collision kernel in which pure scattering also leads to multiplication. This leaves some freedom to the user in defining the absorption, scattering, and multiplication probabilities. This question will not be investigated here, but should be considered in practical applications. In what follows, we use the general form of the collision kernel as given in Equation (5.74) (omitting the irrelevant absorption term). Accordingly, the transformed kernel from Equation (5.152) reads C(P' ,P") = C,(P')es(P' ,P") + of(P') = V(P") [cs(P')Cs(P',P") + cf(P')

E

fl(P' ,P")

n=1

E

n=

nqn(P')Cn(P',P")]/U(P')

(5.155)

for every P' and P" inside the domain of simulation. We shall now derive an unbiased nonanalog game which is played by the kernels in Equations (5.151) and (5.152) and in which the statistical weight of a particle is not changed. In a sense, this game is very unusual: it is nonanalog, since the kernels differ from the analog ones; nevertheless, it resembles an analog game because of the constant statistical weights. Such games will be referred to as transformed games. They also differ from the nonanalog games treated in the previous section in that they are not expected to give unbiased "pointwise" estimates, i.e., the expected score due to a single starter will not necessarily be the same as in the analog game. Instead, we require that it produce an unbiased estimate of the final score, i.e., of the integral quantity of interest. Let M,(P) denote the expected score due to a unit weight starter from P and let Q(P) be the (yet undefined) nonanalog source density. The transformed game is unbiased if

i

dPO(P)1VA1,(P) = f dPQ(P)M,(P) = R

where lir is the statistical weight of the starter from P in the transformed game. This weight is defined by the equality itself in the sense that proper choice of it will eventually ensure the unbiased final estimate, as will be seen below. Again let f(P,P'), is(P',P"), and i„(131 ,P") be the contribution functions assigned to a transition, absorption, scattering, and n-tuple multiplication event, respectively, in the transformed game. Obviously, the transformed contribution functions are not expected to be partially unbiased and, in general, will be different from the analog estimators. The equation that governs the expected score due to a unit weight starter follows from Equations (5.129), (5.130), (5.151), and (5.152) as (P) = =

1

(P)/V(P) + fdPri(P,P') f dP" (P ,P")

,(P")

,(P)/V(P) + fdP' T(P,P') dP"C(P' ,P") V(P")

1(P")/V(P)

(5.156)

where (P) = f dP' T(P,P')U(P') [i(P,P') + Ca(P') ia(P') e's(P') .1-dP"es(P',P")is(P' ,P") + af(P')

E

n =1

,P") in(P',P")

(5.157)

205 The conditions under which the transformed game is unbiased are given in the following theorem. Theorem 5.10 — The transformed game results in the same final expected score as the analog game if: 1.

The source term in Equation (5.157) satisfies 41(P) = 1(P)

2.

I (P) being the expected analog score in a flight as given in Equation (5.133). The nonanalog source density is chosen according to Q(P) = Q(P)V(P)/fdP' Q(P') V(P')

3.

(5.158)

(5.159)

The weight of every starter is 14r = JdPQ(P)V(P)

(5.160)

Proof. Comparison of Equations (5.132) and (5.156) shows that if .11(P) = I(P), then V(P)Ait,(P) = M,(P)

(5.161)

i.e., V•At, is just the expected score due to a starter in the analog game. Furthermore, if Equations (5.159) and (5.160) hold, then 0(P)111A4 i(P) = Q(P)V(P)Ati(P) = Q(P)M (P)

fdPQ(P)14rht,(P) = fdPQ(P).4,(P) = R as stated.

The simplest way of satisfying condition (5.158) is the choice f(P,P') = f(P,P')/U(P') = w(P')f(P,P') ia(P') = ca(P')f„(P')/[ea(P')U(P')] = w(P')wa(P')fa(P') ,P") = c,(P') Cs(P' ,P") f,(P' ,P")/rCs(P') es(P' ,P")U(P')] = w(P') w„(P' ,P") f„(P' ,P") I'„(P',P") = cf(P')q„(P')C„(P',P")f„(P',P")/ lef(P') EL(P')

,P") U(P' )1 = w(P') w„(P1 ,P") f„(P' ,P")

(5.162)

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Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

where f, fa, f„ and f,, form a set of partially unbiased estimators in the analog game as defined in Section A. Alternatively, if one applies an expectation-type estimator as f(P,P') = I(P)/V(P)

(5.163)

and sets all the other estimators to zero, then Equation (5.158) is obviously satisfied since in this case L(P) = fdP'T(P,P')U(P')f(P,P') = I(P)fdP"f(P,P') = I(P) If one wants to make life still more complicated, one can further distort the transformed game by introducing nonanalog transformed kernels and, concurrently, statistical weights. The weight generation rules of the previous section then apply, with the transformed kernels in place of the analog ones. What is to be kept in mind is that distorted or not, the transformed game, in general, is not directly related to the physical phenomena. It simulates the histories of hypothetical particles, and only the final expected score coincides with the analog one. This fact is also reflected by condition (5.158) and by Equations (5.162) and (5.163), which imply that different transformed estimators are to be used in different transformed games, in contrast to the partially unbiased nonanalog games where the estimators can be chosen independently of the kernels. Looking at the transformed kernels from another point of view, they also can be interpreted as the kernels in a partially unbiased nonanalog game with statistical weights. If one plays this nonanalog game, it will most naturally yield the same expected score as the transformed game if the weight generation rules of Section B are applied and the particles are started from the analog source density. The particularity of such a nonanalog game is due to the specific forms of the nonanalog kernels in Equations (5.151) and (5.152), and manifests itself in the fact that the weight of a particle depends on the coordinates of the two phase-space points between which the actual transition or collision takes place, but is independent of the prior history of the particle. It is this feature of the nonanalog game that makes it unusual, and makes it possible for the weights, once they have been absorbed into the contribution function (cf. Equation (5.162)), to seem to disappear. Thus, the transformed game is not something radically different from a usual partially unbiased nonanalog game, but, rather, is a reformulation of a specific nonanalog game. The advantage of the transformed game is twofold. First, if the function V(P) is chosen such that the transformed kernels assume higher values in regions from where more detailed information is required and are lower in less important regions, then the simulation is concentrated on the important regions with high probability. As a result of this, reduction of the variance of the score is expected. This is why selection of coordinates from the transformed kernels in Equations (5.151) and (5.152) is called importance sampling. *6,12,18,42 On the other hand, even if one constructs a partially unbiased game that results in the same first and second moments of the score as in the transformed game, the latter has the advantage of not requiring the calculation of statistical weights. The price to be paid for importance sampling is that the starters' coordinates are to be selected from the transformed density in Equation (5.159) rather than from the analog one. Also, the determination of contributions may involve extra computations. In practical cases, it usually pays off. The equation that governs the second moment of the score in a game with importance sampling will not be detailed here. On the basis of the results in Section 5.III.B, its construction is straightforward. It will be shown, however, that importance sampling, in prin* The function V(P) is often called the "importance function". Since the same name is applied to the adjoint collision density tti* (P) and sometimes also to the expected score M,(P), we would rather not use this terminology.

207 ciple, may result in a game of zero variance. It will be proven in Section 5.VIII.A that the choice V(P) = M,(P) provides a zero variance scheme, i.e., the better the function V(P) approximates the expected score, the smaller the variance expected. D. GENERALIZED EXPONENTIAL TRANSFORMATION Importance sampling has first been applied in a simple form for increasing the efficiency of the calculation of reaction rates deep in matter. Special steps are to be taken in such estimations because of the approximately exponential attenuation of particle density with increasing depth having the consequence that the number of particles reaching very deep in matter will be rather low. It is reasonable to assume that playing a hypothetical but unbiased game in which the number of particles decreases slower (with increasing distance from the source) than in the analog game, the statistical accuracy of the estimates may be improved. Such a hypothetical game may be defined in two ways. In the first, we distort the transition kernel in such a way that the mean free flight toward the deep-laying region is increased, thus forcing the particles to penetrate deeper than they would in the analog game. This method is called path stretching and will be defined in the next Section. Alternatively, we define a hypothetical particle density as the product of the analog density and some function increasing approximately exponentially with depth. Then we transform the transport equation (and the simulation procedure corresponding to it) so as to describe the hypothetical density. The latter idea was the starting point of the exponential transformation. The first such game was introduced for slab geometry in the early works by Kahn" and Leimdorfer."-" In this section, we show that exponential transformation in its most general form is identical to the general importance sampling procedure. In the next section, we prove that path stretching also is equivalent to exponential transformation. Let us start from the first-moment equation of an analog game as given in Equations (5.126) and (5.127) and let b(P) be some, for the moment, arbitrary function. Exponential transformation of the analog first moment in a general sense means the introduction of a transformed function ht,(P) according to the relation ht,(P) = exp[b(P)]M,(P)

(5.164)

Multiplying Equation (5.126) by exp [b(P)], it is seen that the transformed moment satisfies the equation Ati(P) = P11 (P) + f dP' exp[b(P) — b(P')1T(P,P') tf dP" exp[b(P') — b(P")] C(P' ,P")

(P")

(5.165)

where ‘7,(P) = exp[b(P)]I,(P)

(5.166)

and I,(P) is given in Equation (5.127). Let us now introduce the transformed kernels as T(P,P') = exp[b(P) — b(P')1T(P,P')A(P')

(5.167)

208

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

and (P' ,P") = exp[b(P') — b(P")1C(P',P")/A(P')

(5.168)

where A(P') is such that T(P,P') is normalized to unity. Taking into account the identity P'

b(P') — b(P) =

dto)Vb(r + tw)

(5.169)

(where w is the directional unit vector component of P) and defining a "stretched cross section" as &(P) = Q(P) + wVb(P)

(5.170)

it follows from Equations (5.32), (5.167), and (5.169) that the transformed transition kernel is T(P,P')dP' = A(P') cr(P')exp [ — I dtii(r + to.),E)] dD for

P' = (r + Do),E)

(5.171)

Hence, the normalization factor becomes A(P') = 15-(P')/cr(P')

(5.172)

Accordingly, the transformed scattering kernel in Equation (5.168) reads C(P',P") = cr(P') exp[b(P') — b(P")1C(P' ,P")/(3-(P')

(5.173)

It may be noted again that the transformation (5.173) leaves some freedom in defining the probabilities of the different (transformed) collision events, similar to the case of the collision kernel with importance sampling (cf. Equation (5.152)). Proceeding in full generality, we assume that the collision kernel takes on the general form of Equation (5.74). The source term of Equation (5.165) is rewritten with the aid of Equations (5.166), (5.127), and (5.171) as 3-1(P)

= fdP' T(P,131)o-(P')0(')[f(P,P') + c,,(P')fa(P') + c,(P')IdP/Cs(P',P")fs(P',P") + cf(P')

E

n=1

nqn(P')IdP'C„(P',P")f„(P',P")]/a(P')

(5.174)

At this point, Equation (5.165) can be cast into a form which contains the transformed kernels in the second term on the RHS, yet the analog collision kernel and contribution functions appear in the source term y,. In order to obtain an equation with the transformed

209 functions only, we define the transformed contribution functions f, (a = a,s,n) by the equation fdP' i(P,P')14P,P') + a(P') fJP') + 6,(P') f dP" + ,(P')

E

nein(P')IdP' an(P' ,P")

,P")] = 21-,(P)

,P") fs(P' ,P") (5.174)

n=1

Note, however, that Equation (5.174) becomes identical to condition (5.158) in Theorem 5.10 in the previous Section if one substitutes U(P') = ii(P')exp[ — b(P')Fo(P')

(5.175)

This also means that with this substitution, the transformed contribution functions can again be constructed according to Equations (5.162) and (5.163). Having defined the transformed kernels and contribution functions, we have fully described a transformed game, i.e., established a game which yields the expected score ,M,I (P) in Equation (5.164). Furthermore, starting the particles from a density Q(P) = Q(P)e -bm/f dPQ(P)e -bm with a weight 11r = fdPQ(P)e -w) the exponential transformation yields a transformed game which is unbiased with respect to the final score, i.e., fdk)(p)iAt,(p) = fdPQ(P)Mi(P) An immediate observation is that when deriving the moment equation for the game with exponential transformation, we have simply reversed the derivation of the moment equation for the importance sampled game. Consequently, the following theorem has been proven. Theorem 5.11 — Importance sampling and general exponential transformation are equivalent if the biasing functions in the two procedures are related as V(P) = e -b(P) Then U(P') takes on the form in Equations (5.175) and (5.170).

Exponential tranformation was originally proposed for improving the statistical properties of the simulation in slab geometry, particularly in estimating transmission through thick slabs. For the sake of illustration, let us consider a homogeneous, nonmultiplying slab

210

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

situated between x = 0 and x = X perpendicular to the x axis. The analog transition kernel, Equation (5.32), reads* T(P,P') dP' = cr • exp[o-(x — x')/µ] dx 71[..LI

for

ii(x — x') < 0

0

x' < X

(5.176)

while the collision kernel has the specific form csCs(p„µ')d[il

if

0

otherwise

(5.177)

C(P',P")dP" =

where 1.1, is the cosine of the angle between the particle's direction and the positive x axis. Note that we have introduced the vacuum-equivalent purely absorbing surroundings as defined in Section 5.I.B. Assume that the only nonvanishing contribution is what is assigned to an intercollision free flight and it has the form f(x,x',µ). Then the analog first-moment equation [Equations (5.126) and (5.127)1 reads =

x•) 0

0

otherwise

(5.179)

=-

* Here and in further applications, the delta functions and the corresponding arguments in the kernels will be omitted, as if the integrations with respect to them had been performed.

211 The simplest contribution function is the weighting function itself F(x,x',11) = F(x',µ) which yields a score only if the particle leaves the region x < X. Alternatively, the simplest expectation-type partially unbiased estimator follows from Theorem 5.7 and Equation (5.176) as = f dP' T(P,P')f(x'

f(x,x',µ) =

exp[o(x — X)/II]

if µ > 0 (5.180)

0

otherwise

From the point of view of estimation, the reaction rate in question reads R=

dµ dxQ(P)M1(x,P) = M,(0,1)

The expected score M1 (x,µ) due to a starter at x in positive directionµ decreases approximately exponentially as x is moved from X toward zero. [This is also reflected by the firstflight expected score in Equation (5.180).] Therefore, it is expedient to use an exponential transformation of the form Att,(x,µ) = exp[b(p)(X — x)11V1,(x,µ)

(5.181)

that makes the expected score in the transformed game more uniform along the x axis. It is reasonable to choose the function b(p.,) such that it assumes larger values for positive directions (pointing toward the important region) and has lower values in the opposite directions. Practical realizations will be overviewed in Section 7.III.0 in connection with the optimization of path stretching. Now the transformed game will be played with the following functions. In view of Equations (5.170) through (5.173), the transformed kernels become T(P,P')dP' = O-(µ)exp[o-(p..)(x — x')/p.] dx 'dill for µ(x — x') > 0

(5.182)

'e(P',P")dP" =6-(14 — expl[b(µ) — b(p.,')](X — x')IcsCs()1,µ')dp,'

(5.183)

and

where &

= cr

11,17 (µ)

(5.184)

and in order to keep the stretched cross section meaningful, we must require that 1b(µ)1 0

(5.186)

and zero otherwise. It is interesting to note that the estimator in Equation (5.186) is again the expected value of the leakage estimator (5.185) over a flight from x in the transformed game; i.e., the transform of the analog expectation estimator is the expectation estimator in the transformed game. E. PATH STRETCHING As was mentioned in the previous Section, distortion of the transition kernel in such a way that the mean free flight of the particles is increased toward more important regions ("stretching" of the particles' flight) may also increase the number of particles reaching these regions. In general, the term path stretching (or track-length biasing) means that the analog transition kernel is replaced by a nonanalog kernel of the form am t(P,P') dP' = Cr(P') exp [

0

dt 6-(r + to) ,E)] dD; P' = (r + Do),E)

where &(P) is to be chosen small along important directions. If the scattering kernel is not biased, the first moment of the score in a partially unbiased nonanalog game with the transition kernel above satisfies the equation M i(P) = 1,(P) +

T(P,P1 )f dP"C(P',P")

Mi(P")

that follows from Equations (5.129) through (5.131). The source term of the equation is written

11(13) = f dP t(P,P')

f(P,P') + Fe(P',P)]

where Fc(P',P") is a shorthand notation of the (analog) expected contributions from the first collision as detailed on the RHS of Equation (5.130). The game is partially unbiased, i.e.,

213 M i(P) = M,(P) if the weight generation rules of Theorem 5.8 are applied. In this specific case, these rules become = W • o-(P')expt fdt lcr(r + to) ,E) — er(r + to.),E)]}/Ci-(P')

and W" = W' Now, denoting wVb(P) = &(P) — Q(P)

(5.187)

we find that the nonanalog transition kernel in the case of path stretching is identical to the kernel in Equation (5.171), introduced in connection with the exponential transformation. The difference between the two games is that in path stretching, the bias in the transition kernel is compensated for by statistical weights of the form W' = Wo-(P')exp[b(P') — b(P)1/6-(P')

(5.188)

and a partially unbiased nonanalog game is played, while in the game with exponential transformation, the factor multiplying the statistical weight in Equation (5.187) is included in the collision kernel and the transformed moment. This leads to the transformed collision kernel in Equation (5.173) and to the transformation of the source distribution. As a result of this, the exponential transformed game is played without statistical weights. We have therefore proven that importance sampling, exponential transformation, and path stretching represent different realizations (or just formulations) of the very same nonanalog simulation procedure. Path stretching is applied if, for some reason, distortion of the collision kernel is not advisable. The usefulness of path stretching in a given problem depends on the choice of the biasing function b(P). In slab geometry, the distorted total cross section follows from Equation (5.187) as a ax b(x,p,,E) 6-(x,R,E) = o-(x,E) + µ —

(5.189)

In problems of deep penetration along the positive x axis, the mean flight path of a particle is increased toward the deep-laying region if b has a negative gradient. Various possible forms of the function b will be investigated in Chapter 7. It should, however, be emphasized that although the methods introduced in the previous sections are expected to decrease the variance of the score, nothing was said about their effect on the computing time. Optimization of all the variance reduction techniques is to be performed so that the resulting efficiency is the highest possible. To estimate the efficiency of a game, the computing time per history must also be guessed. Equations describing this quantity are investigated in the next Section. F. COMPUTING TIME AND NUMBER OF EVENTS PER HISTORY Consider a general nonanalog game described by the kernels T(P,P') and C(P',P"). For the sake of simplicity, we shall assume that no splitting is played in the game. Inclusion of the time necessary for performing splitting in the considerations below is straightforward.

214

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

Let Mt(P) denote the expected value of the computing time necessary for the simulation of a nonanalog history that is started from the point P. We shall suppose that the time of playing any event is independent of the coordinates of the event. In the majority of the computer applications, this assumption is justified. Furthermore, we assume that playing an absorption does not necessitate extra computing time and the times for selecting from the various kernels in a collision (scattering, n-fold multiplication) are equal to each other. In view of the assumptions above, computing time is spent for two types of events: a time tf is necessary for playing a free flight (including the check of a possible escape and the selection of the type of collision) and a time tc is spent by the selection of the postcollision coordinates. Obviously, these time values are independent of the weight of the particle. Note that these times can be considered as contributions of the respective events in a nonanalog game, while the weight of the particle is kept unchanged. Summing up these contributions, an estimate of the computing time per history is obtained and the moment equations can be used for the analysis of the estimate. More rigorously, Mt(P) satisfies Equations (5.129) through (5.131) with the contribution functions = tf,

ia(P') = 0

fs(P',P") = fo(P',P") = to and with the weights W' = W" = Wa = NNT = W Hence MP) = tf + to fdP' T(P,PTC(P') + f dP' T(P,P')IdP"

,P")1;At(P")

(5.190)

This Equation may play a role in the a priori estimation of the efficiency of various games. In most cases, it is satisfactory to work with a simpler form of the equation where we assume that the computing time per history is simply proportional to the number of flights to be simulated Mt(P) to N(P)

(5.191)

where N(P) is the expected number of flights in a history started at P. This is the case if the time of playing a flight is tf = to, while to = 0. Obviously, N(P) is the solution of the equation N(P) = 1 + f dP' T(P , P ' ) fdP" a(P ' , P") &(P") 1 + fdP" Il(P,P") N(P")

(5.192)

On the basis of Equations (5.190) and (5.192), it is easy to establish conditions under which a nonanalog game requires less computing time than a corresponding analog game. For the sake of simplicity, let us consider the number of flights as a true measure of the computing effort and consider the analog equation corresponding to Equation (5.192): N(P) = 1 + fdP"L(P,P")N(P")

215 Let n(P) = N(P) — N(P) Then n(P) satisfies the equation n(P) = f

[L(P,P") — L(P,P")]N(P") + idP"L(P,P")n(P")

Now if the nonanalog game is feasible, then this equation has a unique solution and, according to Theorem 5.2, n(P) > 0 if L(P,P") > L(P,P") at every point of the domain of simulation. Thus, we have the following. Theorem 5.12 — A nonanalog game with the kernels T(P,P') and e(P',P") results in a lower expected number of collisions (flights) than the analog game if the nonanalog kernels satisfy the inequality

f

dr T(P,131)C(P' ,P") > fdP' t(P,P')e(P' ,P")

for every P,P" such that N(P") > 0.

Note that this simple condition is sufficient but not necessary, and in certain cases it may be needlessly strict. Nevertheless, in many practical situations it gives a reliable indication of the possible gain in computing time. For the sake of illustration, let us examine how the introduction of the equivalent nonmultiplying game instead of a multiplying game, as defined in Section 5.III.C, affects the computing time/history. Let us consider an analog multiplying game with the kernels T(P,P') and C(P',P"). The transition kernel of the equivalent nonmultiplying game is the same as that in the analog game, while the collision kernel, by Equations (5.88) and (5.83), is C(P',P") = C.(P',P") = C(P',P")/c(P') with c(P') = cJP') + cf(P') E nqn(P') n=1

This shows that C(P' ,P") > e(P',P") whenever c(P') > 1, and we conclude that the equivalent nonmultiplying game certainly reduces the computing time if the mean number of secondaries per collision in the analog game is greater than unity. We remind the reader that Equation (5.192) describes the expected number of collisions in a history where no splitting or Russian roulette is applied. Naturally, splitting increases this number, while Russian roulette decreases it. We shall see in the next section that this property of Russian roulette may even influence the feasibility of a nonanalog game.

216

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

G. FEASIBILITY OF A NONANALOG GAME Sufficient conditions of the feasibility of an analog game were derived in Section S.I.B. Under these conditions,the expected number of particles present in the domain of simulation tends to zero with a probability 1 if the number of collisions in the history tends to infinity. The conditions concern the norm of the analog transport kernel L(P,P") in Equation (5.14) and are formulated in Theorems 5.1 and 5.2. Most naturally, if the nonanalog transport kernel defined as L(P,P") = JdP'T(P,P')C(P',P") satisfies the same conditions, then the number of nonanalog particles will also tend to zero as the simulation goes on and consequently Equation (5.192) will also have a unique bounded solution. In other words, under the stated conditions, the expected number of collisions in the nonanalog game without Russian roulette will also be finite with a probability 1. Therefore, it might seem reasonable to impose the conditions of Theorem 5.1 on the nonanalog transport kernel L as well in order to ensure the feasibility of the nonanalog game. There is, however, a little problem with this reasoning; namely, nonanalog games not conforming with these conditions are widely used and are found feasible. The resolution of this apparent contradiction is that it is not the number of collisions but the total statistical weight of the particles present in the system that determines the feasibility. This is so because particles with low weights can be eliminated from the system with high probability by the application of Russian roulette and this elimination will leave the estimation unbiased. In the analog game, the statistical weights of the particles do not change and therefore the number of particles in the system is equivalent to the total weight of them. A finite number of collisions per history then implies the elimination of all the particles, i.e., the total weight of the particles. In a nonanalog game, however, the number of particles has no direct connection to their total weight. In order to illustrate the situation, let us consider a simple example. Let the quantity to be estimated be the absorption rate due to a monoenergetic particle migrating in a nonmultiplying homogeneous infinite medium. Let the total cross section of the medium be unity and let the scattering probability be c. Accordingly, the reaction rate that is estimated reads R = JdP'(P')(l — c) i.e., the weighting function is f(P) = 1 — c This weighting function will serve as the estimator scoring at every collision. Assume that survival biasing is used (cf. Chapter 3.11), i.e., that absorption is replaced by weight reduction. Hence the, nonanalog kernels are T(P,P')dP' = T(P,P')dP = e'dD

for P' = (r+Dw),

and a(P' ,P") = C(P' P" )lc = Cs(6),to')

D 0

217 The weight of a particle undergoing a collision, according to Equation (5.147), is multiplied by ws(P',P") = c Thus, if the weight of a starter is W0 = 1 and Wn denotes the weight of the particle after the n-th collision, then W„ = W n _ l • ws(P',P") = c"

(5.193)

The expected score follows from Equation (5.56) as M1 (P) = (1 — c)f dP1 T(P,P') + cidP' T(P,P')IdP"

,P") KI (P")

This equation has the solution =1 a rather trivial result. Now, in the analog simulation (where no survival biasing is applied), the expected number of collisions follows from Equation (5.192) N(P) = 1 + JdP"L(P,P")N(P") Its solution is N = 1/(1 — c) This means that the number of collisions is finite with probability 1, which also follows from Theorem 5.1 since the condition dP"L(P,P") = c < 1 is satisfied. On the other hand, the expected number of collisions in the nonanalog game is unbounded, and the conditions of Theorem 5.1 also fail to hold for fdPi(P,P") = JdP"

=1

However, it is seen from Equation (5.193) that the total weight of the particle tends to zero with an increasing number of collisions. Assume that Russian roulette is applied in the nonanalog game in the following way. If the weight Wk of the particle becomes less than some threshold value wth, it is turned to unity with a probability Wk and is set to zero with the complementary probability. Now if wth is such that Ck < Wth < Ck - 1

the Russian roulette is played after every k-th collision and the particle survives with a probability W, = ck. Hence, the probability that the particle is alive after N = m • k collisions is c", which means that the particle is removed from the system with a probability

218

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

1 if the number of collisions tends to infinity. Therefore, we conclude that although the nonanalog kernels do not satisfy the conditions of Theorem 5.1, the game is feasible provided Russian roulette is applied. Two comments are to be made here: 1. Notice that from the point of view of the number of collisions, Russian roulette manifests itself as an artificial extra absorption, thus making the otherwise unfeasible game feasible. 2. Strictly speaking, the nonanalog kernels with and without Russian roulette are different and therefore when examining the feasibility of the nonanalog game, the kernels should be modified to account for Russian roulette (as introduced in Section 5.III.D). The modified kernels then satisfy the conditions of Theorem 5.1 in most practical cases. Nevertheless, we shall investigate the conditions of feasibility of a nonanalog game without supposing explicit inclusion of Russian roulette. More precisely, we first examine the conditions under which the total weight of particles in a nonanalog game tends to zero. Next, we prove that if the weights tend to zero, then Russian roulette yields a finite game with a probability 1. It turns out that any partially unbiased nonanalog game that corresponds to a feasible analog game is also feasible. Theorem 5.13 — Given an analog game which is feasible in the sense of Theorems 5.1 and 5.2. If the statistical weights in an arbitrary nonanalog game are chosen according to the selection rules of Theorem 5.8 (i.e., if the nonanalog game is partially unbiased) and the weights so generated are bounded at every point of the nonanalog domain of simulation, then the total weight of the particles present in the domain of simulation tends to zero as the number of collisions in a history tends to infinity. Proof. For nonmultiplying games, the theorem is proven in Reference 29. Here we give the proof for a general multiplying game. Let yk(P,W)dWdP be the probability that a particle leaving its k-th collision in dP about P has a weight in dW about W. (The number k also comprises all the collisions suffered by the ancestors of the particle back to the source.) For convenience, source particles are considered to leave their 0-th collision. Thus 4:).(1),W) =

wq(P)] 0(P)

(5.194)

where wq(P) = Q(P)/Q(P)

(5.195)

the statistical weight of a starter at P, as shown in Theorem 5.9. It is heuristically obvious that Yk is the sum of the probabilities assigned to the various collision events. More precisely, let q4,$) (P,W)dWdP be the probability that a particle leaving its k-th collision via scattering in dP about P has a weight in dW about W. Similarly, let cp;,'"(P,W)dWdP be the same probability if the particle is the i-th in an n-fold multiplication. Then (Pk(P,W) = 40P,W)

E E 4''')(P ,W)

(5.196)

n = 1 i =-- 1

On the other hand 4)(P",WNW" = f dP (Pk - i (P,W)f dP' i(P,P') Cs(P') s(P' ,P") dW

(5.197)

219 where, according to the weight generation rules in Equations (5.142) through (5.147) W = W"/[w(P,P1 ) ws(P' ,P")] = W"T(P,P1 ) Cs(P')

,P")/1T(P,P') cs(P')Cs(P' ,P")

(5.198)

The balance Equation (5.197) is interpreted as follows. The probability of a given weight after a scattering is the integrated product of the probabilities of having a weight conforming with the weight generation rules after the previous collision and of making a free flight followed by a scattering. Similarly, for the i-th progeny in an n-fold multiplication (p;:''')(P",W")dW" = fdP q),(P,W)fdP1 T(P,P') of(P') ein(P') 6,;)(P' ,P") dW (5.199) where a;;)(P',P") is the density of the postcollision coordinates of the i-th particle, which, in our treatment, is assumed to be independent of i. The weights in Equation (5.199) are related as W = W"/[w(P,P')w,,(P',P")] = W"t(P,P') CXP') ein(P') „(P' ,P")/1T(P,P') cf(P') qn(P')Cn(P' ,P")]

(5.200)

Now, if 4k(P)dP denotes the expected value of the weight of the particle when leaving its k-th collision in dP about P, i.e., if 4.k(P) = f dW W(pk(P,W) and k)(P)dP and kr''')(P)dP are the corresponding eventwise expectations, then from Equation (5.196) cc

a>k(P) = ae(P)

n

+E E (!)t" n=1=

Multiplying Equations (5.197) and (5.199) by W", making use of the relations in Equations (5.198) and 5.200), and integrating with respect to W, we obtain the recurrence it'k(P") = fdP (1)k -i(P)L(F,P")

k = 1,2,...

(5.201)

where, as before L(P,P") = idP' T(P,131)C(P1 ,P") the transport kernel. For k = 0, Equation (5.194) gives 09 ( P ) = Q(P) Repeated use of Equation (5.201) and insertion of Equation (5.202) yields 4k(13") = fdPQ(P)Lk(P,P")

(5.202)

220

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

where the iterated kernel Lk is defined in Equation (5.18). The total expected weight of the particles that leave collisions anywhere in the domain of simulation is 41, = fdP"4)k(P")

fdPQ(P)f dP"Lk(P,P")

The salient point of the proof is that the expected weight is expressed by the analog iterated kernel Lk. Now, if the analog game is feasible, then the conditions of Theorem 5.1 hold, i.e., if k = m•N + i where N is the threshold number in Equation (5.17) and 0 i< N, then

f

dP"Lk(P,P") b"B'

i.e., 4k bmB'

and since b < 1 and B < + co limek = 0 as stated.

The theorem above does not yet establish the feasibility of a partially unbiased nonanalog game. It remains to prove that a history in which the statistical weights tend to zero can be terminated in a conservative (unbiased) way after a finite number of collisions. We have already referred to the Russian roulette procedure as an unbiased termination procedure that preserves the expected value of the particles' weights. In the proof below, we consider Russian roulette as a special case of collisionwise splitting. We suppose that a particle leaving a collision at P with a weight W is killed with a probability z„(P,W) and is left alive with a probability z,(P,W) = 1 — zo(P,W) In view of Equation (5.97), the survivor leaves the roulette with a weight W/z,(P,W). We note that extension of the considerations below to more general splitting procedures is laborious, but leads to conclusions similar to those of the following. Theorem 5.14 — If the probability of a survived Russian roulette, z,(P,W), is such that lim z,(P,W)/W = Z < + w—o

(5.203)

then the number of collisions is finite with probability 1 in any history where the statistical weights tend to zero. Proof. It is to be shown that the probability of an infinite number of collisions is zero. Let „ be the probability that a particle survives n successive Russian roulette procedures. If

221 P„ and WI , W2, " • Wn, respectively, denote the coordinates and weights of the P1, P2, particle after the respective collisions (but before roulette), then 9 n = 11 z,(P„W,) =1 On the other hand, if would be the weight of the particle after the i-th collision in the same game without roulette, then W, = W',' and

-

Wn = W1,/ II z,(P,,W,) i=

n = 2,3,...

Inserting this relation into g"„, the n-fold survival probability reads 9'„ = W,z,(Pn,Wn)/W„ Now, according to the assumption of the theorem, the weight in the game without splitting tends to zero as n increases, i.e., lim W = 0 and, by Equation (5.203), z,(Pn,Wn)/W„ remains finite even if W„ vanishes. Hence lim 9'„ = (lim

zn)(Pn,Wn)/W„ = 0

thus establishing the theorem.

Note that the simplest form of the Russian roulette as introduced in Section 3.11 and Section 5.III.D conforms to the conditions of the theorem as there W/wsp(P)

if W w,h(P)

1

if W > w„,(P)

z,(P,W) =

with ws,,(P) > w,h(13) and therefore zi(P,W)/W for every weight value W.

1/wth(P) < c")

222

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

H. DELTA SCATTERING In certain applications, it is desirable to generate a greater number of collision points in a history than results in an unaltered analog game. This may be reached for example, by using track-length biasing (cf. Section E) if the biased total cross-section is chosen higher than the analog one (path shrinking). The same effect can, however, be obtained in a modified analog game as follows. Let us define a modified total cross section instead of the analog cross section u(P) as cr8(P) = cr(P) + cro(P) where o-o(P) is some given nonnegative function. Let Ts(P,P') = cr,(P') exp [ — ir, dt o-s(r + tw ,E)] be the modified transition kernel to be used in the simulation. So far, the procedure is identical to that in path stretching. Now let q,(P) = o-o(P)/cr,(P) Obviously, q8(P) is between zero and one. Let the game be played in the following way: Let the particle leave an analog collision at P. Let us select a free flight from T,(P,P') and move the particle to P'. With a probability 1 — q,(P'), let us play an analog collision and go to step 5. With a probability qJP'), the directional and energy coordinates of the particle are left unchanged (a delta scattering is played) and we return to step 2. 5. The scores are determined according to the analog events between two analog collisions. 1. 2. 3. 4.

First, it is to be proven that the delta-scattering game above is unbiased, i.e., that the analog collision density produced by it is identical to that in a game without delta scattering. This follows from the following. Lemma — The probability density of the distance between two analog collisions is identical to the analog transition kernel. Proof. Let T*(P,P') denote the probability density that a particle starting from the point P has its next analog collision in the delta-scattering game about P'. Then it is to be shown that T*(P,P') = T(P,P') Now, T.(P,P') satisfies the equation P'

T*(P,P') = T,(P,P')[1 — q,(P')] +

dP, T,(P,Pi)q,(131)T*(P„P')

(5.204)

since according to steps 2 through 4 above, having selected a flight from T,(P,P1 ), the particle will either suffer an analog collision [with a probability 1 — qs(P'); this is expressed by the first term] or the selection procedure is restarted at the point of delta scattering [with

223 a probability qs(P'), represented by the second term on the RHS]. Let us write T. in the form T,(P,P') = o-(P')e-'•(P.P') and also Ts as Ts(P,P') = us(P')e -'&("') where, according to the definition of Ts P'

P' Ta(P,r) = j"

dto-s(r +tw,E) = J dtlo(r +to.),E) + cr„(r + tw,E)]

T(P,P') + To(P,P1 ) Then Equation (5.204) reads P' = o-(P')e--r8(P•P') +

dP,o-,(P,)e—ra("0 o °(P ) o-(P')e-'-(PI•P') ga(Pi) P'

= cr(P')e-'0.P") + o-(P') f dP, o-„(P,)expl — TAP,P1)

T*(P1,P')}

where we inserted the explicit value of qs(P'). Now, if P'

dt o(r + tw,E) = T(P,P')

T*(P,r) =

i.e., if T*(P,P') = T(P,P') then —Ts(P,P,) — T*(P„P') = —T(P,P,) — To(P,P,) — T(P1,P') = —T(P,P') — To(P,P1) and the Equation reads e -'("') =

e—'8(P'P') + e —T(PY.) fP P'

= eTgP'P')

+

dPI uo(P,)e-'°"')

e -"r(P•P')I1 — e - T°(P.P')1

=

e -'("')

i.e., Equation (5.204) indeed holds with T.(P,P') = T(P,P'). In order to complete the proof, it remains to show that Equation (5.204) has a unique solution. This is easily proven by realizing that the equation satisfied by the difference of any two possible solutions has only the trivial zero solution.

Thus, we proved that if the scoring procedure in the delta-scattering game is identical

224

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

to that in the analog game (i.e., if only analog events contribute to the score), then the delta-scattering game is unbiased. One may, however, also wish to assign scores to flights ending in delta scatterings. This can be done in an unbiased manner by examining the firstmoment equation of the delta-scattering game. The transition kernel in this game is T,(P,P'). Although delta scattering is defined through an alteration of the analog transition kernel only, let us realize that by requiring a straight-ahead scattering whenever delta scattering occurs, we also have implicitely altered the collision kernel to Co(P' ,P") = [1 — q,(P')] C(P' ,P") + q,(P') 8(P" — P') With these kernels, the expected value of the score due to a particle starting from P in a delta-scattering game satisfies an equation of the form M JP) = IV(P) + fdr Ts(P,P') dP" Cb(Pi ,P") MI (PH) where I,(P) represents the expected score due to a single flight and collision (irrespective of whether it ends up in an analog collision or in a delta scattering). The expected partial score I,(P) is to be chosen such that the solution of the moment equation is identical to that of the analog first-moment equation. The first-moment equation above is detailed as MI(P) = Is(P) + f dP'Ts(P,131 )q,(r)M i lln + f dP' T,(P,P') [1 — q,(P')]1 dP"C(P ,P")M,(P") On the other hand, the analog moment equation is given in Equation (5.126) as Mi(P) = I,(P) + fdP' T(P,P') f dP"C(P' ,P")M,(P") Comparing the two Equations, we have Ib(P) = I,(P) + f

IT(P,P') — T8(P,P')[1

q,(P')ilfdP"C(P',P")M,(P")

— fdP' T,(P,P')q,(P')M,(P') The second term on the RHS can be rewritten according to Equation (5.204) (with T. = T, in view of the lemma). Thus P'

I,(P) = (P) + f dP' dP, T,(P,Pi)cls(PI)T(P„P') dP" C(P' ,P") NI (r) — fdP, Tb(P,P1)(101)Mi(Pi) d P"C(P',13")Mi(P") = (P) + JdP,Ts(P,P1)gs(PI){J~dP'T(Pl P')f Finally, according to the analog moment equation, the terms in on the RHS reduce to —I,(P,), i.e., 8(P) = I,(P) — f dP, T,(P,P,)q,(P,)I,(P,)

1

(5.205)

225 Equation (5.205) determines the relation between the expected scores in the two games due to a flight-collision event pair. Thus, if the partial score in the delta-scattering game satisfies Equation (5.205), the game is unbiased. In the special case of the simplest collision estimator, i.e., if Ii (P) = f dP' T(P,P')f(P') and IS(P) = fdP' Ts(P,P')f,(P') Equation (5.205) yields T(P,P' )F(13' ) = fdP1 T,(P,P')F,(P') + fdP,T,(P,P,)q,(P,)1 dP1T(P,,P')F(P') Multiplying Equation (5.204) by f(P') and integrating with respect to P', it is seen that f,(P') = f(P')[1 — qb(P')] satisfies condition (5.205). The result is heuristically obvious: the probability of an analog collision at P' is 1 — q,(P'); therefore, in any collision (analog or delta scattering), the score is equal to the analog score times the probability of an analog collision. It is equally obvious that if the contribution function in the analog game is additive, i.e., if f(P,P') = f(P,P,) + f(Pi,r) (this is the case with the track-length estimator), then fs(P,r) = f(P,P') This can again be proven by multiplying Equation (5.204) by f(P,P'), integrating over P', and comparing the result to Equation (5.205). A special application of delta scattering can substantially simplify the simulation. If the artificial cross section cro(P) is chosen such that the modified cross-section cr,(P) is independent of the position of the particle, then free flights can be simulated without regard to possible crossings of boundaries between two different media. '1'45 This may make the tracking much faster, since there is no need to calculate geometrical and optical distances between collision points and region boundaries. Note, however, that the gain in computing time by this trick again is deteriorated by the loss due to the increased number of collisions. To minimize this loss, it is advisable to choose Q6(P) = cr max(E)

cro(P) = o-max(E) — if(P) where o-ma„(E) is the maximum cross-section of the material, composing the system, at the given energy E.

226

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

The idea of delta scattering has served as a theoretical tool for introduction of the tracklength estimator.42 It also has recently been applied in practical calculations in connection with correlated games.19'4" The second-moment equation of the game follows easily from the analog moment equation derived in previous chapters by simply substituting T, and C, in place of the analog kernels, and will not be detailed here. Similarly, the number of collisions to be played in a history will satisfy Equation (5.192) with the respective kernels. The heuristically obvious fact that delta scattering increases the number of collisions per history can also be seen on the basis of Theorem 5.12.

VI. PARTIALLY UNBIASED ESTIMATORS We have seen in Section 5.V.A that a set of estimators Slf(1),13'),fa(P'),f,(P',P"),{nfe(P',P")},7=

S{f,f.,f„{nf.}}

(5.206)

result in an unbiased estimate of the reaction rate R = fdPtti(P)f(P) in an analog game governed by the kernels T(P,P'), C,(P',P"), and {Cn(P',P")}:_ , if the relation

J

dP'T(P,P')[f(P,P') + ca(P') fa(P') + cs(P') fdP" C,(131 ,13")f,(P' ,P")

+ cf(P')

E

qn(P') f dP"Cn(P' ,P") nfn(P' ,P")] --= f dP' T(P,P f(P') = I(P)

(5.207)

n=1

holds. It has also been seen in Section 5.V.B that any nonanalog game that satisfies the weight generation rules of Theorem 5.8 is also unbiased with the same estimators. Estimators in the set (5.206) that satisfies Equation (5.207) were called partially unbiased estimators as any such set results in the same expected partial score in a flight followed by a collision. Obviously, the simplest partially unbiased set is Slf(P'),0,0,1011 Stf,fa,Unfnlen% ,} is a shorthand notation of the estimation procedure in which f(P,P') is scored when a free flight from P to P' is played, fa(P') is the score assigned to an absorption at P' , and f,(P' ,P") and fn(P',P") are the scores if a particle emerges at P" from a scattering or from an n-fold multiplication at P', respectively. [This means that if in an n-fold mulP'('„), respectively, then the score from tiplication the secondaries emerge at P'(',), this event is

E fn(P',V(%))

i=1

as detailed in Section 5.III.A.1 In an analog game, we have a certain freedom in defining the separate scores; namely, the score assigned to a free flight can also be attached to the scores due to the different

227 possible events in a collision, provided the expected score in a free flight + collision process remains unchanged. This idea is a simple illustration of the possible variety of the partially unbiased estimators and also a motivation for seeking transformation procedures that leave the expected score unaltered. A simple example of the heuristic arguments above is that the basic estimator Slf(P'),0,0,{0}} is equivalent to the estimator set SI0,f(P'),f(P1 ),IfOn1:-11 since both sets give the same score, f(P'), at every collision point P'. Note, however, that the two sets do not necessarily yield the same variance, as follows from the results of Section 5.III.B. Let us introduce the following notations fEs(F) JdP"Cs(P',P")fs(P',P")

(5.208)

fEnan = nfdP"C„(P' ,P")f„(P' ,P")

(5.209)

and

Evidently, fEs(r) is the expected score from a scattering at P' and it will be called the expected scattering estimator. Similarly, fEn(r) is the expected n-fold multiplication estimator. With these notations, Equation (5.207) becomes fdP' T(P,P') [f(P,P') + cJP')f(P') + cg')fEs(P') + OP') E ch(P')fEn(P')] n=1

= fdP' T(P,P')f(P')

= I(P)

(5.210)

Accordingly, the set Stf,fa,f,s,{f„}:= j that satisfies Equation (5.210) is partially unbiased. For convenience, in the following derivations let us denote p„(P') = ca(P') ,

Pi(P') = cs(P')

p„+ ,(P') = cf(P')q„(P') ,

n = 1,2,...

(5.211)

and go(P,P') = fa(P') + f(P,P'), g„,,(P,P') = f,(P') + f(P,P'),

81(13,P) = fEs(P') + f(P,P') n = 1,2,...

(5.212)

With these notations, the set of estimators in Equation (5.210) is rewritten as S{0,{g,};_ 0} and the Equation itself, which expresses the condition that the set be unbiased, is rewritten as fdP' T(P,P')

E p,(P')g,(P,P1) = 1(P)

i=0

where, according to Equation (5.211)

E P,(13 ') = 1

i=0

(5.213)

228

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

In the following Section, transformations are derived that allow us to generate an arbitrary number of partially unbiased estimator sets from a known such set. Since S{f(P'),0,0,{0},7_ ,} is known to be partially unbiased, the transformations make the derivation of new estimators possible. It will be shown in Section B that the most frequently used estimators follow from the transformations. The variance of the score by the most common estimators is estimated in a simplified transport model in Section C. A. TRANSFORMATION THEOREMS Notice that Equations (5.207), (5.210), or (5.213), which define the partially unbiased estimators, are built up by taking successive expectations. First, the expectation of the contributions over the possible postcollision coordinates is calculated in Equations (5.208) and (5.209). Next, these values are averaged with the probabilities p, [in Equation (5.211)] over the next events and finally, the expectation over the next flight is taken according to the density T(P,P'). The transformations of the estimators to be derived below are based on the fact that several random functions may have the same expectation with respect to a given distribution. In the following theorems, recipes are given for the generation of random functions with identical expectations. Theorem 5.15 — Given a function g(P,P') of deterministic coordinates P and a random vector variable P'. If the random coordiantes P' are distributed according to the density function F(P,P'), then with an arbitrary function X(P,P,,P'), the random function g(P,P') = f dP, [F(P,P,) X(P,P„P') g(P,P, )/fdP2 F(P,P2) X(P,Pi ,P2)]

(5.214)

has the same expectation as g(P,P'), provided the integrals in Equation (5.214) exist. Proof. The proof is elementary. Taking the expectation of k(P,13'), we have

f

dP' F(P,P')g(P,P') = f

i

[F(P,131 ) g(P,P, )

dP F(P,131) X(P,P„P')/fdP2 F(P,P2)X(P,P„P2)]

= f dP, F(P,P,) g(P,P,)

If we insert F(P,P') = CJP,P') or F(P,P') = Cn(P,131), then this theorem is applicable for the construction of scattering and multiplication contribution functions that are transformed to the same expected scattering estimator fEs in Equation (5.208)] or expected multiplication estimator [in Equation (5.209)], respectively. A special form of transformation (5.124) was first proposed by Maiorov and FrankKamenietzky35 and was generalized in Reference 26 to the form above. Obviously, this theorem makes it possible to generate partially unbiased estimators that depend only on the starting and end points of an intercollision flight (P and P', respectively), but are independent of the type of event in the collision terminating the flight. Indeed, putting F(P,P') = T(P,P') and g(P,P') = f(P'), the transformed estimator g(P,P') defines the partially unbiased set Sfk,{0},_ 01. The commonly used estimators follow from this transformation. Nevertheless, for specific purposes (e.g., for the estimation of energy deposition or fission energy), reaction-dependent estimators are also required. A transformation for the general estimator set in Equation (5.12) is defined in the following theorem.

229 Theorem 5.16 — Given a set of unbiased estimators SIO,{g,};_,,I, let X,(P,P,,P') (i = 0,1,...) be arbitrary functions. Let a„(P,Pi) (i = 0,1...) be some functions that satisfy the relations

E

=0

j = 0,1,...

aJP,P1) = 1),(131)

(5.215)

where the probabilities pi are introduced in Equation (5.211). Then the transformation g,(P,P') = fdP, IT(P,Pi )X,(P,P„FlE aij(P,PI)g,(P,PI)]/ i=0 fdP, T(P,132) X,(P,131,P2)P,(P2)}

(5.216)

yields a partially unbiased estimator set, S{0,{g,};_ 0}, provided the integral in Equation (5.216) exist. Proof. It is to be shown that I,(P) = f dP' T(P,P')

E pi(pygg,r) ]=0

= fdr T(P,P') E p,(P')g,(P,P') = I,(P) =0 Multiplying Equation (5.216) by T(P,P')p,(P'), integrating with respect to P', and summing over i yields fi(P) = fdPiT(P,131)

EE

i-oi-o

a„(P,Pdgi(P,P,)

where the integral in the denominator of Equation (5.216) has dropped out the same way as in the proof of the previous theorem. Making use of Equation (5.215), we have I,(P) = fdP, T(P,131 )

E

,=0

R(P,)gj(P,P,) = Ii(P)

as stated.

Note that it was assumed in the derivation that the order of summation in the double series can be interchanged. This would necessitate some further assumptions on a„; however, in practical cases, the summation is never extended to infinity (the upper limit equals the maximum number of progenies in a multiplication) and this problem vanishes. The simplest way to satisfy Equation (5.215) is to choose a, as a.„(P,131 ) = b,(P,P1)13;(131)

(5.217)

230

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

with some b, that satisfy

E=0 b,(P,P,) = 1

(5.218)

Combining the results of the theorems above, it is seen that given a partially unbiased l can be generated by transestimator set Sfg,{gr„1, a new set of estimators forming g (the score in a flight) according to Equation (5.214) and the reaction-dependent scores g, with the aid of Equation (5.216). For the sake of completeness, we show that any partially unbiased estimator is the transform of the simplest estimator f(P'). Theorem 5.17 — If SIO,{g,};_ 01 is a set of partially unbiased estimators, then there exists a transformation of the form in Equation (5.216) that brings the basic estimator S{O,f,f,{f}:= ,} into S{0,{g,};=0}. Proof. It is to be shown that there exist functions X,(P,P,,P') and b,(P,P,) (i = 0,1,...) such that

war_ o

g,(P,P') = JdP, {T(P,P,)X,(P,P,,P')b,(P,PORP,)/ dP2 T(P,P2) X,(P,PI,POIVP2)}

J

(5.219)

where the b,'s are related to the quantities in Equation (5.215) as b,(P,P,) =

E ao(P,P,)

j=0

and

E b,(P,P,) = 1

i=0

Simple substituions shows that if the g,'s form a set of unbiased estimators, then they follow from transformation (5.219) by putting X,(P,P,,P') = g,(P,P') and b,(P,P,) = JdP' T(P,P')g,(P,P')p,(P')/1 dP' T(P,P') f(P')

Finally, a trivial feature of the partially unbiased estimators is stated.

231 Theorem 5.18 — Any linear combination of partially unbiased estimator sets is also partially unbiased, provided the linear combination coefficients add up to unity, i.e., if (k = 1,2,...,N) are sets of partially unbiased estimators, then so is the set

E akg(k), E N

N

s tk=,

gck)

tk_,

Ii_01

if N

E

k=I

ak =

In the following Section, the commonly used estimators are introduced through transformation of the weighting function f(P). B. COMMONLY USED ESTIMATORS The simplest unbiased estimator was seen to be the weighting function f(P) in the reaction rate (5.2). The estimator is called the collision estimator and it scores f(P,) at the collision points 13,. If one is only interested in an unbiased estimate of the reaction rate irrespective of the computational effort involved, then this estimator serves the purpose well and there is no reason to seek other possible contribution functions. In practical cases, however, the amount of computation necessary to reach a given accuracy is almost as important a factor in the construction of a Monte Carlo game as the unbiasedness. Therefore, it usually pays to choose appropriate estimators for a special-purpose Monte Carlo computational scheme. In certain cases, the collision estimator is obviously inadequate to the problem. If, for example, some reaction rate is to be estimated in a region whose characteristic dimensions are small compared to the mean free path of the particle (e.g., estimation in thin layers), the probability of having a sufficiently large number of collisions in the region is small. In such cases, an estimator which also scores if the history crosses the region but no collision occurs in it would be more useful. Similarly, estimation of reaction rates in regions which are reached by only a small fraction of the histories (e.g., deep-penetration problems) is very inefficient with estimators that score only at points actually crossed by the history. Instead, the use of estimators that score "from far" would be more advantageous. Such estimators were introduced in Section 3.II.D and, on a more general level, will be generated in this section. In early Monte Carlo applications, estimators that score only in case of absorption were frequently investigated. Such estimators are called last-event estimators. They also are special cases of transformation (5.219) by putting x,(P,P,,P1 ) =

—13')

and b,(P,PL ) = 8,,„ Then from Equation (5.219), the transformed estimators take on the form g,(P,P') = 8,,„,f(P')/c(P')

(5.220)

232

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

This means that the set S{0,f/ca,0,101:_,} is also partially unbiased and it scores f(P')/ca(P') in an absorption at P' and zero in any other reaction. Similarly, the whole estimation procedure can be assigned to any of the various possible collision events, the probability of which is different from zero. Thus, by putting = 8(13, and b(P,P,) = 81,k we have g,(P,r) = 8,,kf(13')/Pk(r) where, according to Equation (5.211), p, = cs and p„ = cfq„ (n = 1,2,...) and the transformed estimator scores in scatterings if k = 1 and in n-fold multiplications if k = n + 1. The opposite statement is also true: given a set of reaction-dependent estimators SIO,{g,} _ 01, there exists a transformation which makes the estimation procedure independent of the type of collision, i.e., that results in a set of the form so,{0,_,j. To see that, let us again choose = 8(131 — P') and let aii(13,131) = pi(P1)13031) Then, from the transformation in Equation (5.216), we obtain the new estimators gi(p,r) =

E pi(P')g,(P,P') = g(P,P')

j 0

independently of i. Such estimators are called composed estimators. g(P,P') is the expectation of the estimators g,(P,P') over the possible postcollision events. Substituting the reaction probabilities in place of p, according to Equation (5.211), and the original estimators according to Equation (5.212), the composed estimator reads g(P,P') = f(P,P') + ca(P')fa(P') + cs(P')fEs(r) + cf(P')

E

n=i

qa(r)fE„(P')

(5.221)

Since g(P,P') is independent of the type of events, the estimator set S{0,1-gr_ o} is equivalent to the estimator set sfg,017 01, which scores g(P,P') in a flight from P to P' and zero in any other event. (Equivalence again means the equality of the expected scores with possibly different variances.) The commonly used estimators are of the type SIg,{0};_ 0l, and in what follows they will be derived from the collision estimator f(P') through the reaction-independent transformation of Theorem 5.15, Equation (5.214).

233 Let us consider transformation (5.214) with the estimator g(P,P,) = f(P,) and with the density function F(P,P,) = T(P,P,): f(P,P') = f dP,[T(P,P,)X(P,P,,131)f(P,)/1 dP,T(P,P2)X(P,P„P2)]

(5.222)

The explicit form of the transition kernel is given in Equation (5.32) as T(P,P')dP' = cr(P')exp{ —

dto-(r + tw,E)}8(E — E')dE'dD

0 and zero otherwise. Obviously, it only depends on the if P' = (r + Dw,E) with D distance D of the point P' from P and is nonzero only along w from D = 0 to D = + c. Let D, and D2 be the distances of P, and P2 respectively, from P along w, i.e., let P, = (r + D,63,E) and P2 = (r + D2w,E), and let us introduce the shorthand notations* T(P,P')dP' = T(D)dD and f(131) = f(D1), X(P,P„P') = X(D,,D) Then the transformation in Equation (5.222) reads f(D) = I dDi [T(D,) x(D, ,D) f(D,)/f dD2 T(D2) x(D 'DO]

(5.223)

Now, choosing the free function X as X(D,,D) = a,

if D

d

(5.224)

with some arbitrary constant a and distance value d, the transformed estimator follows from Equation (5.223) as f(D) = f dD,T(DI )f(D1 )/[1 — e —T(d)]

(5.225)

where T(d) = I dtcr(r + tw,E) is the optical distance from P to Pd = (r + dw,E). For d = + co, Equation (5.225) defines the well-known expectation estimator f(D) = fE(P) -= f dP' T(P,P1 )f(P') = 1(P)

*

Cf. footnote to Equation (5.176).

(5.226)

234

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

which gives the expected value of the score over the next flight from P. In a homogeneous medium, if f(P,) = f for DI < L and zero otherwise, the expectation estimator scores fE(P) = f(1 — e-') On the other hand, if d in Equations (5.224) and (5.225) is the distance to the boundary of the region, inside which the reaction rate is to be estimated, from the starting point P inside the region, then f(D,) = 0 if D, > d and the transformed estimator scores I(P)/[1 — e - r( '"[

if D

0

if D > d

d (5.227)

f(D) = fE,(P,P') =

i.e., it gives a contribution independent of the site of the next collision point, provided this collision takes place inside the region of interest, and the score is zero if the particle leaves the region. Combined with a nonanalog game which forces the particle to stay inside the region, this estimator is used in the expected leakage probability method proposed by Kschwendt and Rief.2" This method will be reviewed in Section 5.VIII.D. Let us now consider a function opposite to that in Equation (5.244), i.e., let if D d

X(D,D i) = a , and zero otherwise. Then the transformation yields

d

I(P)e (d>

if D

0

if D < d

(5.228)

f(D) = fE2(P,P') =

This estimator scores only if the free flight from P is longer than d. According to Theorem 5.18, any normalized linear combination of the estimators in Equations (5.227) and (5.228) is also partially unbiased, i.e., for arbitrary function a(P), the estimator fsg,13 ') = a(P)fEl(P,r) + [1 — a(13)] f E2(P,P') is also partially unbiased. This estimator has the explicit form fsi(P,P r ) = {a(P)X(Pd,P') + [1 — a(P)] X(13',Pd) [e"*" — 1]} I(P)/[1 — e --r"d"]

(5.229)

where if d D x(Pd,P') = 1 — X(P',Pd) =

0

if d < D

The combination coefficient will be chosen in Section 5.VIII.H such that the resulting estimator will be a good approximation of the minimum-variance partially unbiased estimator.

235 The general form of track-length estimators is obtained from the transformation in Equation (5.223) by putting X(D,D i) =

if D, > D

and zero otherwise. In this case, the transformed estimator in Equation (5.223) becomes f(D) = fT(P,P') = J dD, f(DO[T(D1)/1 dr:02T(D2)] Di

(5.230)

With the explicit form of the transition kernel, it is written fT(P,P') = f dp f(r + D, ,E) cr(r + Dio.),E)

(5.231)

Again, in a homogeneous medium, if f(P,) = f for D, < L and is zero otherwise, the tracklength estimator scores fT(P,P') = fo • min(D,L) Hence, the score is proportional to that part of the flight length which lies inside the region of interest. Obviously, the track-length estimator gives a contribution whenever a part of a flight is traveled in the region, irrespective of whether the next collision is inside or outside the region. The estimators obtained through transformations of the simplest estimator f(P) in this Section were all introduced by heuristic arguments during the long history of the transport Monte Carlo methods. Several other special estimators occasionally used in practice are also special cases of the transformation (e.g., the special track length-type estimators proposed in Reference 42). The transformation, however, may also provide new estimators which do not follow from obvious heuristic reasoning. For example, consider the estimator which is the track length-type transform of the expectation estimator. Inserting the expectation estimator I(P) in place of f(P) in Equation (5.230), we obtain a new estimator of the form f(D) = fT„(P,P') = I(P)T(D)

(5.232)

Here, •r(D) is the optical distance between P and P'. This estimator was called "trexpectation estimator" in Reference 26 since it is a hybrid of the track-length and expectation estimators. One may contemplate whether it unifies the advantages or the disadvantages of these estimators. It turns out that for light absorbers in optically not-too-small regions, it results in a lower variance than both the track-length and expectation estimators. For optically thin regions, it is much worse than any of them; otherwise, it resembles the track-length estimator. A very important point should be emphasized here. The estimators introduced in this Chapter were derived under the assumption that the transition kernel is normalized to unity. This assumption has been repeatedly exploited in the various forms of the transformations. It has been seen in Chapter 5.I.0 that assuming a vacuum-equivalent black absorber around the domain of simulation, the transition kernel can always be normalized to unity. When applying the above estimators, this normalization should always be performed or, if for some special reason (e.g., for the introduction of some special nonanalog game) it is not convenient, the estimators must be used in their proper form, which accounts for the finite probability of an endless free flight. The respective formulas are derived in Reference 26. The variety of partially unbiased estimators, of course, has its own theoretical interest.

236

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

Nevertheless, from a practical Monte Carlo point of view, their usefulness lies in the fact that in different problems their relative efficiency is different. Investigation of the relative variances of the estimators is postponed to Chapter 5. VIII. Nevertheless, in order to obtain some insight into the main characteristics of the various estimators, the following Section is devoted to investigation of the variances of various estimators in a highly idealized transport model. C. ANALYSIS OF VARIANCES IN THE STRAIGHT-AHEAD SCATTERING MODEL The straight-ahead scattering model is a favorite tool of approximate analytical Monte Carlo and transport theoretical calculations. In this model, the particles are assumed to propagate along a straight line and a collision may result in either an absorption or an emission of one or more particles with a direction identical to the direction of the incident particle. In nonmultiplying cases, it is also called the delta-scattering model." It is a model one step simpler than the Fermi scattering model introduced in Section 5 . II . D. Its main advantage is that most of the equations appearing in our treatment can be solved analytically in the straight-ahead approach; at the same time, the solutions reflect the basic characteristics of the exact ones. We shall here consider the first and second moments of the score provided by the different estimators when the absorption rate is estimated in a finite homogenous slab. Let the particles start at x = 0 in a positive direction along the x axis and let the slab be situated between x = 0 and x = X. Assume that the total cross section of the material in the slab is unity, the probability of absorption is c., and the mean number of secondaries per collision is c. Let the slab be surrounded by a purely absorbing medium of total crosssection 1. We do not require that the medium be nonmultiplying; however, for the sake of simplicity, we assume that the nonmultiplying game equivalent to the multiplying process is applied (cf. Section 5.III.C). (Remember that for c < 1, it is equivalent to survival biasing.) Otherwise, the game is assumed to be analog. Then the expected score, according to Equation (5.86), satisfies the equation M,(x) =

dx'

+ cM,(x')]

= I1(x) + cl dx' e-(x' -")M,(x')

(5.233)

where f(x,x') is any of the partially unbiased estimators. As the absorption rate is to be estimated, R=

dx ili(x)ca

the simplest estimator is the weighting function f(x) = ca

if 0 x X

and zero otherwise. With this estimator, the expected partial score (the expected score in a free flight and collision) is Jx

I, (x)

ca e (x" x)dx ' = call — e (x x)]

(5.234)

237 The details of the solution of Equation (5.233) are given in Appendix 5C. It follows from Equation (C.7) of Appendix 5C that the first moment in Equation (5.233) has the form M,(x) = call — e-('-exx-")1/(1 — c)

(5.235)

Thus, the expected absorption rate due to a starter at x = 0 is °xi/(1 — c)

WO) = call — e

The second moment of the score, according to Equation (5.92), is the value at x = 0 of the solution to the equation x

1

M2(x) = f dx' x + c2 f dx' e (x" -- ') M2(x')

= I2(x) + c2f dx'e a ( x' -x) M2(x') As is shown in Appendix 5C, the solution of this equation reads M2(x) = 12(x) + c2.1" dx' e --(1-`2)(x" -x) I2(x')

(5.236)

Now let us consider the source term 12 of the second-moment equation with various estimators. I.

The collision estimator coincides with the weighting function in the reaction rate, i.e., fc(x1 ) = Ca

if 0

x' ,X

With this function, the source term becomes Ir(x) = ca[Ii (x) + 2cf dxe-("' -x) M1(01 = ca[2M,(x) — Ii(x)] 2.

(5.237)

The expectation estimator has the form fE(x) = I1 (x) and the source term reads I(E)(x) = I,(x) • Ir(x)/ca = I,(x)[21‘41 (x) — I,(x)J

3.

(5.238)

The track-length estimator follows from Equation (5.230) as fT(x,x') = ca • min(x — x',X — x) and therefore

f2T)(x) = 2ca[I,(x) — c,,(X — x)e-(x - x) + K(x)]

(5.239)

238

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations where K(x) = cl dx'(x' — x)e -(x— x) M,(x')]

A little algebra shows that K(x) = ca(X — x)e-(x-x) — [(1 + c)Ii(x) — Mi(x)]/c with which the source term becomes IST)(x)= 2ca[M,(x) — I1 (x)1/c 4.

(5.240)

Finally, the trexpectation estimator from Equation (5.232) takes on the form fTx(x,x') = Ii (x)(x' — x) and the source term becomes I(TX)(x) = 2I,(x)II,(x) + K(x)] = 2I,(x)lMi(x) + ca(X — x)e-(x-x) — Ii(x)]/c

(5.241)

Now, it is apparent from Equation (5.236) that if for two estimators fa and fb the source terms satisfy ISa)(x) < IV(x) for every x then the variance of the score with the estimator fa is smaller than that with the estimator fb. This fact will be exploited when comparing the merits of the various estimators. As a first observation, we note that according to Equation (5.234) ca > I,(x) and therefore the collision estimator results in a greater variance than the expectation estimator. Comparing the collision and track-length estimators, it is seen that IT(x) < Ir(x) if Mi(x)

IA) < elMi(x)

1 /21[1(x)]

for 0

x

X

for 0

x

X

i.e., if i/2 < e -(x-x){`/2 + [e.(x-x) — 11/0

Accordingly, the track-length estimator is more efficient than the collision estimator if c > 1. Furthermore, if c < 1, for small values of X the RHS of the inequality is (1 + X — x)/ 2, i.e., for thin slabs, the track-length estimator is again more advantageous than the collision estimator.

239 In view of Equations (5.238) and (5.240), the track-length estimator is superior to the expectation estimator if I i (x)[2M,(x) — I,(x)] > 2ca[M,(x) — I,(x)]/c and is inferior to it if the opposite relation holds. This inequality can be rewritten as (cI, — ca)M, > (cI,/2 — ca)I, Arguing quantitatively, we conclude that with an increasing number of secondaries and also with increasing slab thickness, the total expected score, M,(x), becomes much greater than the first-flight expected score, I,(x), and therefore in sufficiently large regions with a large number of secondaries per collision, the track-length estimator gives a more efficient estimate of the absorption rate than the expectation estimator. Finally, let us compare the track-length and trexpectation estimators on the basis of the source terms in Equations (5.240) and (5.241). The expectation estimator will be more efficient if I,(x)[M,(x) + ca(X — x)e (x-"' — I,(x)] < ca[M,(x) — Ii(x)] After insertion of the explicit forms of M, and I„ the inequality becomes [1 — e -('-')(x -1/(1 — c) > [1 — e-(x-x)][X — x + 1] Obviously, for a sufficiently large number of secondaries per collision, the inequality holds and the expectation estimator is better than the track-length estimator. This also means that for large c, the expectation estimator seems to be the most efficient of all the estimators considered here. One should not forget, however, that the conclusions of this section were only drawn by comparing the source terms of the second-moment equations instead of comparing the second moments themselves. The latter also could have been done on the basis of Equation (5.236) and the results in Appendix 5C. Nevertheless, the conclusions so obtained would not be more valuable than the above results, for the game investigated here is highly idealized. The chief merit of the above analysis is that it gives a picture of the method of variance comparison to be followed in Chapter 5.VIII. Nevertheless, it is to be noted that the results reveal quite accurately the qualitative merits of the different estimators, and not much more can be deduced from a more rigorous analysis as will be seen in Section 5.VIII.G.

VII. APPROXIMATE SOLUTIONS OF THE MOMENT EQUATIONS It is apparent from the results of Chapters 5.1 through 5.IV that the Monte Carlo moment equations have at least as complicated a structure as the transport equation that describes the collision density of particles. Since Monte Carlo methods are used in just such cases when the solution of the transport equation is prohibitively complicated or cannot be determined at all by deterministic methods, it is even more hopeless to try to solve the moment equations governing the Monte Carlo scores. Nevertheless, approximate solutions of these equations may reflect adequately the main characteristics of the exact solutions and may be of great help in evaluating the efficiency of a Monte Carlo strategy or in comparing the relative merits of various methods. The approximate model should be sufficiently simple to make analytical or fast numerical

240

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

calculation possible. The conclusions drawn from such approximate investigations are to be checked in realistic Monte Carlo simulations. Two of the simplified transport models, the Fermi scattering model and the straight-ahead scattering model, were introduced in Sections 5.II.D and 5.VI.C, respectively. In this Chapter, a less idealized situation is considered and approximate solutions of the moment equations are established for monoenergetic, isotropic, homogeneous cases. Let us mention that besides the method discussed here, almost all the standard methods of approximating solutions to the transport equation can be adopted for the approximate calculation of moments. Thus, diffusion and S, calculations have been successfully applied for Monte Carlo moment calculations."16 41 A. THE SIMPLIFIED MODEL Let V be a simply connected convex region containing a homogeneous medium. Consider a monoenergetic transport process in V with isotropic postcollisional direction distribution in the laboratory system. In this model, every reaction rate is proportional to the expected number of collisions in the region; therefore, we shall assume that the latter is the quantity to be estimated by Monte Carlo. This means that the weighting function, f(P), in the reaction rate is unity inside V and vanishes outside, i.e., the reaction rate reads R

= fv

(5.242)

dP tir(P)

The kernels describing the transport model above have the forms T(P,P')dP' =

(5.243)

—

and (5.244)

C(P' ,P") dP" = — dw' 4-rr

where P = (r,w) P' = (r + Dw,w), and T = crD is the optical distance between P and P' along w. D is the corresponding geometrical distance, u, is the total cross-section of the region, and c is the mean number of secondaries per collision. Let V also denote the volume of the region V and, similarly, let S denote both the surface and the area of the surface of V. Two kinds of source densities will be considered. One is isotropic in the whole solid angle and uniform over V: Q,(P) = 1/4rrV,

if r e V

(5.245)

the other is uniform over the surface S and is isotropically distributed over the inward directions: Qs(P) = 1/27rS,

if

r E S and

nw > 0

(5.246)

where n is the inward normal to S. Finally, we suppose that the score moments due to a particle started from outside V is zero, i.e., Mr(P) = 0,

if 1-0/US

This assumption can be interpreted in two ways. Either it is assumed that the region is

241 surrounded by a black absorber and there is no return from outside V, or the particles entering V are assumed to be independent of any particle previously left V and they are thought to be part of the uniform surface source in Equation (5.246). The assumption will be relaxed in Section D by introducing an albedo-type quantity that accounts for the returning particles. The present form of the model allows us to examine the score moments in V without regard to the surroundings. B. THE SEPARATION ASSUMPTION It is well known from transport theory that even in the specialized model above, only a very limited number of problems can be exactly solved, and these solutions involve deep mathematical foundations. Therefore, some further approximation is to be introduced in order to obtain an easily treatable method of solution. The approximation is first demonstrated in the case of the analog first-moment equation M,(P) = Ii(P) + f

T(P,P') fdP"C(P' ,P") M,(P")

where, when the collision rate in Equation (5.242) is estimated I,(P) = lvdP' T(P,P') With the kernels in Equations (5.243) and (5.244), the moment equation takes on the form D(P)

M,(P) = 1 — e -'°(P) + o-

dDe'

4,-rr

idw'M,(r+Do3,co')

(5.247)

where D(P) is the geometrical distance of the surface S from the point P along co. [For the sake of curiosity, we note in passing that since L(P , P") = fdP'T(P,P' ) C(P' ,P") = fdP' C(P",P') T(P' ,P) = K(P",P) T(P,P') = T(P',P) and Q(P) = Q

(a = V,S)

the expected score also satisfies the equation M,(P) = — f dP' Q(P') T(P' ,P) + f dP" M,(P") K(P",P) Q. i.e., in view of Equations (5.4) and (5.5) that define the collision rate, it is seen that MI(P) = 45(P)/Q. This means that the expected collision rate due to a starter from a volume or surface source at P is proportional to the collision density at P.]

242

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

Let us now consider the case of a volume source. Multiplying the equation by Qv(P) in Equation (5.247) and integrating with respect to P, the resulting equation reads D(P) M, = I +

47ry dP I

1

&re-' — f dw' M,(r + Do) ,co') 47r

(5.248)

where M, = 47rV JdPM,(P) = R

(5.249)

and I, =— 1 f dPI,(P) 47rV v The last and most important approximation in our model is that M,(P") in Equation (5.248) is replaced by its average value M, in Equation (5.249). Doing so, Equation (5.248) becomes M,

I, + cPcM,

M,

I,/(1 — cPc)

(5.250)

where 1

Pc

r

f

D(P)

— f dP [1 — e -'D(P)] ki 47rV ivdP1 dT e =47rV 1 1 4Try

dr f tho [1 — e -'13(r•")] 4nr

(5.251)

Pc is the first-flight collision probability' in V. More precisely, Pc is the probability that a particle started in V from an isotropic, uniform distribution suffers its next collision in V. The approximation above is called the separation assumption, referring to the fact that the average of a product is approximated by the product of the separate averages. It can be seen that in the simple form above, the separation assumption is equivalent to the assumption that the collision points are uniformly distributed over V. An analogous approximation was introduced through an invariance principle by Stuart' for escape probability calculations. Note that the first-flight collision probability Pc for a body of a given shape is a unique function of the optical mean chord length T = Crf

in the body,' where

= 4v/S is the geometrical mean chord length.

(5.252)

243 In the case of the collision rate estimation, the source term of the first-moment equation I, in Equation (5.248) equals the first-flight collision probability, I, = Pc and hence the expected total score in Equation (5.250) reads M, = R = Pc/(1 — cPc)

(5.253)

In the case of the surface source, Equation (5.246), the expected total score due to the incident particles is =

1 f dP M,(P) 21TS s

and it is similarly approximated as

I, +

M

(5.254)

cPcM,

where 1 = 27rS dP I(P) s and D(P)

1

j5

C

= — f dP f dT e 2TrS s

It can be seen8 that Pc is the probability that a neutron incident isotropically and uniformly on the surface of V will suffer its next collision in V, and it is related to the first-flight collision probability Pc as Pc = T-(1 — Pc) Inserting Equation (5.250) into the RHS of Equation (5.254), the expected total score due to the surface source becomes M, --- I, +

— P0)/(1 — cPc)

Again, for collision rate estimation, I, = Pc, I, = M,

Pc,

(5.255)

and

T-(1 — Pc)/(1 — cPc)

(5.256)

In the expressions of the approximate total score, Equations (5.253) and (5.256), the value of the first-flight collision probability P, appears, whose calculation for irregular bodies may be difficult. For regions of regular shape, analytical expressions and tabulated values of 130 are presented in Reference 7. A shape-independent approximation to the first-flight collision probability was proposed by Wigner8 as Pc(I)

T-/(1 + T)

In practical cases, this approximation deviates from the exact values by no more than 15%.

244

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

In the case of a general monoenergetic moment equation, the separation assumption will be used in the following form fdP Q„(P)IdP' T(P,P')A(P,P') fdP"C(P' ,P") M(P") D(r,m)

a„[Idr f (Ito f x

4Tr

o

dT e A(r,o);r + Do.),(o) c(r + Do.))1 M

(5.257)

Where we put X = V, ax = 1/47rV for a volume source and X = S, a„ = 1/27rS for a surface source. A(P,P') is some given function, c(P) is the mean number of secondaries per collision at P, and M stands for the total score moment in question M= 47rV f dPM(P) C. ON THE QUALITY OF THE APPROXIMATION The form of the separation assumption as introduced above suggests that it may only be used in monoenergetic, isotropic transport in bodies of more or less regular shape. Although the assumption might be generalized to more realistic cases, such a generalization does not seem to offer any advantage, for two reasons. First, in complex problems, the evaluation of the integrals on the RHS of Equation (5.257) would be almost as laborious as the application of more precise deterministic solutional schemes. Second, the approximations proposed here are primarily intended for semiquantitative (and usually comparative) analysis of various Monte Carlo methods and, except for very special cases, such an analysis can be performed on elementary transport problems with satisfactory results. Therefore, when judging the quality of the approximation, the most meaningful question is, how well does Equation (5.253) approximate the exact collision rate of monoenergetic particles in a homogeneous medium with isotropic scattering? In other words, what can be said about the difference of the exact and approximate solutions? Extensive numerical tests show' that the approximate and Monte Carlo values of collision rates and related quantities are in very good agreement for a not-too-high number of secondaries per collision (c < 1), and the approximation consistently underestimates the exact collision rates. Although no general proof has so far been found, the following arguments make it more than probable that underestimation is an inherent feature of the approximation. Let us introduce the following notations Mi(r) = 7r ido.) M,(P) 4 and 1 1 I,(r) = — Ida) fdP1 T(P,P1 ) = — '&01 — 477 47r

- oD(P)1

Thus, from Equation (5.251), the first-flight collision probability reads Pc =

1

fdrI,(r)

(5.258)

245 Integrating the first-moment equation (5.247) with respect to w we have, with r' = — r/o) + r 1f M1 (r) = 1-1 (r) + — j ckocr f 47r

ri cMi(r +tole — r1)

or substituting thodle — ri = de/ir' — rI2 the moment equation becomes M1 (r) = Ii (r) + cf dr' G(r,e)Mi(e)

(5.259)

where cr — 47r

G(r,r') = G(r',r)

— r12

Accordingly 11(r) = tdr' G(r,e) = tdr'G(r' — r)

(5.260)

The solution of Equation (5.259) is written in the Neumann series from M1(r)

=E

ck-iik(r)

(5.261)

k=7

where fk +1(0 = tdr'G(r,r')Ik(e)

(5.262)

Finally, denoting P, = Pe and

Pk

=1 — fdrIk(r) Vv

(5.263)

it follows from Equations (5.261) and (5.263) that the total expected score becomes M1

=E

k=1

ck—Pk

Now this quantity is approximated in Equation (5.253) as M,=

E k=7

Ck

(5.264)

246

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

Obviously, if one proves that Pi, % Pck = P';, then it is also established that the approximate collision rate underestimates the exact one. For k = 2.n, it follows from Equations (5.263) and (5.264) that Pen =

1

Ldr

1

2

%- [v 1v- dr In(r)] =

Repeated use of this inequality yields Pk

for k = 2' (i = 1,2,...). For values of k different from powers of two, the inequality has not yet been established; nevertheless, it seems reasonable to assume that it holds for every integer k. Based on these arguments and on numerical results, we consider the separation assumption as an approximation safely underestimating the collision rate, i.e.,

M, % Pc /(1 — cPc) Experimental and approximate collision rates for various bodies and dimensions are compared in Figure 5.3 as functions of the mean number of secondaries.3' It is seen that up to about c = 0.8, the agreement is excellent in all the cases studied. The error in the approximation increases slowly with increasing dimensions and drastically with an increasing number of secondaries. Finally, we note that a rough overestimation of the collision rates in the simplified model follows from the fact that the first-flight collision probability is greater than the multiple collision probabilities P„ and all are less than unity. Therefore, it follows from Equation (5.264) that

M,

IV(1 — c)

The separation assumption will be repeatedly exploited in approximate comparison of the variances and efficiencies of various estimators and procedures. D. EFFECT OF SURROUNDINGS In the introduction of the separation assumption, we have supposed that a particle that leaves the region V will not re-enter it. If we introduce an albedo-type quantity that accounts for the particles returning due to the escape of one particle, approximate solutions also can be established for such moment equations, the exact solution of which effectively depends on the surroundings of the region considered. Based on the formalism of Amouyal and Benoist,' an approximation to the number of collisions in a region is proposed by Maiorov and Frank-Kamenietzky35 which turns out to be a special case of the separation assumption. A short description of the formalism to be used for general moment equations is given below, along with an application to the estimation of the number of collisions. Let the moment equation at hand be of the form M(P) = I(P) + fdP L(PP")M(P") and assume that the source density to be used in the calculation is a linear combination of volume and surface sources of the form in Equations (5.245) and (5.246): Q(P) = (1 — Qv(P) + «QM')

(5.265)

247

M1

0.14

SPHERE

SLAB

CYLINDER

R-0.1

0.12 0.10 0.08

(M1/2)

0.06 0.04 0.02

1.4

R-1

1.2 1.0 0.8 0.6 0.4 0.2

14.

R=5.

12. 10. 8.0

I /

6.0 4.0 2.0

,c(

0.2 0.4 0.6 0.8

0.2 0.4 0.6 0.8

0.2 0.4 0.6 0.8

C FIGURE 5.3. Collision rates in various bodies of characteristic dimension R as a function of the mean number of secondaries c. Continuous line: exact values; broken line with circles: approximation by Equation (5.253). (For a slab of R = 0.I, one half of the rate is plotted.)

248

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

Let My

f dP Q,(P) M(P) 47rV .iv dPM(P)

Ms

1 r fdP Qs(P)M(P) 277S isdP M(P)

and

Similarly, denote 1 f 1 r I = 417V ivdPI(P) 47rV ivdPivdP"L(P'P")' v

Lv = and Ls =

1 s dP dP"L(P,P"), v 27S

1 — f dP I(P) 27S s

Then, according to Equation (5.257), the total scores due to the volume and surface sources become

My

Iv/(1 - Lv)

and Ms ~ Is + LsIviv = Is + LsIv/(1 — Lv) respectively. Now, since the total expected score is the linear combination of the sourcewise scores M = idPQ(P)M(P) = (1 — a)Mv + aMs therefore, with the explicit forms of My and Ms, we have the approximate solution: M = [1 — a(1 — L,)]Iv/(1 — Lv) + aI,

(5.266)

This approximation is only useful if one knows the ratio of the volume and surface source densities. On the other hand, assuming that the physical source of the particles is situated entirely inside V and the surface source is due to particles reentering it after an escape, the parameter a can be fixed and the effect of the space surrounding V can be accounted for. This is demonstrated by the following. Let N- be the number of particles escaping from region V (per unit time) due to a physical source that produces Nv particles in V (per unit time). Let the number of particles entering V be Ns and assume that Ns is proportional to N Ns = IN-

(5.267)

249 The quantity F characterizes the surroundings, and in simple geometries it can be related to the collision probability Pc . Obviously, it defines the number of particles entering V due to one particle leaving it. Now the particles representing the effective volume source in the region originate either from the physical volume source or from collisions in V, i.e., (5.268)

NE ---- Nv + cR where R is the number of collisions in V. On the other hand

(5.269)

R = PCNE + f',N,

for the obvious reason that the collisions occur either because a particle emerges from the effective source NE and suffers its next collision in V or because a particle just returning to V suffers a collision in it. P, and P, are the collision probabilities defined in Section B. Finally, the number of escaping particles is expressed as (5.270)

N - = (1 — Pc)NE + (1 — Pc)Ns

The solution of the algebraic equation system [Equations (5.267) through (5.270)1 is the following: R = [P, + F(PC — Pc)]Nv/{1 — F(1 — P) — [P, + Ns = [1 — F(1 — P,)]Nv/11 —

F(1 —

'Vic —

P,)[0. (5.271)

P) — [P, + F(13, — P,)C1

and NE =

F(1 — pc)Nvio — r(1 —

Pc) — [P, + F(P, — P,)]c}

(5.272)

It can be easily seen that in the derivation of the results above, the separation assumption has been implicitly used. This is also seen from the fact that by putting F = 0, Equation (5.271) reduces to Equation (5.523). The relation among the number of particles originating from the surface and volume sources is defined by Equation (5.272). Hence, the parameter a in the expression of the source density in Equation (5.265) becomes a = Ns/(Ns + Nv)

= co —

P)/{1 — [P, + F(P, — P,)]c}

In conclusion, if the albedo value F representing the effect of the surroundings on the particles' distribution is known, Equation (5.266) gives an approximate expression of the total collision rate in the monoenergetic, isotropic, homogeneous model. The approximation becomes exact in purely absorbing medium and is expected to yield reasonable results for moderate values of the mean number of secondaries per collision.

VIII. ANALYSIS OF SECOND MOMENT EQUATIONS The elaboration of the theory of moment equations was mainly motivated by the fact that with the aid of the equations that describe the second moment of the score, the variance of specific games, estimators, or biasing schemes can be qualitatively and, to some extent,

250

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

quantitatively analyzed. It has been mentioned that there is little hope of finding exact solutions to the second-moment equations in general, but very often we are only interested in the relative merits of some procedures, i.e., we intend to decide whether a particular game results in lower variance than another game. Usually, this can be done without effectively solving the equations. On the other hand, in certain applications, approximate calculation of the variances (e.g., by the method proposed in the previous Chapter) gives sufficient practical information. The theoretical variance of a game is connected to the score moments as follows. Let M,(P) be the i-th moment of the score due to a unit weight starter at the point P in the nonanalog game in question. Let Q(P) be the source density of the starters and let W(P) = W be the statistical weight of the starter from P. Obviously, the expected final estimate is R = fdP()(P)WM,(P) and the variance of the estimate is MR] = fdPO(P)W2 M2(P) — R2 In this Section, zero-variance Monte Carlo schemes are first reviewed. Such schemes result in final scores which have no statistical fluctuation, i.e., every history yields the very same final score. Conditions are established in Section B under which the variance of a feasible game is bounded. Variance reduction capabilities of nonanalog games in general and of special biasing schemes in particular are investigated in Sections C and D, respectively. The variance and efficiency of the equivalent nonmultiplying game are examined in Section E. Minimum-variance partially unbiased estimators are derived in Section F. Comparison of the variances of partially unbiased estimators is followed by the derivation of a new, effective, self-improving estimator. Some remarks concerning the effect of variance reduction on the efficiency of a game in general conclude the analysis of the second-moment equations. Finally, optimum biasing of the source density is addressed. A. ZERO-VARIANCE SCHEMES Discussion of a Monte Carlo scheme that has no statistical error seems absurd since such a game would give an exact answer in one history, i.e., in a quasideterministic way, while, in general, Monte Carlo is used only in cases where deterministic methods fail to work. It will be seen below that the suspicion concerning the practical feasibility of such games is justified indeed. Nevertheless, besides their theoretical interest, such games do have practical importance in the sense that they represent the "best of all" Monte Carlo games that, in principle, can be arbitrarily closely approximated, and their structure indicates the directions of the approximations. Zero-variance schemes were first derived through a special importance sampling procedure, and these schemes involve last-event (absorption) estimators. 12 '8 Zero-variance biasing schemes with a collision estimator were introduced by Ermakovw and Hoogenboom.' 4 Schemes with arbitrary partially unbiased estimators were derived from the moment equations by Dwivedi9 and were generalized by Gupta." Both derivations concern nonmultiplying games. One might think that the form of the estimator is irrelevant if it is about a zero-variance game. This, however, is not so since the practically realizable games may only be approximations of the ideal one and will

251 therefore result in finite variance. In this case, the form of the estimator will, in turn, influence the resulting variance. In the first part of this section, we define the kernels of a general multiplying, partially unbiased nonanalog game that results in pointwise zero variance, i.e., that define a game in which the scores in a history depend only on the starting point of the history, but are identical for every starter from a given point. Then we introduce a nonanalog source density that, together with the pointwise error-free game, yields a zero-variance estimate of the reaction rate in question. The schemes proposed in the works referred to above are special cases of the game to be derived here. In the derivation below, the estimators will be fixed and nonanalog kernels leading to zero variance will be determined. The opposite way, i.e., derivation of suitable contribution functions with fixed kernels, seems equally reasonable and, indeed, it will be seen in Section F that for any analog game, partially unbiased estimators exist that yield zero variance. For the sake of simplicity, let us assume that scores result from intercollision flights only, i.e., that the estimator used in the simulation is of the form WA}}. Without loss of generality, we can write the analog collision kernel in the form C(P',P") = ca(P')8(P"—P) +

E

ncn(P')C„(P',P")

n=1

where c„(P') is the probability that n secondaries are emitted in a collision at P' and the density function of the postcollision coordinates in an n-fold multiplication is Cn(P',P"). It is easy to see that by choosing c,(P') = ca(P') + cf(P')q,(P') cn(P') = cf(P')q„(P') and replacing C,(P' ,P") in Equation (5.74) by [c,(P')Cs(P',P") + cf(P')q1(P')C1(P',P")1/c,(P') the collision kernel in Equation (5.74) (for an analog game) takes on the form proposed here. Similarly, let us write the nonanalog collision kernel (to be determined below) as e(P' ,P") = Ca(P') 8(P" — P) +

EI nen(P') &(P' ,P")

Consider an analog game with the estimator f(P,P') = 0 that results in the required expected score. The first moment of the analog game is governed by the equation M1 (P) = fdP'T(P,P')f(P,P') + f dP'T(P,P')

E

n=1

nc„(P')fdP"Cn(P,P")M i(P"). (5.273)

Now, let H(P,P') and Hn(P') be (for the moment) arbitrary functions. Simple manipulations yield a first-moment equation equivalent to Equation (5.273): M,(P) = fdP' T(P,P'){f(P,P') +

E

ncn(P')[1 — H(P,P')Hn(P')DT-i n(P')}

n=1

f

dP' T(P,P')H(P,P') n=1

nen(P') Hn(P') f dP"Cn(P' ,P") M I (r)

(5.274)

252

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

where = f dP"C„(P',P")M,(P")

(5.275)

This form of the moment equation will be used for the definition of the nonanalog zerovariance kernels below. Let us consider a nonanalog game played by the transition kernel T(P,P') and by the collision kernel C(P',P") above. Assume that the analog contribution function f(P,P') and the weight generation rules in Equations (5.142) through (5.149) are applied; then this game is partially unbiased. The second moment of the score due to a starter with unit weight at P reads, according to Equation (5.81), as M2(P) = JdP'T(P,P'){(W')2f2(P,P') + 2W ' f(P,P')

nCn(P') „(P' ,13")WN,(P") n I

+E n=1

f

n(n — 1)C,,(P') [fdP"e„(P',P")WNI(P")] 2}

d131

,Pi n --=

naa(P1 ) fdP" „(P' ,P") (WD21;42(P")

(5.276)

where, in view of the weight generation rules (5.277)

W' = T(P,P')/T(P,P') and = Wic„(P')Cn(P',P")/C„(P')Cn(P',P")

(5.278)

Equation (5.276) follows from Equation (5.81) by putting W = 1 and =

=0

Then, reordering the terms in Equation (5.81) according to the powers of f(P,P') and making use of the relation above of the quantities ei and C‘, (i = 1,2,...) to the reaction probabilities cr and kernels C„ Equation (5.276) is obtained. In the derivation, the identities Ca(P) +

E asp)

=1

and fdP" i(P' , P") = 1 have also been exploited, The game will result in zero variance if MAP) =---- [MOW

253 i.e., if (5.279)

M2(P)/M1(P) = M1(P)

Dividing Equation (5.276) by M,(P) and making use of Equations (5.275), (5.277), and (5.278) in the first term in the RHS the equation becomes [M2(P)/M1(P)] = f dP' t(P,P1 )1W'f2(P,P1 ) + W'

E

ncn(P')ifin(P')

n=1

12f(P,P') + (n — 1)c„(P')rrin(P')/Cn(P')1}/M,(P) + f dip' T(P,P') 13(13) i nC (P') f dP" „(P',P") 0(P) M(P) n= I n 1 (WD2[K12(P")/MI(P ')]

(5.280)

where 0(P) may represent, at the moment, any function with which the RHS of the equation exists. A sufficient condition of Equation (5.279) is that the integrands in the two terms on the RHS of Equation (5.280) be equal to the respective integrands in Equation (5.274). For the second term, it means that T(P,P') 0(P') C (P') M,(P) n

n

(P',P") Ml(r) (W")2 0(P') "

= T(P,P')H(P,P')cn(P')Iin(P')C„(P',P") Inserting Equations (5.277) and (5.278) into the LHS, the equation is satisfied with T(P,P') = T(P,P')0(P')/IH(P,P'/M1(P)1

(5.281)

C„(P') Cn(P',P") = cn(P')C„(P',121M1 (P")/10(P')I-InUni

(5.282)

and

for arbitrary functions 0, H, and Hn. The first two functions are restricted by the requirement that T(P,P') be normalized to unity. Accordingly

i

dP' T(P,P') 0(P')/H(P,P') = M,(P)

Comparison to the analog moment Equation (5.273) shows that by choosing 0(P')/H(P,P') = f(P,P') + i nen(P') f dr Cn(P',Pn)Mi(P") n=i the normalization condition is satisfied. With the notation of Equation (5.275), it is rewritten as 0(P')/H(P,P') = f(P,P') +

E n= I

nc„(P')Tri(P') -. f(P,P') + M,(P')

(5.283)

254

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

[Note that any partially unbiased estimator in place of f(P,P') would make T(P,131) normalized. This choice, however, will make the further derivation simpler.] Further restrictions on the functions so far undefined will follow below. It remains to ensure the equality of the first terms on the RHS of Equations (5.274) and (5.280) in order to meet Equation (5.279). The two expressions are certainly equal if f(P,P') +

E

ncn(P')ifin(P')[l — H(P,P')H„(P')]

n=1

= If2(P,F,)

+E

ncn(P')Iiin(P')[2f(P,P')

n=1

+ (n — 1)cn(P') iTin(P')/Cn(P')] W/M, (P)

(5.284)

On the other hand, integrating Equation (5.282) with respect to P", we have &'.(r) = cn(r) ITVP')/Ifin(r) 0(lnl

(5.285)

while, according to Equation (5.281) W' = T(P,P')/T(P,P') = H(P,P')M,(P')/e(P') Inserting the last two equations into Equation (5.284), the equality to be fulfilled becomes f(P,P')

+E

o=1

ncn(P')Iiin(P')[1 — H(P,P')H„(P')]

= f(P,P') [ f(P,P') +

E

nc„(P')Fan(P')] H(P,P')/O(P')

n=1

+ E ncn(P') iTin(P') [f(P,P') + (n — 1)Hn(P') 0(P')] H(P,P')/0(P') n=1

Now in view of Equation (5.283), f(P,P') on the LHS is equal to the first term on the RHS, i.e., the equation to be satisfied is 1 — H(P,P')Fln(P') = [f(P,P') + (n — 1)Hn(P')0(P')]H(P,P')/0(P') for every n. Again from Equation (5.283), we obtain that this is equivalent to the equation Hn(P') = M1(P')/nO(P')

(5.286)

where Mi(r) =

n=1

ricn(P')rTin(P') = idP" C(P',P)M1(r)

(5.287)

255 We have thus completed the construction of the nonanalog kernels that result in a game with pointwise (or, better, starterwise) zero variance. The transition kernel follows from Equations (5.281) and (5.283) as T(P,P') = T(P,P') [flP,P1 )

MI(Fil/MI(P)

(5.288)

The multiplication probability is obtained from Equations (5.285) and (5.286) as e,,(P') = nc„(P')in„(P')/M1(13')

(5.289)

while the postcollision densities are defined by Equations (5.282) and (5.285) as „(P',P") = Cfl(P',P")M,(P")/iTin(P')

(5.290)

where tTri„(P') = fdP"Cr,(P',P")M,(P") We note in passing that according to Equations (5.9) and (5.12), the transition kernel (5.288) can be expressed by the adjoint collision density (VW') as T(P,P') = T(P,P') [f(P,P') — f(P) + tfc*(P')]/M,(P) where f(P) is the weighting function in the reaction rate [Equation (5.2)1 to be estimated. It is apparent that the absorption probability in the nonanalog game is zero since co

1—

=

E on(F) = E nen(r)ffinwym,(r) = 1

n=i

n=1

and the nonanalog kernels are normalized to unity. We have thus proved the following. Theorem 5.19 — Given an analog unbiased game with a nonnegative contribution function assigned to the intercollision flights. If one chooses the kernels of a nonanalog game according to Equations (5.288) through (5.290), then the game with the analog contribution function will be partially unbiased and any starter will produce an estimate with zero variance.

Two comments are to be made here. First, since the kernels are defined through the expected score which is unknown at the time of the simulation, such a game, of course, cannot be realized. Nevertheless, approximations to the kernels above may substantially reduce the resulting variance, as will be seen in Section 7.111. Second, the game so defined gives a zero-variance estimate of the reaction rate due to a starter from any source point. However, it does not garantee zero variance of the total score R unless the nonanalog source density is specifically chosen. Indeed, the variance of the total score reads D2[R] = idk)(P)W2 M1(P)

[fdPQ(P)M,(P)12

256

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

where, according to Theorem 5.9 W = Q(P)/Q(P) is the starting weight of the particle at P. Now, choosing 0(P)W2 = Q(P)1 dP' Q(P')MI(P')/Mi(P) = Q(P)R/Mi(P) we have Q(P) = Q(P)Mi(P)/R

(5.291)

and D2[R1 = 0, i.e., this form of the nonanalog source density ensures a zero-variance estimation of the reaction rate R. It is to be emphasized that the hypothetical game defined above is partially unbiased, i.e., it results in the same pointwise score per unit starter weight as the corresponding analog game (and zero variance is reached by position-dependent starting weights). This is why the considerations above are not only of theoretical interest, but can also be used for the construction of approximately optimum feasible games. In contrast to that, the zero-variance schemes proposed in early works12 J8 are based on a special importance sampling of a nonmultiplying procedure with last-event estimators where both the starting weight and the final score of every starter are just the required quantity R, thus obtaining zero variance. In order to illustrate how importance sampling leads to zero-variance schemes, we give here the outlines of the classical derivation in terms of the moment equations. Let R = f dPf(P)*(P) be the reaction rate to be estimated in a nonmultiplying game. We have seen in Section 5.I.A that the simplest first-moment equation that describes this problem reads MI (P) = fdr T(P,P')[f(P') + fdP" C(P ,P")MI(P")1 Let us introduce functions as U(P') = f(P') + fdP"C(P',P")M1(13") and V(P) = JdP'T(P,P')U(P') = MI(P) According to Equations (5.151) through (5.153), U(P') and V(P) define an importance sampling procedure which is governed by the transformed kernels T(P,P') = T(P,P')U(P')/M,(P)

(5.292)

257 and (P' ,P") = C(P',P")M,(P")/U(P')

(5.293)

Recalling the relation between the score moment and the adjoint collision density [Equation (5.12)] M,(P) = JdP'T(P,P')I,*(P') it is seen that U(P') = f(P') + fdP" K(P' ,P") ili*(P") i.e., by Equation (5.9) U(P') = ip*(P')

(5.294)

and T(P,P') = T(P,P')LP*(r)/MI(P) The absorption probability in the transformed game is = 1 — fdP" 'C(P' , P") = f(P')/U(P') In view of Theorem 5.10, the transformed game with a last-event estimator is unbiased if the transformed contribution function satisfies the equation U(P')Ca(P')fa(P') = f(P') i.e., if ia(P') = 1 Furthermore, if the source density and the starting weight in the transformed game are Q(P) = Q(P)V(P)/JdPQ(P)V(P) = Q(P)M,(P)/R and =R respectively. Since the game is assumed to be feasible (in a theoretical sense), the probability of an endless history is zero, i.e., sooner or later every particle is absorbed, scoring exactly unity in every history. This means that the expected score due to a unit weight starter is unity in every transformed history. [The same follows from Equations (5.156), (5.292), and (5.293) since the first-moment equation of the transformed game reads i(P) = f dP'T(P,P')[f(P')/M,(P) + f dP"C(P' ,P")M1(13")Ati(P")/MI(P)]

258

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

which has the unique solution 41,,(P) = 1.] As every starter of unit weight scores unity and the final score in every history is lirAt i(P) = R the variance of the final estimate is zero. It is interesting to note that the form of the transformed source density is in fact, immaterial since the final score of every starter is identical. Quasi-optimum biasing schemes derived from the ideal zero-variance procedure will be considered in Chapter 7.111. B. ON BOUNDEDNESS OF THE VARIANCE Theorems 5.7 through 5.9 in Sections 5.V.A and B establish conditions under which a nonanalog game yields the same expected score as the analog game. We have also seen in Section 5.V.G that any nonanalog game that corresponds to a feasible analog game and satisfies the conditions of the theorems above is also feasible in the sense that the probability of an endless history is zero. From a statistical point of view, however, a game in which the second moment of the score is not bounded is dangerous since, in general, there is no guarantee that, by increasing the number of histories, the empirical mean of the estimates tends to the expected value of the quantity in question. (Nevertheless, in special cases it is possible to construct games that give reliable estimates with unbounded variance. This matter will be discussed in Section 6.IV in connection with the estimation of the particles' flux at given points.) In reaction rate estimation, the boundedness of the second moment (variance) of the score is generally almost as important a requirement as the feasibility of the game. Let us consider a partially unbiased multiplying nonanalog game that corresponds to a feasible analog game. The second moment of the score due to a unit weight starter from P satisfies Equation (5.81), which, by making use of the weight generation rules in Equations (5.142) through (5.149), can be written in the form M2(P) = I2(P) + fdP"I(P,P")[1%:42(P")

(5.295)

M;(P")]

with )

fdP' T(P,P') w(P,P') [cs(P') f + cf(P')

E

n=I

Cs(P' ,P") ws(P' ,P")

nqn(P') fdP"Cn(P' ,P") wn(P' ,P")]

(5 . 296)

and 12(P) =

fdP'T(P,P')w(P,P')[NP,P') + 2i(P,P1 ):4,(131) + j-'2(P')]

(5.297)

where °s- 1(13') = ca(P')i.a(P') + c(P') f + cf(P')

E

n=I

Cs(P'

,P") +

(P")]

nqn(P') fdP" Cn(P' ,r)[in(P' ,r) + Mi(r)]

(5.298)

259 and ‘j472(P') = ca(Pr)wa(r)iar) + cs(P')fdP"Cs(P',P")ws(r,r) [is(Pi ,P") + MI (p„)]2 +

CP')

E

nqn(P')fdP"C,,(P',P")w„(P',P")[ic,(P',P") + M,(P")]2

E

n(n — 1) q„(P') witn(P') fidP"C„(P',P")[fn(P',P") + Win] 12

n=

+

CP')

(5.299)

n=1

The weight factors in Equations (5.298) and (5.299) are those defined in the weight generation rules of Theorem 5.8, i.e., w(P,P') = T(P,P')/T(P,P')

(5.300)

wa(P') = ca(P')/ea(P')

(5.301)

ws(131) = cs(13')/Os(r)

(5.302) (5.303)

ws(P',P") = ws(P')Cs(P',P")/ s(P',13") = ws(P')wc(P',P") wf,n(P') = cf(P') qn(P')/Cf(P') 4n(P')

(5.304)

and wn(P' ,P") = wf,n(P') Cn(P' ,P")/ n(P' ,P") = wf.„(P') w,(P' ,P")

(5.305)

The kernels and reaction probabilities in Equations (5.296) through (5.299) are the same as those that govern the analog game. Equation (5.295) is known to have a unique bounded solution if (cf. Theorem 5.1): 1. 2. 3.

The supremum of the integral of .2(P,P") with respect to P" is bounded The norm of a sufficiently large-order iterate of is less than unity The source term I2(P') is bounded

In the analog game, Y(P,P") = L(P,P") and conditions 1 and 2 coincide with the first two conditions of the feasibility. Thus, if an analog game is feasible, its variance is bounded whenever the source term of the second-moment equation is bounded. This source term, in turn, is certainly bounded if the contribution functions are bounded. In the opposite case, there is a real danger of an infinite variance, which is a problem in the estimation of the collision density (or flux) at a point. In nonanalog games, it is usually difficult to check the fulfillment of condition 2 in general. For nonmultiplying games, it is sometimes expedient to replace conditions 1 and 2 by the stricter condition supf dP"Y(P,P") supfdP"fdP1 T(P,r)cs(P')C,(131,P")w(P,Pi)ws(P' ,P")

b 0 if the factors are positive. The reasoning above is summarized in the following theorem. Theorem 5.20 — A partially unbiased nonanalog game has a lower variance than the corresponding analog game with identical contribution functions (estimators) if for every event sequence starting from P, entering a collision at P', and leaving the collision at P", the following inequalities hold: T(P,P') > T(P,P')

(5.308)

for every P and P' such that f(P,P') > 0 t(P,P') '6s(13') s(P',P")

T(P,P')cs(P')Cs(P',P")

(5.309)

for every P, P', and P" such that f,(P',P") > 0 or MAP") > 0 i(P,P')C(P')Cin(P') an(P',P")

T(P,P')cf(P')qn(P')C„(P',P")

(5.310)

for every P, P', and P" such that f„(P',P") > 0 or M2(P") > 0 t(P,P')Ca(P')

T(P,P')ca(P')

(5.311)

T(P,P') cf(P')qn(P')

(5.312)

for every P and P' such that fa(P') > 0 t(P,P') Cf(P')4a(P') for every P and P' such that

f

dP"Ca(P',P")[fa(P',P") + M,(P")]

0

and in some finite region of the domain of simulation, strict inequality holds in one of the relations above. Proof. If conditions (5.308) through (5.311) hold, then according to Equations (5.300) through (5.305), w(P,P') < 1, w(P,P')w.(P',.) < 1, and thus every term in the expression of A(P,P') is positive, hence, so is A(P) for every P.

The conditions above are sufficient but not necessary for variance reduction; in fact, they are rather strict. For instance, a strict inequality in condition (5.308) certainly will not hold over the entire phase space since both the analog and nonanalog transition kernels are

262

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

normalized to unity and their difference may have no constant sign. On the other hand, the theorem implies that a distortion of the kernels, which increases their values in important regions (where the contribution and the expected score is large) and decreases them at less important points, will probably yield variance reduction. The theorem gives conditions of variance reduction in terms of the kernels that describe a transition from a starting point (or a postcollision point) to the next postcollision point. For symmetry reasons, it is expected that similar conditions can be constructed in terms of the transition probabilities between two successive precollision points. This is indeed so, and it is demonstrated by the example of a nonmultiplying game in which only free flights between two collisions contribute to the score, i.e., fa(r) = fs(P',P1') = 0 Let .7(P,P') = T(P,P')w(P,P') and %(P' P") = cs(P')Cs(P',P")ws(P',P") Then we write Z(P,P") = f dP',7(P,P')%(P',P1') Furthermore, let us introduce the notations m,(P') = f dP"C(P',P")M,(P")

i = 1,2

and 612(13")

= fdP"%(P' ,P")K12(13")

Then the difference of the analog and nonanalog second moments follows from Equations (5.295) through (5.299) as A(P) = fdP'T(P,P')11 — w(P,P')11f2(P,P') + 2f(P,P')m,(P') + + fdP',/(P,P') [m2(13')

612( 3')]

It is seen that A(P) > 0 if w(P,P') < 1 and m2(P') > th2(P'). On the other hand, it follows from Equations (5.295) and (5.297) that =

dP' T(P,P')IdP" 9-(P' ,P") [f2(P' ,P") + 2f(P' ,r) m,(P") + 1112(13")] (5.313)

and similarly for the analog moment m,(P) = f dP' C(P,P')IdP"T(P' ,r) ff2(P' ,P") + 2f(P' ,13") m (P") + ni2(13")] (5.314)

263 Subtraction of Equation (5.313) from (5.314) shows that m2(P) > m2(P) if C(P,P') T(P' ,P") > `C (P ,P' ) 9-(P' ,P") and we have the following. Theorem 5.21 — A partially unbiased nonmultiplying nonanalog game in which scores are assigned to intercollision flights only has a variance not greater than the corresponding analog game with the same contribution function if for every sequence of the events of entering a collision at P, leaving the collision at P', and reaching the next collision at P", the following inequalities hold: 't(P',P")

T(P',P")

(5.315)

for every P' and P" such that f(P',P") > 0 or m2(P") > 0 C(P,P')T(P',P") -% C(P,P')T(P',P")

(5.316)

for every P, P', and P" such that f(P', P") > 0 or m2(P") > 0.

Generalization of the theorem to multiplying games goes along the lines followed here and brings up another condition analogous to condition 5 in Theorem 5.20. Obviously, the theorems above can also be used for comparison of any two nonanalog games when the kernels representing the two games should be inserted into the two sides of the inequalities. D. EXAMPLES: SURVIVAL BIASING AND ELP AND MELP METHODS In spite of the relative weakness of Theorem 5.20, it has a very important immediate consequence concerning the most common biasing procedure. We have seen in Chapter 3.11 that survival biasing (the nonanalog procedure where absorption is replaced by weight reduction) is expected to decrease the variance in quite general circumstances. In a nonanalog game, in which survival biasing is the only distortion, all the kernels are identical to the analog ones, while the absorption probability is zero and the scattering and fission probabilities are multiplied by 1/[1 — cJP')] 1, i.e., T(P,P') = T(P,P'); C,(P',P") = C“(P',P"), 's(13') = cs(r)/[1

(a = s,n)

ca(r)] = cs(PMcs(13') + cf(P')] % cs(r)

and = cf(P')/[cs(P') + cf(P')] % cf(P') Now, if no score is assigned to an absorption in the analog game, then the conditions of the theorem hold. Thus, if there are regions of the domain of simulation where the absorption probability is different from zero, then survival biasing decreases the variance. This matter will be reexamined in a broader framework in the next Section. Another example where the theorems above lead to definite results is the expected leakage probability (ELP) method.' Let V be a subset of the phase space and assume that

264

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

some reaction rate in V is to be estimated. Assume also that no particle can enter V from outside, i.e., M, (P)= 0

if P

v

In the ELP method, the particles are supposed to start from V and the nonanalog transition kernel is such that the probability of leaving the spatial region of V is zero, i.e., if we denote 15(P) = dP' T(P,P') v

then the nonanalog transition kernel reads T(P,P') = T(P,P')/15(P)

if P,P'

E

V

(5.317)

and zero otherwise [cf. Section 5.VI.B, Equation (5.227)1. In the simplest form of the method, the collision kernel is not distorted, i.e., 'a(P',P") = C(P',P") Now, if the contribution functions in the analog game are such that they give scores only if their arguments belong to V, then T(P,P') T(P,P') whenever f(P,P') > 0 and conditions (5.308) through (5.312) are satisfied. Finally, if V does not contain the whole geometrical space, then 15(P) is definitely less than unity for certain points P and a strict inequality holds in Equation (5.308). Accordingly, the ELP method results in lower variance than the analog game. Note, however, that variance reduction here also does not necessarily call forth an increase in efficiency. In fact, it can be seen32 that the ELP method is likely to decrease the efficiency in the estimation of monoenergetic reaction or leakage rates because the gain in variance does not compensate for the loss in computing time, which is due to the increased number of collisions. Nevertheless, use of the ELP method is justified in problems where the spatial region of V is small (i.e., the probability of leakage is large) and effects of multiple scattering (e.g., slowing down) are to be investigated in V.2° The ELP method decreases the variance of the score because it keeps the particles inside the important region. It can, however, be shown that the majority of the collisions are played in the vicinity of the boundary of V, and the weights of the particles entering such collisions may be very different. This problem may be remedied by distorting the collision kernel in such a way that direction pointing toward the inner part of the region is more probable. This biasing is expected to level the statistical weights and thus decrease the variance of the score. A method to this effect was proposed by Borgwaldt and is called the modified ELP (MELP) method. The basic idea of the method is formulated as follows. Let P' = (r,E')e V be a phase-space point adjoint to the point P' = (r,E')e V such that C(P,P') = C(P,P') In other words, the probabilities of the postcollision coordinates P' and P' are equal. Let il(P) again be the probability that a particle starting from P has its next collision inside V.

265 Let the nonanalog transition kernel be the same as in the ELP method [cf. Equation (5.317)] and let the nonanalog collision kernel be C(P,P') = C(P,P')2,0(P')/0(P') + 0(P')];

P,P'

E

V

(5.318)

This means that selecting P' from C(P,P') is accepted as a postcollision coordinate with a probability. *(P')/[D(P') + *(P')] and the adjoint point P' (or one of them if there are more) is accepted with the complementary probability. Obviously, the postcollision direction which determines a longer (optical) track inside V has the greater probability of acceptance. Now t(P',P") = T(P',P")/'0(P')

T(P',P")

for P',P" E V

and e(P,131)i(P',P") = C(P,P')T(P',P")2/[0(P') + '0(P')] C(P,P')T(P',P")

for P,P',P" E V

and, therefore, the conditions of Theorem 5.21 are satisfied, provided the game is nonmultiplying, the only contribution functions is of the form f(P',P"), and there are no particles entering V from outside. Most of these limitations can be removed, and it is found that the MELP method decreases the variance, compared to the analog game. Comparison of the variance of the ELP and MELP methods is not possible on the basis of Theorems 5.20 or 5.21 since the weight factor w,(P,P') in the MELP method assumes values both greater and less than unity. It can, however, be seen that if the domain of simulation is such that M,(P) > M,(P)

(i = 1,2)

whenever the optical distance from P to the boundary of the domain is greater than this distance from P, then the MELP method has lower variance than the ELP method. The simple method of variance comparison followed in this section can be successfully applied to the investigation of the variance of the equivalent nonmultiplying game introduced in Section 5.III.C. This will be seen in the next Section. E. VARIANCE AND EFFICIENCY OF THE EQUIVALENT NONMULTIPLYING GAME It has been shown in Section 5.III.0 that any multiplying game can be replaced by a hypothetical, nonmultiplying game that results in an unbiased estimate of the required reaction rate. This game is expected to have the advantage of not producing branching histories and thus requiring a lower amount of computing time than the multiplying game. We have seen in Section 5.V.F that if the mean number of secondaries per collision is greater than unity, then it is indeed so. In order to have an idea about the efficiency of the game, its variance should be compared to that of the multiplying game. Let us consider a multiplying analog game and introduce the equivalent nonmultiplying game, as in Section 5.111.C. The second

266

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

moment of the score in the analog game is given in Equations (5.295) through (5.299) after setting all the weight factors to unity, i.e., M2(P) = f dP'T(P,P')[P(P,131) + 2f(P,P'),Ti1 (P1 ) + ,72(P')] + idP'T(P,P') f dP"C(P',P") [M2(P") — W(P")1

(5.319)

with 7',(P') = cn(P') fn(P') + cs(P')fdP"Cs(P' ,P")[fs(P' ,P") + M1 (P")]

+ efon E

n=1

nqn(P')IdP"C„(P' ,P") [fn(P' ,P") + Mi(P")]

(5.320)

and 7 '2(P')

= ca(P i )f 2i(P') + cs(P')IdP"Cs(P',P")[fs(P',P") + M1(P")12

+ cAP') E

n=1

+ cf(P')

E n=--1

nqn(P')IdP"C„(P',P")[f„(P',P") + M1(P")12

(5.321)

n(n — 1) cln(P') fidP"Cn(P' ,P") [fn(P',P") + MI(P")112

The second-moment equation for the equivalent game follows from Equation (5.92) as At2(P) = idPI T(P,r)lf2(P,P') + 2f(P,P')$1(P') + $2(P')] + f dP' T(P,P')c(P') fdP"C(P',P")1.42(P") — MI(P")]

(5.322)

where JI(P') = f dP"C(P',P")[f*(P',P") + M1 (P")1

(5.323)

J2(P') = c(P') fdP" C(P' ,P") ff*(P' ,P") + M1(P")12

(5.324)

and c(P') = cs(P') + c1(P')

E

nqn(P')

(5.325)

n=1

the mean number of secondaries per collision in the analog game. f.(P',P") is the contribution function in a collision from P' to P", and the equivalent nonmultiplying game is unbiased if f. satisfies Equation (5.88) with unit weights, i.e., if fdP"C(P' ,P")f*(P' ,P") = ca(P') + c1(P')

E n=1

+ cs(P')IdP"Cs(P',P")fs(P' ,P")

nqn(P') f dP"Cn(P' ,P") fn(P' ,P")

(5.326)

267 An immediate consequence of this equation is that the quantities ,Tf(P') and j,(P') in Equations (5.320) and (5.323), respectively, are equal. [Note that they represent the expected final score due to a particle entering a collision at P'. This quantity was denoted by Ni(P) in Chapter 5.1111 Introducing the notation A(P) = M2(P)

At2(P)

the difference of the variances in the analog and equivalent games, it satisfies the equation A(P) = f dP' T(P,r) ri°2(13')

I 2(P' )]

+ fdP' T(P,P') [1 — c(P')[f dP"C(P' ,P") [MAP')

Mi(P")1

+ JdP' T(P,P') c(P') f dP" C(P' ,P") A(P") Now, A(P) > 0 if c(P') < 1 and .12(P') > J2(P'). In order to keep the derivation as short as possible, we do not proceed further in full generality, but we assume that fa(P') = 0,

fs(P',P") = f„(P',P") = fe(P',P")

in the analog game, i.e., we consider a game in which no score is assigned to an absorption and the scores in a collision do not depend on the type of reaction in the collision. This simplification is also justified by the fact that the great majority of the commonly used estimators are such indeed. Then f*(P',P") = fc(P',P") satisfies Equation (5.326) and 5-'2(13')

12(P') = [1 — c(P')]1dP"C(P' ,P") M;(r) + cf(P')

E

n(n — 1) q„(P') f dP"C„(P',P") [fc(P',P") + M,(P")]

2

(5.327)

n=

is positive if c(P') < 1. Thus, we have the following theorem. Theorem 5.22 — The variance of an analog game with the contribution functions f(P,P') and fc(P',P") is not lower than that of the equivalent nonanalog game with the same contribution functions if c(P')

1

A number of comments are proper here. 1. Since 5-, — oT2 in Equation (5.327) usually is definitely positive for c(P') < 1, it is expected that the variance of the equivalent game is lower than that of the analog game even if c(P') is slightly greater than unity.

268

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

2. As was shown in Theorem 5.12 in Section 5.V.F, the number of collisions to be played in the equivalent game is greater than that in the analog game if c(P') < 1 i.e., the variance and computing time antagonize, a phenomenon characteristic of most variance reduction techniques. 3. The equivalent game coincides with the survival-biased nonanalog game for nonmultiplying analog games since then c(P') = cs(P') = 1 — ca(P'). 4. For multiplying analog games, the equivalent game is a generalization of survival biasing in the sense that not only absorption, but also multiplication, is replaced by alteration of the statistical weight. It is reasonable to ask what gain is expected in efficiency by introduction of the equivalent nonmultiplying game, instead of the survival-biased multiplying game. By analogy to the derivation of Theorem 5.22, it can be shown that the condition of variance reduction by the equivalent game would be 00

cs(P') + cf(P')

E n=

nqa(P') = c(P') < 1 — ca(P') = cs(P') + cf(P')

Although this condition never holds, it is only sufficient, not necessary, and comment 1 above applies here too. On the other hand, the number of flights played in the equivalent game is certainly lower than the corresponding number in the game with survival biasing, and the efficiency ratio of the two games, which is determined by the product of the respective variance and number of flights, is still uncertain. Let us compare the efficiencies of the survival-biased analog and equivalent nonmultiplying games in the approximation of the separation assumption (cf. Section 5.VII.B). Let us consider the problem of a monoenergetic isotropic transport in a homogenous medium in V with an absorption probability ca. Assume that any collision results in either an absorption or an n-fold multiplication. Denoting E = 1 — Ca

the man number of secondaries per collision in the game is c = En and the collision kernel reads En C(P',P") = — 8(E' — E) = EnCs(P',P") 47r The survival-biased collision kernel, according to the definitions in Section 5.VIII.D, is C(P',P") = nCs(P',P") and therefore the weight factor to be applied after every collision is ws(13' ,P") = wn(13',13") = wf,n(r) = E Let the collision rate be the quantity to be estimated and, for the sake of simplicity, let us use the collision estimator. Then f(P,P') = 1,

= i'a(P',P") = in(P',P1') = 0

269 and according to Equations (5.295) through (5.299), the second moment of the score in the survival-biased analog multiplying game satisfies: MA(P) = tdP'T(P,P'){1 + 21dP"EnCs(P',P")Mi(r) + e2n(n — 1) [fdP"Cs(P' ,P")M,(P'')

2 ] 1 + f dP' T(P,P')IdP"enCs(P',V)M2(13") v

(5.328)

Taking into account that the first-moment equation (5.273) becomes Mi(P) = I dP'T(P,P') + f dP'T(P,P')IdP"EnCs(P',P")MI(P") v

(5.329)

the first two terms on the RHS of Equation (5.328) are rewritten to yield M2(P) = 2Mi (P) — 1 dP'T(P,P') + I dP' T(P,P')E2n(n — 1)M7(r) v v

+

dP' T(P,P')fdP"enCs(P',P") MAP") ' 1v

(5.330)

with Mi(P') = fdP"Cs(P',P")Mi(P")

(5.331)

Now, in the approximation of the separation assumption Equation (5.257), the averaged solution of Equation (5.328) is

M 2 = [2M1 —

PC + en(n — 1)m(2)]/(1 — enPc)

where P, is the first-flight collision probability in V and m = (2)

1

„Tv L

dP' &I-2(P')

i

(5.332)

Similarly, from Equation (5.329) M, = P/(1 — enPc) (provided EnP, < 1) and therefore the total variance in the survival-biased analog game is 2 = iA2 NV _

P,(1 — Pc) , (1 — EnP,)2 (1 — E- nP,)

+

13,E2n(n — 1) ID [I11(2) — IQ 1 — enr,

(5.333)

It follows from Equation (5.192) that the number of flights to be played in this game satisfies the equation N(P) = 1 + fdr T(P,P').fdP"nCs(P' ,P")&(P")

270

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

i.e., the average number of flights with the separation assumption is = 1/(1 — n13,) The product of the variance and the number of flights in the survival-biased analog game becomes 154i = Pc(1 — Pc)/[(1 — EnPc)2 (1 — € 2n13c)(1 — nPc)] + a (5.334) where a = Pc€ 2n(n — 1)[(1T1(2)

Ri/R 1 — € 2n13c)(1 —

(5.335)

The second moment of the score in the equivalent nonmultiplying game follows from Equations (5.322) through (5.324) as ht2(P) = 2M1(P) — dP' T(P,P') + dP' T(P,P') f dP"e2n2Cs(P',P").42(P") i.e., the average second moment in the separation assumption is A2 =

(2M1 — Pc)/(1 — € 2n2Pc)

Hence, the variance reads g2 =

13,(1 — PA(1 — EnP,)2 (1 — € 2n2P,)]

The mean number of flights satisfies the equation X(P) = 1 + fdP' T(P,P').fdP"Cs(P',P"),N(P")

X = 1/(1 — Pc) Finally, the inverse of the efficiency in the equivalent game reads = Pc/R1 — EnPc)2 (1 — € 2n2Pc) Now the efficiency of the equivalent game is higher than that of the survival-biased analog game if E-1 > W -1. Notice that a in Equations (5.334) and (5.335) is nonnegative for n 1 since

1n

=

2 2 1 f dP' [fdP"Cs(P' ,P")M,(P")] [— fdP' fdP"Cs(P',InMi(rd = 47rV J 47rV

Thus, the efficiency relation above certainly holds if E --1 — a > 'V', i.e., if (1 — 13)4(1 — € 2n13c)(1 — nI3c)] > 1/(1 — e2n2Pc)

271 After rearrangement, this relation becomes (n — 1)(1 — On) > 0 which holds if 1 < n < 1/E2 or, equivalently, if 1 — ca < c < 1/(1 — Ca)

(5.336)

Numerical experiments show33 that the gain in efficiency predicted by the considerations above is actually realizable. Figures 5.4 and 5.5 present the efficiency ratios of the equivalent nonmultiplying game and the survival-biased multiplying analog game when estimating the collision rate in homogeneous slabs of different thicknesses, x, in a monoenergetic isotropic simulation. The absorption probability is ca = 0,2 and 0,3, respectively [n is the number of secondaries per collision, c = n(1 — c2)]. It is seen that the efficiency ratio increases with increasing dimensions and also with increasing absorption probability.33 F. ZERO-VARIANCE PARTIALLY UNBIASED ESTIMATORS: THE MINIMUM VARIANCE-COMPOSED ESTIMATOR Zero-variance nonanalog games have been derived in Section A. We have noted there that zero-variance simulation can also be performed (in principle) with any given kernels by the introduction of proper contribution functions. Somewhat more will be shown in this Section, namely, that there exists a zero-variance partially unbiased estimator set for any analog game. Although such a set may have little direct practical use, it will be seen in Section H that approximation to the optimum partially unbiased estimator yields efficient and practically feasible estimators. The minimum-variance partially unbiased estimator set follows from the conditional minimization of the second moment of the score with respect to the contribution function under the condition that the estimators satisfy Equation (5.207). According to Theorem 5.2, the second moment of the score is minimum if the source term of the second-moment Equation (5.319) is also minimum. Therefore, the minimum-variance partially unbiased estimator set follows from the extremum problem

f dP'T(P,P'){[f2(P,P') + 2f(P,P').°1,,(P') +

2(P')1

+ 2X(P)[I,(P) — g(P,P')]} = minimum

(5.337)

where g(P,P') = f(P,P') + ca(P')fa(P') + cs(P1 )1 dP"Cs(P',P")fs(P',P") + cf(P') E nq„(P').1dP" C„(P' ,P") f„(P' ,P") .=i

(5.338)

272

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

2.6

-

2.5

A

2.4 2.3 2.2 2.1 A

2.0

X=2.0

1.9

,-.

A

a u

A

....., ......„.

1.8 1.7

A

1.6

•

A

1.5 1.4

A

X '` 1 . 0

1. 1 1. 0

. ..• • .

•

.

V

•

•

1.3 1. 2

i

A

X-0.5 0 . .

. 0

• •

•

.

0

0.9

0

0.8 0.7 0. 6 O. 5

I

I

a

i 1.4

1.2 1.3 1.0 1.1 (0.80) ( 0.88) (0.96) (1.04) (1.12) n (c)

1 1.7 1.5 (1.20) (1.28) (1.35) , 1.6

FIGURE 5.4. Efficiency of the equivalent nonmultiplying game relative to that of the survival-biased analog game in slabs of optical thicknesses X at absorption probability c, = 0.2.

is the composed estimator defined in Equation (5.221). 9-,,(P') and J2(P') are given in Equations (5.320) and (5.321), respectively, and X is the Lagrange multiplier of the problem. Note that according to Equations (5.320) and (5.338) J,(P) = g(P,P') - f(P,P') + M,(P') with

M,(P')

= fdrC(P',P")M,(P")

(5.339)

273 4.0 3.8 3 .6 3.4

•

3.2 3.0

•

2. 8 2. 6

U L) .-I ._. -.. .-

• •

2.4

X-2.0

2.2

•

A

2.0 •

1.8 • •

1.6 1.4 •

1. 2 1.0

•

• •

X=1.0

X `' 0.5

•

•

.

•

o

..d• A

• • • .

1. 8 1.6

1 1 • i • 1 1 1 I i i 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 (1.54) (1.40) (1.26) (1.12) (0.98) ( 0.84) (0.7) A (c )

FIGURE 5.5. Efficiency of the equivalent nonmultiplying game relative to that of the survival-biased analog game in slabs of optical thicknesses X at absorption probability c, = 0.3.

M,(P') is the expected score due to a particle entering a collision at P'. Solution of the extremum problem in Equation (5.337) is easily obtained by equating to zero the variations of the LHS of the equation with respect to the contribution functions. Variation with respect to f(P,P') yields g(P,P') + M,(P') = X(P) Multiplying this Equation by T(P,P') and integrating over P', we have X(P) = fdP'T(P,P1 )[g(P,P') + M,(P')] = 11 (P) + fdP'T(P,P1 )1 drC(P',P")MI(P")

274

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

A(P) = Mg) Hence g(P,P') = M,(P) — M,(P')

(5.340)

i.e., the optimum composed estimator is the difference of the expected scores at the starting and end point of a flight." Variations of Equation (5.337) with respect to the other contribution functions provide the equation system f(P,P') + fa(r) = Mg) f(P,P') + fs(Pi ,P") = M,(P)

M,(P")

and f(P,P') + fr,(P',P") + (n — 1)1dP"Cn(P',P")[fr,(P',P") + M,(P")] = M,(P) — M,(P") It is easily seen that the solution of the equation system that also satisfies Equation (5.340) is f(P,P') = M,(P),

fa(r) = 0

(5.341)

and fs(P',P") = fn(P',P") =

MR')

(5.342)

Equations (5.341) and (5.342) define the minimum-variance partially unbiased estimators and the following theorem holds. Theorem 5.23 — The minimum-variance partially unbiased estimator set in Equations (5.341) and (5.342) yields a zero-variance estimate. Proof. Substituting the estimators into Equations (5.319) through (5.321) of the second moment, it can be seen that M,(P) = M1.(P). Instead of the formal proof, however, let us realize that the contribution functions in Equations (5.341) and (5.342) score M,(P) — M,(P; ,) in the i-th flight (that starts from the collision point P,) if it is followed by a real collision and M,(P,) if it is the last flight (followed by an absorption). Therefore, the final score in a history started from P is always M,(P), with no fluctuation.

Note that we have also shown in passing that the minimum variance-composed estimator has the form in Equation (5.340). Naturally, this estimator is no more feasible than the zerovariance estimators above. Nevertheless, it will be seen in Section H that there exists a partially unbiased estimator which approximates rather well this optimum. It can be seen from Equations (5.319) through (5.321) and (5.339) that in a game where the optimum composed estimator g(P,P') scores in the intercollision flights and zero contributions follow from the collisions, the variance of the score satisfies the equation D2(P) = f dP' T(P,P') V(P') + fdP' T(P,P')f dP"C(P' ,P")1Y(P")

(5.343)

275 with V(P') = f dP"C(P',P")Mi(P") — [f dP"C(P',P")M,(P") ( ')

+c, P

E

n(n — 1)q„(P')[1dP"Cn(P',P")/V1,(P")]

2

(5.344)

n=1

Another interesting consequence of the derivation above is that although the composed estimator is the expectation of the reaction-dependent contributions over the possible outcomes of the collision and, as such, one might expect a lower variance, it does not necessarily decrease the variance of the score, compared to the reaction-dependent estimators. Nevertheless, in most practical cases, the estimators do not depend on the type of collision (cf. Section 5. VI. B) and therefore approximations to the minimum variance-composed estimator have lower variance than the usual estimators. In the following section, we compare the variances of the commonly used estimators in an analog simulation. A corresponding analysis for nonanalog game may be performed in a similar way. G. RELATIVE MERITS OF THE COMMON ESTIMATORS We consider here the estimators derived in Section 5.VI.B. Let us first notice that (except for the last-event estimator) all these estimators are the composed type, i.e., they do not depend on the type of reaction at the end of the free flight and they only depend on the starting point, P, and end point, P', of the flight. Comparison of the variances of reactiondependent estimators in nonmultiplying games is reported in References 28 and 31. The second moment of the score in an analog game with the estimator f(P,P') follows from Equations (5.319) through (5.321) as M2(P) = fdP' T(P,P')[f2(P,P') + 2f(P,P').1dP"C(P',P")M I(P")] + f dP' T(P,P')1 dP"C(P' ,P") MAY)

(5.345)

We shall follow the usual procedure of variance comparison, i.e., we examine the difference of the source terms of Equation (5.345) for estimator pairs. If ZI(P) denotes the difference of the variances with the estimators fi(P,P') and f2(P,P'), then A(P) = A(P) + fdP"L(P,P")A(P") where A(P) = f dP' T(P,P')Ifl(P,P') — fi(P,P') + 21f ,(P ,P ' ) — f2(P ,P')]dP" C(P' ,P") M, (P")}

(5.346)

The variance of the score by the estimator f, is greater than that by f2 if A(P) > 0 for every point P of the domain of simulation (sufficient condition). Let us first compare the variance of the score with an arbitrary estimator of the form f,(P,P') = f(P,P')

276

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

and the variance with the expectation estimator corresponding to it f2(P,P') = fE(P) = f dP' T(P,P')f(P,P') The difference of the source terms in this case reads A(P) = f dP' T(P,P')f2(P,P') — [f dP' T(P,P')f(P,P') + 2fdP'T(P,P')[f(P,P') — fE(P)IMI(r)

(5.347)

Or

A(P)

f dP1 T(P,P'){[f(P,P') + 1A,(Ini 2

[fE(P) + M,(P1 )]2}

with Mi(P') = f dP"C(P',P")Mi(P") The difference of the first two terms in Equation (5.347) is certainly nonnegative and therefore the following theorem holds. Theorem 5.24 — A partially unbiased estimator yields greater variance than the corresponding expectation estimator if fE(P) < f(P,P') for every point P, P' such that

f

dP"C(P',P")M,(P") > 0

Application of the results of the theorem is illustrated by the problem of estimating the collision rate in an arbitrary bare body V. The collision estimator is f(P') = 1

if

P'

E

V

while the corresponding expectation estimator reads fE(P) = 1— e-r

if

P

EV

where T is the optical distance from P to the surface of V. Obviously, f,(P) < f(P') inside V and if no particle starts outside the body, the expectation estimator has a lower variance than the collision estimator. One is tempted to believe that the expectation estimator should always reduce the variance, compared to any other partially unbiased estimator of the form f(P,P') and therefore Theorem 5.24 might seem rather weak. It is easy to see, however, that although the variance per collision of the expectation estimator is zero, games exist where the variance of the final

277 score by the expectation estimator is definitely greater than that of certain other partially unbiased estimators. 28.42 For example, for a not-too-large probability of absorption (not-too-small number of secondaries per collision), the track-length estimator has, indeed, a lower variance than the expectation estimator.28 On the other hand, it is also seen from Equation (5.347) that with increasing probability of absorption, the third term in the expression of A(P) decreases in modulus, i.e., for sufficiently strong absorption, the positive difference of the first two terms is dominant and the expectation estimator becomes more efficient than any other partially unbiased estimator. In the limiting case of a purely absorbing medium, the expectation estimator will be identical to the optimum composed estimator in Equation (5.340), i.e., fE(P) = M i (P) = fdP' T(P,P') f(P,P')

(5.348)

with zero variance. Although in most practical cases the expectation estimator provides the smallest variance of all the partially unbiased estimators, it should be taken into account that when using it, one has to evaluate an exponential function at every collision. Since this procedure is rather time-consuming, the track-length and collision estimators may find their application even in such problems where their variances are expected to be somewhat higher than that of the expectation estimator. Next, we consider the variances of the collision and track-length estimators in collisionrate estimations. Let us assume again that the collision rate is to be estimated in a bare body, i.e., no particle enters the body from outside. In this case, fi(P,P') = 1 T(P,P') for P' inside the body f2(P,P') =

T(P,130 for P' outside the body

and, according to Equation (5.346), the source term of the variance difference equation reads A(P) = 1 —

e --r(P'PS) —

2 + 2[1 + T(P,P,)] e —r("s)

Ps

dP'T(P,P')[1 — T(P,131)]M,(P')

=

2T(P,Ps)I e'"8) — 1} + IPsdPi T(P,P')[1 — T(P,In]Mi(r) (5.349)

since

f

dP' T(P,P')QP,P') = 2 — 2[1 + T(P,P,)] e —(P•Ps)

The following theorem is established on the basis of Equation (5.349). Theorem 5.25 — The variance of the track-length estimator in collision rate estimation

278

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

is less than the variance of the collision estimator if the maximum optical extension, Tm, of the region (where the estimation is performed) satisfies Tm

1/2

for

P,

(r + tw,to,E),

t

0

then condition (6.17) may only be satisfied for finite systems. It is to be emphasized again that conditions (6.19) and (6.20) are sufficient, but not always necessary for a bounded variance of the correlated score difference. Nevertheless, in certain cases these conditions are also necessary. In what follows, we briefly outline two examples of such cases. Let us consider an infinite homogeneous medium as the unperturbed system. Assume that we are interested in the change of some reaction rate due to the change of the density of the material by a factor 1 — a (perturbed system). The scattering kernels are identical

314

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

in the two systems, i.e., 'y = 1, e s = c, and I-. = co. Accordingly, conditons (6.19) and (6.20) become a < 1/2 and sup c,(P) < (1 — 2a)/(1 — a)2 P

It is easy to see that if in a monenergetic transport problem these conditons are violated, then the variance of the score difference is indeed unbounded.8'67'83•84 If in some region the survival probability c,(P) is altered so that the total cross section remains unchanged, then 'y = 1 and a = 0. If the maximum optical extension of the region is Trn, then Tm = Tn, and condition (6.20) gives supraP)/cs(P)1(1 — e —rm) < 1 For an infinite homogeneous medium in momoenergetic approximation, this condition reduces to Cs

< Vcs

which again is not only sufficient, but also necessary for a finite variance.8 In contrast to the example above, it may also happen that condition (6.19) or (6.20) fails to hold for a problem which seems feasible. In such cases, either the conditions are too restrictive for the specific problem or the game is not defined properly. In practice, one has to examine the effects of the approximations applied in the derivation of the conditions and one also has to consider the possibiltiy of using a perturbed analog game (where the game is played analog in the perturbed system) instead of an unperturbed analog game. It will be seen in Section D that in certain cases this change in the simulation does indeed reduce the variance. C. CORRELATED DIFFERENCE ESTIMATORS In the derivation of the moment Equations (6.1) and (6.9), we assumed that both the unperturbed and perturbed scores result from the same contribution function f(P,P'). Thus, if f(P,P') is a partially unbiased estimator of the unperturbed reaction rate, the change in the reaction rate due to the perturbation is estimated via the contribution function Af(P,P') = (W' — 1)f(P,P')

(6.21)

where W' is the weight of the nonanalog particle after the flight from P to P'. If P' = P'„ the point where the particle enters its i-th collision, then W' = W'(i), as given in Equation (6.8). The partially unbiased estimator of the unperturbed reaction rate, f(P,P'), may be any of those introduced in Chapter 5.VI. The most commonly used are the collision and tracklength estimators. It is, however, not necessary that the perturbed and unperturbed scores be estimated by the same estimator. It can be easily seen48 that the derivations in Sections A and B remain valid if the score vector Wf(P,P') (Wf(P,P'),f(P,P'))

315 introduced in Equation (6.1) for a general correlated game is replaced by f(P,P',W) = (Wf,(P,P'),f,(P,P'))

ue37

where f, and f2 are two arbitrary partially unbiased estimators. For the estimation of the score difference, it means that the estimator in Equation (6.21) is replaced by Af(P,P') = W'f,(P,P') — f2(P,P')

ue38

Use of different estimators for the perturbed and unperturbed scores has not yet been investigated; nevertheless, it is reasonable to assume that in specific problems it may yield a lower variance than a common estimator of the two scores. On the other hand, if the contribution function depends on the material composition of the system (which is the case with the most common estimators), then it is inevitable that one would use different estimators in two games which differ only in the material-dependent quantities. D. VARIANCE OF THE CORRELATED SCORE DIFFERENCE Two important aspects of the correlated variance are investigated in this section. First, it is shown that the correlated game has the advantage of reducing the variance, compared to the corresponding uncorrelated estimation. This is seen by proving that the relative variance of an uncorrelated estimate tends to inifinity as the perturbations vanish, while that of the correlated game generally remains finite. Second, it is to be decided which system (the perturbed or the unperturbed) has to be chosen as the domain of simulation. Finally, the possiblity of playing the correlated game in an "intermediate" reference system is briefly discussed. In the previous Sections, we assumed that the game is an unperturbed analog, i.e., that it is played analog in the perturbed system. This, however, is only a matter of convention and the same score difference can also be estimated in a perturbed analog game. Usually, one prefers using the scheme with the lower variance. In the second part of this section, sufficient conditions are derived under which the unperturbed analog game results in lower variance than the perturbed analog game. We assume that both the analog and nonanalog particles start their random walk with unit weights and also that the contribution function f(P,P') assumes nonnegative values. The derivations are based on the second form of the moment equation as given in Equation (6.7). The weights W'(i) are given in Equation (6.8) by putting W = 1. Let us denote Ln(P„,P;,.• • ,P,1,+

= [

Ln(P.,Pi,... ,13:,±

= [ 11 'hi); i=1

T(P,_ 1 ,1):)C(P:,P,)] T(P„,P,', +

(6.22)

and (P:,13;)] t(P.,P:,±

(6.23)

Then the second form of the moment equation in the unperturbed analog game with F(s) (s, — s2)2 is written m{(s, — s2)2}(P0) =

E

c

r

d13; dP,

n=0

f dr„ ±

1 X

[

( W'(i) — 1) f(i)1 2

[

(W '(i)

WWI}

(6.24)

316

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

where, for the sake of brevity, we put f(i) = The term in braces is rewritten as An+ l(P„,P;,•••,13:,±1) -= [W'(n + 1) — l[f(n + 1) {2

[W'(i) — 1]f(i) + [W'(n + 1) — 1]f(n + 1)1

(6.25)

Finally, it follows from Equations (6.3), (6.4), (6.22), and (6.23) that i) = W'(n + 1) Ln(130,13;

Ln(130 ,13;

(6.26)

In order to compare the variances of the independent and correlated games, let us introduce the relative variance as the ratio of the variance and the square of the expectation of the score: d2(P) = D2{s, — s2}(P)/M2{s, - s2}(P) — sJ(P) tends to zero if the perturbation

In the case of independent simulations vanishes, while the limit of the variance is D2Is — S21(P) = M2(P)

M2(P)

it(P)

Mi(P)

2[1‘42(P)

WP)]

which is usually different from zero. Hence, the relative variance of the score difference as estimated in independent games tends to infinity if the perturbation is vanishing. On the other hand, in a correlated game d2(P) =

— s2)2}(P)/M2{(s, — s2)} — 1

and from Equations (6.9) and (6.22) through (6.24) d2(P) =

n-0

n=0

fdP; JdP,

fdP; idP,

JdP~ + ,1-,a(P„,P'

idK+, Ln(P.,Pi • •

i)

•••

I)/

+ )[W(n + 1) — 1]f(n + 1)

1 12

Obviously, the kernels T and C are characterized by certain material and geometrical quantities (such as cross sections and geometrical distances) and, thus, so is the kernel Ln in Equation (6.22). The perturbations in the system appear in the kernels through alteration of some of these quantities. If ak (k = 1, 2, . . . , K) represent these quantities in the unperturbed system and ak are the same quantities in the perturbed system, then we can write Ln(130,1311, • • • >13:, + 1) = Ln(P.,P; • • • ,13.'n + Rad)

317 and +illakI) Let ak(a) = ak + a(ak - ak)

+ mak,

Then ak = ak (0) and ak = ak (1). Obviously, the perturbation vanishes as a tends to zero. Thus, the limit of the relative variance in the correlated game while the perturbation vanishes reads lim d2(P)

n=0

=

n =0

idP; fdP,

t ) lim An±1(13`"P 2'• • • '13"± ')/ a 2 W'(n + 1) - 1 i) lim f(n + 1)] - 1 a ct—,0

fdP; fdP,

where both the numerator and the denominator have been divided by a2 before taking the limit. Let us denote Vsf,)(n) = lim n,-.0

W'(n) - 1

Then, according to Equation (6.26) W1, „(n + 1) = lim

Ln(P„,E;,. • •

o

k = I a ak

1-„(1)„,E;,...,P:,+11{4}) aLn(P0 ,P;,. • • ,K

log Ln(13„,P;,...,13:, ± ,Raj)

and from Equation (6.25) lim An+I(Po,P;,a—.0

et 2

NAT I)(n + 1)f(n + 1)[2

=i

V1- „(i)f(i) + \\T I)(n + 1)f(n + 1)]

Now, in most practical cases the derivatives of the kernels with respect to the physical and geometrical parameters ak are bounded and therefore both the numerator and denominator tend to finite values, provided the unperturbed game is feasible (i.e., if it terminates after a finite number of collisions with a probability of one). Thus, omitting a rigorous proof, we conclude that in the majority of practical cases, the relative variance of the score difference as estimated by the correlated game remains finite as the perturbation vanishes. Similar conclusions were reached for specific perturbations by Dejonghe et a1.8 A more detailed proof of the assertion above is given by Dubi and Rief.14 The conclusion above also means that in the case of small perturbations, the correlated game is expected to yield a lower variance than the corresponding independent games that are played with the kernels T, C, and T, C, respectively, and with the contribution function

318

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

f(P,P'). It does not mean, however, that the score difference in a correlated game has a lower variance than in any other two independent unbiased games. To see this, it is sufficient to note that in the idealized case when both the unperturbed and perturbed scores are estimated in zero-variance schemes, the variance of the score difference is also zero, which cannot be outperformed by any correlated game. Let us now turn our attention to the proper choice of the domain of simulation. The second moment of the score in the unperturbed analog game is described by Equations (6.22) through (6.25). By analogy to Equation (6.24), the second moment of the score difference in the perturbed analog game (when the perturbed reaction rate is the one which is estimated in an analog game) reads m*{(si — s2)2}(130) =

E fdP; JdP,

f dP:,±

n=0

Al; +1(1),P;

>13:,+1)

(6.27)

with +1(Po ,Pi

• • ,13n + 1) = [1 — W*(n + 1)]f(n + 1){2E [1 — W*(i)]f(i) ,=1

+ [1 — W*(n + 1)]f(n + 1)1 (6.28) Equation (6.28) differs from Equation (6.25) in that 1 — W* appears in the former in place of W'(i) — 1 in the latter. This is so because in the perturbed analog game, the particle migrating in the perturbed system has a permanent weight of unity and the particle simulated in the unperturbed system carries a statistical weight that is changed from collision to collision. According to the weight generation rules in Equations (5.142) through (5.146) W*(i) = [RP; _1,1):)/i(P, _ 1 ,PD] [C(P: ,12%)/(P: ,P;)1W*(i — 1) and comparision with Equation (6.8) shows that (6.29)

W*(i) = 1/W'(i)

Introducing Equations (6.29) and (6.26) into Equations (6.24) through (6.28), we note, after subtraction, that the variance difference in the perturbed analog and unperturbed analog games reads

m*{(si — s2)2}(P.) — mf(s, — s2)21(P.) = E fdri fdPi fdrn± n=0

i)B.,

319 where = A*(P„,13;,.= [1 — W'(n)]f(n){[ 1 — W'(n)12 f(n)/W'(n) n- 1 + 2 E t i - W'(i)]2 f(i)/W1 (i)}

(6.30)

Now, if Bn is positive for every n, then the variance of the unperturbed analog game is lower than that of the perturbed analog game. On the other hand, Bn > 0 if 0 < W'(n) < 1

for

n = 1, 2, ...

(6.31)

and in view of Equations (6.3) and (6.4), we have the following theorem. Theorem 6.2 — The variance of the unperturbed analog game is not greater than that 1, i.e., of the perturbed analog game if 0 < w(P,P') 0 < T(P,P')

T(P,P')

(6.32)

and 0 < w(P,P')w,(P',P") 0 < t(P,P') C(P' ,P")

1, i.e.,

T(P,P') C(P' ,P")

(6.33)

for every point of the domain of simulation. It is to be stressed here, too, that condition (6.31) is sufficient but not necessary for a positive variance difference. Moreover, conditions (6.32) and (6.33) are again sufficient but not necessary for the fulfilment of inequality (6.31). Nevertheless, there are some important points that follow from the theorem. First, if conditions (6.32) and (6.33) hold, then the correlated game results in finite variance, provided the analog game is feasible. Indeed, if the weight factors w and ws are less than unity and the analog game is feasible in the sense of Equation (6.5), then the integral kernels in Equations (6.11) and (6.14) satisfy the inequalities L(2)(P,P") < L(o)(P,P") = L(P,P") < 1 and therefore the condition of Theorem 6.1 is satisfied. Second, if the weight of the nonanalog particle, W'(i), is less than unity, then the score difference estimator in Equation (6.21) is negative and so is the final estimate. Accordingly, the score in the unperturbed system is larger than that in the perturbed system. Thus, the considerations above suggest that the correlated game is to be played analog in that system where the higher score is expected. In certain problems, neither the unperturbed nor the perturbed system is acceptable as the domain of simulation. This is the case, for example, with geometrical perturbations when the two systems are overlapping but both have regions not common with the other. (Obviously, in such a problem the statistical weight of the nonanalog particle may become infinite, whichever game is played.) In such cases, the easiest solution is to define an extended domain of simulation in which both the unperturbed and the perturbed particles are processed nonanalog.

320

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

In general, one might expect that playing the game in a system which is neither unperturbed nor perturbed, but, rather, "somewhere in between", the variance of the estimate may be smaller in any of the unperturbed or perturbed analog games. The practical advantage of such an "intermediate" reference system depends on programing considerations as well as on the possibility of defining a reasonable reference system. Mathematical description and a priori investigation of a correlated game in a reference system is analogous to that followed in this Chapter. Thus, it is easily seen that the two forms of the moment equation in Equations (6.1) and (6.7) remain valid if T and C denote the kernels in the reference system; the only alteration is that the weights of the correlated particles are to be chosen different from those in Equations (6.2) and (6.8). Namely, if T,(P,P') and C,(P' ,P") denote the kernels in the perturbed system and T2(P,P') and C2(P',P") are those in the unperturbed system, the weight vectors in Equation (6.1) read W' = (Wc,WD,

W" = 1W7,WD

with Wk = WkTk(P,P')/T(P,P') and W; =

,P")/C(P' ,P")

for k = 1 or 2, by analogy to Equations (6.3) and (6.4). Similarly, the weight factor corresponding to Equation (6.8) in the second form of the moment equation reads w'()) (w;(i),w;(0)

with

=„I'D] J=

1

x [

ck(P:,13,)/C(P:,13;)]

7= 1

for k = 1, 2, i.e., for the particles representing the perturbed and unperturbed systems, respectively. Specifically, the second moment of the score difference when the game is played in the reference system has a form analogous to that in Equations (6.24) and (6.25):

m{(si — s2)2}(P0) = E fdr f dP, f drn „ n=0

with An+

= [W;(n + 1) — W;(n + 1)1f(n + 1)12

[W;(i)

— W;(01f(i) — 1W;(n + 1) + W;(n + 1)1f(n + 1)} Optimization of the reference system may be rather troublesome in general. In fact, it is not even certain that the optimum reference system is "between” the two systems in any sense.

321 E. EXAMPLES AND SPECIAL TECHNIQUES Let us first consider an application of Theorem 6.2. Assume that the perturbation introduced into the system consists of an alteration of the cross sections in a finite region V. Let 6(P) = [1 — a(P)]0-(P) where a(P) = 0

if

P V

and may be different from zero inside V. Then condition (6.32) is met if 0 < [1 — a(P')]exp[TJP,P')]

1

with

Teg,r) = fp

ct(1),)o-(Pt)dt;

Pt = (r +to.),(0,E)

Similarly, condition (6.33) holds if 0 < [1 — a(P')] exp[T(P,P')[ C,(P')/cs(P') < 1 where es(P') is the survival probability at a point P' inside the perturbed region V and is equal to c(P') if P' is outside V. For a constant value of a, these inequalities become 0 < (1 — a)e—,

min[1, inf c,(P')/es(P')]

(6.34)

P'

where T, is the maximum optical extension of the perturbed region. If the perturbed system differs from the unperturbed system only in the values of the survival probability, then condition (6.34) becomes 0 < Cs(P)

c,(P)

i.e., the game with the higher scattering probability is to be played analog. Next, we consider a delicate example of the correlated simulation. Let the perturbed system be constructed from the unperturbed one by complete voiding of a region (geometrical perturbation), i.e., let a region of the unperturbed system be replaced by a vacuum. Then T(P,P') T(P,P') since I = 0 in the void. Then Theorem 6.2 seems to suggest that the game should be played analog in the unperturbed system. In this case, however, the weight of the nonanalog particle would become infinite whenever a collision is played in the region voided in the perturbed system, and one might think that this example contradicts the theorem. This, however, is not the case since conditions (6.32) and (6.33) also state that the theorem applies only to nonvanishing nonanalog kernels, which is clearly violated in the example. Because of the possibility of infinite weights, the game obviously may not be unperturbed analog and therefore finite variance can only be obtained in a perturbed analog simulation

322

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

unless other tricks are introduced. An alternative solution proposed by Rief and Fioretti67 involves the introduction of delta scattering into an unperturbed analog game. Yet the problem is not completely solved by either method. Let us realize that the estimated effect of the perturbation will be biased to some extent, since the scattering (and possibly multiplicative) properties of the material, removed from the unperturbed system, will have no effect on the final result (no scattering is played there). An exact treatment of such problems cannot be performed by completely correlated histories. A possible solution is that the particles' histories are separated if a collision occurs in the perturbed region, the perturbed particle suffers a delta scattering, or does not suffer a collision at all, whereas the unperturbed particle enters the collision and then follows an independent path. This makes the simulation unbiased, but weakens the correlation between the two histories (which, in turn, tends to increase the variance of the estimated perturbation). A procedure of this type was proposed by Takahashi' for the calculation of reactivity perturbations in pulsed fast reactors. A satisfactory solution of the problem is expected from further investigations.28 The danger of a biased correlated estimate also is real in cases where the perturbed region is not completely voided, but its optical dimensions are very small, and thus the probability of a collision inside the region is low. In this case, only a minute fraction of the histories will carry information on the difference of the scattering properties of the perturbed and unperturbed regions and therefore a bias, universally associated with the estimation of rare but important events, is expected. This difficulty may be remedied by the method of forced collisions." In this method, the history of the correlated particles is split into two branches when it enters the perturbed region for the first time. The probabilities of a collisionless free flight through the region are calculated for both the perturbed and unperturbed particles, and the first branch of the common history is restarted from behind the perturbed region; the weights of the restarted particles are multiplied by the respective freeflight probabilities. In other words, the fractions of the particles that are expected to cross the perturbed region without collision are deterministically transmitted through the region. The particles in the second branch will carry weights equal to the difference of the original weights and those of the transmitted particles, and they are forced to enter a collision inside the perturbed region. This is performed by selecting a free flight from a truncated transition kernel concentrated on the perturbed region, as is done in the ELP method (cf. Section 5.VIII.D). Let us note in passing that care must be taken in properly altering the weight of the perturbed particles when they are restarted from the boundary of the perturbed region since their weights were altered after the previous collision according to a free flight interrupted at the boundary. (cf. Sections 5.IV.A and B). In this method, particles migrating in the second branch of the history will carry information on the scattering properties of the perturbed region. Therefore, these particles are to be kept in existence until they contribute to the final score. They should not, for example, be split or Russian rouletted any more.' It may be questionable whether scattering should be forced at the first crossing only or perhaps also at repeated entries to the perturbed region. Nevertheless, it seems logical to limit the forced collision to the first entry as repeated forced collisions are likely to deteriorate the efficiency of the method, for two reasons. First, every repetition increases the number of histories and thus the number of collisions to be played per starter. Second, the weight of a particle that has been forced to collide several times tends to be very low, for the method is used for optically transparent regions when the probability of a free flight through the region is close to one. As mentioned in the previous section, in certain applications, a reference system is used where the game is played analog in order to make the estimation feasible in both the perturbed and unperturbed systems. A special technique used in conjunction with correlated games in reference systems is discussed briefly below.

323 The weight factors associated with a transition from a point P to P' in the reference system of a transition kernel T(P,P') were seen to have the form wk = Tk(P,P')/T(P,P')

for k = 1 (perturbed system) and k = 2 (unperturbed system). Making use of the explicit form of the transition kernels P'

wk = [ffk(P ' )/0-(P)1 eXpt

dt[0-k(Pt)

k = 1, 2

cr(Pt/l}

where Pt is a point situated at a distance t from P on the Spatial line connecting P and P'. Obviously, whenever cr > o-, for any of the systems (k = 1 or 2), the weight factor for the system may become very large for large free flights. Large fluctuations in the statistical weights, on the other hand, may lead to a large variance. A possible resolution of the problem proposed by Riefm was the introduction of delta scattering in all three systems. Let the delta-scattering cross sections (cf. Section 5.V.H) cro and o-„,„ (k = 1, 2) have the following properties: cro(P) a 0, o-,(P)

0-0,k(P) % 0,

k = 1, 2 k = 1, 2

cro ,(P) = Q(P) + o-o(P) = o-s(P),

for every P. Then, obviously, the weight factors due to transitions become unity, *k = 1, and (if no other biasing is applied) the weight of the particles is changed at collision points only (i.e., at sites of delta scatterings or real collisions). Now, the probabilities of delta scatterings in the various systems are (cf. Section 5.V.H) q= o-o/o-s,

qk =

cro.kiffs

k = 1, 2

Crs.k/Crk

k = 1, 2

and, if cs =

Cs,k

denote the nonabsorption probabilities in the various systems, then the weight factors in a collision, the ratios of the scattering probabilities in the perturbed or unperturbed systems to that in the reference system, become ck(1

ch,)/c(1 —

= cr,,kicrs for a real scattering

w'ts' = qk/q = o-0,„/cro for a delta scattering In the scheme above, two quantities are freely chosen, viz., Q(P) and o-o(P), the real and delta-scattering cross sections of the reference system. (The delta-scattering cross sections of the perturbed and unperturbed systems are then determined by the conditions above.) It is heuristically obvious that the fluctuation of the statistical weights is decreased in the scheme by increasing the number of simulated flights (i.e., computing effort). In practical application (as it so happens in most cases), common sense, practice, and test calculations must determine the values of the free parameters.

324

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

F. PERTURBATION SOURCE METHOD An approach different from the correlated Monte Carlo procedure was proposed by Matthes for estimating the effects of small perturbations on reaction rates.5"1 The procedure is based on the assumption that the quantities of second order in the perturbations can be neglected. The method proposed assumes a coupled direct-adjoint game where the direct game provides a perturbation-independent response function (weighting function) for the adjoint game. This original method can be converted to a purely branching direct game. Let us consider again the variation of a reaction rate R due to some perturbation in the system: SR = fdP[tii(P) — tli(P)1f(P) = fdP&P(P)f(P)

(6.35)

According to Equations (5.4) and (5.5), the unperturbed collision density satisfies the transport equation tkP) = fdP'Q(P')T(P',P) + fdP"Ili(P")K(P",P)

(6.36)

Let the differences of the perturbed and unperturbed kernels and sources be denoted as ST(P',P) = t(P',P) — T(P',P) SK(P",P) = K(P",P) — K(P",P) and Q(P) = Q(P) Q(P)

8

Then the perturbation in the collision density reads 41(13)

'

= SK(P)

f dP"SVPH)K(Pll,P)

(6.37)

where = fdP'SQ(P')T(P',P) + sidPi(P')ST(P',P) + JdP"

(6.38)

Neglecting the terms of second order in the perturbation (such as SQST and S4fi8K), the equation becomes = f dP'SQ(P')T(P' ,P) + fdP'Q(P')ST(P' ,P) + idrtli(P")8K(P",P)

(6.39)

325 It is seen from Equation (6.37) that 8t1i(P) satisfies a transport equation analogous to Equation (6.36) with a first collision density 40(P) defined in Equation (6.39). Rewriting Equation (5.10) in terms of the collision density perturbation, we have a relation between the direct and adjoint collision densities as 8R = f dP8ili(P)f(P) = f dPOP)8ilio(P)

(6.40)

This relation suggests the following procedure. In the first step, let us simulate the unperturbed collision density in a direct game with the unperturbed kernels and determine the quantity 8450(P). In practice, integrals of 8450(P) over small phase-space regions are estimated, which implies the assumption that 8ilt0(P) is approximately constant over the separate regions. If the perturbations are small, this approximation causes a negligible error. The first two terms in Equation (6.39) can often be determined analytically. If analytical integration is not possible, they can be estimated as first-flight collision densities. To see this, let us write the first two terms in the form JdP

6T(P',P)] Q(p,)[8Q(P;) Q(P) T(P',P) 1.(13' 'P)

Obviously, this expression defines the first-flight collision density at P due to a starter at P' selected from the source density Q(P') when the starter has an initial weight W—

8Q(P')

8T(P',P)

Q(P')

T(P',P)

This contribution to 40 (or better, to its integral over the small phase-space regions) can be estimated parallel to the simulation of the first flights in the unperturbed direct game. The integral of the third term in Equation (6.39) over a small region is an ordinary reaction rate in the unperturbed game with a weighting function equal to the integral of the kernel 8K(P",P) over the region. Such integrals can be estimated in the course of the direct simulation. In the second step, an adjoint game is played again in the unperturbed system which simulates tlt*(P) (cf. Section 4.VII) and the adjoint reaction rate of the RHS of Equation (6.40) is estimated. The weighting function in the adjoint reaction rate is 40(P), calculated in the first step. Although the considerations above are valid only in first-order perturbation approximation, this limitation can be easily removed. Indeed, if the direct game is played in the perturbed system, then the exact value of the perturbation in the collision density, Stlio in Equation (6.38), can be estimated and used as the weighting function of the reaction rate in the adjoint game. The same result is obtained in the alternative method, where the direct game is played in the unperturbed system and the adjoint game in the perturbed."'" These methods have the common drawback that if the perturbed system is considerably more complicated than the unperturbed (i.e., if small but complex structures are inserted into a simple geometry), the simulation of the perturbed game may require considerably more effort than that of the unperturbed game. The techniques treated in this Section were introduced mainly for estimation of reactivity perturbations. This concern will be revisited in Section 6.111. An alternative method that does not require the introduction of an adjoint game is based

326

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

on the fact that the perturbation of the collision density satisfies the transport equation (6.37), and this equation contains the unperturbed kernels. Therefore, starting particles from the first-flight collision density in Equation (6.39), the reaction rate (6.35) can be estimated in an unperturbed direct game. To show this, let us write Sklio(P) in Equation (6.39) in the following form: 811J0(P) = f dP'Q(P')T(P' ,P)

Q (P' ) [ Q(P')

8T', (P P)1 T(P' ,P)

[" + fdPildP"tlr(P")C(P"P')T(P'P) 8C(P ,P') C(P",P')

8T(P',P) T(P',P) 1

(6.41)

Equations (6.41) and (6.37) show that the unperturbed game provides the source of another game which in turn, simulates the perturbation. The scheme is analogous to a coupled multiparticle simulation (cf. Section 5.IX.D) and can be played in the following way: 1. Let an unperturbed particle be started from the unperturbed source density Q(Po) and let its next collision be selected from the unperturbed transition kernel T(Po,P;). 2. Let the selected collision point P; also be the first collision point of a "perturbation particle" i.e., a particle that simulates the collision density perturbation Sir in Equation (6.37). The weight of the perturbation particle at the collision point P; due to the starter at Po will be =

6Q(Po) Q(P0)

ST(P ,P;) ° T(Po,P;)

3. Let us follow the simulation of the history of the perturbation particle according to the unperturbed kernels and accumulate the contributions to the reaction rate in Equation (6.35) (including the contribution due to the first flight from Po to P;). 4. At the end of the history of the perturbation particle, let us continue the history of the unperturbed particle by playing a collision (from PI' to some P,) and, if it survives the collision, by selecting a new flight from P, to P. 5. Again, let us consider P, as the source point of a new perturbation particle and let 13; be its first collision point. The weight of the new particle at 13; will be W = 8C(P; ,P,) + 8T(P1 ,PD ' C(P;,P,) T(P„PD Return to step 3. Naturally, if the starting weight of the "simulation particle" is zero, i.e., if the collision and transition of the unperturbed particle take place through a phase-space point where no perturbation is introduced, then the history of the perturbation particle is not to be simulated. This method of perturbation estimation in most cases is less effective than the correlated game presented in the previous section. However, in problems where the perturbed regions cover only a small fraction of the whole domain of simulation, the game above may outperform the correlated game. Coupling the two methods proved very efficient.' In the coupled game, a correlated technique is applied for the first few collisions of every starter and separate histories of perturbation particles are started from subsequent collisions. The variance of the perturbation source method in general is investiaged by Noack.62

327 G. PARAMETRIC PERTURBATIONS: INTEGRAL MONTE CARLO The methods of perturbation calculations presented in the previous sections are efficient only if the perturbed system does not differ essentially from the unperturbed one. In case of drastic differences, the statistical weights in the correlated game tend to fluctuate, thus resulting in a large variance of the final estimate. The first-order perturbation source method is limited a priori to small perturbations, and the exact perturbation source methods also have the limitations discussed in the previous Section. A special method was proposed by Usikov82 for estimating the change of a reaction rate due to finite perturbations. The method is based on the assumption that the perturbation may be characterized by the change of certain system parameters. Let us here consider the simplest case, when some change in a system parameter a represents the perturbation. (a may be, for example the number density of an element in the material composition of the system, a characteristic geometrical distance, or any other quantity playing a role in the kernels of the simulation.) Let the parameter value a, represent the unperturbed system and let a2 be the value of the parameter in the perturbed system. Let R(a l) and R(a2) denote the reaction rates of interest in the unperturbed and perturbed systems, respectively. Then the perturbation of the reaction rate is SR = R(a2) — R(a,) If R(a) is a differentiable function of the parameter, the perturbation is written as er2 dR SR = f da. — da

(6.42)

The estimation of SR is performed in a two-step Monte Carlo procedure. The first step consists of a simple selection of the parameter a between a, and a2 . In the second step, the derivative dR/da is estimated in random walks. Let p(a) be a probability density function defined over the interval [a, ,a2 ] and let us rewrite Equation (6.42) in the form .2

1 dR

SR = f da p(a) [— p(a) da

(6.43)

Selecting an a value from p(a) and then estimating dR/da in a simulation procedure where the initial statistical weight of every starter is multiplied by 1/p(a), obviously we obtain an estimate of SR. Estimation procedures for the parametric derivatives of reaction rates are detailed in the next Chapter. Note that the probability density p(a) in Equation (6.43) is arbitrary and its proper choice may reduce the variance of the estimate. Let us realize that optimization with respect to p(a) here is analogous to the problem of source density optimization discussed in Section 5.VIII.J. If M2(a) denotes the second moment of the score in the random-walk estimate of dR/da with unit weight starters and with a given, then the optimum probability density p(a) follows from Theorem 5.26 as .2

P(a) = VM2(a)/1 da VM2(a) 0„ If the form of M2(a) can somehow be guessed, then p(a) can be chosen close to the optimum. Otherwise, the easiest way is to select a from a uniform density. It can be seen82 that in the latter case, the variance of the estimate is minimum if every random walk is started with a newly selected a value.

328

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

The integral Monte Carlo method can be easily extended to problems where the change in the reaction rate is due to simultaneous changes of several system parameters. In this case, however, it is necessary to estimate the derivatives of the reaction rate with respect to all the parameters involved, which requires either several independent differential Monte Carlo estimations or a correlated differential game.

II. DIFFERENTIAL MONTE CARLO: SENSITIVITY ANALYSIS Analysis of the effect of uncertainties in some characteristic parameters on given reaction rates, or determination of the sensitivity of reaction rates to small changes in the system parameters, play an important role in nuclear design and operation. In complex systems, the only way of performing such an analysis is through Monte Carlo simulation. Correlated Monte Carlo presented in the previous Chapter is a possible tool for examining such effects. An alternative method is the estimation of the sensitivity of the reaction rates on some system parameters that characterize the uncertainties (perturbations). To be more specific, let us consider a reaction rate R depending on the parameter a. Assume that we are interested in the variation of the reaction rate when the parameter varies around a given a value. If R can be expanded into a Taylor series around this value, the variation due to a change of Act in a is expressed as R(a + Aa) — R(a) 8R(a) =

d"R = 1 clan

2, - (Aa)"

n

(6.44)

where dn R/dan is the n-th derivative of R(a) at a. If the change Aa in the parameter a is sufficiently small and R(a) is a suitably smooth function of a, then the first few terms on the RHS of Equation (6.44) approximate well the change in the reaction rate. 8R(a) =

d"R n a (Aar

(6.45)

Parametric derivatives of the reaction rate are estimated in differential Monte Carlo games. Use of the Taylor series form of the change in the reaction rate has the advantage over the correlated Monte Carlo technique of providing this change for arbitrary (but small) variations of the system parameter(s), while correlated games determine the perturbation due to a given variation of the parameter(s) (cf. Section G). In other words, the derivatives are characteristic of the sensitivity of the reaction rate in question to the variation of the parameters. In fact, sensitivity is defined as the ratio of the fractional changes in the reaction rate and in the parameter when the parameter perturbation tends to zero,44 i.e., SR Da a dR S = lim — — = — — A„,-0 R R da Equations (6.44) and (6.45) concern the case when the reaction rate changes due to the variation of a single system parameter. In the case of several parameters, the corresponding multivariate Taylor series applies and the partial derivatives of R with respect to the parameters in question have to be estimated. For the sake of simplicity, we shall constrain our derivation to the single-parameter case; extension of the considerations to several parameters is straightforward. An unbiased procedure of estimating the first derivative was first proposed by Mikhailov57 and independently by Miller" and Takahashi.8' The schemes were proposed for the calculation of reactivity changes in nuclear reactors due to small variations in system parameters.

329 Hall' gave a constructive derivation of a multiparameter second-order derivative estimation procedure. The relative merits of the correlated and differential games in perturbation calculations are compared for special problems by Rief,"-" and a concise description of a game that estimates parametric derivatives is given by Matthes.52 An unbiased game estimating the first parametric derivative is derived in Section A as the limiting case of the generalized correlated moment equations delineated in the previous Chapter. Special weight generation rules to be applied in differential games are discussed in Section B. It is shown in Section C how the system parameters are to be adjusted on the basis of measured reaction rates and calculated sensitivities. The considerations are extended to estimation of higher-order derivatives in Section D. A simple analytical example given in Section E illustrates the procedure. Finally, the derivations are extended to problems with contribution functions which also depend on the system parameters. In the discussions below, the game that estimates the parameter-dependent reaction rate at a given parameter value will be called the unperturbed game; the system at the given parameter value is the unperturbed system. The differential game estimates the derivative of the unperturbed reaction rate with respect to the parameter at its given value. A. ESTIMATION OF FIRST-ORDER DERIVATIVES Let us first consider a correlated unperturbed analog game and recall the second form of the generalized moment Equation (6.7) that gives the moment of a correlated score function F(s) = F(s„s2): n+1 M{F(S)}(P0,1) =

(F[ i

=

(i)f(i)1

F[

i=

W'(i)f(i)])

(6.46)

where pointed brackets stand for (Fn[...]) =

[i

E

n= o 11

f dP; fdP,JdP~f dPn

dP:,,

T(P,_ ,,PDC(P:,Pi) JT(P„,K+I)F„[• •

and for the sake of brevity, we put f(i) = fa); the contribution function assigned to a flight from P;_, to P.. It is tacitly assumed in Equation (6.46) that F[ia,] =0 We shall first assume that this contribution function is common to both (correlated) particles. Consider a correlated game in which the unperturbed system corresponds to a parameter value a and the analog particle scores s2 in this system. Let the perturbed system be characterized by the parameter value a + Da and let the nonanalog particle give the score s, . This means that the transition and collision kernels in the two systems differ due to the

330

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

difference of some parameter value characteristic to the systems, and the unperturbed kernels T and C are taken at the parameter value a. Denoting this dependence in the form T(P,P') = T(P,P'Ia) and C(P',P") = C(P',13"1a) the perturbed kernels read T(P,P') = T(P,P' + Act) and C(P,P') = C(P,P' + Act) Assuming a common contribution function of the form f(P,P'), we also postulate that the contributions do not depend on the parameter a. This is the case, for example, with the track-length estimator if flux integrals are estimated, but is certainly not the case with the expectation estimator. The considerations below will be extended to parameter-dependent contributions in Section F. Let us consider score functions of the form F(s) = GC' — s2 ) Da

F[

=1

W'(i)f(i)] = L

(6.47)

i — 1 f(i)] i= W(Aot

Obviously, by taking the limit of Da —> 0, the argument of the score function tends to the derivative of the score, the quantity of interest in the differential game. We shall assume without further investigation that the order of taking the limit za —> 0 and the expectation can be interchanged, i.e., that lim G(s) = G( lim s) Act-00

and

M

ds Uot)

m i Goim t

Act—>0

st — 52)1 Act

= lim

+CI

52)l Act

(6.48)

We do not discuss this assumption, but, rather, note that for games that are feasible and have a bounded variance, this assumption comes true at least for linear and quadratic function of s in place of G(s). Note that the expectation in Equation (6.46) is taken with the unperturbed

331 kernels, which remain unchanged while Act tends to zero. Thus, taking the limit in Equations (6.46) and (6.47), we have MIGCN(P,„1) = (Gril d W(i)f(i)] — G[ dot

11 W'Wf(01)

(6.49) Act = 0

It is apparent that a flight from P,_, to P: contributes to the differential score by the quantity d — W'Wf10110=0 da and the statistical weight that multiplies the contribution function f(i) in the i-th flight is (6.50)

= da — WWIA0=0

versus W'(i) in an ordinary nonanalog game. Comparison of Equations (6.46) and (6.49) shows the very important consequence that, apart from the difference between the statistical weights, the differential games and the ordinary nonanalog games are identically played. (Note immediately, that the situation is somewhat different if the estimator f also depends on the parameter a, as will be seen in Section F.) Taking note of the expression of W'(i) in Equation (6.8), the statistical weight in the differential game in Equation (6.50) has the explicit form =WT ( c; L1-1, T(P,_

x

[ =1

C(P',

+ Aa.)/T(Pj _

lad

+ Au)/C(P; Act = 0

Va logJ(P, „P;la) + E — loge C(13;,plot)} = w{= 1— ,=, da da

(6.51)

after the i-th flight, where W is the initial (starting) weight of the particle. Now if M,(P) and M,(P) denote the expected scores due to a unit weight starter from P in the unperturbed and perturbed systems, respectively, and if we put G(s)=s then, from Equation (6.49), the expected score becomes m dda s }(p)

{

s, — 52}(P)= 1[Kii(p) Act = — M (P) da 1

lim

veto

Act

mi(p)]

332

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

On the other hand, if the source density of the particles does not depend on a, then the parametric derivative of the reaction rate at hand reads

, mg)

dRd — a dPQ(P)Mi(P) JdPQ(P) cv da = d

=

ds1 — ot j(P) l')1‘4t d

We have thus proven the following theorem. Theorem 6.3 — An unbiased estimate of the parametric derivative of a reaction rate is obtained from an ordinary analog game, with contributions from free flights only, if the statistical weights of the particles are chosen according to Equation (6.51).

B. DISCUSSION OF THE GAME Let us now investigate more closely the weight generation rules in Equations (6.50) and (6.51). Let, as before, W'(i) and W"(i) denote the weights of a particle before and after its (i) be i-th collision in an ordinary nonanalog game, respectively, and let W:,)(i) and the corresponding weights in a differential game. If the nonanalog game simulates the effect of a finite parametric perturbation and the differential game gives the corresponding derivis to be ative, then W'(i) and W:,,(i) are related according to Equation (6.50), and expressed by W"(i). According to the weight generation rules in Theorem 5.8, the ordinary nonanalog weights are related as W"(i) = W'(i) • 1A4P:,P,)

(6.51)

W'(i + 1) = W"(i) . w(P„P„)

(6.52)

ws(13:,P,) = C(13:,P,la + Aa)/C(P:,Pilot)

(6.53)

w(P„P: + ,) = T(PoP: + ,la + Aa)/T(Pi ,P; +11a)

(6.54)

and

where

and

Therefore, from Equation (6.50) d d wi)(i + 1) = — da W"(i)L =0 + — log T(P ila) da " This means that the weight of the particle after its i-th collision in the differential game is w7i)(i) = — det

333 and from Equation (6.51) VsT,t(i) + — loge C(P:,13,1a) da Hence, we conclude that while the ordinary nonanalog weight generation rules [Equations (6.51) through (6.54)1 are multiplicative, the corresponding rules in a differential game are additive: \'(o(i) = V4/ 1)(i)

(6.55)

w5 )(P: ,P1)

and w1( i)(i + 1) = \

row +

w")(Pi,P: ,)

(6.56)

where w(2 )(P:,Pt) = — log C(P:,Pila) da e

(6.57)

d ww(P„P: ,) = — dalog T(P ,P' ±11a)

(6.58)

and

Recalling the explicit forms of the kernels in Equations (5.32) and (5.33) 1r-el

T(P,P') = o-(P')expf

dto-(13,)};

Pt = (r + teo,E)

and C(P',P") = cs(P')Cs(P',P") the additive weight factors in Equations (6.57) and (6.58) become dc(P') dC (P' P") w(,')(P',P") = da /cs(P') + /Cs(P',P") da and wo)(p,p,)

da(P7

da

cr(P )J

dt

do-(p) da

Specifically, when the total cross section cr depends on the parameter a in the form cr(Plot) = ao-(P)

334

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

then w")(P,P') = 1 —

it 0

dtcr(Pt) = 1 — T(P,P')

Several other special cases are reported, for example, in References 67 to 69. As a consequence of the additive weight generation rules, it may happen that the statistical weight of a particle becomes negative, a phenomenon preferably avoided in ordinary nonanalog games. Another consequence of the additive weight generation rules is that in an analog differential game (which is played according to the analog kernels), particles temporarily having zero statistical weight must not be eliminated from the simulation (as is done in an ordinary game). Such particles may obtain nonzero weights in a later stage of the simulation. In other words, statistical weights in an analog differential game determine the actual contributions of the particles to the score, but are not measures of the "size" of the particles, which is the case in an ordinary nonanalog game. A further pecularity of the differential games is that in a nonanalog differential game, multiplicative and additive weights are present in parallel. They are to be generated independently, according to the respective generation rules. The score is to be multiplied by the product of the two weight factors, but all the weight-dependent biasing schemes like splitting and Russian roulette are to be applied according to the multiplicative weight value. (Note that the additive weight is sometimes considered simply as part of the contribution function. In our formalism, this weight is a direct descendant of the multiplicative weight of an ordinary nonanalog game, while the definition of the contribution function is unchanged. Therefore, we think it more logical to call it a weight rather than a contribution function.) In what follows, we discuss briefly the nature of the moment equation. Since the specific form of the weight generation rules has not been exploited in the derivation of the moment equation in Chapter 5.11, the relations in Equations (5.56) and (5.58) apply here too. Thus, the r-th moment of the differential score satisfies the equation wrom f

ds \ cliat/ jr

+E

(1)

dP'T(P ,P')ca(P' )[V/ f(P,P'

(ri) fdP'T(P,13')IdP"C(P' ,P")[W f(P,P')]r- i [NA/1W i MI (da i}(P") (6.59)

where W(1) is the weight of the particle at P, NA.7 and AP'„ are related to W(1) according to Equations (6.55) through (6.58), and we have omitted a from the arguments of the kernels. In spite of the formal equivalence of the ordinary nonanalog and differential moment equations, they are essentially different. This difference is due to the fact that the former, after making use of the (multiplicative) weight generation rules, does not contain information about the previous history of the particle, while the latter will always "remember", due to the additive weight factors. This property can be best visualized through the first-moment equation. From Equation (6.59) with r = 1 ds \7%11) f(p,p,) Mt -cT I t l(P) = f dP'T(P,P') W (I) + fdP'T(P,P')f dP"C(P',P")

MI ds l(P") W(i) daf

(6.60)

335 According to Equations (6.55) through (6.58) W (1) Woo

=1+

w ( 1 1( p p ) W(1)

and )(P,P') + W(2 )(P',P") = 1 + w" W(1) W(1)

w7i)

still containing W(1), the weight accumulated during the history preceding the collision at P. In contrast to that in an ordinary nonanalog game, the corresponding weight ratios follow from Theorem 5.8 as W' W = w(P,P') and

=

w(P,P')w,(P',P")

independently of W. It is interesting to note that an alternative form of the differential first-moment equation can be derived from the analog unperturbed moment equation. Let us write the ordinary first-moment equation (5.57) into the following form M{s}(P) = idP'T(P,P')f(P,P') + fdP'T(P,P')IdP"C(P',P")MIsl(P") Since ds (p) M{da

da Is/11s")

differentiation of the first-moment equation above yields

mt

(11 fdP"T(P,P)w")(P,P')f(P,13') + fdP'T(P,131)1 dP"C(P',P")[w")(P,P') + w(,i)(P',P")1M{s}(P") + fdP'T(P,P').1 dP"C(P'

da

}(P")

(6.61)

This equation is clearly different from Equation (6.60) and is similar in form to the moment equations investigated in Chapter 5, i.e., it is indeed of the Fredholm type. This, however, is reached at the expense that the source term of the equation contains the expected score in the ordinary game. On the other hand, Equation (6.61) suggests an alternative method of estimating parametric derivatives. This method is analogous to the perturbation source method discussed in Section 6.I.E and will not be repeated here.

336

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

Returning to the realization of a differential game, we have seen that Equation (6.59) is not appropriate for the investigation of feasibility. On the other hand, since the simulation goes according to the unperturbed kernels, the differential game is feasible if the corresponding analog game in the unperturbed system is also feasible, provided the analog kernels are such that the differential weight factors in Equations (6.57) and (6.58) are bounded. This is also seen from Equation (6.61), which shows that the integral kernel of the Fredholmtype equation is just the analog transport kernel. Investigation of the conditions of a finite variance is a little more complicated since the second moment of the differential score does not follow from that of the analog game by simple differentiation. This is evident from the relation f ds )21 ds M{s2} = 2M{ s v} 1\41 da) da J The variance of the differential score was investigated by Zolotukhin and Usikov,86 who give conditions under which the variance of the differential score is bounded. C. DATA ADJUSTMENT WITH SENSITIVITIES Measured reaction rates (called integral experiments) are usually known to be more or less inaccurate due to uncertainties in the material characteristics playing a role in the weighting function of the reaction rates. Differential Monte Carlo can be applied to the adjustment of the material characteristics (parameters). A method of parameter adjustment proposed by Hall32 is based on comparison of the results of integral experiments and Monte Carlo estimates of the reaction rates in question that are dependent on the material parameters to be adjusted. Let a = (a „a2, . . . ,a,)T be the (column) vector of the parameters which are to be adjusted, and on which the integral measurements and the Monte Carlo estimates depend. (One may think, for example, of a set of cross sections at given energy values.) As the material parameters are usually determined experimentally, they are more or less biased by statistical and experimental errors. Assume that the covariance matrix of the parameters, denoted by A, is also given. Let = (m1 , m2,

Inn) r

denote a set of integral measurements and let r = r(a) = (r„ r2,

rn)T

be the Monte Carlo estimates of the measured quantities at the actual values of the parameters. Finally, let R be the estimated covariance matrix of the Monte Carlo estimates r. The purpose of the procedure is to find an adjusted parameter set a instead of a which "best fits" the measured and calculated values. The adjustment is based on the assumption that both the calculated reaction rates and the parameters follow multivariate normal distributions with expectations and covariances m, R and a, A, respectively, the the maximum likelihood estimates of the parameters give the best-fitting set. This means that the optimum estimate a of the parameter vector minimizes the log-likelihood function logL = [r(a) — m]TR -i(a)[r(a) — m] + (a — a)TA -1(a — a)

(6.62)

337 Minimization is performed by setting the derivatives of logL with respect to the parameters a equal to zero. The resulting equations read T(a)R t(a)[r(a) — m] + A -- `(a — a) = h(a) = 0

(6.63)

provided the derivatives of R ' can be neglected. Here a(a) =

,(a)/aa jl

is the sensitivity matrix of the reaction rates. Equation (6.63) is nonlinear in the parameters and therefore its solution is not straightforward. Hall proposes a Newton-Raphson iterative solution of the form a n ± = an — 1-1 -'(an)h(an)

(6.64)

where H(a) = fah (a)/aaj and, neglecting again the derivatives of R-' and

a,

differentiation of Equation (6.63)

yields H(a) = a T(a)R -1(a)19(a) + A

(6.65)

The iteration consists of the following steps: 1. Let n = 0 and a„ = 2. Calculate the Monte Carlo estimates r(an), R(an), and

a (an).

3 Determine the value of logL from Equation (6.62) and H(an) from Equation (6.65). 4. Let n = n + 1 and determine an , from Equation (6.64). 5. If the value of logL or the parameter set an is converged, the a = an; otherwise, return to step 2. It can be seen that the covariance matrix of the adjusted parameters a is approximately expressed as32 ASH-'(a) The adjustment procedure clearly requires a great amount of Monte Carlo calculations. On one hand, the reaction rates and their covariance and sensitivity matrices are to be determined. These quantities can be estimated parallel in a correlated game (ordinary weight generation rules for the first two quantities and differential rule for the third). On the other hand, because of the iterative solution of Equation (6.63), the estimation procedure must be repeated several times using successive values of the adjusted parameters. Note, however, that by neglecting the derivatives of the covariance matrix R and sensitivity matrix 0 , we have tacitly assumed at certain points that they are independent of the

338

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

parameter values. Taking full advantage of this assumption, the Taylor series of the reaction rates around a has the form r(a) = r(a) +

a • (a — a)

and (6.63) reduces to OTR -'[r(a) — m +

a • (a — r)] + A

(a — a) = 0

(6.66)

After rearrangement, the difference of the measured and calculated rates reads m — r(a) = [Rai -'A + 01(a — a) Taking into account the identity [ROT - A

= na T [R +

anna

the adjusted parameters are expressed as a = a + naIR + aa]- i[m — r(a)]

(6.67)

In this expression, the covariance and sensitivity matrices are taken at the original values a of the parameters. Similarly, the covariance matrix of the adjusted data follows from Equation (6.65) as A = H -- i(a) = A — A0'IR +

ana ri an

(6.68)

Equations (6.67) and (6.68) were derived by Dejonghe et al.' driectly from maximum likelihood principles. This approach seems to have the advantage that no iteration is necessary for the adjustment, i.e., the reaction rates and related matrices have to be estimated only once. Note, however, that the two approaches are not equivalent in spite of the fact that both are based on the approximation that at certain points the derivatives of R and a can be neglected. In fact, Equations (6.63) and (6.64) are also valid if a depends on the parameters and its derivative is neglected solely in the expression of H in Equation (6.65). Thus, the iterates a„, are evaluated with updated values of a(a. ) . In contrast, Equation (6.66) assumes a constant a . It can be easily seen that the adjusted parameter set given by Equation (6.67) is just the first iterate a,, as follows from Equations (6.63) through (6.65). Hence, we conclude that the second approach defines the first-order adjustment of the parameters, and further iterates by the first approach give adjustments depending on higher-order derivatives of the reaction rates with respect to the parameters. D. ESTIMATION OF HIGHER-ORDER DERIVATIVES Higher-order derivatives of the reaction rate in the expansion (6.45) can also be estimated by Monte Carlo. The weight generation rules of higher-order differential games follow from repetition of the line of thought of the previous and present Chapters. Let us denote s(1)

ds dot

339 and let ds S(1) = du

et+

va

Let us consider a correlated game estimating a function of so) and go) in which the correlated particles are associated with the systems of parameter values a and a + Da. Let 1C1) and NNT„ denote the statistical weights of the particles (which multiply the contribution function, while the first derivative of the score with respect to a is estimated according to the procedure in Section A. Finally, let the score function estimated in the correlated differential game be of the form sm G(

— s(1) ) Act

Then by analogy to the derivation of Equations (6.7), (6.46), and (6.47), it is easily seen that the expectation of the score function has the form

mtc(.)

«

5"))}(po,1) —

[ NVi)*(1)(i) Li=1

0, then lim (gm s — (1))/Aa =

ds , des = da d dal

and lim (W*1) — Wi))/Acit = d 11%7;1,

2 'W'

(6.70)

Thus, Equation (6.69) in the limiting case of Aa. —> 0 becomes n+ 1 M{G(B) }(Po ,1) = (G[

>

N'Y2 )(i)f(i)]

G[

W;2)(i)f(i)])

Notice that the statistical weight in Equation (6.8) has the form W' =

+ Da)/II(a)

where II (a) is an abbreviated notation of the product of the kernels. With this notation, the statistical weight in the first-order differential game reads

W"' = da

Fd = [— 11(0)1/1-1(0) da

340

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

and its derivative is 2 c12 ] [-da 2111(a) ir") {[— da Muir")}

, da W`

Therefore, the statistical weight in a second-order differential game, according to Equation (6.70), is d2 d2 _ [d-1 -(adll(a w(2) ) d(Ac)2 " Successive use of the arguments above shows that the generalized moment equation of a function of the r-th parametric derivative of the score has the form n+ I M{G(7 drS) it }(P0 ,1) = (G[

where

WWi)f(i)]

G[

‘ 11 0(i)f(i)1)

r) (i) is the particle's weight after the i-th flight ddr ot IC— 0(i) + \11 00/Wr_1,(i) = d(Aa)r W(011-0 Wr)(i) = d—

(6.71)

Specifically, for the second derivative, the weight generation rule becomes NA/ 2)(i) =

d2

d2

2, — log,T (I) _ „P;la) + L — logeC(P',,P;la) + ENV,)(1)12 (6.72)

j=1

da 2

j=i

da2

and for the third derivative NNT 3)(i) =

(13

d3

2, -- log T(P _„110.) + 2, — togeC(P; ,PJla) da3 " = da3

+ 31C1>I0 W 2)(0 + PAP(1 )(013 It is stressed again that, apart from the differences in the weight generation rules, there is no difference between the estimation procedures of an ordinary reaction rate and its parametric derivatives. Therefore, in principle, the derivatives up to any order can be estimated along with the reaction rate itself in a single game. E. A SIMPLE EXAMPLE For the sake of illustration, let us consider the estimation of the derivative of the collision rate with respect to the total cross section in an infinite homogeneous medium of total cross section r and survival probability (mean number of secondaries per collision) c. Let the transport in the medium be monoenergetic and isotropic. Then the kernels of the simulation are ,P;)

crexp( — crti )

and C(P:,13) c/4ir

341 where ei is the length of the flight between the points P; _, and P. It is easily seen that for any function Fe[ . . . [ II T(P, _ „P:) C(P; ,P,)] T(Pe,P:,± I ) F„1• • •1 =1 .±1 ...f clfe+ , en cr."' exp( — E f, 0

f dP ; f dP =

0

f

(6.73)

When estimating the derivative with respect to Q, the weight of a particle in its i-th flight follows from Equation (6.51) as =

d 2 -(logecr — rre,) =

— CrE ei)/a ,=i

;=,

(6.74)

(Obviously, the weights assume both positive and negative values.) Consider the case when the collision estimator is applied, i.e., when f(13, „P;) = f(i) = 1 The expectation of a function G of the derivative is deduced from Equations (6.49), (6.73), and (6.74): ds G (det) =

c" G" n=0

with ,so

G„ = de, Idt, 0 0

X

n+1

f clee+i o-n'exp(—o- E 0 ,=,

e,)

n+

{

G[

W;i)(i)]

G[E W(1)(i)] i=1

1=1

(6.75)

Let us denote ti = a E f,

(6.76)

J= 1

Then the differential weight reads W:1)(i) = (i — ti)/o-

(6.77)

Let us apply here the generating function technique introduced in Section 5.IX.C, i.e., let G(s) be the moment-generating function G(s) =

(6.78)

342

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

Then the moment-generating equation becomes

m{ (dclu s )r} =

M {4d) ds ff}

X=0

1 dr = Q` L cn dX r C =

GniX=0

Inserting Equations (6.76) through (6.78) into Equation (6.75), we obtain n+

Gn = dt, dt2

1 dtn± ,e'rtex p[X

E (i — ti)] — exp[X =,

i=,

•—

= k + , — An with An = exp[X,

n(n +

dt, iti dt2 ... _ idtn exp[ — E xt, — tn]

2

Simple integration yields the expression of A„ as An

= exp[X

n(n + 1)] / 2 ,

(ix +

Hence, the expected score reads m rds1

=

do-f

cn i- dA„+ , o- n = 0 L dX

dAni dX x=0

Now dAn dX

n(n ± 1) L 2

'± ny i=i iX + 1 A

=1

and thus dAn dX

= X =0

Accordingly, the first derivative of the collision rate with respect to the total cross section is zero. The result is trivial since the number of collisions in an infinite homogeneous medium with constant cross section is independent of the value of the cross section. The second moment of the score in the estimation of the first derivative follows from the second derivaties of A„ with respect to X. Since d2An dX2

X =o

= i=i E i2

—

n(n + 1)(2n + 1) 6

Thus d2Gn = 1[(n + 1)(n + 2)(2n + 3) — n(n + 1)(2n + 1)1 = (n + 1)2 dX2 x=o 6

343 and from Equation (6.79) s)2} mt(ddcr

12

Cr

=0

(n + 1)2 c" = 1 1 + Cr2 ( 1 -

F. EXTENSION TO PARAMETER-DEPENDENT ESTIMATORS In the derivations above, we assumed that the contribution function (estimator) f(P,P') is independent of the system parameters with respect to which the reaction rate is differentiated. The considerations can be easily extended to the case when the contributions do depend on a. If we again define the differential game as the limit of a correlated game and make use of the results of Section 6.I.C, it is seen that the differential score in the i-th flight of the history is s = lim [W'(i) f(ila + Da) - f(ila)]/Act Ckee-,•0 where f(ila) = P:la) is the (parameter-dependent) contribution in the i-th flight (the flight takes place between P„ and PD. Taking the limit, we have si = vvi)f(ilet) +

f(ila)

(6.79)

where is the statistical weight of the particle after the ith flight, as given in Equation (6.51). It is remarkable that only a part of the score is proportional to this weight; the other part is independent of it. This also means that the final score in this differential game is the sum of the scores in a "pure differential game" (a game where the contribution does not vary with the parameter) and in an ordinary game with a contribution function dflda. The same interpretation also follows from the explicit form of the reaction rate R = fdPtp(P)f(P) since after differentiation we obtain aR as

—=

dP

[ a a — 110(P)] f(P) + f dP Llt(P) — aa f(P) aa

The first term is estimated in the "pure differential game"; the second, in the ordinary game. The differential score in Equation (6.79) can also be interpreted in an alternative way. Taking note of the weight generation rule in Equation (6.56), the score s, in Equation (6.79) is rewritten as si = 1,r1 ,(i — 1) nil«) + w(1)(Pi_DP:)f(ilet) + — Oa) as Now, since f(ila.) is the abbreviated notation for the contribution f(P, ,,13:1a) and, according to Equation (6.58), the additive weight factor reads a w")(P,_,,P:) = — loge TO', -1,13:100 as

344

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

we have s; = NA,P(' „(i — 1) f(P,_

+ da fT(Pi-1,P:la)f(Pi -1, P:la)VT(P, ,Pia)

Thus, the expected partial score resulting from a flight from P,_ in the differential game is — 1) I ,(P11) + — act 1,(P-,)

1,(P, _ 1 ) = f dP T(P, _ „P: la) s, =

(6.80)

where =f is the expected partial score in the corresponding ordinary game. Equation (6.80) is interpreted as follows. The score due to a flight from in the differential game is the sum of two contributions. One is the contribution by any of the partially unbiased estimators of the ordinary game multiplied by the differential weight at (i.e., by the weight attained after the scattering to P,_,). The other is the contribution of an estimator that has an expected partial score equal to al ,(P, _ Oa). This estimator may be either the one given in the second term of s, as

aa

[T(P _ „P!Ic)f(P -1,P:)/T(P,-1,P:10 )

or any other estimator that follows from it by the transformation theorems established for partially unbiased estimators in Section 5.VI.A. The estimators applied so far in practical realizations of differential games all follow from Equation (6.79), with well-known ordinary estimators in place of Oa). The theory of partially unbiased estimators, when applied to differential contribution functions, still may result in new, efficient estimators specific to differential games. Finally, differentiation of the first-moment equation gives an equation corresponding to Equation (6.61) in the case of a parameter-dependent estimator: wi t ddot

} (F)

f dP'T(P ,F)[w("(P ,P')f(P ,131) + f(P,131)] as + JdP'T(P,P')JdP"C(P',P")[wW"(P,P') +

,P")] M{s}(P")

)m f ds I (p„)

+ JdP'T(P,P') dP"C(P',P"

J

G. PERTURBATION ESTIMATION BY DIFFERENTIAL GAMES: THE TAYLOR SERIES APPROACH Let us consider again the estimation of the effect of small parametric perturbations on a given reaction rate. Specifically, assume that the change of a reaction rate due to the alteration of the parameter a by Da is the quantity to be estimated. The correlated Monte Carlo method investigated in Chapter 6.1 is a possible tool for such estimations. There is, however, an alternative approach based on differential Monte Carlo which has certain advantages over a correlated calculation.

345 Let us assume that we are interested in the expectation of some function of the score difference due to a perturbation of a. In formula, let the score function in the moment equations (6.7) and (6.46) have the form F(s) = G(s, — s2) where G(s) is some given function, s, is the score in the perturbed system (characterized by the parameter value a + ha), and s, is the score in the unperturbed system (at the parameter value a). The expectation of G(s, — s2) in an unperturbed analog correlated game follows from Equations (6.46) and (6.7) as n+1 M{G(S

[

s2)}(P0) =

W(i)

G[ (W'(i) i -1

1) f(01

I

1)fW1

where W'(i) = W'(ila + Act) is the statistical weight of the perturbed particle at the collision point at P:, while =1 for the weight of the unperturbed particle (cf. Chapter 6.1). The correlated game that defines this moment equation can be substituted by an equivalent differential game in the following way. Let us expand W'(ija + Au) into a Taylor series around Da = 0 as W'(ila + Da) =

d'W'(ila + Act) ,= 0 d(Act)'

(Aot )v Act = O

In view of Equation (6.71), the v-th derivative of W'(ila) is just the weight W;,(i) in the estimation of the v-th derivative of the score. Therefore + Act) — W'tila) =

E

W;(i) (Da )' v!

and the moment equation reads M{G(s, — s2)}(P,) = (G[

(Act)v " +1 , . . , E w, (Imo] „-, ")

G[2, v =,

(A a)' v,, ,=,

.

w(o(i)f(1)1

The above two forms of the moment equation express the fact that the expectation of any function of the score difference in the two systems can be equally estimated either in a correlated ordinary game or in a differential game where the parametric derivatives up to a desired order are estimated concurrently. The score gathered in a history in the latter case is the sum of the estimates of the derivatives multiplied by the respective power of Act and divided by the factorial of the order of the derivative. The Taylor series approach has a particular advantage estimating expected score differences [i.e., in the most common case, when G(s) = s]; namely, it is possible to estimate the various derivates, Mfdvs/dal , of the reaction rate in question without specifying the value of the parameter change (la), and the results can then be applied to an a posteriori estimation of reaction rate perturbations at

346

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

different da values. In a parametric study of reaction rate perturbation, a single differential game may be substituted for a large number of distinct correlated games. In a simple reaction rate perturbation, if denotes an estimate of the v-th derivative of the reaction rate, then N

SR = R(ot + Act) —

(a)

„

ko,) k"ai' vl

is an estimate of the reaction rate perturbation, where N is the number of terms kept in the Taylor series expansion. Numerical experiments show that two terms in the expansion already give satisfactory estimates in typical perturbation calculations.67•68 Note that in order to give an a posteriori estimate of the variance of the result, it is also necessary to estimate the covariance matrix MI(dsvidetv)(d/doc!')} of the final scores in the differential game. If V„µ denotes an estimate of the (v,p.) element of the covariance matrix, then the variance of the estimated reaction rate perturbation due to the parameter change Aa is NN

E E v (°"."` v! t! III. CRITICALITY CALCULATIONS One of the most important quantities characterizing a nuclear reactor is the effective multiplication factor, keff, a measure of how far the fissile system is from the critical state. If keff = 1, the system is critical; if keff < 1, it is subcritical; and if keff > 1, the system is in a supercritical state. This factor is an eigenvalue-type quantity appearing in the transport equation in the form tli(P) = drk!)(PIK,(P",P) + — 1 K(P",P)] keff

(6.81)

where Ks(P'',P) = dP'cs(P")Cs(P",P')T(P',P) and Kf(P",P) = fdP1 c,(P") E nq.(1311)C„(P",P')T(P',P) n= while c, and cf are the respective scattering and fission probabilities, and C, and C, are the respective probability densities of the neutron's coordinates after a scattering or fission. qn is the probability that n neutrons are produced in a fission and T is the transition kernel. All these quantities were defined in Chapter 5. Equation (6.81) is a homogeneous equation, i.e., it describes a system with a steadystate neutron flux (collision density) without an external source. If the system characterized by the kernels in Equation (6.81) is critical, then the equation has a solution with keff = 1. In the opposite case, the system can be hypothetically altered to critical by changing the number of neutrons produced in a fission by an appropriate factor. This factor is just like„.

347 Introduction of this factor also means that when talking about collision density, we may address two different quantities, such as the collision density resulting from the Monte Carlo simulation with the physical kernels, and the hypothetical density defined by Equation (6.81). This duality will not be confusing in the derivations below; a distinction will be made between the two quantities when necessary. It can be shown that Equation (6.81) has an everywhere positive solution belonging to a positive eigenvalue, and this eigenvalue is the largest among the possible eigenvalues of Equation (6.81). Calculations aimed at determining the largest eigenvalue, k„, will be called criticality calculations. We abandon here the reactor physical discussion of the transport equation (6.81); the interested reader is referred to Reference 2. Most naturally, the necessity of using Monte Carlo methods in criticality problems arises at least as frequently as in the case of reaction rate calculations. Application of Monte Carlo methods to such problems dates back to the early 1960s. Comprehensive lists of early works are given in References 9, 55, 60, and 65. Monte Carlo estimation of the multiplication factor in Equation (6.81) differs from an ordinary reaction rate estimation mainly from two points of view. First, keff is an eigenvalue and not a reaction rate; second, the transport equation (6.81) has no first-collision term. Therefore, criticality calculations by Monte Carlo methods involve specific problems, which will be reviewed in this Chapter. A. PRINCIPLE OF THE SIMULATION: THE SOURCE ITERATION The method of Monte Carlo estimation of keff is based on the most common deterministic solution procedure of the criticality transport equation, called the source iteration or the method of successive generations. The idea behind the method is that the transport equation (6.81) is set in a form analogous to the inhomogeneous transport Equation (5.4), except that the first collision term contains the unknown collision density and the multiplication factor. The resulting equation is then solved by iteration. A number of slightly different formulations of the method are given below. 1. First Method Let us define the following functions 4i(P) = 41(1))/W

(6.82)

and Q(P') = f dP"tli(r)cf(P")

E

nq„(P")Ce(P",P')/k,„

(6.83)

n I

where kl,(P) is the collision density in Equation (6.81) and W is chosen such that

i

dPQ(P) = 1

i.e., Q(P) can be considered a source density of unit total strength. We start our discussion with this method because in all previous derivations, the (fixed) sources were assumed to be normalized to unity. The quantity W, satisfying the normalization condition, reads W = idP"tlft(P")cf(P")v(P")/ke„

(6.84)

348

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

and, accordingly, the artificial source density reads Q(P') = fdP"Ill(P")Cf(P",P')/1 drtli(P")cf(P")v(P")

(6.85)

where C1(P",P') = cf(P")

E

nqn(P")Cn(P",P')

E

nqn(P")

n=1

and v(P") =

n=1

the mean number of secondaries per fission. Dividing Equation (6.81) by W and making use of the definitions in Equation (6.82) and (6.83), we obtain an equation for the renormalized collision density as tii(P) = f dP"tli(P")Ks(P",P) + tlin(P)

(6.86)

i°(P) = f dP'Q(P')T(P',P)

(6.87)

where

The solution of these equations has interesting properties. By integrating Equation (6.86), we have

i

dPtp(P) = f dPVP)cs(P) + 1

i.e., since cs + ca = 1

f

dPcn(P)I15(P) = 1

(6.88)

From Equations (6.82) and (6.84), it is seen that the effective multiplication factor reads keff = idP tlfr(P)cf(P)v(P)

(6.89)

Equations (6.86) and (6.87) are formally eqivalent to an inhomogenous transport equation that describes the renormalized hypothetical collision density 111(P) in a nonmultiplying medium as a result of the source density Q(P). An iterative solution of the equation is defined as follows: 1. Let Q(°)(P) be some initial (guess) source density and let LOP) = fdPQ(')(P')T(P',P)

349 2. Let ifJ("(P) denote the solution of Equation (6.86) with k)(P) in place of the first collision density 3. Insert this solution into Equation (6.87) to define the first iterate of the source density, Q0)(p).

4. Repeat the procedure until convergence is reached. The n-th iterate of the collision density satisfies the equation 41(n)(13) = f dP"tiMP")Ks(P",P) + 111(on)(P)

(6.90)

with

f

dP'qn-1)(P')T(P',P)

(6.91)

and Q(°(P) = f dP"4,(°(P")Cf(P",P)/fdP"‘11(°(P")cf(P")v(P")

(6.92)

The n-th iterate of the effective multiplication factor follows from Eqaation (6.89) as k0 = fdP Iii(n)(P)Cf(P)v(P)

(6.93)

Combining Equations (6.91) through (6.93), Equation (6.90) reads 41(°(P) = f dInli(")(P")Ks(P",P) + 17.1 idni("--1)(P")K,(P",P)

(6.94)

Obviously, if the iteration converges as n tends to infinity, then limLI-J(0(P) will be proportional to 44P), the solution of Equation (6.81) provided kn tends to the eigenvalue ke„. The proof of the convergence of the method is outlined in Section B. The procedure above is based on recursive calculation of the first collision (or source) term of the transport equation (6.86). This is why it is called the method of source iteration. On the other hand, Q(n -1) represents the source density of neutrons born in fissions that are triggered by the collision density of tr -", while 4, is the collision density produced by the neutrons originating from (Y" -1). In other words, fission neutrons brought into existence by neutrons in the (n — 1)st generation represent the n-th generation in the procedure. This interpretation calls forth the name of method of successive generations. The n-th iterate of kn„ in Equation (6.93) is the fission rate of the n-th-generation neutrons, a quantity that can be estimated by Monte Carlo. The source density of the (n + 1)-th generation is Q(° in Equation (6.92), and it consists of the neutrons that emerge from fissions induced by the nth generation. The source is normalized to unity via division by kn. This is in accordance with the role of kn„ in making the system (hypothetically) critical, as discussed in connection with Equation (6.81).

350

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

2. Second Method The procedure above can be reformulated in an alternative way.65 Let us assume that the physical system under investigation is either critical or subcritical. Then a steady-state (physical) collision density can only be reached if an external source is present in the system. Let 4,(P) denote the (physical) collision density produced in the system by an external source S(P). Then •I;•(P) satisfies the equation (i)(P) = fdP"(1)(P")[Ks(Pn,P) + Kf(P",P)] + 4.0(P)

(6.95)

with the first collision density il),(P) = f dP'S(P')T(P',P)

(6.96)

Let us now consider the iterative solution of the following equation system: on)(P) = fdP"(1)(n)(P")Ks(P",P) + (1)(: )(P)

(6.97)

(VP(P) = i dP"On - "(P")1(f(P",P)

(6.98)

and

with C(P) = 4.(P) given in Equation (6.96). In this procedure, (1:0)(P) is the collision density due to the external source S(P) if collisions leading to fission do not contribute to the collision density (first generation). 4:1)(n)(P) is the collision density of neutrons created in fission that are provoked by the (n — 1)st-generation neutrons. Let the n-th iterate of the multiplication factor be defined as k„ = f dPck;'+1)(P)/fdPi:le(P) ' S./So) 1) or, since S("_ 1) = fdP4:1e(P) = f dP"i:Vn - ')(P'')cf(P")v(P") kn reads k„ = JdP •f•(")(P)cf(P)v(P)/f dP (1)(n - "(P)cf(P)v(P)

(6.99)

Clearly, kn is the ratio of the number of fission neutrons created by the n-th and (n — 1)st generations. This also means that kn can be estimated as the ratio of two reaction rates. Assume again that the iteration converges, i.e., that lim 4(n)(P) = (r)(P),

lim kn = k n-..

351 Now, if 4.(') > 0, then k = 1 and 44;•(-) satisfies Equation (6.81) with keff = 1, i.e., it is the true physical collision density in a critical system. On the other hand, if 41(-) = 0, then it can be seen that the sum a(P) = E .4)(n)(P) exists and satisfies Equation (6.95), i.e., it is the collision density in the subcritical system with the external source S(P). It can also be proven that, in this case, k (the limiting value of kn) is just keff, the effective multiplication factor of the system (the proof is sketched in the next Section). Finally, denoting Ifiw(P)

(1)(")(13)/1dP 0,,n)(P) 4,(n)(P)/fdP0"-- "(P)cf(P)v(P)

it follows from Equations (6.97) through (6.99) that 0")(P) = dP"0")(P")1QP",P) +

fdP"On- `)(P")K,(Pn,P)

Comparing this equation to Equation (6.94), it is seen that On)(P) = (n)(P)

i.e., the iterates of the renormalized hypothetical collision densities in the first and second methods are identical. Accordingly, the two iterative methods are equivalent; the only difference between them is that the first method assumes normalized source densities in every iteration, while in the second method the collision densities (1)(n) in the different generations are not renormalized and add up to the true density of the source-driven system, Equation (6.95). This also means that in supercritical systems the first method is expected to work, while the iterates of the collision density in the second method are to diverge. 3. Third Method By making use of the specific forms of the kernels Ks and Kf, the transport equation (6.81) can be converted into several different integral equations that connect directly the subsequent generations of the fission neutrons. These equations are equivalent and can be transformed into each other. Here we derive two of them, the equation that connects the spatial source density of the fission neutrons in subsequent generations and the equation describing the connection between the phase-space distributions of the fission points. According to Equation (5.6), the scattering kernel Ks in Equation (6.81) has the form Ks(P",P) = cs(P")Cs(E'—>E1e)T(r'—>rIE) where P = (r,E) and P" = (r', E'). Assume that the fission kernel can be written as Kf(P",P) = cf(P")v(r)x(Elr')T(r1-->riE) where x(Elr') is the energy-direction distribution of the fission neutrons. With this form of

352

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

the kernel, we have assumed that the energy and direction of the fission neutrons are independent of the energy and direction of the neutron that gave rise to the fission. This assumption is justified in most practical cases. Let z(r,E,r0) be a function that satisfies the equation z(r,E,ro) = f dP"z(r',E',ro)1{,(P",P) + x(Elro)T(r,,-->rIE)

(6.100)

with P = (r,E),

P" = (r',E')

Obviously, z(r',E',ro) is the collision density at P' = (r',E') due to a neutron that emerged from a fission at ro. Furthermore, let us define the function S(r) = idEtP(P)c,(P)v(P)/ke„

(6.101)

S(r) is the density of fission neutrons emerging from around the spatial point k. [More precisely, it is the fission density in the hypothetical system where the collision density is IKP).] Let us multiply Equation (6.100) by S(i-0) and integrate with respect to e„. Then it is seen that the resulting equation is identical to Equation (6.81) if we put tli(P) = f dez(r,E,e)S(e)

(6.102)

Z(r,r') = IdEcf(P)v(P)z(r,E,e)

(6.103)

Finally, let us denote

It follows from the interpretation of z(r,E,r') that Z(r,r') is the density of fission neutrons emerging from around r due to a fission neutron started from r'. An equation containing only the fission density S and the kernel Z is obtained if Equation (6.102) is multiplied by c,v and integrated with respect to E. Doing so and taking note of Equation (6.101), we have S(r) = I j-dr'Z(r,e)S(e) ke,

(6.104)

This equation has the form of a conventional eigenvalue problem. Its solution is sought by iteration: S(n)(r) = ideZ(r,e)S(n -')(e)

(6.105)

The iteration is equivalent to the method of successive generations since S(") (r) represents the fission source density in the n-th generation of the neutrons and is related to the (n — 1)-th-generation fission source through the kernel Z(r,r'). Let S(n) = fdrS(")(r)

(6.106)

353 and k = S'")/S(n - "

(6.107)

Then k„ is equivalent to the n-th iterative of the multiplication factor in the second method, as defined in Equation (6.99). Assume again the convergence k„ —> keff and s(-)(r)Is(n)

S(r)

Then it is easily seen that S(r) satisfies Equation (6.104). This method is essentially equivalent to the second method introduced above. The difference in the formulation of the two methods, however, is important from a simulational point of view. The fission density formulation of the third method emphasizes that neutrons are followed from (birth in) a fission to (death in) a fission, thus giving a well-defined population of a generation. The successive populations are correlated through the spatial distribution of the fission neutrons only. (This is due to the special form of the fission kernel Kf.) The simulational aspect is reflected even better by an alternative formulation of this method. The equations above determine the connection between the number density of fission neutrons in successive generations. Similar relations can be established for the number density of fission points or, in brief, the fission density. Let (PAP) = tP(P)cf(P) 'Then clearly pf is the density of fissions about the phase-space point P. If we denote U(P,P") = cf(P)z(r,E,e)v(P")

(6.108)

then from Equations (6.101) and (6.102), the following equation is deduced: (pf(P) =

i idP"U(P,P")(Pf(13") keff

(6.109)

Obviously, U(P,P") is the density of particles entering a fission at about P due to a particle causing a fission at P". The iterative solution of Equation (6.109) is defined as pr(P) = fdP"U(P,P") 41" "(P)

(6.110)

In fission-neutron formalism [Equations (6.101) through (6.107)1, the effective multiplication factor turned out to be the limit of the ratio of the total number of fission neutrons in two consecutive generations. In the fission-point description [Equations (6.108) through (6.110)1, it is expected that the ratio of the number of fission points will define an estimate of k„f. It is, indeed, so. If N(n) denotes the total number of fissions in the n-th generation, then IN1(") = fdPpr(P) and taking the ratio kn = N(n)/N(^ -"

(6.111)

354

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

it is seen that in the case of convergence, the function (PP) = Yr)(P)/N(-) satisfies Equation (6.109), with k„, = Note that with any function f(P) such that

f

dP yr(P) f(P) 0

an iterative estimate such as = R(")/R("- 4)

(6.112)

= f d1344")(P)f(P)

(6.113)

with

will also converge to k„,. Finally, for the sake of completeness, we also note than an equation similar to Equations (6.104) and (6.109) can be formulated that describes the collision density in the system. It follows from Equations (6.101) and (6.102) that an ordinary eigenvalue equation equivalent to the transport equation (6.81) can be constructed in the form tIJ(P) = 1 dP"A(P,P") kJ(P") k,, where A(P,P") = z(r,E,e)cf(P")v(P") 4. Fourth Method We have seen that the iteration schemes of the third method are essentially equivalent to that of the second method. This also means that the schemes of the third method define increasing neutron populations in supercritical systems and dying out populations in subcritical systems. In practical realizations, however, it is desirable to control the number of simulated particles. In the opposite case, i.e., if the number of particles per generation is allowed to increase or decrease exponentially (as happens in the previous schemes), there is little hope of obtaining a converged population distribution within reasonable computing time or statistical accuracy. Therefore, in practice, a renormalization of the population size is performed in every generation. With the renormalization of the third method, we obtain schemes essentially analogous to the first method (in a formalism more adequate to the simulation). Here we consider the fission-density scheme defined by Equations (6.108) through (6.113). Let f(n — I )(P) denote the fission density produced by the (n — 1)-th generation in a controlled game. Let the control be such that before generating the n-th generation, the source is renormalized to unity with respect to some weighting function f(P). Then the nth generation is started from the fission density (pr 1)(P) = (1)(fn -1)(P)/ fdPf(P)(W-1)(P)

(6.114)

355 Let kn denote the normalization factor, i.e., fc„

=

JdPf(P)(pf" .)(P)

According to the physical meaning of the kernel U(P,P") in Equation (6.108), the (normed and unnormed) densities in the two generations are related as (6.115)

Cp(f.")(P) = f dP"U(P,P")y(f"- ')(P)

It is heuristically felt that the fission densities in the third and fourth methods are related. To see the relation, let us assume that both methods start from the same initial density , which is normalized according to Equation (6.114). Then it follows from yr = Equations (6.110) and (6.115) that 44')(P) = cpV)(P) i.e., the normalized first iterate reads cp`ri>(p) = (PV

)(P)/ki

Repeating the reasoning, we have for the n-th iterates n -1 (i4.1)(P) = e)(P)/I1

i=1

n ,f4n)(P)

= (P(f")(1))/ II ki

Finally, the n-th iterate of the effective multiplication factor in the third method follows from Equations (6.112) and (6.113) as the ratio Icr, = R(')/R(n -1), where n1 lt")

=

fdPf(P) 0

Then the proof is performed in two steps. 1. It can be shown that under condition 1 through 3 A is a completely continuous positive operator and, consequently, has a positive eigenvector and a simple positive eigenvalue k„f. Furthermore, k,, is the greatest (in absolute value) of all the eigenvalues of the operator A

357 2.

It is then proved that with the arbitrary function (1)0 which is not orthogonal to the eigenvector i, the limiting relation lim

liAnc13.011

=k

'ff

holds in any Lp norm. Now, choosing the L, norm and taking into account that AN). = (Von) the n-th iterate of the first collision density in the second method [Equation (6.98)] we have

11(13(:±"11/11(1)(:)11 = Ussachoff85 gives an alternative proof based on the recurrence in Equation (6.105). The proof assumes that the integral operator in Equation (6.104) has an infinite spectrum of eigenvalues with a complete orthogonal set of eigenfunctions. The n-th iterate of keff is then expressed in terms of kw/k"), where 10) is the i-th eigenvalue and le') < k") = keff. The expression shows the convergence of the iterates to keff and also gives an estimate of the error in the iterates. What we have outlined in this section is the proof of the convergence of the source iteration method when the quantities in question (functions and integrals of them) can be exactly determined. In other words, the proof concerns deterministic (analytical or numerical) methods. In the Monte Carlo approach, however, none of the integrals involved are calculated exactly; rather, random quantities with expectations equal to the integrals are used in the estimation of Ice. Furthermore, the realizations of the collision points in a generation are strongly influenced by the collision points sampled in the preceding generation. These facts make a Monte Carlo procedure essentially different from a deterministic source iteration. They have the unfavorable effect that, although the iteration converges, systematic errors appear in the estimates, as will be seen in the next sections.

C. PRACTICAL REALIZATIONS The method of source iteration or successive generations serves as the basis of a Monte Carlo estimation of the effective multiplication factor. It is, however, obvious from the alternative formulations of Section A that the simulation and estimation procedure can be realized in several ways. In what follows, we review a number of various estimation schemes. The simplest method of simulation would be to let the neutron population evolve according to the transport equation, i.e., to play analogously the physical processes taking place in the reactor. Then the neutrons start from an arbitrary source distribution, generate the collision density of the first generation, and establish the source for the next generation by producing new neutrons from fissions. The muliplication factor is then calculated as the ratio of the number of fission neutrons in successive generations. Such a procedure would simulate the second iterative method introduced in Section A. The drawback of such a direct approach is the same as that discussed in the previous section, i.e., if the effective multiplication factor differs essentially from unity, there is a high probability that the generations gradually die out (if keff < 1) or become overpopulated (if keff > 1) before reaching a sufficiently well-iterated value of One version of this method uses normalized source densities in every iteration. This modified scheme corresponds to the first method in Section A. In this case, every generation may consist of an arbitrary number of starters, the distribution of which is determined by the fission source produced by the previous generation

358

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

[cf. Equation (6.92)]. Although this procedure is feasible, it has the disadvantage that, although the fission neutrons that leave fissions are generated in order to establish the fission source of the next generation, all of them will not necessarily be processed. Though this problem is not too serious and can be avoided by skillful programing, in practical realizations, use of fission sites as the initial characteristics of a generation is preferred to the fissionneutron scheme. Equation (6.109) in the third method connects the fission points of successive generations. Similarly, Equation (6.104) connects the spatial density of fission neutrons. The simulation procedure applied in practice conforms with these interpretations of the transport equation and with the iterative schemes following from them, as detailed in the fourth method. A strategy with a fixed number of fission points is described below. '7'4° Let every generation be started from exactly N fission points. Then the estimation procedure consists of the following steps: 1. Let the initial fission points (to be chosen by guesswork) be Pi'', 13(2'), . . . Let n = 1. 2. From the points PI,— (i = 1, 2, . . . ,N), v(P;"- I )) neutrons are started with energies and directions chosen from the density function x(Elr(n - II). The neutrons are followed until they are absorbed, give rise to a fission, or escape from the system. 3. Let the new fission points be 13;"), . . . ,121[;;) [These points represent samples from the fission density 4)[")(P) in Equation (6.115).] 4. An estimate of the reaction rate in Equation (6.116), i.e., that of the n-th iterate of the effective multiplication factor, is Nn

kn

=

E=1 (1/N) = N„/N

(6.118)

5. If N„ > N, the next generation starts from N fission points out of the N„ points Pr. If N„ < N, some of the new points are used several times or, alternatively, N — N„ fission points of the previous generation are repeatedly used as starting points of the new generation. The total number of starting fission points is N in every generation. 6. Let n = n + 1 and return to step 2 until the iterates kn attain a sufficiently wellconverged value. The converged value is accepted as an estimate of keff. The procudure above can be modified at certain points with no effect on the expected result. Such modifications may make the simulation more efficient. We consider here a number of possible alterations of the basic procedure (steps 1 through 6). First, we note that, although in an analog simulation the number of neutrons starting from a fission point P should be equal to v(P), the mean number of fission neutrons, usually v is not an integer. A possible practical realization of this selection is to start a number of histories, L, and multiply the estimate in Equation (6.118) by v/L. If L is large enough, the probability that the fission points generated by the histories is less than N may be made practically zero. Alternatively, one can select the fission point Pr as the starting point of an (n + 1)-th-generation history with a probability

V(Pr)/E

j= 1

V(Pjn))

359

and start new histories until N new fission points are generated. In this case, no effort is wasted in simulating the histories which would yield fission points after the N-th one.46 The estimate of the (n + 1)-th iterate, ko, „ of the multiplication factor is also modified. In the basic procedure, the ratio of the number of fission points in subsequent generations estimates k. , according to Equation (6.118), and the number is directly obtained in the simulation. In the modified procedure, the number of new, (n + 1)-th-generation fission points is N; however, the number of n-th-generation fission points necessary to yield N new ones is to be estimated. Let L(') be the number of histories processed in the n-th generation until the N-th fission point is produced. On average 1-)

= N- ;=, EN

histories start from a single n-th-generation fission (source) point. If f (" ) denotes the average number of histories necessary to produce a new fission point in the n-th generation, then an estimate of the number of n-th-generation fission points needed to generate a new point is f(")/i), and an estimate of the iterate of the multiplication factor is

k,, = NiusienVid = iVe(n) It might seem logical to put e(") = L(')/N, i.e., to estimate the mean number of histories per new points by the mean number of histories actually processed. However, note that simulation of the n-th generation is terminated as soon as the N-th fission point is generated, and we have no information on whether the (L(n) + 1)-th or further histories would result in fissions. In other words, e(n) might also be estimated by (Li") + 1)/N, (Lt") + 2)/N, etc. What we know for sure is that exactly Li") — 1 histories were generated to produce N — 1 fission points. Thus, an unbiased estimate of e(n) is en) = (11") — 1)/(N — 1) and therefore an estimate of the iterated multiplication factor reads

kn , =

N — 1 x--1 N 2, v(P(^))1(L(n) — 1) N =

Note that this estimator (defined as the ratio of fission points in successive generations) can also be interpreted as the production-to-fission ratio in the n-th generation.16 Another possible modification of the basic procedure follows from the fact that the estimate [Equation (6.118)] of the reaction rate in Equation (6.116) consists of contributions of a last-event estimator that scores unity at the end of a history. According to the results in Chapter 5.VI, this estimator can be replaced by several other estimators that result in the same expected score per history. Working codes offer several possible estimators for the purpose. 45 '" New generations define new estimates of the effective multiplication factor that are essentially independent of the previous estimates in the basic method.* The multiplication factor can also be estimated as an average of the iterates ("ergodic estimate"):17,88 kern =

km, =

E

j=m

kJ(n — m + 1)

(6.119)

Note that the independence of such estimates may not be complete since the starting points of a generation are determined by the collision points of the previous generation. The correlation of the separate estimates is taken into account when estimating the variance of the final estimate. (cf. Section D).

360

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

where 0 < m < n and m 1 is the number of iterations to be discarded from the estimation due to the influence of the arbitrarily chosen initial source distribution. Frank-Kamenietzky" proposes a modification of the simulation procedure in which exactly one neutron emerges from every fission. This is reached by suitable alteration of the reaction probabilities. In the basic procedure, a neutron entering a collision at a point P is absorbed there with a probability ca(P) and results in the emission of v(P) fission neutrons with a probability cf(P). Scattering is played with the complementary probability c,(P) = 1 — ca(P) — cf(P). Let a = maxv(P)cf(P)/[1 — cs(P)] and let us denote =

v(P)cf(P)/a

and ca(P) = 1 — ca(P) — 4(P) These probabilities define a game in which the scattering probability is the same as in the basic procedure, but the fission and absorption probabilities are altered. Let v*(P) denote the mean number of secondaries per fission in the altered game and let us define v*(P) = 1 and Z*(r,r') = f dE z(r,E,e)v*(P)cT(P) Then, according to Equation (6.103), Z* is related to the unaltered kernel Z as Z*(r,r') = Z(r,r')/a Inserting the altered kernel into the fission-neutron density equation (6.104), we have S(r) = 1 f dr' Z*(r,e)S(e) keff with keff = keff /a

i.e., in the altered game (in which exactly one neutron is produced by every fission), the fission density of the original game is reproduced, while the effective multiplication factor (eigenvalue) becomes 1/a times the original one. The iteration procedure based on the altered game is also suitable for the estimation of keff, and it has the advantage that the number of histories per generation is always N, the number of fission points. Several other modifications of the basic procedure exist. The interested reader is referred to the works by Lieberoth,46 Frank-Kamenietzky,' Gast and Candelore,25 and Zolotukhin and Maiorov,88

361 The application of source iteration in Monte Carlo eigenvalue estimations raises a number of specific problems which are not present in a fixed-source, reaction-rate estimation procedure. First, the recursive process assumes an arbitrarily chosen initial distribution of fission points. In principle, it may happen that due to an unfortuante choice of the initial distribution, the procedure does not yield the required final result. However, as was pointed out in Section B, unless this distribution is orthogonal to the desired solution, the procedure does converge to the proper distribution. Starting from an everywhere positive distribution, orthogonality is certainly avoided. Moreover, if, for example, the initial distribution is region wise uniform and proportional to the density of the fissionable material in the regions, the iterated distributions in most practical cases will not be influenced by the actual distribution of the initial fission points after five to six iterations." [This also means that, in such cases, one can put m = 5 in Equation (6.119),] Nevertheless, in unfortunate cases, the effect of the initial distribution may persist after a large number of iterations. Theoretically, the rate at which this effect dies out is determined by the ratio of the second-largest eigenvalue to the largest one (dominance ratio) and, if this ratio is near unity, a persisting initial distribution is expected85 (cf. Section B). A more serious theoretical problem follows from the fact that ke„ is estimated as the ratio of two random variables instead of the ratio of their expectations. To see this more closely, consider the equation defining the third method of source iteration. According to Equation (6.112), the effective multiplication factor can be expressed as keff = JdP JdP" f(P)U(P,P") (pf(P")/1 dP f(P)pf(P) where tpf(P) is the number density of fissions at P and f(P) is an arbitrary function. In the n-th generation ke„

= R(8)/R(8- 1)

(6.120)

where, according to Equations (6.110) and (6.112) R("

= f dP f(P)e -1)(P)

and 12(') = f dPf(P)(p(''(P) = f dP f drf(P)U(P,P") (pr °(P")

(6.121)

Therefore, if (0') is a sufficiently well-converged approximation of the real density (pf , then k„ in Equation (6.117) approximates well the value of ke„ in question. In the limiting case when yr becomes equal to (p„ kr, goes over to ke„. However, note that IV") in Equations (6.120) and (6.121) is the expectation of a reaction rate-type quantity, and thus k, essentially is the ratio of two expectations. In the Monte Carlo procedure, on the other hand, we only determine estimates of the reaction rates It") and it"- and the expectation of the ratio of these estimates, in general, will not be equal to ke„. Correction for this bias is derived in Section 6. V. E. Although application of the fourth method (renormalization by using a fixed number of fission points) seemingly removes this difficulty since kn is expressed as a single reaction rate [Equation (6.116)], in fact, this is not the case. This method avoids the above bias, but it is also biased because of the renormalization procedure of the fission density. Formally,

362

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

it is obvious from the fact that the normalization factor in Equation (6.114) is again a reaction rate (expectation) which is replaced in the simulation by an estimate. In practice, when a fixed number of fission points is used, the bias is due to the fact that only the first N realizations of the fission points in a generation are retained; the rest of the realizations are discarded. As a result of this arbitrariness, the iterated fission points will not necessarily be distributed according to the density. 44") in Equation (6.114). Consequently, the reaction rates, which represent the iterates of the multiplication factor and are estimated according to Equation (6.1 18), will be different from kn in Equations (6.116) and (6.120). Hence, we conclude that even if the iterated fission densities in the third method, 44"), converge to the exact one, the resulting estimate of the multiplication factor will be biased in all methods. The situation is further aggravated by the fact that the third method does not necessarily yield an unbiased estimate of the fission density either. This is because the fission points of subsequent generations are correlated. In fact, it can be seen16,24,25,46 that unless the number of fission points per generation tends to infinity with an increasing number of generations, the converged density and multiplication factor will be biased. Several attempts were made to estimate the bias in the multiplication factor estimated with a constant number of fission points per generation. It is showni° that the bias decreases at least as fast as 1/VN, but most authors guess an 1/N behavior. This conjecture is corroborated by a number of proofs based on not-too-strong assumptions. 41,42,56,87,88 Although these derivations give estimates of the magnitude of the bias, they are not suitable for correcting the estimated values since the formulas contain constants that cannot be estimated. In spite of the apparently serious theoretical deficiencies of the Monte Carlo source iteration methods, realistic test calculations show that, in practice, the bias in k,„ does not exceed 0.4% with N = 12 (12 fission points per generation) and is expectee" to be negligible with N = 50. This means that source iteration can be safely used in practical criticality calculations. There remains, however, the problem of correlated generations which, although it usually does not affect essentially the estimated multiplication factor, may make the estimated variance of it quite unreliable. This question is investigated in the next section. D. VARIANCE OF THE ESTIMATED MULTIPLICATION FACTOR We consider the estimate of the variance of the effective multiplication factor when the factor is determined by the ergodic estimate [Equation (6.119)] in a simulation procedure with a constant number of fission points. According to Equation (6.119), the estimated multiplication factor after the n-th generation is

k=

E

J=--m

k;/(n + m — 1)

(6.122)

where kJ is the estimate of the same factor in the j-th generation (by one of the source iteration methods) and the first m — 1 generations are omitted from the estimate. The variance of the ergodic estimate is D2[k] = ([k — (k)]2) where (0 denotes the expectation of the random variable t. Inserting Equation (6.122) into this expression and denoting kj = (kj)

363 we have D211(1 = [

J=.

(ki

ki)]2)/(n — m + 1)2

1

(n — m + 1)2

E ((k. - k;)2) + 2E ((k, - ki)(k; - 10)} (6.123)

If we assume that the fission-point distribution in generations m, m + 1, . . . , n are sufficiently close to their limiting distributions, the generationwise variances of the iterated multiplication factors are approximately identical, i.e., if V2 denotes the variance of the multiplication factor in a single generation, we can put (1Ci — kj)2) = V2

j = m, m + 1,

,n

Furthermore, let p, denote the correlation coefficient between the estimates of the multiplication factor in the i-th and j-th generation, i.e., let pi; = ((k, — ki)(ki — 10)/V2 Again, if the simulation is sufficiently converged, then the correlation coefficient depends solely on the difference of the generation numbers, i.e., p1, = p ;_ ;. With these notations, the variance in Equation (6.123) reads D21k1 = V2 [1 + 2 2, ,=,

n—m+1—i p,]/(n—m+ 1) n—m+1

(6.124)

This expression of the theoretical variance, however, is not applicable in practice since estimates of the correlation coefficient p, with large lags i are both troublesome for programing reasons and unreliable in terms of statistical fluctuations. An alternative to this expression was proposed by MacMillan." He shows on physical grounds that a conservative estimate of the correlation coefficients of higher lags can be approximately expressed by the coefficient of lag 1 as Pi

I Pi

where d is the ratio of the second-largest to the largest eigenvalue of the transport equation (6.81) d = k(1)/k"

k( ')/ke„

called the dominance ratio. (A rigorous derivation of this approximation is given in Reference 88. Properties of the Monte Carlo eigenvalue calculation in the discretized random-walk model were thoroughly investigated by Gelbard and Prael.27) Now, extending the summation in Equation (6.124) to infinity and making use of the fact that (n — m + 1 — i)/(n — m + 1) < 1

364

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

we have a conservative estimate of the variance D2[k]

V2[1 + 2p,/(1 — d)]/(n — m + 1)

Yet two problems remain to be solved when we estimate the variance according to this expression. First, the dominance ratio, d, is not easy to estimate. Second, numerical experiments show25 that in certain cases this approximation is overconservative, i.e., it results in spuriously high estimates of the variance. Gast and Candelore25 propose an alternative formula with empirically fitted constants: D2[k] ----- V2[1 + 10pV(1 — pi)]/(n — m + 1)

(6.125)

Equation (6.125) can be directly estimated in the iteration procedure as follows. An estimate of the generationwise variance V2 is obtained as

E (kJ -

k)2/(n — m + 1)

=

and the correlation coefficient p, is estimated as n-1 =

E

J=.

(ki — k)(ki+ — k)/[V2(n — m — 2)]

Test calculations show25 that for small systems the correction term [second term in brackets in Equation (6.125)] is negligible and only becomes essential if the multiplying material extends over larger regions. E. A ONE-STEP SCHEME: ACCELERATION OF THE ITERATION It was shown in Sections 5.V.0 and 5.VIII.F that importance sampling may, in principle, result in estimation schemes of fixed-source reaction rates with zero variance. The same technique can also be implemented in the source iteration procedure to obtain an exact estimate of the effective multiplication factor. Most naturally, zero-variance schemes in eigenvalue estimations are no more realizable in practice than those in fixed-source problems. Nevertheless, approximations to these schemes may substantially reduce the variance of the estimated eigenvalue. Importance sampling resulting in zero-variance criticality calculations was first proposed by Goad and Johnston.29 They combined the method with a technique similar to the ELP method (cf. Section 5.VIII.D), as a result of which the starters are not allowed to escape from the system. An alternative scheme proposed by Kalos" is given below. It is based on the method of importance sampling without statistical weights. Consider the fission-neutron density formalism of the third method in Section A. The spatial distribution of the fission neutrons, S(f), is governed by Equation (6.104) S(r)fdr'Z(r,r') S(e) ke,

(6.126)

where Z(r,r') is the density of the fission neutrons emerging from around r due to a fission neutron started from r'. In the course of the source iteration, the n-th-generation fission

365 neutron density, S'")(t), is connected to the density of the previous generation through Equation (6.105): )(e) S(n)(r) = IdeZ(r,e)S('

(6.127)

The n-th iterate of the estimated multiplication factor follows from Equations (6.106) and (6.107) as kn = f dr S(')(r)/fdr S(n- "(r)

(6.128)

Let us introduce an importance function V(r) such that V(r') > 0 for every r' such that

f

drZ(r,e) > 0

i.e., V is definitely positive at every point that can serve as a starting point of fission neutrons contributing to the generation of next-generation fission neutrons. Let 52(r) = S(r) V(r) and Z(r,r') = V(r)Z(r,r')/V(r')

(6.129)

Then Equation (6.126) can be rewritten as

keff

fdr' 2(r ,e) 99(r')

(6.130)

Equation (6.130) is analogous to Equation (6.126) and defines an iterative procedure similar to that in Equation (6.127): Yth)(r) = f dr ' 2(r ,r ') 59(' "(e)

(6.131)

By analogy to Equation (6.128), an estimate of Ice, from the n-th and (n — 1)-th generations reads = fdr 92(n)(r)/fdr92"-"(e)

(6.132)

Now, if the importance function V(t) is the solution of the equation V(r) =

1

, , , f dr V(r )Z(r ,r)

(6.133)

366

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

[i.e., V(r) is adjoint to S(r)1, then it follows from Equations (6.130) and (6.131) that the following integral relation holds fdr 97(n)(r)

f

dr' V(r) Z(r,r') 99(n - ')(r ')/V(rs )

= f dr' [fdrV(r)Z(r,r')] J(a- "(r')/V(e) = keff f dr'97("-"(e) Accordingly, the n-th iterate of the multiplication factor in Equation (6.132) is (6.134)

ke = keff

i.e., the exact value of the effective multiplication factor is reproduced in every iteration. This scheme would have zero variance if the reaction rates (k = n — 1, n)

fdr'J)(k)(e),

were estimated without statistical fluctuations. Although this can also be attained (at least in principle), the main consequence of the scheme is that knowledge of the adjoint function V(k) makes a one-step (iteration-free) estimation possible. Therefore, approximate values of V(r) may accelerate the source iteration. Before turning to a discussion of practical applications, let us realize that it is not necessary to perform the importance-sampled game above in order to obtain a one-step estimation procedure. Instead, appropriate weighting functions of the reaction rates defining Ice may be used in an analog game, as is stated in the following theorem. Theorem 6.4 — Let an analog game be played according to Equation (6.127), i.e., let S(n)(r) be the n-th-iterated fission-neutron density. Let the n-th interate of the effective multiplication factor be defined as k„ = R(n)/R("-"

(6.135)

R(n) = fdr S(n)(r)V(r) = fdr'idrV(r)Z(r,e)S(n "(e)

(6.136)

with

where V(t) is the solution to Equation (6.129). Then kn = keff i.e., the weighting function v(e) yields a one-step estimate of the effective multiplication factor. Proof. It is easily seen from Equations (6.129) and (6.130) that the iterated analog and importance-biased fission-neutron densities are related as J(n)(r) = S(')(r)V(r),

(n = 1, 2, ...)

367 if the same relation holds for the initial distributions. Therefore, Equation (6.136) can be rewritten as R(")

fdr W(")(r)

i.e., comparison of Equations (6.135) and (6.134) shows that k„ = keff

In practical realizations, the usual problem of zero-variance estimations is encountered, namely, determination of the weighting function V(r) [the solution of Equation (6.133)] is no easier than the solution of the original transport equation. Nevertheless, approximations to the exact solution of the equation are expected to accelerate the estimation of ke„ by source iteration. An approximate solution is obtained during simulation in the following way. Let the system of interest be divided into a number of domains In the first stage of the simulation, let us determine the expected number of fission neutrons in the various domains due to starters from different domains in the previous generations. If Z1 denotes the estimated number of fission neutrons in D, due to a starter from 13_, in the analog game, then V(r) may be approximated by a domainwise constant function Vi, which is the eigenvector belonging to the largest eigenvalue of the equation V)

= -k E V; Z;;

Having determined the values V, on the basis of the estimated Z,,'s in the first stage, the second stage of the simulation is performed with importance sampling or, equivalently, the weighted integrals of the original fission density are calculated with the function V(r) =

if r

E. Di

In an alternative method, V, is directly estimated parallel to the iteration procedure. The estimation is based on the physical meaning of V,. It can be seen that V, is the expected number of fission neutrons in the whole system due to a fission neutron started uniformly over 133. Matthes" proposes the following procedure. Let Vic") be the estimate of V„ in the n-th generation and let Nr be the number of fission neutrons started from D, in the (n + 1)-th generation. If the i-th from among the Isic,n3 histories produces v, fission neutrons in 13,, then the estimate of V, after the (n + 1)th generation reads

v(.±.) =

NI.)

E E vd' Vf:‘) NC") j

1= 1

k

The method is slightly refined by Hoffman et al.35 F. REACTIVITY CHANGE DUE TO PERTURBATIONS It was shown in Chapter 6.1 that the effects of small perturbations on reaction rates can be very efficiently estimated by correlated Monte Carlo. The advantages of the correlated method are not confined to fixed-source problems, estimation of reactivity rate perturbations

368

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

in criticality calculations by correlated games may be equally efficient. Estimation of the change in the multiplication factor,* however, is slightly more complicated than estimation of reaction rate changes since k,„ is estimated as the ratio of two reaction rates [cf. Equations (6.99), (6.107), (6.112), and (6.118)]. In this section, we review a number of proposed methods for estimating the change in keff due to some perturbations introduced into the system. Let us consider again the fission-neutron density formalism in Equation (6.104). Let S(r) and Z(r,r') be the fission-neutron density and kernel, respectively, in the unperturbed system, i.e., the eigenvalue equation in this system reads keffS(r) = fdr'Z(r,e)S(e) Let S(r) and 2(r,r') denote the corresponding functions in the perturbed system. If keff is the effective multiplication factor in the perturbed system, then keffS(r) = f dr '2(r , r')S(r' ) Accordingly, for the arbitrary function f(r), the multiplication factors are expressed as keff = f dride f(r)Z(r , r' )S(r')/1 drfir)S(r) and keff = f dridef(r)2(r,r1 )§(r')/fdrf(r)S(r) At this moment, we do not specify the weighting function f(r). The simplest choice is f(r) = 1. On the other hand, we have seen in the previous section that setting f(r) = V(r) the solution of Equation (6.133), makes possible (in principle) a one-step estimation of keff. This choice may also find application in the estimation of the reactivity perturbations to be seen later in this section. Let us now introduce the shorthand notations drf(r)S(r) = (f,S) J

and dr f def(r)Z(r ,e)S(r') = (f,ZS) f

*

Although the considerations and formulas below concern the perturbation of the effective multiplication factor, the subject is often referred to as "reactivity perturbation calculations". The reactivity p is defined as p = 1 i.e, the two perturbations are essentially identical — 1/1t,ff, and the perturbations are related as Sp what justifies the shorter name of the subject.

369 Then the multiplication factors in the two systems read keff = (f,ZS)/(f,S)

(6.135)

keff = (f,i§)/(f,§)

(6.136)

and

The quantity to be estimated is the difference of the two factors 8k = 8k,„ — keff Denoting 8S(r) = S(r) — S(r) and 8Z(r,e) = Z(r,r') — Z(r,r') it follows from the expressions of the multiplication factors above that 8k = [(f,ZS) — (f,ZS) — ke„(f,8S)[/(f,§) = [(f,8ZS) + (f,Z8S) — ke„(f,8S)1/(f,S)

(6.137)

In the majority of practical methods, it is assumed that estimation of the perturbation is started after having reached a reasonably well-converged fission-neutron density, S(r), and multiplication factor, keff, in the unperturbed system.5".6''" This means that the reactivity perturbation estimation reduces to the estimation of reaction rates with fixed sources, which can be performed according to the rules introduced in Chapter 5. The unique feature of the simulation is that a history also terminates if a fission occurs, and it is the fission event that contributes to the estimates. The simplest way of estimating the reaction rate R = (f,ZS) = f dridef(r)Z(r,e)S(e) consists of the following steps: 1. A neutron is started from the source S(r) [with an appropriate weight W(r) depending on the norm of S(r) and the method of selecting from S(r)]. The energy and direction of the starter is selected from X(E1 r). 2. The neutron history is generated according to the transition and collision kernels of the medium until the neutron escapes from the system, is absorbed, or gives rise to a fission.

370

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

3. In the first two forms of termination, the neutron yields no contribution to the reaction rate. If a fission occurs at the phase-space point P" = (r',E') then the contribution of the history is W(r) v(P")f(r') where v(P") is the mean number of fission neutrons leaving a fission at P". A rigorous proof of the unbiasedness of the estimation procedure in steps 1 through 3 for estimating the reaction rate (f,ZS) is given in Appendix 6A. A number of the simulation and estimation techniques introduced in Chapter 5 can also be applied here to modify the above procedure. The most obvious possible alteration of step 3 follows from the theory of partially unbiased estimators. Accordingly, the score W(r)v(P")f(r') per fissions can be replaced by the sum of the scores W(r)cf(P,)v(P,)f(r,) over all the collision points in the history, where cf(P,) is the probability of a fission in a collision at P. Let us now consider some of the possible ways of estimating the perturbation in the effective multiplication factor. 1. We shall first assume that the converged, unperturbed fission-neutron density and the corresponding (but unknown) perturbed density are normalized as (f,S) = (f,S) = 1

(6.138)

Then the perturbation in keff in Equation (6.137) reduces to 8k = (f,SZS) + (f,Z8S)

(6.139)

The first term on the RHS is easily estimated in a correlated game since it is of the form of a reaction-rate perturbation 8R = (f,('Z — Z)S) = idrf(r)(P(r) — fdrf(r)(p(r) with Cp(r) = ZS(r) and (p(r) = ZS(r), which are the fission-neutron densities in the perturbed and unperturbed systems, respectively, due to a source density S(r). In estimating the second term in Equation (6.139), one has to resort to some approximation (unless the converged distribution in the perturbed system is also estimated) since 8S(r) in the integral is not known. If the perturbation is small, it is reasonable to assume that the perturbed fission-neutron density S(r) is appropriately reproduced by the first-generation fission neutrons started from an initial source equal to S(r) in the perturbed system. Under this assumption, we write §(r) = Ide2(r,e)S(e)/fdr1de f(r) Z(r,e)S(e)

(6.140)

where the denominator ensures the normalization required in Equation (6.138). The approximate distribution in Equation (6.140) may be generated parallel to the correlated game that estimates the first term in Equation (6.139). The main point is that the unperturbed starters produce next-generation unperturbed fission neutrons [samples from (Sr)] and, at the very same points, perturbed fission neutrons are also born with appropriate statistical weights [samples from S(r)]. The statistical weights are generated according to the conventional weight generation rules in a nonanalog game, i.e., while the simulation is performed

371 in the unperturbed system, the weights are changed so that they account for a migration in the perturbed system. [In brief, the perturbed fission neutrons are the progenies of the perturbed particles in the correlated game that estimate the first term in Equation (6.139).] The second term is then estimated in the next generation, where the particles are those produced by the previous generation. Their initial weights are equal to the difference of the weights of the perturbed and unperturbed particles born together, and the game is played in the perturbed system, i.e., according to the kernel Z(r,r'). Note that, making use of the approximation in Equation (6.140), the perturbation in Equation (6.139) can be rewritten as 8k --- (f,ZS) — (f,ZS) + (f,Z2S)/(f,ZS) — (f,ZS) = (f,Z2S)/(f,ZS) — (f,ZS)

(6.141)

One might object that it is unnecessary to bother with correlated games to estimate the two terms in Equation (6.139) when the two terms in Equation (6.141) can be directly estimated in two successive generations. However, the objection is groundless since Equation (6.141) is only an approximate reformulation of the expression 8k = keff — keff, where keff is replaced by (f,Z2S)/(f,Z S) and, thus, even if the approximation in Equation (6.140) is very good, 8k in Equation (6.141) is estimated as the expectedly small difference of two essentially independent estimates, both of the order of magnitude. 1. In contrast, the two terms in Equation (6.139) are both of the same order of magnitude as the final estimate 8k. Note that the assumption of normalized densities, as formulated in Equation (6.138), is not essential. The third term in the expression (6.137) of the reactivity perturbation (which vanishes in the case of normalized densities) can also be estimated in the second step of the simulation procedure above.6' 2. An important feature of the above method is that it may only work for small perturbations since assumption (6.140) certainly fails to work if the fission-neutron densities in the two systems are very different. Polevoi63 proposes a scheme which is valid for arbitrary perturbations. The price one pays for a more exact treatment is that more than one generation must be simulated to determine the perturbed fission-neutron density. This, however, is done in such a way that every generation gives a contribution to the estimate of 8k. Let us assume again that an asymptotic, unperturbed fission-neutron density, S(r), is reached in a preliminary stage of the simulation. A specific scheme is followed from this point. Let a first-generation "distribution" be determined according to the relation A,(r) = fdr'8Z(r,e)S(e)

(6.142)

and let the distribution, A, (r), of the successive generations be simulated according to the equation A„ ,(r) = ideZ(r,r')A,,(e)

(6.143)

The last equation defines a procedure analogous to the iterative simulation of the perturbed fission-neutron density defined in method three of Section A. The main difference between the two simulations is that in the procedure defined by Equations (6.142) and (6.143), the initial distribution is not necessarily everywhere positive [A,(1.) is the difference of the

372

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

fission-neutron densities resulting from the random walk of particles started from S(r) and migrating in the perturbed and unperturbed systems, respectively]. Therefore, simulation of the game between the n-th and (n + 1)-th generations defined by Equation (6.143) may involve particles with negative weights. Let us observe that no renormalization of the number of particles is assumed in the procedure, and therefore the perturbation of the multiplication factor will be determined according to Equation (6.137). If Zn(r,r1 ) denotes the n-th iterate of the integral kernel Z(r,r'), then Equation (6.143) can be rewritten as An± ,(r) = fdrni f

1 ...1 dr,2(r,r.)2(rn,rn _,)...2(r,,r i)A,(r,) (6.144)

dr'Zn(r , r' ) ,(r' )

If the perturbed and unperturbed densities were iterated independently, then their n-th iterate would read §n(r) = IdeZ„(r,e)S(r)

(6.145)

Sn(r) = fdr7„(r,r1 )S(r) = IQ„S(r)

(6.146)

and

i.e., the n-th iterate of the density perturbation would take on the form 8Sn(r) = 8,,(r) — len'„S(r)

(6.147)

Now, it follows from Equations (6.142) through (6.144) that the "density" in the iteration procedure above is related to the iterated real densities as

An + i(r)

= fdriZn(r,e)f dr"8Z(r' ,r")S(r") = fdr'Zn ± (r,r' )S(e) — fdr'2„(r,e)fdr"Z(r' ,r") S(r") (6.148)

= S,,+1(r) — keffSn(r) Consequently, we have An „(r) + kei 4n(r) =

— Ic!„§„_ ,(r)

(6.149)

and since the perturbed iteration is assumed to start from the converged density S(e), i.e., S0(r) = S(r)

(6.150)

we have

E

m=I

Kfim Am(r) = k(r) — k'n',S(r) = 8Sn(r)

(6.151)

373 The perturbation of the effective multiplication factor, Equation (6.137), is estimated in the (n + 1)-th generation as 8k„ = Rf,Zik) — (f,ZSn) — knif(f,8Sn)]/(f,k) The first two terms in the numerator can be simplified, in view of Equations (6.145) and (6.146), as (f,IS„) — (f,ZSn) = (f,§n+ 1)

(f,Sn+i) = (f,8Sn+1)

(6.152)

i.e., the numerator, after making use of Equation (6.151), becomes (f,8S.+1)

keff(C8SO = (f40±1)

(6.153)

The denominator follows, again from Equation (6.151), as (f,gn) =

E keft m(f,A,n) + Kff(f,S)

(6.154)

m=1

Combining the expressions in Equations (6.153) and (6.154), the perturbation of the effective multiplication factor is expressed in the (n + 1)-th generation as 8kn = (f,A,)/IIQX,S) +

E

KYf m(f,A„,)]

(6.155)

m=1

The estimation of 81( is thus reduced to repeated estimations of the reaction rates (f,A,n) during the simulation of the successive generations. Note that the reaction rate (f,S) and the unperturbed multiplication factor are supposed to be known at the beginning of the procedure, since the estimation of the perturbation is started after an asymptotic, unperturbed fissionneutron density has been reached. The iteration is continued until 8k. is converged at a predetermined level. It is interesting to examine the estimate of 8kn„ after a single iteration. For n = 1, Equation (6.155) gives 8k, = [(f,Z2S) — (f,i4S)]/(f,iS) = (f,i2S)/(f,iS) — k, Comparing this expression to Equation (6.141), it is apparent that the previous approach is a special case of Polevoi's method with n = 1. Note that an unbiased estimation of the "density" A, in Equation (6.142) is not always trivial. A critical analysis of the estimation is given in Reference 28. 3. Alternative methods follow from a specific form of the weighting function f(r) appearing in the expressions so far derived. Let f(r) = V(r)

(6.156)

374

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

where, again, V(r) is the function adjoint to the unperturbed fission-neutron density, i.e., it is the solution of the equation keffV(r) = IdeV(e)Z(r',r) Using this weighting function, the last two terms in Equation (6.137) vanish and the perturbation of the effective multiplication factor reduces to 8k = (V,6ZS)/(V,S)

(6.157)

This expression suggests a simple estimation procedure that is exact for small perturbations. If we are given a well-converged, unperturbed fission-neutron density S(r) and a sufficiently accurate approximate adjoint distribution (e.g., by one of the methods presented in the previous section), then for sufficiently small perturbations, one can neglect the difference of (V,S) and (V,S) when estimating the denominator of Equation (6.157). An estimate of the reactivity perturbation exact up to the first order of the perturbation is then obtained as 8k ----- (V,6ZS)/(V,S)

(6.158)

by correlated sampling of the numerator. This method has the advantage that an estimate of 8k is obtained in a single generation, but also the drawbacks that it necessitates the calculation of the adjoint density and is limited to small perturbations. 4. As an alternative to the method above, the perturbation source method" introduced in Section 6.I.E can also be applied here. The perturbation kernel 8Z(r,r') follows from Equation (6.103) as 8Z(r ,r ') = IdE8z(r,E,r') cf(P) v(P) + idEz(r,E,r') 8 [cf(P) y(P)] where z(r,E,r') is the collision density at about P = (r,E) due to a fission neutron emerged from r', and 8F denotes the difference of the function F in the perturbed and unperturbed systems. With this partition, the numerator of Eqaution (6.158) reads (V ,8ZS) = f dP V(r)Icf(P) v(P) tp,(P) + 8 lc,(P) y(P)] tli(P)}

(6.159)

11,(P) = idez(r,E,e)S(e)

(6.160)

where

is the collision density due to the fission source S(r') and = fdro8z(r,E,ro) S(r.) The perturbation of z(r,E,r') follows from Equation (6.100) as 8z(r,E,r0) = f dP"8z(r',E',ro)Ks(P",P) + idP"z(r',E',r0)8ks(P",P) + 8[x(Eiro)T(r,„—>r1E)]

375 where P" = (r',E'). Multiplying this equation by S(r0), integrating with respect to re,, and taking into account that, in view of Equation (6.101), S(r,,) can be written as S(r') = idE't11(P")cf(P")v(P")/keff we have the following equation for tji,(P) th(P) = f dP"Ili,(P")Ks(P",P) + fdP"LP(P") x

Sk,(P",P) +— 1 cf(r) v(r) S[x(Elr') T(r'—>r1E)] Keff

(6.161)

where we have made use of the relation in Equation (6.160). Equation (6.161) is a transport equation defining the collision density tli,(P) in a nonmultiplying system. The first-flight collision density in the equation is represented by the second integral on the RHS, and it contains 4i(P). The simulation of this collision density consists of two steps. In the first step, the original collision density t[i(P) is determined in an ordinary nonmultiplying game with a source S(f), and the first-flight collision density of th(P) defined in Equation (6.161) is also established. In the course of this step, the second term of Equation (6.159) is estimated as a reaction rate due to the collision density Ili(P). In the second step of the simulation, the first-flight collision density [second term in Equation (6.161), determined in the first step] is used to simulate li,(P) in Equation (6.161), and the first term in the reactivity perturbation, Equation (6.159), is determined through an estimate of the corresponding reaction rate with th(P). Note that both steps involve nonmultiplying games since 111(P) is produced by the fixed fission source S(r), according to Equation (6.160), and t[J,(P) obeys the nonmultiplying transport equation (6.161). More details of the practical realization are given by Matthes" and Hoffman et al. 35 5. The perturbation source method presented above is exact up to the first order of the perturbation because (V,S) in Equation (6.157) was approximated by (V,S) in the expression (6.158) of the reactivity perturbation. Hoffman et al.' propose a modification of the procedure not limited to first-order perturbations. The modification is based on the fact that the multiplication factor of the perturbed system is also an eigenvalue of the adjoint perturbed equation ke, V(r) = f dr' V(e)2(e,r) and therefore, for arbitrary function f(r), it is expressed as

keff = (T,szo/(1,1,i)

(6.162)

Now, putting f(1) = V(f) (the adjoint density in the perturbed system) and f(r) = S(r) in Equations (6.135) and (6.162), respectively, the perturbation of the effective multiplication factor reads Sk = keff — keff = [(V,ZS) — (V,ZS)]/(V,S) = 6'i,SZS)/(V,S)

(6.163)

376

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

Estimation of this quantity involves determination of the perturbed adjoint distribution V(r) and the estimation of the perturbation-related reaction rate in the numerator. A possible realization of the estimation by the perturbation source method is described in detail in Reference 35. Obviously, Equation (6.163) contains no approximation and therefore, in principle, it makes an exact estimation of 8k possible. The methods presented in subsections 3 through 5 have the common feature that their accuracy (unbiasedness) depends on the accuracy of the approximation to the adjoint densities used as weighting functions in Equations (6.158) and (6.163). In contrast to this, the exactness of the methods in the first two subsections does not depend on the choice of the weighting function in Equations (6.139) and (6.155). f(e) only influences the variance of the estimates, while their unbiasedness depends upon how well the perturbed fission density is reproduced. Finally, we note that all the methods presented in this section can also be formulated in terms of the fission-point density (pf(P) and kernel U(P,P") introduced in Equations (6.108) and (6.109) in Section A. Zolotukhin and Usikov give the expression corresponding to Equation (6.163) in the fission-point density formalism, along with a number of applications and approximations'. G. PARAMETRIC DERIVATIVES OF keff Let us assume that the effective multiplication factor depends on a system parameter a. Let Sk be the perturbation of the effective multiplication factor due to a change Aa of a. In this section, we give an estimation method of the parametric derivative dk/da as the limit of Ak/Aa, while Aa tends to zero. Consider Polevoi's method for estimating the reactivity perturbation 8k (Section F, subsection 2). This method is based on successive determination of the "difference distribution", An(r), according to Equations (6.142) through (6.144). Let An(r) denote the "difference distribution" produced by a parameter change Zia and let us define the limit a n(r) = lim An(r)/Aa We shall assume that taking the limit Act —> 0 and integration are interchangeable; then it follows from Equation (6.142) that

a a ,(r) = f [— Z(r,r1 )] S(r') aa This equation defines a Monte Carlo game which starts from the equilibrium fission-neutron density S(r), is played according to the derivative kernel aZ/aa, and results in the "fissionneutron density" a ,(t) . The simulation and estimation of the reaction rate (f, a 1) is analogous to the reaction rate estimation in a fixed-source differential game (cf. Chapter 6.11). Further iterates of an(r) follow from Equation (6.143) after division by Aa and taking the limit Aa —> 0 as a,(r) = f driZ(r,e)an(e) Accordingly, a,(r) is the hypothetical "fission-neutron density" due to the "source distribution" a rso in a game governed by the unperturbed kernel Z. The simulation poses no special problems, compared to a fixed-source game, except for the possible appearance of negative statistical weights. The reaction rates (can) can be easily estimated, and the n-th

377 iterate of the derivative of the effective multiplication factor follows from Equation (6.155) as dk„ da =(f, ,)/k:„(f,S)

(6.164)

since lim 8„ = 0. In practical cases, it is reasonable to assume that —0 dkeff lim dk da da

(6.165)

which means that Equation (6.164) can be considered as an estimate of dklda in the n-th generation. Equation (6.164) and expressions of the higher-order derivatives of keff were derived by Mikhailov57 as early as 1967. An alternative method proposed by Takahashi' is based on a direct estimation of the reaction rate

a dkn = da 8a

tli(n)(P) cf(P) v(P)

which follows from Equation (6.93) by differentiation. The simulation is equivalent to a fixed-source differential game.

IV. ESTIMATION OF FLUX AT A CERTAIN POINT In certain (mainly radiation shielding) problems, it is necessary to estimate the neutron flux at the sites of a number of detectors. If the finite dimensions of the detectors are accounted for, the Monte Carlo procedure can be (in principle) based on the reaction-rate estimation methods reviewed previously. In the majority of such problems, however, it is desirable to neglect the presence of the detectors, and the particle's flux at given spatial points is to be determined. At first glance, estimation of pointwise quantities would appear to be inconsistent with the capabilities of transport Monte Carlo methods. Nevertheless, these methods can be made suitable to flux-at-a-point estimations. A possible method of pointwise calculations, the adjoint Monte Carlo simulation, was introduced in Chapter 4.VII. If the particles originate from an extended source and the flux (or some related quantity) is to be estimated at a single point, then the adjoint game fits the problem. However, if the flux has to be determined at several points, then an adjoint game is to be played as many times as the number of detector points. Moreover, if the flux at a point due to a point source is the quantity of interest, then the adjoint simulation faces the same difficulty as the direct game (the detector in the adjoint game is equivalent to the source in the direct game). A special estimator of flux at a point was first proposed by Kalos.37 This estimator can be easily incorporated into any direct game without considerable alterations of the simulation. The estimator is derived in Section A, and it is shown to have the disadvantage of resulting in an unbounded variance. Divergence of the variance, in turn, yields a slower convergence of the average score to its expectation than in estimates with finite variances (Section B). In order to avoid or relax the difficulties related to the singular variance of the simplest estimator, several alternative estimators were proposed, some of them using special nonanalog estimation procedures. 22,23,58,69,76-80 Details of newer developments along with an extensive list of earlier results are given in the report by Kalli and Cashwel1.39

378

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

In this Chapter, we give a systematic derivation of a number of analog estimators that follow from each other by taking expectations over more and more distant events. The successive estimators have better and better convergence properties (Sections C and D). Practical modifications of the various estimation procedures are discussed in Section E. All the considerations are presented for the case of a single detector point. Generalization of the procedures to multiple detectors, although it presents certain practical questions, is straightforward. It is our belief that in spite of the considerable effort invested by numerous researchers in the elaboration of more and more refined procedures, the estimation of flux at a point is not yet solved satisfactorily. A. THE NEXT-EVENT POINT ESTIMATOR The basic difficulty in estimating flux at a point follows from the fact that the probability of playing a history that has a collision at the selected point is zero. It brings us a little closer to the solution of the estimation problem if we remember that expectation-type estimators also gather scores from points which are not reached by the actual history. Let us consider what this trick involves in the case of point estimation. The flux at a point r* can be written in the form of a "reaction rate" as q)(r*) = fdP 4,(P) 8(r — r*)/cr(P) i.e., the contribution function in a direct simulation would be f(P) = 8(r — r*)/o-(P) Now, the expected score in a free flight with this contribution [and with the notations in Equation (5.7)] is f,(P) = f

T(P,131) f(p') = T(r-->r *IE)/o-(r*,E)

Making use of the explicit form of the transition kernel in Equation (5.32), the estimator reads ir r.I

flap) = exp [ —J

dt o-(r* +

8 (co

r* — r1 ir

r*) l Ir

1 r*I2

(6.165)

Unfortunately, f,(P) is nonzero only if the direction of the flight in which it is scoring goes through the point r*. The probability of selecting such a flight is again zero; f,(P) would score after a particle emerges from a collision at P. It is felt heuristically that by taking the expectation of the possible postcollision scores before the collision is actually played, an estimator with more favorable properties is obtained. To see this, let us consider the transport equation (5.4) that describes the collision density due to a source Q(P). The flux at the detector point r* follows from this equation as cp(r*) = IdE*ili(P*)/cr(P*)

IdE*IdP'Q(P')T(P',P*)/cr(P*)

+ idE*IdP"ili(P")1 dP'C(P",P')T(P',P*)/cr(P*)

379 where P* = (r*,t.o*,E) = (r*,E*). With the explicit forms of the kernels, the equation reads (P(r*) = f dr' t IdEQ(e4.0* ,E)T(Ir' — r*Ilr',04,E)/[le

r*I2 0-(P*)]}

+ f del dE'Vr',E'){1dEC(63',E'—w,1„Elr')T(Ir' — r*Ile,w,1„E)/ (6.166)

[Ir' — r*I2 cr(P*)[}

where T(Ir' — r*IIr',63*,E) is a shorthand notation for the integral of the transition kernel according to the relation T(Ir'

r*I2 0-(P*)]

= f dE'f dto'T(P',P*)/cr(P*) = Idw'T(r'—>r*Iw',E)/cr(r*,E) = exp d[

0

tcr(r' + ao*,E)1/Ir' — r*I2

(6.167)

and (.0* = (r * — r')/Ir' Equation (6.166) defines an estimation procedure in which the source particles contribute to the score according to the first term of the RHS, and the collided particles produce the reaction rate in the second term. In this reaction rate, the weighting function (after interchanging w,E with w',E') has the form fEs(P') = IdE'C(u),E—>w*,E'lr')T(Ir' — r*IIr',04,E1 )/[Ir'

r*I2 cr(r*,E')]

(6.168)

and P' = (r',w,E). This estimator scores in every collision and its contribution is equal to the expected score over the collision and the free flight following it. It is easily seen that fEs in Equation (6.168) is the expected scattering estimator [cf. Equation (5.208)] associated with f, in Equation (6.165) fEs(P') = JdP"C(P' i.e., it is, indeed, the expectation of f, over one more event. This estimator no longer has such a singularity as f and f, above have; analytical calculation of it, however, is usually cumbersome. Therefore, in practical applications, a one-sample Monte Carlo estimate of the integral in Equation (6.168) is generated. This is performed in the following way.

380

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

1. At the collision point P' = (r',63,E), the number of secondaries and their energies E' are selected from the marginal density C,(E-->E'le,co) = 2. For a secondary of energy E', the quantity — r*Icr(r*,E')] (6.169)

f„(P',P") = C2(co—*co*Ir',E',E)T(Ir' — is scored, where =

Obviously, the expectation of the contribution f„(P',P") over the possible postcollision energies and number of secondaries is just fEs(P') in Equation (6.168). The contribution of the source particles to the score [first term on the RHS of Equation (6.166)] is either calculated analytically or estimated in a way similar to the collided contribution: the starting site r and energy E is selected from the marginal source density fclo3Q(P) and the starter contributes to the final score by fQ(P) = [Q(r,co*,E)/IdtoQ(r,w,E)] T(Ir" — r* Ijr,04,E)/[lr" — r*I2 cr(r*,E)] = [Q(r,co*,E)/f dcoQ(r,co,E)] exp[—I

I

dt o-(r +tco* ,E)Jilri

r*I2

(6.170)

Note that the estimator f„ in Equation (6.169) also follows from the transformation in Theorem 5.15, Equation (5.214), if we put X = 1 and F(P,P') = L(P,P'). C2 (co-->co*Ir',E',E) in Equation (6.169) is the probability of a scattering from the direction w to w* if the scattering takes place at r* and the energy is changed from E to E'. In most practical cases, this probability does not depend separately on w and w*, but, rather, only on the scattering angle r* - r

11* = w • co* = 0,3 Ir — r

*I

and therefore C2 becomes 1 C2(co—>co* Ir',E,E) = 2,Tr ^1(11*Ir' ,E' ,E) The resulting estimator is called the next-event estimator and has the form fNE(P',P") =

1 —

-y(i.c*Ir',E',E)exp[ — f 0

dtcr(r' +tco*,E'd

(6.171)

381 This estimator, unlike f, in Equation (6.165), gives a contribution to the final score in every collision (from which a secondary emerges) irrespective of the actual postcollision flight direction. It has, however, the very uncomfortable property that its contribution is unbounded as the collision point r' approaches the detector point r*. It will be seen that the divergence of the score does not impede an unbiased estimation of the flux, but it does result in an infinite variance of the estimate. The fact that the expected final score due to the estimator fNE equals the collided part of the flux at 1* follows directly from the derivation of the estimator. This also means that the expected score by fNE is bounded whenever the flux we wish to estimate is bounded. A formal proof of the finiteness may also be given, which will help us in further considerations. The second (collided) term in Equation (6.166) is rewritten as (1),(r*) = fd(r — r*)Ir — r*I -2 [IdE41(r,E)IEs(P)Ir' — where fEs is the expected scattering estimator in Equation (6.168). The bracketed term is finite for every t, while the rest of the integrand can be written in polar coordinates as d(r — rOlr' —

=

—

resulting in a finite integral. The finiteness of the expectation of unbounded scores in practical terms means that although certain histories may yield arbitrary high scores, the probability of such histories is sufficiently small. (Note that here we take advantage of the very same fact that makes a direct point estimation impossible.) Let us now estimate the variance of the score in a similar way. If M2 denotes the second moment of the total score, then M2 is certainly greater than or equal to the first moment of the score with a contribution function f2,,,E(P',P"), i.e.,

M2

fdP'VP')[J. dP"C(P' ,P"KE(P' — r*Idfile — r*1 -2f 1dE'Lli(P')[dP"C(P',r)f7iE(P',Pn)le

=f

1'4424

Since the quantity in braces is generally nonvanishing at r' = r*, the integral with respect to I r' — r*I diverges, i.e., the variance of the total score is infinite. This fact does not necessarily imply that the flux estimated with the aid of the next-event estimator is unreliable. In fact, we show in the next section that the average of the estimates from several histories does converge to its expectation, although the convergence is slower than it would be if the variance were finite. To conclude this section, we note that the next-event estimator introduced in Equatiims (6.169) and (6.171) contributes to the score only for particles that effectively leave the collision point. In practice, it is customary to use the next-event estimator in such a way that the score is attributed to the particle that enters the collision, and the average number of secondaries multiplies the score. In this case, the estimator in Equation (6.168) is sampled in the following way: 1. A possible postcollision energy is selected from the density

f

clt.o'C(o),E—>co',E'lr')/c(P') = Ci(E—>E'lr',03)/c(P)

382

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

where c(P') = f dP"C(P',P") = idco'l dEC(0),E—>ai',E'le) the mean number of secondaries in the collision at P'. 2. The quantity fNE(P',r) = c(P' )fNE(P',PH) is scored irrespective of the number of actual secondaries. B. CONFIDENCE LIMITS FOR SINGLULAR ESTIMATORS Because of the unbounded variance of the next-event estimator, the classical Central Limit Theorem does not apply to the empirical mean of several estimates by the next-event estimator, i.e., the unbiasedness of the empirical mean and its rate of convergence to its expectation do not follow from the considerations valid for estimates of finite variance (Chapter 3.111.). Kalos has investigated these questions in connection with the next-event estimator,' and Dubi et al. amplified the considerations to cover other estimators with less severe singularities.15 Here, we take a more general approach based on a special case of a theorem on stable attracting probability distributions!' Theorem 6.5 — Let C„ C2, . . . ,to be identically distributed random variables with a common density function pc(x) defined on the semi-infinite interval ( — xo, + co). Let

xo

dy yPc(y) = 0

and define the function U(x) = f X dy y2134(y)

(6.172)

Let L(x) be a slowly varying function, i.e., let it be such that limL(xt)/L(x) = 1 If, for large values of x, the function U(x) behaves like U(x) x2-°`L(x)

(6.173)

with 1 < a 2 and if there exists a sequence an such that for some constant C nL(a„)/4 C with increasing n, then the distribution function of the random variable Sn

= E cjan

(6.174)

383 tends to some stable attracting distribution function Gn(x) as n tends to infinity. If a = 2, the stable attracting distribution is normal. The characteristic function corresponding to G«(x) is

cut) =

e'“«)

(6.175)

with Walt)

= tC

F(3 - a) _ a(a - 1)

e -"a-N12

and the upper sign applies when t > 0; the lower, for t < 0. The theorem provides us with confidence limits and also with convergence rates of the estimates by singular estimators in the following way. Let µ„µ2, . . 4.1..n be estimates of the reaction rate (in fact, flux at a point) R indifferent histories. In the previous section, we have seen that they are unbiased i.e., (111,) = R Since the next-event estimator contributing to p, has an I r

r*I -2-type singularity, the

final score ix, will have the same type of singularity. We shall show that the random variables = (µ, - R)/R (i = 1, 2, . . . , n) satisfy the conditions of the theorem, and therefore the distribution function of the variable S„ =

E

i=i

4/an =

n [1 v, " — 2, an n i =

-

with suitably chosen constants an tends to the stable attracting distribution G. This means that n (1 " - - E [Li - R)/R < €} n->00 G«(E) n ,=, Probt or equivalently Prob{

—E— 1

n

n

R < ERan/n}

n-> 00

- G«(- €) = 13(0 (6.176)

Equation (6.176) amounts to saying that with a probability 13(e), the estimated total score

differs from its expectation R by no more than e(Ran/n) for a sufficiently large number of histories. The convergence rate is of the order of a„/n.

384

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

This statement is meaningful only if lim an/n = 0. Now if the random variable in question satisfies the conditions of Theorem 6.5, then from Equation (6.174) lim

L(a ) a " = lim , n

with 0 < a 1. Thus, it remains to show that for a slowly varying function L(x), the limiting relation lira L(x)/x" = 0 holds for a > 0. Assume that the opposite statement holds, i.e., suppose that for some nonzero A lien L(x)/x" = A 0 xym

Then obviously lim L(tx)/(tx)" = A for any t > 0, and therefore L(tx)/(txr lim —1 , L(x)/x" On the other hand, since L(x) is slowly varying L(tx) (tx)" (tx)" lim = tim — = , L(x) x"

1

which contradiction is resolved if and only if A = 0, i.e, if an/n tends to zero with increasing n. It is remarkable that the confidence limits are expressed in terms of the expected value, R, in contrast to the nonsingular case where the standard deviation appears in the confidence limits. It remains to determine the values of an. Let us consider the general case when the singularity of the final score in a history has the form µ(r) where r = Ir

R/[(k + 1)rk]

nkl, the minimum distance of the collision points in the trajectory from the

detector point. [We leave the exponent k undefined instead of setting k = 2 because of the estimators with a singularity 1/r (k = 1) to be investigated in the next Section.] For the sake of simplicity, we assume that the probability density function of the minimum distance r is = (k + 1)rk,

0

r

1

385 In the case of the next-event estimator, this assumption is roughly equivalent to the assumption that the collision points are uniformly distributed in a sphere around r*. Obviously =

0

ldrp.,(r)pr(r) = R

and (p2) =

0

drii2(r)p,.(r) = +co

where, again, brackets denote expectation. Under the simplifying assumption on the distribution of r and the score ix, the probability density function of the score reads dr

=

1 = - [R/(k + 1)]"k k

(2+ 1/k),

R/(k + 1)

E1r,wo).1dwC2(coo--631r,E,E0)I(r,u),E)

(6.188)

C L(E0—>E1r,coo) = idcoC(coo,E0—>w,Elr)

(6.189)

C2(coo-->wir,E,E0) = C(wo,Eo—>o),Eir)/C,(Eo-->E1r,coo)

(6.190)

where again

and

392

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

FIGURE 6.2. Geometry in scoring with once more collided flux estimator.

Let us assume that the direction w is expressed in terms of the angles relative to the line connecting r and r*, i.e., with the notations of Figure 6.2. w = w(0,x) Then dw = sin0d0dx and b = Ir — r*Isin0 = r • sine

dw = bdOdx/r If we rewrite Equation (6.188) with the angular variables 0 and x, the expected score reads 1(Po) = 1 — r dX dO

f

27r o 7r

=

1 r

27r2 C2(630-->w(0,x)lr,E,Eo)b1(r,w(0,x),E)1} 2.7r

idECI(Eo--Eir,w0) f f dx docAto„--063(0,x*,E,E0) 0 0

x [fdDT(Dir,w,E)f(r,E,r + Dw,E)]

(6.191)

393 Now, since bI(P) = g(P) is bounded, the expected score I(P0) has a 1/r singularity and thus a finite variance. Therefore, i(Po) itself may serve as an estimator of the flux at r*. In most practical cases, the integrals in Equation (6.191) cannot be evaluated analytically, and therefore 1(P0) is estimated by the one-sample method and with the aid of an estimator f(P,P') with 1/b singularity in the following manner: 1. The number of secondaries and the postcollision energy E of every secondary in a collision at Po is selected from the marginal scattering kernel Ci(E0-->Elr,coo) Then for each secondary, the next steps are to be repeated. 2. The angles x and 0 are selected from a uniform distribution over [0,27r] and (0,711, respectively, and the direction w(0,X) is determined. Let P = (r,w,E). 3. A possible next collision point P' = (r + D w,E) is selected from the transition kernel T(P,P'). 4. The contribution si from an estimator f(P,P') of singularity 1/b is determined according to steps 1 through 4 in the previous section. 5. The quantity so = 27r2 C2(coo—>co(0,x)lr,E,E0bs,/r 21T 2(32

iii)C,( 00—>6.)(0 ,X)Ir,E,E.)k(D,D,Ir,E)/r

(6.192)

is scored. Certain possible practical modifications of the procedure above are obvious. Thus, if the angles x and 0 are sampled from some density function h(0,x) instead of the uniform density in step 2, then the score becomes so = [C2(000-->co(0,x)lr,E,E0)/h(0,x)] bs,ir

(6.193)

Furthermore, if one prefers scoring before the number of secondaries in the collision is determined, then step 1 is modified thusly: l' a possible postcollision energy E due to a collision at Po = (r,coo,E0) is selected from the marginal density C,(Eo-->E1r,coo)/c(P) Accordingly, the score in step 5 becomes so = 21T2 C2(wo-->o)(0,x)lr,E,E0)c(Po)bs,ir

(6.194)

The estimator introduced by the above procedure is called the once-more collided flux-ata-point estimator since it scores from events which are two collisions ahead. Note that when using the once-more collided estimator, source particles also contribute to the collided part of the flux. In the case of source particles, the first collision in step 1 is replaced by the selection of the initial coordinates. Thus, if Q(r,63,E) is the source density, then the expected score corresponding to Equation (6.191) reads 1 " dx d0 IQ = — dridEQ,(r,E) — — In2Q2(0)(0,x)Ir ,E)lbI(r,(0(0,X),E)l 0 27r Jo

394

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

where Q,(r,E) = IdwQ(r,w,E) and Q2(wlr,E) = Q(r,w,E)/Q,(r,E) In this case, step 1 above is replaced by the step 1" The site r and energy E of a starter is selected from Q,(r,E). Furthermore, C2 in Equations (6.192) through (6.194) is to be replaced by Q2 above. All the considerations of this Chapter concern analog games and no attempt was made to introduce nonanalog kernels appropriate for eliminating the singularities. In many of the point-flux estimation methods, the 1/r2 singularity of the next-event estimator is transferred to the transition kernel, thus reaching a bounded variance.39•76." These methods have the common disadvantage that special measures are to be taken to avoid unwanted fluctuations of the statistical weights. We will not discuss such schemes; the interested reader is referred to the literature quoted. To conclude this section, we note that two problems remain unresolved. First, the direct (uncollided) distribution of the source particles to the flux at a point [cf. Equation (6.170)] still has an unbounded variance if the estimation point (detector) is embedded in the source region. This problem cannot be avoided by tricks similar to those yielding bounded-variance estimators of the collided part and seems to be persistent in any (nonadjoint) scheme. Second, although (one-sample) averaging of the score over future events eliminates the singularity of the variance, higher moments of the score will be singular, which brings up the same difficulties in estimating the variance as arose in estimating the mean with the next-event estimator. Higher moments can be made bounded by averaging the score over further events; however, every averaging step involves duplication of the simulation procedure in the sense that determination of a contribution is performed through Monte Carlo procedures similar to those played for the continuation of the history. In practical realizations of the once-more collided flux estimator, such duplication can be avoided in the majority of events, as will be seen in the next section. E. PRACTICAL MODIFICATIONS OF THE BASIC METHODS The motivation for all the effort invested in the derivation of new point estimators originates from the singular behavior of the next-event estimator near the detector point. Far from the detector, the next-event estimator behaves regularly and there is no need to apply more sophisticated estimators. For example, if the detector is situated in a vacuum surrounding, i.e., if no collision occurs in the vicinity of the detector point, the next-event estimator can be safely applied. The same is also true for collisions far from the detector. On the other hand, for distant collision points, the contribution of the next-event estimator is small because of the rapidly decreasing exponential and 142 functions in it. Consequently, for such collisions, a considerable amount of computing time is spent with essentially negligible influence on the final result. Iida and Seki36 propose a very simple method for economizing computational time. The idea is that a quantity f(P',P") = f,,,E(Pr ,P")/p(e)

395 is scored at every collision with a probability p(r') and zero with the complementary probability. Obviously, the expected score with this estimator in a collision from P' to P" is p(e)i.(P',P") + [1 — p(0] • 0 = f,,E(P',P") The probability p(r') is chosen as 1 p(r') —

a2/Ir'

if Ir' — r*I < a r*I2

if Ir' — r*I % a

where a is some given distance value. Obviously, nonzero contributions from points far from the detector are rare. The point estimators discussed in this Chapter are expressed as integrals with respect to one or more phase-space coordinates, and contributions by the estimators are determined through one-sample Monte Carlo evaluation of the integrals. The statistical reliability of the individual estimates may be improved if the integrals are determined by several independent samples, as was discussed in Section C. A similar but more elaborate method of improving statistics is used in the MCNP code.7•43 In this method, pseudo particles are generated in every collision of an ordinary particle and are deterministically transported to a spherical neighborhood of the detector point. The pseudo particles contribute to the integrals in question; the weights of the particles are determined by the probabilities of the transitions from the actual collision points to the spherical neighborhood of the detector. These particles are later processed as any other ordinary particle. Heuristically, the generation of the pseudo particle can be considered as an inner Monte Carlo simulation of that part of the ordinary history which would reach the sphere around the detector. Consequently, the ordinary particle (which created the pseudo one) is processed further from the collision point independently of the pseudo particle, but is killed if it enters the sphere. This procedure is discussed in more detail in Reference 43 in connection with the DXTRAN routine. Although the method provides more detailed information about the neighborhood of the detector point than the simple one-sample Monte Carlo estimate, the introduction of an artificial branching process is very time-consuming. Fraley and Hoffman23 propose a simple approximate method for handling the singularity of the next-event estimator. In their approach, the contributions of collisions in the neighborhood of the detector point are substituted by some averaged value. The idea is that a constant value b is defined for which the reaction rates

R = f dPtli(P)fa(P) and

R = f dPtil(P)fEs(P)Ir — r*I2/b2 are equal. The spatial integrals in the reaction rates above are extended over a sphere of radius a. The constant b in general depends on the collision density ili(P), but it can be shown that in monoenergetic isotropic cases for small radii a, it is fairly independent of the

396

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

flux shape. For energy-dependent problems, it is proposed to tabulate the values of b as a function of the energy and to score fEJP/Ir — r*12/b2(E) in every collision inside the sphere around r*. The perturbation source method introduced in Section 6.I.E is also applicable to the estimation of the flux at a point. In this procedure, the conventional next-event estimator is used for collisions outside a sphere around the detector point, and contributions determined by the adjoint flux are obtained when crossing the sphere and also if collision occurs inside the sphere. All these contributions are finite, thus avoiding the singularity of the next-event estimator. The disadvantage of the method is that knowledge of the adjoint flux in the neighborhood of the detector point is required. A homogenized one-dimensional calculation procedure of the adjoint flux is proposed in the original communication.23 In all the methods so far mentioned, different estimation procedures are applied outside and inside a region (usually a sphere) around the detector point. Such separate scoring may also yield some improvement in the determination of the confidence limits even if the singular next-event estimator is also used inside the sphere. The improvement is due to the fact that the part of the score that originates from outside the sphere has no singularity and therefore tends toward its expectation at a rate of 1/V/T1. Let Rout and Rm denote the estimates of the flux at the detector point as obtained by the next-event estimator from collisions outside and inside a given sphere around r, in n histories, respectively. Furthermore, let V2 be the estimated variance of the score from outside the sphere and let Rout and Rm be the expectations of the corresponding estimates. Since Rom is regular, the distribution of Rom — Ron, is approximately normal, and with a confidence level

R = (RE.) — 4)(—Eout) the relation

gout — Rout'

EoulVV2/n

€ o„,/V2/n

holds. [41)(x) is the normal distribution function.] On the other hand, in view of the results in Section B, with the same confidence level

I where

Ein

k, —Rin

Rin

Ein

n"3 n"3

k, n113

is such that 13 G312(Ein) G312(

Ein)

and G3,2(x) is the stable attracting distribution corresponding to the random variable 1i,„ according to Section B. The values of G3,2(x) are tabulated in Reference 15. The total score is

397 and its expectation is R = Rio + Rom Hence, with a probability 13, the difference of the score and its expectation is limited as

IR — kl

Eout

VV2 /n + €01

Eout

Vn

+ Ein 11'n —

n'"

Note that in the considerations above, it was implicitly assumed that the collision density inside the sphere is approximately uniform. [This assumption was exploited in constructing the probability density function pr(r) in Section B).1 Uniformity can be checked, for example, by scoring the average collision rate in the sphere and comparing it to the estimated collision density at the detector point. More rigorous checking is obvious, but needs more reliable, detailed information on the collision density in the sphere. Use of the once-more collided flux estimator can also be restricted to certain regions of the domain of simulation. Let us again define a sphere around the detector point r*. Let P. = (r,w,„E„) be the coordinates of a particle entering a collision at r. If r is inside the sphere, then the once-more collided flux estimator is applied with no condition. If r is outside the sphere, then the once-more collided flux estimator is combined with the simple next-event estimator as follows. Let Om be the bevel angle of the cone determined by the sphere and the scattering point r. Let us divide the integral with respect to 0 in Equation (6.161) into two parts, one from 0 to Om and the other from 0„, to Tr. Rewriting the integration variable co in place of 0 and x in the second integral, the expected score by the once-more collided flux estimator in Equation (6.191) can be rewritten as 2-rt lem d0C2(wo--w(0,x)Ir,E,E0)bI(r,w(0,x),E)} idEC,(E0—>E1r,too){1 dx

I(P0) =

+ dE dwC(630,E0—>o),Elr) dDT(Dir,w,E)f(r,E;r + DO),E) J osem

(6.195)

Let us change the variables x and 0 to Xi and 0, and denote 0)(010(1) = 01 Finally, let us insert the expression dcoC2(63.-->oalr,E,E0)/[C2(6).-->colr,E,E.)

J0em 0 Note that f(P,P') in the second term on the RHS of Equation (6.195) is replaced by fEs(P') in Equation (6.199). This can be done since both estimators are partially unbiased. Equations (6.197) through (6.199) suggest the following estimation procedure (cf. Figure 6.3): 1. The number of secondaries and their postcollision energies and directions, E, and w, are selected from the original collision kernel C(wo,E0—>w,Elr) For each secondary, the angle 0 between w and r* — r is determined and the steps below are executed. 2. A free-flight length D is selected from T(131 rm,E). 3. If 0 < 0„„ then the once-more collided flux estimator is applied in the following modified form: x, and 0, are selected uniformly in [0,711 and [0,0„,], respectively, in step 2 of the previous section, and the quantity scored in step 5 there becomes so

C2(coo—>o)(0,,x,)1r,E,E0) 1 — cosOm C2(0),, —*(0(0,X)Ir,E,E.)

bs,/r

(6.200)

where s, is the score due to next event estimation as described in step 4 in Section C. The simulation is then continued by displacing the particle from P = (r,w,E) to P' = (r + Dw,w,E); the next collision point of the particle is P'.

399 4. If 0 > Om, then the particle is immediately displaced to P' = (r + Dw,w,E) and the score is determined by the next-event estimator at P'. 5. The simulation is continued by playing the collision of the particle at P'. Obviously, in the modified procedure, the once-more collided flux estimator is applied with low probability if the collision takes place far from the detector. The gain in computing time is due to the fact that the once-more collided flux estimator is applied only if 0 < Om. Notice that the estimation procedure under the condition 0 < Om would be more complicated than the one formulated in Equation (6.200) without the trick of inserting the identity (6.196) into the integrand. This trick is analogous to the reselection procedure proposed by Steinberg and Kalos .76 It should be emphasized that the above procedure gives the once-more collided score in a collision at P.. This means that the procedure is to be repeated at the next collision point P' even if the next-event estimator has been used in the current scoring, i.e., in spite of the fact that a score has already been detected from P'.

V. SPECIFIC PROBLEMS IN STATISTICAL EVALUATION We have seen in Chapter 5 that a given quantity (e.g., reaction rate) may be estimated by several different estimators and methods. The majority of the production-type Monte Carlo programs offer a great number of estimators and methods for parallel determination of the quantity of interest. As a result of such calculations, the user obtains a set of unbiased estimates which may be either correlated or independent. By repeating the Monte Carlo calculations several times, a number of independent sample sets are obtained, and the separate estimates are combined in such a way that the combined estimate is also unbiased and has the smallest possible variance. (This is the case, for example, when the very same reaction rate is estimated by several different estimators concurrently.) Usually, no a priori information on the statistics of the estimates is known, and therefore the combination coefficients of the separate estimates are also determined from estimated data. If the number of independent results (sample sets) is large enough and their distribution is not very far from Gaussian (which is assumed in most practical cases), the combined mean and variance can be determined according to the classical weighted-average formulas as they follow from maximum likelihood principles:3'33'54 The respective results are outlined in Section A. On the other hand, if the number of sample sets is small, the classical weighted average does not give an unbiased estimate of the combined variance. A corrected variance estimate of the optimum combined mean33.49 is derived in Section B, on the assumption that the separate estimated samples are normally distributed. In simulating rare events (such as transmission of particles through thick layers), the normality assumption certainly fails to hold, and only a small fragment of the histories accounts for the event investigated. An unbiased combination of rare sample sets and estimation of the variance of the combined mean49 are discussed in Sections C and D, respectively. Estimation of quantities defined as ratios of two reaction rates poses specific problems since, in practice, the ratio of two expectations is estimated by the ratio of two sample averages and the expectation of the latter quantity does not necessarily coincide with the ratio of the expectations. Such is the case with the estimation of the effective multiplication factor (Chapter 6.111). These questions are discussed in Section E. Finally, efficient estimation of the theoretical variance of a random variable is considered in Section F. A. OPTIMUM COMBINATION OF SAMPLE MEANS In the derivations below, we assume that the result of a history is a k-tuple (x(", x(2), . . . ,x(k)), which will be called a data set. The quantities x(') are realizations of k random

400

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

variables, all having the same expectation m. The random variables may be either correlated (which is the case with parallel estimation by several estimators) or independent (e.g., when calculating a particular quantity in different schemes). If the realizations in a set are correlated, the set is called a correlated set; in the opposite case, the set is an independent set. (Note that since the samples are assumed to be Gaussian, uncorrelated sets are also independent.) The k-tuple of the sample is denoted by the sample vector x = (x"), x( 2 ),

x(k )T

(6.201)

(x is a column vector, -r denotes transpose). The independent simulations result in a number of sample vectors x„ x2, . . . Note that independent sets do not necessarily contain the same number of data elements, (e.g., because one or more data elements in the k-tuple were not calculated in certain sets). Now let us assume that x in Equation (6.201) is normally distributed, let m be the common expectation of the components, and let S denote the covariance matrix of the components. This property will be denoted as 5""( : N(me,S) where e = (1,1,...1) 1 2

k

The density function of x then has the form f(x) = K expl — (x — me)T 9(x — me)/2) where Q K is a normalization factor and aTb =

E a(i)b(" i=1

denotes the scalar product of the vectors a and b. Let us first consider correlated sets (in this case, the number of estimates in a set is always k) and assume that we are given n independent realizations x,, x2, . . . , xn of the random variable x. (For example, a realization may be the result of a history.) We seek a set of vectors c1, c2, . . . , cn such that the combined estimate

_

m

-n ,E-1 c,T x,

is unbiased with respect to the common mean and has the lowest possible variance. One immediately recognizes that the separate realizations are interchangeable (i.e., none of them

401 plays a distinguished role), and therefore ci cannot depend on j. Let us denote c, = c and let x be the empirical (or sample) mean of the realizations, i.e., n

= n- ;_E1 Xi

(6.202)

Then the problem reduces to the determination of a vector c such that = cTir

(6.203)

is unbiased with respect to m and its variance is minimum. If brackets denote expectation, we have ()0 = (50 = me = (m,m,...m)T I2

and therefore, from Equation (6.203), it is seen that (rri) = m, i.e., the combined estimation is unbiased if cTe = 1

(6.204)

The variance of the combined estimate reads* D2[m] = (/T12) - m2

= (CTR RTC) — m2

= cT((i" - em)(Ti - em)T)c = 1 - cTSc = V2 n

(6.205)

Now, under the condition in Equation (6.204), MITI] = V2 is minimum if

ac,

[v2 - kcTe] = 0

i = 1, 2, ..., k

where k is the Lagrange multiplier of the problem. The solution of this equation, also satisfying Equation (6.204), is c = S-le/eTS-'e = Qe/eTQe

(6.206)

In view of Equation (6.205), the vector c can also be written in the form c = nV2Qe

*

(6.207)

Here we make use of the well-known and easily verified fact that the covariance of the empirical mean is 1/n times the theoretical covariance of the random vector (n is the number of samples).

402

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

Before discussing this result, we show that the maximum likelihood estimate of the common mean is identical to the optimum combination* in Equations (6.203) and (6.206). The logarithm of the likelihood function corresponding to n sample sets is logL =

E logf(xj)

J=1

where f(x) is the normal density function of x given above. Differentiation of logL with respect to the expectation m yields -a 0 =

am

a

x-, n

logL = — — 2, (x, — me)TQ(x, — me)/2 = n[eTQR — meTQe] am

in = eTQR/eTQe

(6.208)

in accordance with the expression in Equations (6.203) and (6.206). The main difficulty with the combined mean in Equation (6.208) is that the covariance matirx of the data is very seldom known exactly and therefore the coefficient vector c in Equation (6.206) cannot be determined. In practice, one can only use empirical values expressed in terms of the realizations. It is reasonable to replace the covariance matrix S in Equation (6.206) with its empirical value S

= E (xi - )0(x; -

R)T/(n — 1)

(6.209)

=l

Thus, denoting Q = s- i we can define a combination coefficient vector as =

Qe/e0e

(6.210)

and accept (6.211) as a quasi-optimum combined estimate of the mean. Two problems arise immediately. The first follows from the fact that é is no longer independent of the sample values, and therefore it is not certain that the combined mean is unbiased. Second, if the mean is unbiased, then how can we construct an unbiased estimate of its variance? The first problem is reassuringly settled by the following classical theorem.'

*

This is, in fact, a special case of the more general theorem which states that if there exists a minimum variance estimate of a parameter, then the maximum likelihood estimate is that estimate.

403 Theorem 6.7 — If z and s denote the respective sample mean and covariance of the realizations as defined in Equations (6.202) and (6.209), respectively, then i is normally distributed with a mean me and covariance S/n, i.e., : N(me,S/n) and Ti is independent of S. Furthermore, the empirical covariance can be decomposed in the form n—1

(n — 1)S =

E

J=1

yjyr = A

where yi:N(0,S ) and yi are independent of yi (i # j). Now, since e in Equation (6.210) is expressed in terms of Q = S `, the theorem states that e is independent of 31, i.e., (111) = (c x) = (eT)(x) = m(eTe) = m as eT e = 1. Accordingly, th in Equation (6.211) is an unbiased estimate of the common mean m. As for the theoretical variance of rri, by analogy to Equation (6.205), one shows that = (cTSc)/n

(6.212)

The easiest way to approximate this variance is to accept the quantity V2 = eTSe/n = [neTQe] -

(6.213)

as the empirical variance. With this notation, e = n 1:[20e by analogy to Equation (6.207). It will be seen in the next section that for a large number of sample sets, i.e., if n > 1, Equation (6.213) is indeed unbiased with respect to D2[fh] and therefore it can be used as an estimate of the combined variance. On the other hand, it will also be seen that if n is not large enough, the estimate must be corrected. So far, we have examined correlated sample sets, each consisting of an equal number of estimates. The considerations can be easily generalized to the case when the individual samples in a set are independent and different sets may consist of a different number of samples. Let, as before, a set consist of, at most, k samples x = (x(",x(2) . . . ,x(k)) and, given a number of realizations of x, assume that in some of them one or more samples x(') are missing. Further assume that the component x(') is distributed as x(i' : N(m,s1),

i = 1, 2, ..., k

404

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

Let xi( ') be the j-th realization of x(') and let n, be the number of such realizations. By analogy to the notations used with correlated sets, we introduce the following quantities: n,/s,

(6.214)

c(i) = V2n,/s,

(6.215)

V2 = [

i=1

and

where s, is the theoretical variance of x(`). The component TO) of the empirical mean vector x reads _ x(. 0 =1 n1;= 1

(6.216)

while the empirical covariance matrix is diagonal with elements n.

=

2 (x10 -

J=1

Tc(l))2/(n, — 1)

(6.217)

Repeating the derivation in the first part of this section, we find that the optimum unbiased estimate of the common mean is again rTi = where c is the vector composed of the elements c(') in Equation (6.215). The theoretical variance of in is just V2 in Equation (6.214). Again, if c(') is estimated by e(i) = V2n,/i,

(6.218)

with k

1 :T2 =

[E n,/s,

then with the notation e = (ow, e2), . . . ,e(k)) e n = eT -3 1

(6.219)

is an unbiased estimate of the mean. Furthermore, if n, > 1 (i = 1, 2, . . . ,k), then V2 is a reliable measure of the variance of eh. In the opposite case, a correction factor, to be derived in the next section, is applied. It should be emphasized that a quasi-optimum averaging of the n = /;'=, n, independent samples is possible only because we have the a priori information that the realizations x)(') of an x(`) originate from a common normal density N(m,s,) (i = 1, 2, . . . ,k). This knowledge makes it possible to order the realizations into groups of various statistics (s,), the ordering in turn, results in a reduction of the (theoretical) variance of the combined mean. Without

405 this a priori information, the best thing one can do is to take the arithmetic mean of all the realizations as 1

k

E

E

=_ n

n X(i)

In this case, the variance of x is estimated as

E

— 502/[n(n — 1)]

Since the optimum combination coefficients were derived by minimizing the variance of the combined mean, it is evident that D2[R] D2lini. Nevertheless, it may be instructive to see the proof of this particular inequality in detail. The expectations of the empirical variances (and thus the variance of R) reads k

() =

D2[X]

tn

[E E ((x;') — 1=1 j=

E E si — —

i=1;=1

-

m)2) — n((Tc — m)2)]/[n(n — 1)]

E

—

n i=1;=,

k

k

k

E nisi = E nisi/(E ni

m)2)]/[n(n — 1)]

)2

It will be seen that this variance is not greater than that of the optimum combined mean as given in Equation (6.214), i.e., that k

DITTRI = [

i=i

n1Is.l ]

k

E nisi/(E

i=1

Let us denote a, = (ai/Si)/E (a,/Si) i= I

and bi = ni/E ni i=1 Then the inequality to be proven reads

E bi2 /ai =

1

Taking into account that

Ea = E

i=i

i=1

bi = 1

2

n) = D2[z]

406

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

we have b. 2 E b; /a; = E () 1 a; i = I a, i=1 k

1' i = I a,

=1

as stated. Note that the equality holds only if b;/a; is independent of i, i.e., if s, = s for every realization. This means that the variance of the optimum combined mean is lower than that of the arithmetic mean unless all the samples originate from the same distribution. B. UNBIASED ESTIMATION OF COMBINED VARIANCE FROM SMALL SAMPLE SETS In this section, we investigate the bias introduced when the empirical value V2 defined in the previous section is used as an estimate of the variance of m, the quasi-optimum combined mean of separate sets. In the first part of this section, correlated sets are considered; extension to independent sets is given in the second part. First, we express V2 in terms of the theoretical values V2, S, and c. Next, the true variance D4r11] is given in the same terms, and comparison of the two expressions will give an estimate of the bias. Correcting V2 with the aid of an estimate of the bias yields an estimate of the combined variance unbiased up to the order (k/n)2. Let us write the empirical combination coefficients and covariance matrix of the correlated sets as sums of the exact values and some unknown corrections, i.e., let =c+8 and S=S+ Then it follows from Equation (6.213) that the empirical combined variance reads 072 T C Se

= CTSC

2(cTS6) + (CAC) +

2(cTA8) + (STS8) + (8TA8)

Taking into account that the empirical covariance matrix S is an unbiased estimate of the theoretical one, i.e., (A) = 0 we have for the expectation of V2 the following expression (nV2) = nV2 + 2(cTS(8)) + 2.(cTA8) + OTS8> + (8TA8) where V2 is the theoretical optimum variance given in Equation (6.205). Similarly, expanding the theoretical variance of in in Equation (6.212), we have nD2[61] = nV2 + 2(cTS(8)) + OTS8>

407 Thus, the bias introduced into the estimate of the variance of nil is D2[ria] = [2cTA8) + (8TA8)]

(v2)

(6.220)

Thus far, this result is exact, but useless since the expectations on the RHS are not known. The will be approximated by empirical values. Since Q = S ' and Q is symmetric

Q = Q — 949 + 0[9(o9)21 Inserting this expression into Equation (6.210), we have [e FQ(!

49)/eT9se][1 + eTQA9e/eT9e1 + O[cT(AQ)2]

or, after rearrangement — c = 8 = nVIc(eT949e) — 1)4()e] + O[c(49)2]

(6.221)

This expression relates the random vector 8 to the random matrix A = S. The bias in Equation (6.220) can then be expressed in terms of A. We have, from Equation (6.221): cTA8 = nV2[cT(49eeTQ4)c — cT(49)2e1 + OIcT(49)3e1

(6.222)

where we have made use of the identity cTAc(eT949e) = (cT49e)(cT49e) = cT(4QeeTQ4)c that follows from Equation (6.207). From now on, we neglect the third and higher-order terms in (AQ). It is shown in Appendix 6B that by doing so, the largest neglected term is of the order (k/n)2 (k is the number of samples in a set and n is the number of sets). This approximation also means that the second term on the RHS of Equation (6.220) is negligible, i.e., (V2) —

= 2(cTA8)/n

This equation can be further simplified. Introduce the notation E = eeT and realize that AQ

(S — S)Q =

1 A9 — I

n — 1 —

(6.223)

408

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

where A is defined in Theorem 6.7. Inserting this relation into Equations (6.222) and (6.223), we obtain — D2[6.1] = 2V2[cT(AQEQA)c — cT(AQA)9e1/(n — 1)2 since cT e = 1. Finally, since according to Equation (6.206) 9e = (eT9e)c the expected bias can be written in the form (V 2) _= [ 2V2[cT(ARA)c]/(n — 1)2

(6.224)

with R=

QEQ -

Q(eTQe) = QeeTQ — Q(eTQe)

(6.225)

As is shown in Appendix 6C, (ARA)/(n — 1)2 = SRS + [SRTS + SS,(RS)]/(n — 1) for an arbitrary, deterministic k by k matrix R, where Sp(RS)

=

E (RS)ii

the trace of the matrix (RS). Applying this result to the matrix in Equation (6.225), we have (ARA)/(n — 1)2 = [E — S(eT9e)]n

— 1

SS,[QE — (eT9e)]/(n — 1) (6.226)

When evaluating this result, a number of identities will be used that follow from Equations (6.206) and (6.207). It is easily seen that = nV2e;

cTe = eTc = 1

cTSc = nV2;

clEc = 1

CTS

and

Furthermore Sp(9E) = eTQe = 1/(nV2)

409 In view of Equation (6.226) and the above identities, the bias in Equation (6.224) reads (V2) — D2[ 2n] = 2V2(cTSc)Sp[QE — (eTQe)1/(n — 1) = — 2V2(k — 1)/(n — 1) Thus, the following theorem is established. Theorem 6.8 — An estimate of the variance of the combined mean tit, unbiased up to the order V2(1c/n)2, is = V2[1 + 2(k — 1)/(n — 1)]

(6.227)

where V2 is the empirical maximum likelihood estimate of the sample variance and is given in Equation (6.213).

Accordingly, if k/n < 1, then the usual way of estimating the variance of the sample mean, as given in Equation (6.213), is suitable. On the other hand, if the number of sample sets is low, the correction in Equation (6.227) makes the estimate unbiased (provided k2/n2 is negligible). The considerations above can be easily extended to independent sets. Then Equations (6.218) and (6.219) define the vector c and the variance V2, while the other quantities playing a role in the derivation become S = OA. J S = {bugi};

Q = {8ii/sJ Q=

and A=

— si)}

Repeating the train of thought of the first part of this section, it is seen that the bias in the variance estimation is 2 — D2 [61] = 2V2 (

C(i)461i/ Si )

E c("A/s:)

where A; =g —s; the i-th diagonal element of A, and cW is given in Equation (6.215). The &'s are independent and their distribution is a transformed chi-square distribution of the form — ni + 1)/(n1 — 1)

410

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

Therefore (A) = 0 = 2q/(n, — 1) and the bias is (1:T2)

D2[61] =

2V2 (E c("(1 — c(i)),k/s,) i= I 0)(1 — c°0)/(n, — 1)

= 4V2

Thus, the following theorem is proven. Theorem 6.9 — For independent sets, an estimate of the variance of the combined mean, m, that is unbiased up to the order (k/min n,)2. is + =V2 [1

4

E &0(1 — c°')/(n; — 1)]

(6.228)

Cr72 are given in Equations (6.218) and (6.219), respectively. where 0) and ' As a byproduct of this result, we have an example of how a priori information on the estimated quantity reduces its variance. Let us consider a series of independent sets where every component is realized n times, i.e., n, = n (i = 1, 2,...,k). If we know a priori that the sets contain independent samples, then the variance of the estimated common mean is

=

+ 2a„/(n — 1)]

with = 2[1 —

E wo)2]

If the very same samples are considered to be correlated (i.e., if we do not know of their independence), than the variance of the mean follows from Equation (6.227) as gc = V2[1 + 2ac/(n — 1)] with a=k—1 Clearly au i.e.

c('))2] = 2(k — 1)/k 2[l — ,1 K =1

(k— 1) = ac

411 This means that if we use the knowledge that the sample covariance matrix is diagonal, the estimated variance of the sample mean will be smaller than if we take into account the (usually nonzero) off-diagonal elements of the empirical covariance matrix.S. C. ESTIMATION OF A COMMON MEAN FROM RARE SETS Consider a simulation procedure in which only rare but important events contribute to the score. Then the final score in a history is zero with a high probability and is some nonzero value with a low probability. It is felt that mechanical use of the large-sample statistical formulas in Section A will very likely yield unreliable estimates of both the mean and the variance. This is so because the distribution of the scores is far from Gaussian for the rare occurrence of significant events. Let us first consider the case when correlated rare realizations are simulated, i.e., the data sets contain correlated random variables ("correlated rare sets"). Let z = (z(1), z(2),...,z(k)) be a random vector with an expectation me. Assume that z = 0 with a probability 1 — p and it takes on a nonzero value x with the complementary probability. The vector x represents the rare contributions and will be assumed to be Gaussian, i.e., x:N(me/R,Sr) (Index r in S, and further on refers to the fact that the quantity concerns the rare contributions only.) In general, neither m, p, nor Sr are known. Let z„ z,,..., zr, be independent realizations of z such that z, = x,

for i = 1, 2, ..., v

z, = 0

for i = v + 1,

and n

The likelihood function of the realizations is L=

v

)13'(1 — P)" 'K'expt —

i=1

(xi — me/P)T(Mxi — me/P)/2}

(6.229)

where again Qr = The empirical statistics of the rare events are (6.230) for the mean and

= E (xi — rwx; — R)T/(v — 1) J=1

(6.231)

412

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

for the variance. The maximum likelihood estimate of the common mean m follows from Equation (6.229) by differentiation as Fri = (e rc4i)/(eT9,.e)

(6.232)

P = v/n

(6.233)

while that of the probability p is

Let us denote by = m/P the expectation of the rare contributions. Then it follows from Theorem 6.7 that = (eTQrI(eTQre)

(6.234)

with Qr

=

Sr

i

is a conditionally unbiased estimate of #1, with v given, since the x,'s are Gaussian. Therefore (6.235)

=

is unconditionally unbiased with respect to m (asp is unbiased with respect to p). Thus, if the rarity of the events is realized during the estimation procedure and the conditional mean 11 in Equation (6.234) and the probability f) in Equation (6.233) are separately estimated, then tit in Equation (6.235) is a reliable estimate of the common mean. On the other hand, if the sample statistics (empirical mean and variance) are generated with no regard to the possible rarity, the situation is drastically different, the resulting estimates may be catastrophically biased. This is demonstrated below. Let us consider an estimation process where the sample statistics are produced from the entire sample sets. The empirical mean in this case is

=

E

j=1

zi/n

and the empirical covariance matrix reads n

S =

E (z - i)(;- i)T/(n — 1)

j=1

Again let

=

413 Then the whole-sample and rare-sample statistics are related as i = vi/n =

(6.236)

and S = (v — 1)S,/(n — 1) +

qzx

(6.237)

where q = [v/(n — 1)1(1 — v/n) = P(1 — P) + O(v/n2)

(6.238)

In order to express Q (the inverse of S) in terms of Or (the inverse of Sr), let us realize that, according to Equation (6.237), S and Sr differ only in a diad (apart from a constant factor), and therefore the inverse of S reads Q= n 1 [Qr v—1

qQ,FcierQr/C 1 + qixT9,37)] n —1

(6.239)

Now, if rarity is not realized, the common mean is estimated by the maximum likelihood estimate in Equations (6.210) and (6.211), i.e., it reads m* _ (eTcp)/(eTQe)

(6.240)

The scalar products eTQe, eT0i, and iT0i can be expressed in terms of the rare statistics on the basis of Equation (6.239), and a little algebra yields the expression of the unbiased estimate lb in Equation (6.235) in terms of the whole-sample statistics as = 1:41,

mi*

(6.241)

where r _ fi

n(n v) v(n — 1)

(eT0i)Nel9e)]/ I

-

(6.242)

Obviously, the unbiased estimate di and the whole-sample estimate m* are different, and their ratio is the rarity factor r. It is shown' that the rarity factor can be expressed in terms of the rare statistics as r = 1 + (1 — p)(i — e(1)T(?,( —

+ 0(1/n)

Since Or is positive, definite r is not less than unity and we have the following theorem. Theorem 6.10 — An unbiased estimate of the common mean of correlated rare sets is in Equations (6.233) through (6.235) in terms of the rare statistics or in Equations (6.240) through (6.242) with the whole-sample statistics. The Gaussian maximum likelihood estimate m* in Equation (6.240) underestimates (in modulus) the expectation, i.e., m

414

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

and Iml

1(m*)1

It can also be seen49 that lim (r) = +00

p-'0

(6.243)

and lira p(r) = k/n

(6.244)

Accordingly lira m* = 0 i.e., for very rare events (p < 1), the Gaussian estimate m* predicts a practically zero value of the nonzero common mean m. This bias is often aggravated by human error in that occasional and extremely high contributions among a great number of low ones (zeros in our case) are intentionally discarded as "random fluctuations". Both Theorem 6.10 and the possibility of wrong subjective decisions suggest that whenever the occurrence of rare events is indicated, the obtained data should be checked thoroughly from the point of view of rarity before statistical evaluation. In the case of independent rare sets, the number of realizations as well as the number of rare events v, may be different for different random variables z'. Although in full generality the "rarity probability" p may also be different for different variables, generalization of the considerations to such a case is troublesome. On the other hand, the assumption of a common "rarity probability" p is fairly realistic since in most practical cases the very same phenomenon is simulated independently by different methods and the different estimates (realizations) are combined. Let zu), ,z(k) be mutually independent random variables representing the estimates from independent procedures. Let z(') = 0 with a probability 1 — p and let z(') = x(') with a probability p where x(i) : N(m/p, Let us given ni realizations zip'' (j = 1, 2,...,111) of e) such that =

for j = 1, 2, ..., vi

4) = 0

for j = v, + 1, ..., n;

and

415 According to the results of Section A an unbiased estimate of the expectation of the rare contributions µ=mlp is

=

E

ni

i=1

(6.245)

where (Viigi,r)/E (Virgi,r ) J= 1

(6.246)

and X(i)

_ _ 2

x(i)/v

E(xv — T(0))20, — 1) =, are the rare-sample statistics. It can be easily seen from the likelihood function of the realizations ic') that the unbiased maximum likelihood estimate of the probability p is

=

E vi/E n;

i=i

i=

Therefore, an unbiased estimate of the common mean in terms of the rare statistics reads = pµ = (E vi/ E n1) E emo)

(6.247)

Again, this is a safe estimate if rarity is accounted for when generating the statistics. If, however, the whole-sample statistics z(i)

2 z.,CD/n, = viK")/ni

and si = 2 (zv — yo)) 2 (n1 — 1) J=I

are used, then the combination coefficients can be expressed as 6(,') = (rini/K,)/E (rinj/Ki)

J=1

416

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

with n (n - v;) (2-0))2 ] -1 v;(n, - 1)

v,(v, - 1) r= ' ni(n, - 1) L1 and therefore the unbiased estimate di reads = (E vi/E 10[2

I);

(rinir§,) -i(')]/[E (rinIgi)]

instead of the maximum likelihood estimate corresponding to Equations (6.218) and (6.219). Comparison of the unbiased and maximum likelihood estimates leads to similar conclusions, as in the case of correlated rare sets, and will not be repeated here. D. ESTIMATION OF THE COMBINED VARIANCE OF RARE SETS First, we give an unbiased estimate of the variance of the estimated common mean in terms of the rare sample statistics. Next, we show that if rarity is not realized and the wholesample statistics are used for estimating the variance, then the estimate may be completely unreliable. The considerations are first presented for correlated rare sets Let us denote er eTOrieTOre

Then the unbiased estimate of the mean in Equations (6.234) and (6.235) reads = fleri Hence rn - m =

- me/p) + m(13 - p)/p

with

P

= v/n

The conditional variance of th with v (the number of rare samples) given is D2[thlvi = 132D2[1:Livl + m2(13 - p)2/p2 where D2[1:11v1 = [v(eTcte)i Taking the expectation with respect to v, i.e., summing up with the binomial distribution of v, the unconditional variance becomes

with] =

(v)p°(1 -

On - 1)2[Iiilv]

= p - D2[1.11v] + m2(1 - p)/(n,p) n

(6.248)

417 (Note that vD2[i1.1v] is independent of v; that is why it can appear in the formula after averaging over v.) Now, the results in Section B concerning nonrare sets apply to the variance of fl(since the rare samples themselves are assumed to be Gaussian), i.e., denoting = [v(eT9re)]-1 an unbiased estimate of the conditional variance D2111Iv follows from Equation (6.227) as = ,‘T,211 + 2(k — 1)/(v — 1)1 Since µ

m/p, the second term on the RHS of Equation (6.248) can be rewritten as p)p/n

m2(1 — P)/(110 = il2(1

(6.249)

We have seen in the previous section thatµ can be estimated according to Equations (6.234), i.e., 11, 2

(12 = 612/1j2

(6.250)

To complete the derivation, we need an unbiased estimate of p(1 — p). Simple algebra shows that (P(1 — P))/(n — 1) = P(1 — P)/n

(6.251)

i.e., np (1 — 13)/(n — 1) is an estimate of p (1 — p). Accordingly, the second term on the RHS of Equation (6.248) is estimated as (112(1 — 13)413(n

1)]

and we have the following theorem. Theorem 6.11 — An estimate unbiased up to the order (k/np)2 of the rare-sample variance of the common sample mean di in the case of correlated rare events is §r = m2(1 -

P)/[13(n — 1)] + f.)2" 7 ,211 + 2(k — 1)/(v — 1)1

(6.252)

Naturally, if p = 1 (i.e., if v = n), Equation (6.252) goes over to Equation (6.227). If, however, p is considerably less than unity, Equation (6.252) defines an estimate very different from that in Equation (6.227). This will be shown below. The (corrected) Gaussian maximum likelihood estimate of the variance of m* (i.e., of the whole-sample combined mean) in Equation (6.240) would read S* = [n(eTQe)]-111 + 2(k — 1)/(n — 1)1 Simple but lengthy algebra shows49 that

[e-ve] - = [elle] -

th2( — 13)/(130

where r is the rarity factor defined in Equation (6.242).

418

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

Thus, an estimate of the whole-sample variance becomes s*

[1112(1 — ii)/(f)nr) + ii2N/,2 ][1 + 2(k — 1)/(n — 1)]

(6.253)

Obviously, this estimate differs from the correct one in Equation (6.252) in the appearance of the rarity factor r in the first term. (The difference in the correction factors is negligible in this case.) Now, according to Equations (6.243) and (6.244) lim sr =

while lim s* ,-0

m2/k

i.e., for small p, the whole-sample estimate s* differs drastically from the unbiased raresample estimates and gives completely unreliable underestimations of the combined variance. In the case of independent rare sets, the procedure is similar to that above, except that the estimate of p. = m/p is the one given in Equations (6.245) and (6.246), i.e., E

=

i=

11 is again Gaussian and Theorem 6.9 applies, i.e., an unbiased estimate of the variance of 11 is

$ 4. =

1 +

4 E E;.0(l - i))/(vi - 1)] i=,

(6.254)

where er(') is given in Equation (6.246),

=

E V/g,„]

[1=1

I

and g,., is the rare-sample variance of the i-th random variable as defined in the previous section. Following the line of thought in the first part of the section, we have m = f)(11

1-L)

P)11

and the variance of m reads DIM] =

vkl)

Now

((f) - p)2) = p(1 -

1-L.2(15 — 02)

419 with n=

E

i=1

n,

On the other hand, sµ is an unbiased estimate of the conditional variance D2 flilvi,• • • ,vkl, and it can also be seen that P(1 — p)/n = (15(1 — j5)/(n — 1)) Therefore, the estimate = ria2(1 — ji)43.(n — 1)] + 132Sp.,

(6.255)

is unconditionally unbiased with respect to the variance of the estimated common mean fit in Equation (6.247). An estimate of the probability p is

p=

E vi/n = E vi/ En;

andsµ is defined in Equation (6.254). A discussion similar to that in the case of correlated rare sets can also be given here with identical conclusions. The main consequence of the considerations regarding rare sets is that whenever the danger of occurence of rare but important contributions is real, care must be taken to distinguish between important rare events and unimportant background, and the rare-sample statistics are to be applied in the combined estimation. E. ESTIMATION OF RATIO OF EXPECTATIONS A problem of frequent occurrence in physical reactor Monte Carlo applications is the unbiased estimation of the ratio of two reaction rate-type quantities. This is the case, for example, in the estimation of the effective multiplication factor (Chapter 6.111) and also when peaking factors or moderator-to-core flux ratios are calculated. The difficulty of such estimations is that the expectation of the ratio of two random variables (such as empirical means that estimate the respective reaction rates) does not equal, in general, the ratio of their expectations. Correction for this bias is presented below. Let the ratio to be determined be of the form r = m,/m,

(6.256)

where m, and m2 are expectations of the random variables z and y, respectively: 0 < m, = (TO,

0 < m2 = (s)

(6.257)

These variables are usually sample means, i.e., if x,, and y„ y2,...,y„ are independent realizations of the random variables x and y, respectively, then _ 1 v, " x — x, n

420

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

and _ 1 n Y yi n ,=1 In the derivations below, we shall assume that the variables are indeed simple arithmetic averages, although generalization to more general types of sample means (e.g., such as those introduced in Section A) is straightforward. Let (6.257)

= yh7

is a most reasonable but usually biased estimate of r in Equations (6.256) and (6.257). Now, let f(R,y) be the probability density function of the random variables k and y. If these variables are sample means from a sufficiently large number of realizations (i.e., if n » 1), then (except for unusually extreme distributions) according to the central limit theorem, the density function f(k,y) is dominantly concentrated on a small domain: A=

E [m I — a, ,m, + al];

y € [m2 — a2, m2+ adl; 0 < a,
0

Now, since higher moments of the empirical mean contain higher powers of 1/n (n is the number of independent samples), in most cases they are negligible beside the second moments, and we conclude that the ratio f = si/R is a biased estimate of m2/m,, the bias is given by the quantity in square brackets in Equation (6.263). This result provides us with the key to an unbiased estimate. Obviously, an estimate of the form

r

= (y/41 + (x mi )(Y — m2) + (x mi)21

mim2

(6.264)

would be unbiased up to the order of the relative third moments, but the expectations m, appearing in it are not known. An estimate expressed in terms of the sample statistics is given in the following theorem. Theorem 6.12 — An estimate of the ratio r = /(K) unbiased up to the order of the relative third moments of the sample means, is of the form

r = (y/z)[1 +

+ g„/K2]

(6.265)

with

SI2

( — Tc)(y, — 37)/[n(n — 1)]

(6.266)

422

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

and

= E (xi - K)2/[n(n — 1)]

g„

(6.267)

i =1

Proof. Taking the expectation off and repeating the derivation of Equation (6.260), we have =

([1 + Y— m2] [1

m1

M2

x

C

ir11

m1)21[1 + gi2/07Y+)

MI

g„/K21)

+03 M2 ml

- m1)2

M21(K - MI)

0

M2M1

M2 E

V12

(S12)

/IT .

(5 11

g12

11 X ) + 03 r(4

m 2m1

1111

±03

M2M1

It remains to show that the terms in square brackets reduce to unity. Let us calculate the expectation of the empirical covariance in Equation (6.266): 1 1 (g12) = n—1 ( n

E xiyi -- 2 E xi E

- n E x,yj) = —n[— n

., (xY )

=

n-1 nx

mim2] =

n

—

n

xiy;

Vie

where V12 = ((x, — rn,)(y, — m2)) the covariance of the i-th realizations. (Naturally, it is independent of i since the realizations are identically distributed.) On the other hand, the covariance of the sample means is 1v, V12 =

n

1 = n

2, xi 2, yi) (xiyi) +

1 v,

mim2 = (-2" 2, xiYi n

[n(n — 1)

i.e. (g,2) = V12 Similarly, it can be seen that

which completes the proof.

n

2., xi30

1 = - V 12 n

n2

(SI I)

1 v,

V11

mim2

423 It follows from Equations (6.265) through (6.267) that the correction to yIk decreases like 1/n, i.e., for a sufficiently large number of averaged realizations in )1 and y, the bias in the estimate I- becomes negligible. Having obtained an unbiased estimate of the ratio, it is natural to also seek an estimate of its variance. We shall first determine the theoretical variance of 1-, then give an estimate of it in terms of the sample statistics. Since the density function of )1 and y is assumed to be concentrated in a small region around the expectations, terms of the order x C

- mi \ kly )

(k 0)

M2

are small compared to the second-order quantities and therefore terms proportional to fourth and higher relative central moments will be neglected. Let us first notice that the variance of the corrected empirical estimate? in Equation (6.265) differs negligibly from, that of f in Equation (6.264). Indeed =

\2

—m —2y) ((r — — m2 + m, mi

)

= D2[1-] + 2((i- — 1-)(1- — m2)) +

—

r)2)

= D2[f] + 0, since both f — f and r — m2/m, are 02. Therefore, we consider the variance of r. For the sake of brevity, let us introduce the transformed variables =

— m )1m

and = (y — m2)/m2 Then the variance of I- is DV] = f jdRdyRff)(1 + uv + u2)]2f(R,y) - rri;Im; Expanding l/ into a series of u, we have D2 [r] = (m2/m1)2 k

E„

= V3 1.732

Now, if the number of samples, n, in the average x is large enough, then it follows from the central limit theorem that, irrespective of the actual distribution function of the samples, inequality (6.275) holds with a probability 0.916 if Ek = E„ and with a probability 0.997 if Ek = E„. This means that it is possible to approximate the true probability density function of the mean )1 by a power function of the form in Equation (6.272) such that it preserves the first two moments of k [by choosing a, according to Equation (6.274)] and also such that the probability of occurrence of z values which are not allowed for by the approximate probability density function is negligible. Note that the power k in the approximate density function (6.272) can be fixed, for example, by requiring that it reproduce the fourth moment of — m,). (The third central moment is zero, for the function is symmetric.) Obivously 9(k + 3)2 ((7 — m,)4) = a; k + 1 = D4[K] 5(k + 5) 5(k + 1)(k + 5) and estimates of the second and fourth central moments may determine k.

426

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

F. ON THE DETERMINATION OF THEORETICAL VARIANCES In the majority of estimation problems, expectations of random variables are to be estimated from realizations of the random variable, and the sample variance (variance of the estimated mean) calculated concurrently is only used as a measure of the reliability of the actual estimate. In certain applications, however, estimation of the theoretical variance of the random variable generated is also of interest. This is the case, for example, when different schemes are compared from the point of view of their efficiencies or variances. A special question of variance estimation is addressed in this section. Given N independent realizations x,, x2.....XN of a random variable x. Let m denote the expectation of x, i.e., (i = 1, 2, ..., N)

(x) = (xi) = m

Normally, the realizations are the results of independent histories, and all realizations have identical roles in estimating the quantity of interest. In many production codes, the histories are grouped into "batches" of equal size and the evaluation of the estimated quantity is performed using batch-averaged sample statistics. Batchwise evaluation is used in practice because, with sufficiently large batches, the results will be approximately normally distributed and the corresponding confidence limits are easily obtained. Thus, when estimating the expectation of the realizations, empirical variance is calculated in order to give a qualitative indication of the reliability of the estimate. The reliability of the estimated variance is of secondary importance in this case. The situation is different if the variance itself is the quantity to be estimated, and this question is investigated below. Let us assume that the N realizations are grouped into n batches and the sample mean is estimated in every batch. Then the average n

Mi

=

E x,;_„n±i n i=1

(6.276)

is calculated in the j-th batch, and the final estimate of the mean is 1 k in — k J= ,

E

tirsi

where k = N/n, the number of batches formed from N histories. The rh,'s are obviously independent and = (M) = m irrespective of k or n. The sample variance of th is estimated as

=

E (Fri; — tiri)2/[k(k — 1)],

J=1

(k > 1)

(6.277)

(cf. Section A). First, we show that the estimate in Equation (6.277) is unbiased for any grouping of the histories, i.e., its expectation is independent of the size of the batches. The expectation of the sample variance in Equation (6.277) reads =

(r2)

(

k — 1 (k

m;

=

1 { k — 1 k J =1

( ± ITO2)} k J=1

427 Now 2

1

k k

(E E (171,)(in,))

1

=1 m;)) = ( (/-CE

k2

(E (ffi2))

(6.278) 1 x-1 k

2, (in)' + (k — 1)m2/k

k2 ;_, and therefore

k (V2) =

2

7

(MD - 111

By analogy to the derivation of Equation (6.278), we have from Equation (6.276) that 2 j )_ 1

(x2) +

n — 1 m2 n

1

= D2[x] +

m2

(6.279)

and therefore (V2) =

1 [00 kn

1 Dz[x] m21 = _

irrespective of the number n of histories in a batch. We have thus demonstrated that V2 in Equation (6.277) is an unbiased estimate of 1/N times the theoretical variance of x. Therefore, denoting N s = k — 1 (WI — th)2, '

j = 1, 2, ..., k, (2

k

N)

si represents a sample in the estimation of D2[x]. The corresponding sample mean is.

E Si k ;=, k

S

=

(6.280)

and = D2[x] It is, however, not necessarily true any more that for a given number of histories, the reliability of the variance estimate in Equation (6.280) is independent of the number of batches (or, equivalently, of the number of histories in a batch). Therefore, if the variance of x is the quantity to be estimated, then the scheme resulting in the lowest variance of the estimated variance is to be applied. The most reliable estimate is defined in the following theorem. Theorem 6.14 — The variance of the sample variance is minimum if every batch contains a single history, i.e., if n = 1 and k = N. Proof The variance of the estimates in Equation (6.280) is 2

D2[S] = (g2) - (0

N2 =

(k

1

1)2

k

2

([ E (rTi — 11 )2 ] ) — DU] k J=1

428

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

The optimum value of n = N/k with N given is the one that minimizes the quantity Q(n) =

N2 I(— n, (k — 1)2 Lk ] -1

A 212

(6.281)

In the rest of the proof, we shall assume that the expectation of the random variable x is zero, i.e., we put m = 0. This can be done without loss of generality since, if the assumption does not hold, by putting x, — m in place of x„ all the formulas so far derived remain valid and the new variable has zero expectation. This assumption also means that (Tri) = (rii,tTlitrirrTa,) = 0

(j

(6.282)

1,r,$)

is independent of any other thr. since We shall express Q(n) in terms of the moments of the random variable x, and this expression will be minimized with respect to n. Equation (6.281) can be rewritten as 2 Q(n)

21

[k(k — 1)]

IT-11

12 k. — 1 r N l_k(k — 1)] (( k

--

J2

(_11

j2

_1

)

E k

k =i

Performing squaring, we have Q(n)

N 12 ((k — 1)2 E = [ k2(k - 1)

+ [(k — 1)2 + 1]

E E 'Tow )

as the odd powers of mh drop out according to Equation (6.282). Finally, since, from a are equivalent, we can put statistical point of view, all the (j = 1, 2, ..., k)

= (iTf1 ) = ((

(6.283)

and Q(n) becomes Q(n)

N2 _

=

( (1111)

+

N2 Rk - 1)2 + 1] _2 2

k3 k — 1)

(1111)

(6.284)

It remains to determine the expectations in Equation (6.283) with r = 2 and r = 4. We have already seen that 1 — (m,2) = ((1 x))2) = n (x2), n ,=,

((x) = 0)

429 Similarly, for the fourth moment

= ((-1- E xiy) = IT4 1 ((E xF + n 1 = — (E + 2E n4 j i

EE

;)2

1

E xx;) = — (x4) + 2 n3

n—1 (x2)2 n3

Inserting the above expressions into Equation (6.284), the quantity to be minimized becomes 1 Q(n) = N

n2

[(x4) + (N — 2 — n + N n)(X2)2]

(6.285)

where we have inserted k = N/n. Obviously, Q(n) is minimum with n =-- 1, i.e., if every batch consists of a single history, as stated.

We have thus shown that when estimating the theoretical variance of a random variable (e.g., of the final score in a history) from the realizations x, (i = 1, 2,...,N) the most efficient estimate is =

1

E (x, —

(6.286)

N— 1,-,

as follows from Equation (6.280) with k = N. In other words, for variance estimation, batchwise evaluation is not efficient. In the proof above, we have also established the variance of the sample variance §. Note that the derivation was performed for a zero-expectation random variable, i.e., in the general case, x — m is to be inserted instead of x and the variance of g in Equation (6.286) is —[((x — m)4) — D2[x]] + D2[] = Q(1) — lir[x] = 1

N(N 1— 1) D

2[x] (6.287)

The efficiency of batchwise variance estimation was examined by Dubi.' 2 He proves the statement of Theorem 6.14 for the estimate N

=

E (x — N,

m)2

and shows that D2[s] = — [((x — m)4) — Mx]]

(6.288)

Comparing Equations (6.287) and (6.288), it is seen that the variance of the estimated variance is lower by D4 [x[41\1(N — 1)] if the expectation of the random variable x is known and need not be estimated by m.

430

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

APPENDIX 6A: UNBIASED ESTIMATION OF CRITICALITY REACTION RATES A scheme of estimating reaction rates in source-free systems (hypothetically altered to critical by keff) was introduced on a heuristic basis in Section 6.III.F. Here we prove that this scheme does, indeed, define an unbiased estimate of the reaction rate R = f dridrof(r)Z(r,r0)S(ro)

(A.1)

where S(ro) is the spatial density of fission neutrons in the hypothetical system (in which the number of fission neutrons per collision has been changed to like„ times the real one), f(ro) is an arbitrary weighting function, and Z(r,r') is the density of the fission neturons at r due to a single fission at r'. According to Equations (6.103) and (6.100) Z(r,r„) = f dEcc(P)v(P)z(P,ro)

(A.2)

where P = (r,E), and z(P,ro) satisfies the equation z(P,ro) = f

z(P",r„) Ks(P",P) + idEo x (Po) T(P0,P)

(A.3)

Here, Po = (ro,E,), X(Po) = x(Eolr„) the direction-energy distribution of the fission neutrons emerging at ro, and Ks (P",P) is the number density of the neutrons entering a collision at P due to a neutron colliding at P" in a game where fissions are replaced by pure absorption (nonmultiplying game), i.e., Ks(P",P) = JdP' s(P")Cs(P",P')T(P',P) Now, let us define the function tp(P) = fdroz(P,r0)S(ro)

(A.4)

Then, from Equations (A.1) and (A.2), the reaction rate in question can be expressed as R = idridEci(P)v(P)f(r)f droz(P,ro)S(ro) = JdPtl,(P)ci(P)v(P)f(r) = fdPkIJ(P)g(P)

(A.5)

with g(P) = cf(P)v(P)f(r)

(A.6)

431 On the other hand, multiplying Equation (A.3) by S(ro) and integrating with respect to ro, we obtain an equation on the function 41(P) defined in Equation (A.4) as = idP"VP")Ks(P",P) + f dP'Q(P')T(P',P)

(A.7)

Where Q(P) = x(P)S(r) Equation (A.7) defines t[i(P) as the collision density in the nonmultiplying system due to the source Q(P). Therefore, Equation (A.5) can be interpreted as a conventional reaction rate produced by this collision density. Accordingly, an unbiased estimation procedure of the reaction rate R in Equation (A.5) goes along the lines detailed in Chapter 5, i.e., particles are started from Q(P). They are processed until they escape from the system or are absorbed (recall that fission is also treated as absorption). In the simplest estimation procedure, every collision contributes to the estimate of R by cf(P)v(P)f(r) multiplied by the statistical weight of the particle entering the collision. Alternatively, fission-related absorption [the probability of which in a collision at P is cf(P)] gives a contribution v(P)f(r) times the particle's weight.

APPENDIX 6B: ACCURACY OF THE CORRECTED VARIANCE FROM SMALL SAMPLE SETS The corrected empirical variance of a mean combined from several estimates is derived in Section 6. V.B. Terms of the order D3 = RS — S)Q13

(B.1)

are neglected in the derivation, where S is the covariance matrix of a k-dimensional random vector x, the latter being Gaussian, i.e., x:N(me,S) In Equation (B.1) Q = S-1 and S is the empirical covariance matrix from n samples, as defined in Equation (6.209). Here we prove the following theorem. Theorem — The expectation of D3 in Equation (B.1) is of the order (k/n)2, i.e., (D3) O(k2/n2)I Proof. Recall that, according to Theorem 6.7,

D = SQ — = AQ/(n — 1) — I

(B.2)

432

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

where

yyr.

A= J= I

and yj:N(0,S) independently of ym (m j). Accordingly D3 = (AQ)3/(n — 1)3 — 3(AQ)2/(n — 1)2 + 3AQ/(n — 1) —

(B .3)

Let us denote (B.4)

= YJYTQ then n- I

AQ =

E

Bj

(B.5)

and the following relations hold (AQ)2

= E 131 + EE BjBk k j#k

j

(AQ)3 =

E 13;' + EE (Biw k

+ Bi2+ Bk)

j#k

EEE j k m

BjB,Bn,

j#k,m,m#k

Since the Bj's are identically distributed and independent of each other, we have (AQ) = (n — 1)(B)

(n — 1)I

((AQ)2) = (n — 1)(B2) + (n — 1)(n — 2)I and ((AQ)3) = (n — 1)(B2) + 3(n — 1)(n — 2)(B2) + (n — 1)(n — 2)(n — 3)I where B = YYTQ

(B.6)

and y:N(O,S). Inserting the above expectations into the expectation of Equation (B.3), we have (:03) —

1 (431) — 3(B2) + 21) (n — 1)2 -

In order to conclude the proof, it remains to show that (13"± 1 ) = 0(1MI

433 This will be proven in the lemma below, and thus we have (131) = 0(10I/(n — 1)2 = 0(k2/112)I as stated. Lemma N

(11N + 1 ) = fl (k + 2m)I m=I

Proof. Notice that it follows from Equation (B.6) that BN + I = B(yTQy)N and the density function of the random vector y is py(y) = Kexp( — yTQy/2) Therefore, the moment-generating function of B reads

BN 'tN/N!) = (

G=( N=0

B(tyTQy)N/N!) N=0

= (Bexp(tyTQy)) = KidyBexpr — (1 — 2 t)yTQy/21 Making use of the explicit form of B in Equation (B.6) and putting = 1 — 2t the moment-generator function reads G = Kidy yyTQexp[ — XyTQy/2]

(B.7)

Now, on the one hand, by the definition of the moment-generating function, the following relation holds (BN± 1) = dNG/c1tN1,-0 = ( — 2)NdNG/dX"I x -1 On the other hand, with the notation z = Vky we have dz = V"dy

434

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

and Equation (B.7) becomes G = X -(k 1 2"2 KidzzzTexp[ — zrQz/2] • Q x

2)/2sQ _

Thus N

(B"1) = (_2)NdNx-("2)/2/dxix_II = 11 (k + 2m)I

as stated.

APPENDIX 6C: EXPECTATION OF THE MATRIX

ARA

Theorem — Let n-1

A=

E

1= 1

(C.1)

y-yTJ

where y,:N(0,S) independently of yn, (m 0 j). Then for an arbitrary deterministic matrix R, the following relation holds (ARA) = (n — 1)2SRS + (n — 1)[SRTS + SS,(RS)]

(C.2)

Proof. According to Equation (C.1) n-1 n-1

n-1

ARA =

E

yiyiTRy,y1,1 +

E E yyrktymy-r,„

j=1 m=1 j#m

J=I

Because of the independence of the yi's (ARA) = (n — 1)(yyTRyyT) + (n — 1)(n — 2)SRS

(C.3)

where y:N(O,S). Thus, in view of Equations (C.2) and (C.3), it remains to show that (yyTRyyT) = S(R + RT)S + SSP(RS) Now, let T = yyTRyyT,

Y = (Yi, Y2, •••, Yn)T,

T = {Tim}

(C.4)

435 Then Tim =

E y,yiRjey,yo, J.,

and since y is normally distributed,' we have (yiyjyiym) = SoS, +

+ S,o,So

So being the (i,j)-th element of S. Accordingly (Tim) =

E sii(Ril + R,i)So„ + Sim

SA,

ii

which calls forth Equation (CA).

APPENDIX 6D: EMPIRICAL THIRD MOMENTS Let us consider three correlated random variables u,v, and w. We can assume without loss of generality that the variables all have zero expectations, i.e., = (w) = 0

(u) =

Let u„ v„ and w, ( i = 1, 2,...,n) be independent realizations of the respective variables and let the empirical means of the realizations be

171 = - E u,, V = - E v1 , w = - E IN; n i=

=

n i=i

(D.1)

We wish to construct an unbiased estimate of the third moment T = (TIVW)

(D.2)

Theorem — The estimate

E (u, - c)(v, - v)(w, - W)/[n(n - 1)(n - 2)]

(D.3)

is unbiased with respect to T in Equation (D.2). Proof. It is to be seen that (T) = T Taking the expectation of Equation (D.3), we have (T) = [n(n - 1)(n - 2)]-'

E (u,v,w, - u,v,w - u,vw, i=1

+ uiVW +

+ iTT/w, - UV-vT,T)

(D.4)

436

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

Since the different realizations are independent and all have zero expectation, the following relations hold. 1 1 1 (uvw) = n E (u,viwi) = n (uiviw,) + — 1.1 ,,,,) E ( wi) ; n ;,4i =1 — (uvw)

(D.5)

n

and 1 (uivw) = —

E

n' j.k

(uiviwk ) =

(uiv,w;) =

(uvw)

(D.6)

For the same reasons, we have 1 1 1 T= (TIV W) = — E (uy,w,) = — E (uy,w,) = — (uvw) n- nik n' n2

(D.7)

Making use of Equations (D.5) through (D.7) and their analogons with permutations of the variables, Equation (D.4) becomes (t) = [n(n — 1)(n — 2)1' E (uvw)

3 — + 3

2

—

1) n

=1 — (uvw) E (n2 — 3n + 2)/[n(n — 1)(n — 2)] = — 1 (uvw) n2 n2 i.e., in view of Equation (D.7) (t) = T as stated.

In Section 6.V.E, the moments (fiV2), (CM, and (ti3) were estimated. Unbiased estimates of these moments follow from Equation (D.3) by setting w = v, w = u, and w = u = v, respectively.

REFERENCES 1. Anderson, T. W., An Introduction to Multivariate Statistical Analysis, John Wiley & Sons, New York, 1958. 2. Bell, G. I. and Glasstone, S., Nuclear Reactor Theory, Van Nostrand, New York, 1970. 3. Beer, M., A comparison of maximum likelihood and other estimators of eigenvalues from several correlated Monte Carlo samples, Nucl. Sci. Eng., 76, 295, 1980. 4. Bowsher, H., et al., Magnitude of bias in Monte Carlo eigenvalue calculations, Trans. Am. Nucl. Soc., 45, 324, 1983. 5. Brandt, V., Die Monte-Carlo Berechnung von Quotienten in der Reaktorphysik, KFK 2074 Report, Gesellschaft fur Kernforschung mbH, Karlsruhe, 1975.

437 6. Brown, F. B. and Martin, W. R., Monte Carlo methods for radiation transport analysis on vector computers, Prog. Nucl. Energy, 14, 269, 1984. 7. Cashwell, C. D. and Schrandt, R. G., Flux at a point in MCNP, in A Review of the Theory and Application of Monte Carlo Methods. Proc. Seminar-Workshop, ORNL/RSIC-44 Report, Oak Ridge National Laboratory, 1980, 63. 8. Dejonghe, G., et al., Studies of perturbations using correlated Monte Carlo method, in A Review of the Theory and Application of Monte Carlo Methods. Proc. Seminar-Workshop, ORNL/RSIC-44 Report, Oak Ridge National Laboratory, 1980, 47; see also, Etude de Perturbations Utilisant la Methode de Monte Carlo, in Proc. NEACRP Specialists' Meeting on Nuclear Data and Benchmarks for Shielding, OECD Report, Paris, 1980, 191. 9. Dickenson, D. and Whitesides, G. E., The Monte Carlo method for array criticality calculations, Nucl. Technol., 30, 166, 1976. 10. Drawbaugh, D. W., On the solution of transport problems by conditional Monte Carlo, Nucl. Sci. Eng., 9, 195, 1961. 1 1 . Dubi, A., et al., Monte Carlo aspects of contributons, Nucl. Sci. Eng., 68, 19, 1978. 12. Dubi, A., On the analysis of the variance in Monte Carlo calculations, Nucl. Sci. Eng., 72, 108, 1979. 13. Dubi, A., and Gerstl, S. A. W., Application of biasing techniques to the contributon Monte Carlo method, Nucl. Sci. Eng., 76, 198, 1980. 14. Dubi, A. and Rief, H., A note on some aspects of sensitivity analysis in Monte Carlo, in Proc. NEACRP Specialists' Meeting on Nuclear Data and Benchmarks for Shielding, OECD Report, Paris, 1980, 151. 15. Dubi, A., et al., On confidence limits and statistical convergence of Monte Carlo point-flux estimators with unbounded variance, Ann. Nucl. Energy, 9, 675, 1982. 16. Elperin, T. and Dubi, A., On the Markov chain analysis of source iteration Monte Carlo procedures for criticality problems. I, Nucl. Sci. Eng., 91, 59, 1985. 17. Ermakov, S. M. and Mikhailov, G. A., Course of Statistical Modelling, Nauka, Moscow, 1976, (in Russian). 18. Feller, W., An Introduction to Probability Theory and its Applications, Vol. 3, 2nd ed., John Wiley & Sons, New York, 1971. 19. Feldman, U., et al., Monte Carlo small-sample perturbation calculations, in Proc. Topical Meeting on Advances in Reactor Comp., Salt Lake City, 1983, 124. 20. Frank-Kamenietzky, A. D., Application of Monte Carlo method to multigroup reactor calculations, At. Energ., 16, 119, 1964, (in Russian). 21. Frank-Kamenietzky, A. D., Calculation of multiplication factor of nuclear reactors by Monte Carlo method, in Monte Carlo Method in Radiation Transport Problems, Atomizdat, Moscow, 1967, 212, (in Russian). 22. Fraley, S. K. and Hoffman, T. J., Bounded flux-at-a-point for Monte Carlo, Trans. Am. Nucl. Soc., 23, 371, 1977. 23. Fraley, S. K. and Hoffman, T. J., Bounded flux-at-a-point for multigroup Monte Carlo computer codes, Nucl. Sci. Eng., 70, 14, 1979. 24. Gast, R. C. and Candelore, N. R., Monte Carlo eigenfunction uncertainties, Trans. Am. Nucl. Soc., 14, 219, 1971. 25. Gast, R. C. and Candelore, N. R., Monte Carlo eigenfunction strategies and uncertainties, in Proc. NEACRP Meeting of a Monte Carlo Study Group, ANL-75-2/NEAC-CRP-L 118 Report, Argonne National Laboratory, 1974, 162. 26. Gerstl., S. A., A New Concept for Deep-Penetration Transport Calculations and Two New Forms of the Neutron Transport Equation, LA-6628-MS Report, Los Alamos Scientific Laboratory, 1976. 27. Gelbard, E. M. and Prael, R. E., Monte Carlo Work at Argonne National Laboratory, FRA-TM-64 Memo, 1974. 28. Gelbard, E. M., Unfinished Monte Carlo business, in Proc. Int. Topical Meeting on Advances in Mathematical Methods for the Solution of Nuclear Engineering Problems, Munich, 1981, 145. 29. Goad, W. and Johnston, R., Monte Carlor method for criticality problems, Nucl. Sci. Eng., 5, 371, 1959. 30. Goldstein, M. and Greenspan, E., A recursive Monte Carlo method for estimating importance function distributions in deep penetration problems, Nucl. Sci. Eng., 76, 308, 1980. 31. Hall, M. C. G., Monte Carlo perturbation theory in neutron transport calculations, in Proc. SeminarWorkshop, ORNL/RSIC-44 Report, Oak Ridge National Laboratory, 1980, 47; see also, DUCKPOND — a perturbation Monte Carlo and its applications in Proc. NEACRP Specialists' Meeting on Nuclear Data and Benchmarks for Shielding, OECD Report, Paris, 1980, 205. 32. Hall, M. C. G., Cross-section adjustment with Monte Carlo sensitivities. Application to the Winfrith iron benchmark, Nucl. Sci. Eng., 81, 423, 1982. 33. Halperin, M., Almost linearly-optimum combination of unbiased estimates, J. Am. Stat. Assoc., 56, 36, 1961. 34. Hoffman, T. J., et al., The adjoint difference method and its application to deep-penetration radiation transport, Nucl. Sci. Eng., 48, 179, 1972.

438

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

35. Hoffman, T. J., et al., A Monte Carlo perturbation source method for reactivity calculations, Nucl. Sci. Eng., 66, 60, 1978. 36. Iida, H. and Seki, Y., Reduction of computational time for point detector estimation in Monte Carlo transport codes, Nucl. Sci. Eng., 74, 213, 1980. 37. Kalos, M. H., On the estimation of flux at a point by Monte Carlo, Nucl. Sci. Eng., 16, 111, 1963. 38. Kalos, M. H., Zero variance estimator for reactor criticality, in Proc. Nat. Topical Meeting on New Developments in Reactor Physics and Shielding, Vol. 1, CONF-720901, U.S. Atomic Energy Commission Technical Information Center, 1972, 229. 39. KaIli, H. J. and Cashwell, E. D., Evaluation of Three Monte Carlo Estimation Schemes for Flux at a Point, LA-6865-MS Report, Los Alamos Scientific Laboratory, 1977. 40. Kalil, H. J. and Cashwell, E. D., Evaluation of three Monte Carlo schemes for flux at a point, Trans. Am. Nucl. Soc., 27, 370, 1977. 41. Khairullin, R. K., On a Monte Carlo algorithm for calculation of critical systems, Izv. Vuzov Ser. Mat., 10, 130, 1977 (in Russian). 42. Khairullin, R. K., On the estimation of critical parameters of a class of branching processes, Izv. Vuzov Ser. Mat., 8, 78, 1980 (in Russian). 43. LASL Group X-6, MCNP — A General Monte Carlo Code for Neutron and Photon Transport, LA-7369M Report, Los Alamos Scientific Laboratory, 1979. 44. Lewins, J. and Becker, M., Eds., Sensitivity and uncertainty analysis of reactor performance parameters, in Advances in Nuclear Science Technology, Vol. 14, Plenum Press, New York, 1982. 45. Levitt, L. B. and Lewis, R. C., A Non-Multi-Group Monte Carlo Code for Analysis of Fast Critical Assemblies, Al-AEC-12951 Report, Atomics International, 1970. 46. Lieberoth, J., A Monte Carlo technique to solve the static eigenvalue problem of the Boltzmann transport equation, Nukleonik, 11, 213, 1968. 47. Lichtenstein, H., et al., Progress in the Development of a Reactivity Capability in the SAM-CE System for Validating Fuel Management Codes, EPRI NP-638 Report, Electric Power Research Institute, 1978. 48. Lux, I., Generalized Monte Carlo moment equations with application to correlated and differential games, Int. J. Sci. Eng., 1(2), 1984. 49. Lux, I. and Szatmary, Z., Combined estimation of a common mean from few sample sets and from sample sets of rare events, Nucl. Sci. Eng., 89, 137, 1985. 50. Matthes, W., Calculation of reactivity perturbations with the Monte Carlo method, Nucl. Sci. Eng., 47, 234, 1972. 51. Matthes, W., Simulation of Transport Equation with Monte Carlo, EUR 5347e Report, Joint Nuclear Research Centre, Ispra Establishment, 1975. 52. Matthes, W. K., Comments on perturbation Monte Carlo, ESIS Newsl., 36. 9, 1981. 53. MacMillan, D. B., Monte Carlo confidence limits for iterated-source calculations, Nucl. Sci. Eng., 50, 73, 1973. 54. Martin, B. R., Statistics for Physicists, Academic Press, New York, 1971. 55. Mendelson, M. R., Monte Carlo criticality calculations for thermal reactors, Nucl. Sci. Eng., 32, 319, 1968. 56. Mikhailov, G. A., Calculation of critical systems by Monte Carlo method, Zh. Vychisl. Mat. Mat. Fiz., 6, 71, 1966 (in Russian). 57. Mikhailov, G. A., Calculation of system-parameter derivatives of functionals of the solutions to the transport equation, Zh. Vychisl. Mat. Mat. Fiz., 7, 915, 1967 (in Russian). 58. Mikhailov, G. A., Modification of local particle-flux estimation by Monte Carlo, Zh. Vychisl. Mat. Mat. Fiz., 13, 574, 1973 (in Russian). 59. Miller, L. B., Monte Carlo Analysis of Reactivity Coefficients in Fast Reactors, General Theory and Applications, ANL-7307 (TID-4500) Report, Argonne National Laboratory, 1967. 60. Moore, J. G., The solution of criticality problems by Monte Carlo methods, Adv. Nucl. Sci. Technol., 9„ 73, 1976. 61. Nakagawa, M. and Asaoka, T., Improvement of correlated sampling Monte Carlo methods for reactivity calculations, J. Nucl. Sci. Technol., 15, 400, 1978. 62. Noack, K., Variance analysis of Monte Carlo perturbation source method in inhomogeneous linear particle transport problems, Kernenergie, 26, 227, 282, 1983. 63. Polevoi, V. B., Calculation of large reactivity perturbations by difference-iteration Monte Carlo method, At. Energ., 46, 20, 1979 (in Russian). 64. Preeg, W. E. and Tsang, J. S. K., Comparison of correlated Monte Carlo techniques, Trans. Am. Nucl. Soc., 43, 628, 1982. 65. Rief, H. and Kschwendt, H., Reactor analysis by Monte Carlo, Nucl. Sci. Eng., 30, 395, 1967. 66. Rief, H., The relation correlated tracking and differential perturbation algorithms, ESIS Newsl., 36, 5, 1981.

439 67. Rief, H. and Fioretti, A., Monte Carlo shielding analysis using deep penetration biasing schemes combined with point estimators and algorithms for the scoring of sensitivity profiles and finite perturbation effects, in Proc. 6th ICRS, Vol. 1, Tokyo, 1983, 199. 68. Rief, H., Generalized Monte Carlo perturbation algorithms for correlated sampling and a second order Taylor series approach, Ann. Nucl. Energy, 11, 455, 1984. 69. Rief, H., et al., Track length estimator applied to point detector, Nucl. Sci. Eng., 87, 59, 1984. 70. Rief, H., Monte Carlo uncertainty analysis, in CRC Handbook on Uncertainty Analysis, Y. Ronen, Ed., CRC Press, Boca Raton, FL, in press. 71. Sarkar, P. K. and Prasad, M. A., Estimation of population variance in contributon Monte Carlo, Nucl. Sci. Eng., 87, 136, 1984. 72. Shikov, S. B., Selected problems in mathematical theory of critical reactors, Zh. Vychisl. Mat. Mat. Fiz., 7, 113, 1967 (in Russian). 73. Shikov, S. B. and Shishkov, L. K., On the existence and uniqueness of a positive solution to a critical reactor equation, Zh. Vychisl. Mat. Mat. Fiz., 8, 686, 1968, (in Russian). 74. Smith, K. S. and Schaefer, R. W., Recent developments in the small-sample reactivity discrepancy, Nucl. Sci. Eng., 87, 314, 1984. 75. Spanier, J. and Gelbard, E. M., Monte Carlo Principles and Neutron Transport Problems, AddisonWesley, Reading, MA, 1969. 76. Steinberg, H. A. and Kalos, M. H., Bounded estimators for flux at a point in Monte Carlo, Nucl. Sci. Eng., 44, 406, 1971. 77. Steinberg, H. and Lichteinstein, H., Implementation of bounded point estimators in point cross-section Monte Carlo, Trans. Am. Nucl. Soc., 17, 259, 1973. 78. Steinberg, H. A., Bounded estimation of flux-at-a-point for one or more detectors, in Proc NEACRP Meeting of a Monte Carlo Study Group, ANL-75-2, NEA-CRP-L-118, Argonne National Laboratory, 1974, 281. 79. Steinberg, H., Implementation of improved bounded estimation of flux-at-a-point for several detectors, Trans. Am. Nucl. Soc., 19, 442, 1974. 80. Steinberg, H., Bounded estimation of flux-at-a-point near region boundaries, Trans. Am. Nucl. Soc., 23, 607, 1976. 81. Takahashi, H., Monte Carlo method for geometrical perturbation and its application to the pulsed fast reactor, Nucl. Sci. Eng., 41, 259, 1970. 82. Usikov, D. A., Parametric Integration as a Means of Monte Carlo Calculation of Finite Perturbations in Reactors, FEI-423 Report, Fiziko-Energeticheskij Institute, Obninsk, 1976, (in Russian). 83. Usikov, D. A., On the Variance of Perturbation-Estimation by Monte Carlo Method, FEI-656 Report, Fizilco-Energeticheskij Institute, Obninsk, 1976 (in Russian). 84. Usikov, D. A., Perturbation estimation by solution of inhomogeneous neutron transport problems via Monte Carlo method, At. Energ., 42, 19, 1977 (in Russian). 85. Ussachoff, J. N., Equations for the importance of neutrons, reactor kinetics and the theory of perturbation, in Proc. Int. Conf. Peaceful Uses of Atomic Energy, Vol. 5, P/656, Geneva, 1955, 503. 86. Zolotukhin, V. G. and Usikov, D. A., Estimation of Reactor Parameters by Monte Carlo Method (Perturbation Theory), Atomizdat, Moscow, 1979 (in Russian). 87. Zolotukhin, V. G. and Maiorov, L. V., Estimation of systematic error in Monte Carlo criticality calculations, At. Energ., 55, 173, 1983 (in Russian). 88. Zolotukhin, V. G. and Maiorov, L. V., Estimation of Critical Reactor Parameters by Monte Carlo Method, Energoatomizdat, Moscow, 1984 (in Russian).

441 Chapter 7

OPTIMIZATION OF EFFICIENCY-INCREASING TECHNIQUES As pointed out in the introduction to Chapter 5, the main purpose of constructing artificial (i.e., nonanalog) games is to make the estimation procedure more efficient. Efficiency is defined in Equation (5.1) as the inverse of the product of the computing time and the variance of the estimate obtained during this time. If t denotes the computing time and the quantity scored during t is the random variable s„ the efficiency reads E = {D2[0}The commonly used efficiency-increasing techniques are dependent on a number of arbitrarily chosen parameters, and the procedure is applied in an optimal way if the actual values of the parameters maximize the efficiency. We have seen in Section 5 .V.F. that in many practical cases the computing time is essentially proportional to the number of collisions played during that time. (More rigorously, the time is considered proportional to the number of flights simulated. Both quantities are used in practical applications; in the case of deeppenetration problems considered in the major portion of this Chapter, the difference between the two quantities is usually negligible.) Thus, if s denotes the score gathered in a history and N is the expected number of collisions per history, the efficiency of the game is inversely proportional to the quantity Q = D2[s]N

(7.1)

Q in Equation (7.1) is called the quality factor of the game. Minimization of the quality factor is equivalent to maximization of the efficiency. As was pointed out in Section 5.VIII.I, the variance per history and the number of collisions per history usually vary in opposite directions as the simulation scheme is altered, and this is why in most practical cases a compromise between variance reduction and computing-time reduction has to be found when optimizing a scheme. Note that when a nonanalog simulation is indispensable, the second moment of the score is usually considerably larger than the square of the expected score (otherwise, the nonanalog game would be satisfactorily efficient). Therefore, in many cases Q M{s2}N where M{s2} = M2 is the second moment of the score. Minimization of the approximate expression is often easier than that of Equation (7.1). Since, except for trivial cases, none of the quantities in Equation (7.1) can be determined exactly, practical optimization of particular schemes is always based on approximations. Many Monte Carlo practitioners share the opinion that efficient use of one or another technique is a matter of art, common sense, and professional skill rather than a result of strict mathematical rules. In this Chapter we attempt to give a systematic account on the development in optimization of the two most common efficiency-increasing methods: the splitting and the path-stretching procedures, both used mainly in deep-penetration problems. The subject is far from being settled and important new developments are expected in the very near future. With some arbitrariness, the efficiency optimization methods can be divided into four groups. The most obvious way of optimizing the estimation procedure of a given quantity is to perform a series of preliminary Monte Carlo calculations of the quantity with different

442

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

parameters of the scheme to be optimized and select the parameter value resulting in the highest efficiency in the preliminary runs (experimental-statistical approach). This method is usually very time-consuming and if one wishes to determine statistically reliable parameter values, the computing time required may be even higher than that for determining the target quantity in a nonoptimized scheme. Therefore, it is more expedient to investigate the efficiency of the technique in question in simplified transport models or through an approximate solution of the exact problem. Usually, this can be done analytically (approximate analytical approach). Yet, except for the simplest problems, it is difficult to predict how far the approximate optimum parameters are from the exact ones. Therefore, results of approximate analytical calculations are often only of qualitative merit; they reflect specific features of the optimized schemes. A combination of experimental and analytical methods is expected to yield reasonably well-optimized parameters of the techniques. The idea is as follows. The estimation problem is solved analytically in some approximate model, but certain quantities characteristic of the exact model are left undetermined. These quantities (called "bulk parameters") are independent of the particular nonanalog scheme used in the simulation, but characterize the system and the physical process modeled. The optimum parameters of the efficiency-increasing scheme are expressed in terms of the bulk parameters, which are determined in short preliminary runs. The quasi-optimum parameters of the scheme are than used in production runs and are possibly refined during the calculation. This method is called the direct statistical approach.' Naturally, any scheme that results in zero variance is optimum, and its efficiency is equal to infinity. We have seen that no zero-variance scheme is feasible in practice; nevertheless, approximations to zero-variance schemes may result in efficient estimation procedures. In this optimization method, certain parameters of the kernels are chosen so that the kernels give approximately zero variance in simplified (but realistic) games. The kernels so optimized are then applied in the actual simulation. In this Chapter, we consider the optimization of techniques widely used in deep-penetration Monte Carlo calculations. Simple examples of optimization by approximate analytical and direct statistical approaches are given in Chapter 7.1. The more sophisticated optimization procedures of splitting and path stretching are detailed in Chapters 7.11 and 7.111, respectively. For the sake of completeness, we note that efficiency maximization in a broader sense also comprises the minimum variance combination of different estimators of the same quantity. This matter was addressed in Chapter 6.V and is different in nature from the problems discussed in this Chapter.

I. SIMPLE EXAMPLES OF OPTIMIZATION METHODS Optimization of splitting and path stretching in the straight-ahead scattering model is investigated in Sections A and B. The separation assumption proposed in Section 5 .VII.B is used in an approximate analytical optimization of Russian roulette in Section C. The results of the considerations are very approximate and the numerical results so obtained have little direct practical value. Nevertheless, they reflect the main characteristics of the optimum schemes. More realistic models based on direct statistical approach are given in Section D. In later Sections, it will be apparent how the approximate analytical results help in the understanding and construction of practically applicable optimized schemes. A. OPTIMUM SPLITTING SCHEMES IN THE STRAIGHT-AHEAD MODEL Let us consider the problem of particle transmission through matter. Specifically, let the particles enter perpendicularly the x = 0 face of a homogeneous nonmultiplying slab situated perpendicularly to the x axis between x = 0 and x = X. Let the quantity to be

443 estimated be the number of particles emerging from the slab at x = X. We shall treat the problem with the straight-ahead scattering model introduced in Section 5.VI.C. The model is monoenergetic, and let us assume that the total cross section of the slab is unity. (In other words, all the distances are measured in the optical scale.) Then the transition kernel reads for

T(P,P')dP' =

x' > x

and zero otherwise, while the collision kernel reduces to C(P',P")dP" = c8(11' — 1)dp.' inside the slab and zero outside it (vacuum-equivalent black-absorber surroundings are is the cosine of the assumed). c is the mean number of secondaries per collision and angle between the x axis and the flight direction of the particle leaving the collision. Since 1. the slab is nonmultiplying, c The contribution function in transmission estimation reads if

f(P,P') = f(x,x') = 1

x' > X

and zero otherwise, i.e., the particle scores unity only when it escapes. The first moment of the score (expected number of particles due to a starter at x) satisfies Equation (5.57). With the kernels of the straight-ahead model, the equation reduces to +cf

M,(x) = f

The solution of this equation follows from Appendix 5C as M,(x) = exp[ —(1 — c)(X — x)]

(7.2)

Hence, the expected number of transmitted particles due to a particle entering the slab at x 0 is Tv11(0) =e In what follows, we investigate the two forms of splitting introduced in Sections 5.IILD and 5.IV.B. In the case of collisionwise splitting, the second moment of the score follows from Equation (5.98). Let us consider a splitting procedure where every particle is split into n fragments in every collision. Then the splitting probability and post splitting weight introduced in Section 5.III.D become z,„(P,W) = ar„,„;W~im= 1/n Therefore, the average weight values appearing in Equation (5.98) read = 1,

Wz = 1/n

We seek an n value that minimizes the quality factor of the game. If D2(x) = M2(x) — Mt(x)

444

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

denotes the variance of the score due to a starter from x, then Equation (5.98) with the straight-ahead scattering kernels becomes dx'e -( x' -x)M?(x) + n- f x dx'e -(x' x)D2(x') D2(x) = e-(') - M;(x) + cJ c

= I(x) + -dx'e- (x'-X)D2(xf) nx

(7.3)

If we insert the expression of M,(x) in Equation (7.2) into the source term of Equation (7.3), we obtain I(x) =

1 - c [e -(X — x) 2c

e - 2( 1 — c)(X — x)1

1—

(7.4)

The solution of Equation (7.3) with the source term in Equation (7.4) follows from Equation (C.7) of Appendix 5C as D2(x) = 1 -

1 - c 2(' -`)(x -x)lexp[(1 - 2c + -)(X - x)] - 1} 2c + c/n e

(7.5)

For the calculation of the quality factor in Equation (7.1), we shall also need the expected number of collisions played in a history. This number in a game with splitting is obviously equal to the collision rate in a game where the mean number of secondaries per collision is cn (i.e., the mean number of fragments per collision in the splitting game). By analogy to the absorption rate derived in Section 5.VI.C, Equation (5.235), the collision rate of the split fragments due to a starter at x is seen to have the form N(x) =

1 {exp[(cn - 1)(X - x)1 - 1} cn - 1

(7.6)

Thus, the quality factor of the game is 1 1 Q(n) = D2(0)N(0) = A - (ex - 1) - (ex - 1) a

(7.7)

where A = (1 - c)e- 2(1-c)X =

( 1 — C)Mi(0)

and a = 1 - 2c + -, n

= cn - 1

In order to optimize the splitting procedure, we must find an integer flo , that yields Q(n) = minimum, (n = 1,2, ....) The optimum splitting ratio depends on the mean number of secondaries per collision, c,

445 and on the optical thickness X. Analytical minimization of Equation (7.7) is cumbersome, but it can be easily minimized by simply calculating Q(n) at successive integer values of n. In Table 7.1, optimum splitting ratios are given for selected values of c and X. TABLE 7.1 X

0.1

0.2

0.3

0.4

0.5

0.6

0.7

2.5 3.0 4.0 5.0 10 15 20 30 40 50

1 2 2 2 3 3 4 4 5 5

1 1 2 2 2 3 3 3 3 4

1 1 2 2 2 2 2 3 3 3

1 1 1 1 2 2 2 2 2 3

1 1 1 1 2 2 2 2 2 2

1 1 1 1 1 1 2 2 2 2

1 1 1 1 1 1 I 1 1 1

The numerical results obtained from the model are certainly not realistic; however, they reflect the main tendencies expected from more rigorous models. Thus, it is seen that the optimum number of split fragments per collision increases with increasing absorption (decreasing c) and also with increasing thickness of the slab. It is also observed that the optimum splitting ratio varies very slowly as the thickness of the slab is increased and therefore the real optimum is very likely a noninteger value of n. In other words, collisionwise splitting can only be changed in rough steps and does not seem to be a sufficiently fine tool of efficiency maximization. (One could object that a noninteger splitting ratio can also be realized by allowing for two possible outcomes of every splitting with appropriate probabilites. It can, however, be seen that even in this case, collisionwise splitting is less flexible and efficient than geometrical splitting.) Geometrical (or surface) splitting can also be optimized analytically in the straightahead model. Let us consider again the transmission of particles through a slab of thickness X in which the mean number of secondaries per collision is equal to c. Analytical treatment of the problem is especially simple in purely absorbing media. Efficiency of surface splitting in this case was investigated in Reference 15. However, in most practical cases, optimization of splitting is related to deep-penetration calculations with mild absorption and therefore results from a purely absorbing model are not very enlightening. The straight-ahead scattering model offers a simple tool for investigating the geometrical splitting procedure. Let a splitting surface be situated at x = x., i.e., assume that a particle is split into n fragments whenever it crosses this surface. With the notations of Section 5.IV.B, this means that the only nonvanishing splitting probability is gn(x) and it is nonzero at x = x, only: gk(x)

= t sk,n o

if x = x. otherwise

Furthermore, the weights of the split fragments are Won, = 1/n, (i = 1,2, ..., n)

446

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

and the weight values appearing in the second-moment Equation (5.117) become W=

ox*) E

W2 =

gn(x.)

1/n = 1

and

E

1/n2 = 1/n

at x = x*. The second-moment equation (5.117) with these values reads M2(x) = c f dx'e -('-')M2(x1 ) + e-(x-x)1M;(x.) + - D2(x.)] n

(7.8)

for x < x* and = e " c)(x -

M2(x,,) =

(7.9)

where M I (x.) is the escape rate of a particle started from x = x*, as follows from Equation (7.2). D2(x) in Equation (7.8) is the variance of the score due to a starter from x. Thus D2(x*) = M2(x.) - fs,Vx.) = e a-c)(x-x.)[1 - e

(7.10)

cxx

Substituting Equations (7.9) and (7.10) into Equation (7.8), the second-moment equation becomes M2(x) = Ae-(x-x) + c

x. dx ie -(x-x)M2(x1 ),

x < x.

(7.11)

with A=

_1 DD2(x.)= e-20-0(x-x.)

1 e-o-cxx-x.)[1 - e-(1-0(x-x.)1 _

(7.12)

The solution of Equation (7.11) is easily obtained from Appendix 5C as M2(x) = Ae-(1-`)(1-‘),

for

x < x.

Let us note in passing that the variance of the score is not continuous at the splitting surface x = x*; rather, according to Equations (7.10) and (7.12), it has the discontinuity. D2(x*) - lim D2(x) = e -0-0(x -,11 - e -o-c)(x-.01 -0

n-1 n

D2(x.)

This result is a direct consequence of the fact that Um D2(x) is the mean variance of the score due to n independent starters and is obviously equal to 1/n times the variance due to a single starter.

447 The variance of the escape rate due to a starter from x= 0 is D2(0) = Ae -(1-0(x•-x) _ e - 2( I — c)X = e-(1-c)x[Aeci-c)(x-x,) — e-(1-oxi

Inserting the expression of A into Equation (7.12), we have D2(0) = e

-(1- c)(x -

— e -(1- 0,1

_ [1 — e - (1- cxx - ‘..)11 n

The variance can be expressed in terms of the expected score in Equation (7.9) as 1 D2(0) = M1(0011(0[1 — M,(X —x.)] + — [1 — 1\41(x*)]l n

(7.13)

The expected number of collisions due to a starter is the sum of two terms. The first term is the collision rate produced by the starter between x = 0 and x = x*. The second term is the rate produced by the progeny between x = x* and x = X, provided the starter reaches the surfaces at x*. The collision rates above follow by analogy to Equation (5.235) in Section 5.VI.C, and thus the number of collisions becomes N(0) =

1

[1 — e-(1-`)1 + e-(` -`)x* [1 — e - (' -0(x 1—c 1—c

It can also be expressed in terms of the first moment as N(0) =

1 1—c

{[1 — — xOl + nMAX —x.)[1 — Mi(x0[} M,(X

(7.14)

In order to optimize the splitting ratio, n, again the quality factor Q(n) = D2(0)N(0) needs to be minimized. According to Equations (7.13) and (7.14) the quality factor reads 1

1(x*)[ 1 — M,(X — x*)] + — [1 — Mi(x0[} Q(n) = 1 — 1 c Mi(0){1s4 x 1[1 — M,(X

+ nM,(X —x.)[1 — Mi(x*)[}

or, after rearrangement Q(n) =

1 M,(0)1M,(x*)[ 1 — M,(X — x*)]2 + M,(X — x.)[1 — Mi(x*)]2 1— c

+ [! + nM,(X — x0M1(x•)][1

M,(X —

—

(7.15)

Differentiating Q(n) with respect to n and setting the derivative equal to zero, we have dQ(nn)

d

—

1 + M,(X n2

x.)M,(x.)]

M1(0)

1—c

[1

M,(X — x.)111 — M,(x.)] = 0

448

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

Hence, the optimum value of the splitting ratio is no,„ = [M,(X — x.)M,(x.)1y2 = [e-0

-ocx-..)1-1/2

= e(1--0 X/2 = iM i (0)1 — 1 /2

(7.16)

Accordingly, in this simple model, the optimum number of split fragments is independent of the location of the splitting surface. More realistic simulations indicate that for reasonable splitting surface locations (i.e., not too close to x = 0 or x = X), the optimum splitting ratio is, indeed, position independent.21 This, however, does not mean that the gain in efficiency obtained by the splitting procedure would also be independent of x.. The optimized quality factor follows from Equation (7.15) with n = no,„ as Q(n„p,) =

1 1 — c M,(0){

— M,(X — x.)] + VM,(X — x.)[1 — M,(x.)]12

Now, since VM,(x.) = M,(x./2) it is easily seen that Q(nop,) is minimum if x. = X/2, i.e., if the splitting surface is at the midplane of the slab. The absolute minimum of the quality factor is

Q.,. -

4 c M,(0)M i(X/2)[1 — Mi(X/2)[2 1 —

Note that since the straight-ahead model does not account for backscattering, the results above are expected to be approximate. Realistic tests show that the optimum location of the splitting surface is at a value x. > X/2, but not very far from the midplane, and x. = X/2 is a rather good approximation.2 ' Another interesting feature of the splitting procedure is that the number of fragments starting from the splitting surface at x. = X/2 is e -(1 - c)X/2

nopt = 1

(Remember that the expected number of particles reaching x = X/2 is exp[ — (1 — c)X/2].) Therefore, single-surface splitting virtually cuts the effective thickness of the slab about in half. Furthermore, it follows from the model that by inserting more and more splitting surfaces into the slab, the optimum splitting procedure results in an essentially uniform distribution of the fragments along the slab. It may be interesting to examine the sensitivity of the efficiency (or of the quality factor) to the variation of the splitting ratio, n, in the nieghborhood of its optimum value. For the sake of simplicity, we shall consider the case when the splitting surface is located at x. = X/2. The sensitivity of Q(n) is its relative variation due to small alterations of n, i.e., if S(n) denotes the sensitivity of Q(n), then S(n) =

1 dQ(n) Q(n) do

449 In view of Equation (7.15) and (7.16), the sensitivity reads 1 11041 (0) — 1 S(n) =, n nVM,(0) + 1

1 n — no n n + nop,

where M,(0) is the expected transmission rate through the slab. In most deep-penetration problems, the transmission rate is very small, i.e., M,(0) X

Similarly, let the path-stretched transition kernel be t(p, p I )dp f

=

b)e --""'-x)dx' (' if x' X le-('-b)(x -x)8(X — x')dx' if x' > X

r

(1

Then the particle's weight is multiplied by T(P,13')/4P,P') =

( 1 — b) 'e b(x" x) _ e b(x -x)

if if

x' X x' > X

in a flight from x to x', instead of the factor in Equation (7.20). It is easily seen that the second-moment equations corresponding to the kernels above has the form 1)-,42(x)

e -( i+b)(X-x)

c2( 1

x

b)

- .1" dx'e - (1 +We-Ms:42(x')

The solution of the equation at x = 0 is N42(0) = exp [ (1 + b

c2 1 — 13) j

The path-stretching parameter minimizing the second moment is easily obtained as b=1—c i.e., the optimum second moment is IkA 2(0) = e-2"-c)x = M;(0) Thus, the optimum scheme has a zero variance.8 Zero-variance schemes of a general pathstretching game will be derived in Chapter 7.111. The scheme above is a special case of a general method. This will become clear in light of the general results in Chapter 7.111 if one realizes that the optimum path-stretching parameter, b, satisifes the equation e -b()C - ") = M I (X)

455 or, equivalently, the stretched cross section in Equation (7.18) can be written as d dx logeM,(x) Q = cr — — C APPROXIMATE OPTIMIZATION OF THE RUSSIAN ROULETTE PARAMETER Let us consider the monoenergetic, isotropic particle transport in a homogeneous region V. Let the collision rate in V be the quantity estimated in this game and assume that the collision estimator is applied. We shall consider an analog game, except that survival biasing and Russian roulette is played. Russian roulette will be used in its simplest form, where a particle that has a weight W less than a given value wth obtains a new weight of unity with a probability W and is killed with the complementary probability. The constant threshold value wth is called the Russian roulette parameter. This parameter is to be chosen such that the efficiency of the estimation procedure will be maximum. In the derivation below, we shall use the approximative assumption that the probability that a particle does not escape from V after a collision is always equal to Pc, the first-flight collision probability introduced in Section 5.VII.B. In other words, we assume that the distribution of the collision points is constantly uniform over V. It was seen in Section 5.VII.B that this approximation is equivalent to the separation assumption introduced there. Let c denote the survival probability in V and let the Russian roulette parameter be such that for some integer n el- I

wth

> cn

(7.28)

It is this integer value that is to be chosen optimum in order to maximize the efficiency of the game. Now. if the inequalities in Equation (7.28) hold, Russian roulette is played in every n-th collision. The weight of a particle after the n-th collision is W = c"

(7.29)

i.e., the particle survives a Russian roulette with a probability c". The probability that a particle history consists of exactly i collisions can be written in the factorized form qi = akbJ

(7.30)

where i = kn + j,

k>0,

1 j

n

and ak = (c"P'Dk = (1 — Pc)13e1 + 8„.i131+'(1 — c")

(7.31) (7.32)

Obviously, ak is the probability that a particle survives k consecutive Russian roulettes and still stays in V, whereas is the probability that either the particle escapes from V in its jth collision or is killed in a Russian roulette (if j = n). The weight of the particle before the i-th collision is ci -' (since it has regained a unit

456

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

weight in the kn'th collision and has lost weight during the last j — 1 survival biasing). Therefore, the score in a history consisting of i collisions is nS. =

k

-1

E

=

[1 —

M,

=E

+E

r= o

r=.

+ k(1 —c")[/(1 — c)

(7.33)

The expected score in the history is co;

By inserting Equations (7.29) through (7.33) into this expression, we obtain M1 =

P, 1 — (cPc)" 1 — cPc 1 — c"P2

PC 1 — cP,,

(7.34)

Note that M, in Equation (7.34) is identical to that in Equation (5.253), which was obtained by direct application of the separation assumption. The second moment of the score is

M2 =

E

EE

q,s; =

akbp — 0 k(1 — opio —

An elementary calculation yields W1-2

P,(1 + cPa) 1 — cP, 1 — (c2P,)" (1 — cP,)2 1 — c2P, 1 — (cP,)"

(7.35)

Recall that in an analog game, Russian roulette is played in every collision, i.e., n = 1. Thus, the second moment of the analog score is M2 = Pc(1 + cP,)/(1 — cP,)2

(7.36)

as also derived at the end of Section 5.VIII.G. In order to determine the efficiency of the game, it remains to calculate the expected number of collisions per history. Obviously

=E

iq, =

P, I— 1 — P, 1 — (cP,)"

(7.37)

in the game with Russian roulette, while it reduces to N = M, = P,/(1 — cPa) in the case of the analog game. Now the quality factor of the game with Russian roulette reads Q = CKI2 — NIDI"T = m21'sil l[G„(c,cPc) — H(C,P)1/Gn(C,Pc) where Gr,(x,y) —

— y1— 1 — xy 1 —(xyr Y"

457 and H(x,y) = y/(1 + xy) The quality factor in the analog game is Q = (M2 - MT)N = M2M,[1 - Fl(c,Pc)] Obviously, the Russian roulette procedure increases the efficiency if S(n) = O/Q < 1 i.e., if S(n) = [G„(c,cPc) - H(c,Pc)11/{Gc(c,Pc)[1 - H(c,Pc)]1 < 1

(7.38)

The ratio S(n) was evaluated34'35 for various values of the first-flight collision probability Pc and survival probability c, and it was found that for not-too-large regions (Pc < 0.6), the increase of efficiency offered by Russian roulette does not exceed 5%. This means that for regions with characteristic dimensions less that about 1 to 2 mean free paths, Russian roulette does not pay off. It has also been seen that for such regions, the efficiency is very sensitive to the value of n (i.e., to the number of collisions before Russian roulette), which is connected to the Russian roulette parameter according to Equation (7.28). Therefore, there is a risk that with an improper choice of the parameter, the efficiency is decreased even in cases where an optimum parameter would guarantee a moderate increase. Furthermore, in medium-sized bodies, Russian roulette is not efficient in heavy absorbers and, again, the efficiency varies quite drastically with the variation of the survival probability. The calculations show that survival biasing with Russian roulette may result in a considerable efficiency increase in large bodies (Pe > 0.8) and especially for not-too-strong absorbers (c > 0.4). In these cases, the efficiency is a slowly varying function of the Russian roulette parameter.34 In Table 7.2, the calculated efficiency ratio S(n) in Equation (7.38) is compared to numerical experimental values. The latter were obtained from a Monte Carlo simulation of the collision rate in a sphere of optical radius 3.61 (Pc = 0.8). The calculations and experiments were carried out for selected values of the survival probability c and number of collisions n before Russian roulette. The Russian roulette parameter is deduced from these quantities according to the relation W th = C"

TABLE 7.2 0.3

0.5

0.7

0.9

n

S(n) Exp n S(n) Exp n S(n) Exp n S(n) Exp

2 3 4

0.90 0.98 1.12

0.90 0.99 1.15

2 3 4

0.86 0.82 0.84

0.83 0.80 0.81

3 5 7

0.83 0.79 0.81

0.85 0.78 0.80

6 11 17

0.88 0.85 0.85

0.85 0.82 0.82

There is a rule of thumb well known by practitioners for choosing the Russian roulette parameter. According to this popular wisdom, Wth 0. 1 is a reasonable choice. The approximate results above seem to corroborate this guess in the sense that the optimum

458

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

numbers in Table 7.2 yield the parameter values 0.09, 0.125, and 0.168 for survival probabilities 0.3, 0.5, and 0.7, respectively. For c = 0.9, the value of wth may vary between 0.12 and 0.31 (corresponding ton = 19 and n = 11, respectively) without considerable change of the efficiency. Nevertheless, it should be emphasized here, too, that the above results should be considered more as illustrations of an optimization method than as a definite recipe for determining the Russian roulette parameter. It has also been seen in the simplified example above that, in certain cases, the efficiency of a game with Russian roulette is very sensitive to the choice of the parameter. This may be even more the case in more complicated problems. Competent practitioners say26 about the Russian roulette parameter that "It is very problem dependent and its setting is an art." A systematic study of this common Monte Carlo tool is very much needed, and such a study may lead to surprising results, astonishing even for old practitioners. D. OPTIMIZATION BY DIRECT STATISTICAL APPROACH The approximate analytical optimization models so far investigated in this Chapter were seen to reflect certain characteristic properties of a real simulation and provided us with useful qualitative information on the efficiency-increasing schemes considered. Nevertheless, these models are so oversimplified that the quantitative results they produce may not be considered realistic. The loss of information caused by the application of simplified models can be compensated for to some extent by the application of parameters determined from numerical experiments. In the introduction, this procedure was called the direct statistical approach. In this section, we illustrate the method in two simple cases; more elaborate models will be given in subsequent Chapters. Let us first consider the optimization of a single-surface geometrical splitting procedure by the direct statistical approach. Assume that particles are started from a "source region" and we wish to estimate the number of particles that reach a "detector region". For the sake of simplicity, we suppose that the two regions have no common part and there exists a surface that completely separates the two regions in such a way that any particle that starts from the source region and reaches the detector region must cross the surface. We define a splitting procedure in which a particle is split into n fragments when it crosses the surface for the first time, but no repeated splitting is played in case of a second or further crossing. Optimization of the procedure consists of selecting an n value that maximizes the efficiency of the game. The efficiency will be formulated in terms of "average probabilities" which describe a "typical particle" and which will then be defined through experimental values. Let p, be the probability that a typical source particle reaches the splitting surface and let p, be the probability that a typical fragment starting from the splitting surface reaches the detector region. Let w, denote the average weight of a unit-weight starter when it reaches the splitting surface and w2 denote the average weight of a fragment in the detector region if its weight was unity at the splitting surface. Then the quantity to be estimated can be expressed as R = P1P2w1w2

(7.39)

Now, in an n-for-one splitting, the probability that k out of the n fragments of a starter reaches the detector region after a splitting is qk = PI(Ic)1) (1

P2)n— k

and since the average weight of a fragment is w,/n, its average weight in the detector region

459 is w2w,/n, and the expected score is MI =

k=1

k

11

W2qk PIW1W2P2 = R

i.e., the procedure is unbiased. The variance of the score due to a unit-weight starter is wl

n

D2 = E (k — w2)2clk k= 1 n

(P1w1w2P2)2

= P1\042(1 — pOin+ p1(1 — pi)wiwN i.e. C1

D2 =

—p2

n

pi)pd

(7.40)

with A = PiP2wM In order to optimize the efficiency, we also have to determine the average number of collisions to be played. Let N, be the average number of collisions suffered by a starter before it reaches the splitting surface and N2 be the number of collisions played with a fragment before detection. Then the average number of collisions played in a history of a starter is N = N, + p,nN2 and the quality factor of the game follows from Equation (7.40) as Q = D2N =

P L

n

P2 + (1 — p,)pd(N, + p1 nN2)

(7.41)

Minimization of Q with respect to n yields nopt =

- p2)N1

1/2 1 C(1 - p1)P1P2N2J (1

(7.42)

Therefore, in order to optimize the splitting procedure above, we have to estimate the parameters p, and N, (i = 1, 2). This can, for example, be done in a short preliminary run by scoring the average number of particles reaching the surface and detector and the number of collisions played meanwhile. In certain cases, these parameters may be expressed by parameters more directly characterizing the system, and then these parameters are estimated in preliminary runs. For instance, if transmission through a thick slab between x = 0 and x = X is estimated, the number density of the particles is rather well-approximated by an exponential of the form e -", with some X value. If the splitting surface is located at x = x., then one can put approximately P1 =

P2 = e

x.)

(7.43)

460

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

whereas the number of collisions is expressed as N, = N„(1 — e -x

N2 =

1—e

-

(7.44)

The optimum splitting ratio then follows from Equations (7.42) through (7.44) as nop, = (131132) - "2

=

(7.45)

e - "12

In this case, we have a single parameter, X, to be estimated in preliminary calculations. Notice that the optimum n value in Equation (7.45) is analogous to that in Equation (7.17), obtained from the straight-ahead model, since X = 1 — c in this model. The similarity is due to the approximation in Equation (7.43), which implicitly neglects the effect of particles leaving and reentering the respective half-slabs, as in the straight-ahead model. A slightly more general presentation of the method outlined here can be found in Reference 14; an elaborated theory based on the same principles was worked out by Dubi et al.9-" The considerations will be extended to several splitting surfaces in Chapter 7.11 Let us also note that this simplified model reflects the danger of oversplitting. The quality factor in Equation (7.41) decreases from n = 1 to n = n„, and then it increases essentially linearly with increasing n. Therefore, when choosing a splitting ratio n less than n„,„ the worst that can happen is that the efficency of the game will not be considerably higher than that of the analog game. On the other hand, if the splitting ratio is much larger than its optimum value, the efficiency of the game may be even lower than it would be without splitting. This is due to the fact that for low values of n the variance of the score remains finite, whereas the number of collisions tends to infinity with increasing n. A direct statistical approach to optimization of the path-stretching parameter can be deduced from the original idea of the exponential transformation. Assume that we wish to emphasize the free flights of the particles along a given direction and also that the position of a particle can be characterized by a unique distance value measured along this direction. For example, in the case of penetration through a slab, the direction is the one perpendicular to the surfaces of the slab and the distance is the depth of the point in the slab. In calculating the escape from cylindrical or spherical bodies, the favored direction may, for example, point outward along the radius, and the position is characterized by the radial coordinate. Let co, denote the favored direction for a particle at P = (r, to, E) = (r, E) and let x be the distance value corresponding to r. Specifically let x be the projection of r on the direction to,. The expectation of the score in the analog game satisfies the now very familiar Equation (5.80). With the notation of Equations (5.7) and (5.8), this equation reads M,(r,E) = f dr'T(r + r' lE)f(r,r1E) (7.46)

+ fdr1 T(r —> r' lE) f dE'C(E —> Elr')M1(r' ,E) where, according to Equation (5.32) T(r —> rlE)dr' = cr(r',E)exp{ —dta(r + to.),E)}8((r 0 Ir'

—r

(7.47)

Let us now apply the exponential transformation in the form Ati(P) = e -b'Ml(P)

(7.48)

461 where the parameter b is for the moment undefined and should be chosen such that the variance of the estimate is considerably lower than that in the analog game. Multiplying Equation (7.46) by e -", it is seen that Att,(r,E) = fdr'5-(r —> elE)[wf(r,r1E)e -"] + fdr'5-(r —> elE)1 dE[wC(E -> E'lr)].tt i (r',E')

(7.49)

where ler] (r' .5-(r -* r'IE)dr' = 6-(r',E)exp{ - f - dtO-(r + to),E)}8(

-

(7.50)

and w = o-(r',$)/O-(t',E) with fir',r1 0

dt&(r + tw,E) =

dto-(r + tw,E) - b(x' - x)

(7.51)

0

Now, since x' - x is the projection of (r' - r) to co. x' - x = Ir'

r1/(ww-) = Ir'

r1/1-1-

and the stretched cross section follows from Equation (7.51) as Cr(r,E) = r(r,E) -

(7.52)

where p. is the cosine of the angle between w and w„:

Equation (7.49) defines a transformed game in which the score in an intercollision free flight is wfe -b"", the transition kernel is 5-, and the collision kernel is w • C. As was seen in Chapter 5. V.E, this game is equivalent to a path-stretched game with the contribution function f, kernels '5 and C, and stretched cross section Q, and the statistical weight of the particle is to be multipled by we -boe -.) in the free flight from P to P' = (r',E). So far, we have merely rephrased the derivation of a path-stretched game. Now, the parameter b can be determined from numerical experimental values as follows. The qualitative results obtained from the straight-ahead model in Section B suggest that the pathstretched game will be nearly optimum if the transformed moment At i (P) is independent of the position, or, what is practically the same, if the density of the simulation particles in the transformed game is constant. As a first step, let us determine from preliminary runs a value X such that M,(P) Ae"

(7.53)

462

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

i.e., the (analog) expected score varies approximately exponentially with x. In the majority of practical deep-penetration problems, the spatial drop of the particle density can be approximated quite well by an exponential, thus, approximation (7.53) is reasonable. The value of X in Equation (7.53) can be determined by two ways. One possibility is to start particles from different positions (characterized by different x values) and fit Equation (7.53) to the scores. Alternatively, one may score the flux integral in small regions about selected x values and fit an exponential of the form Be -'`" to the estimated values. Having obtained an estimate of X in Equation (7.53), the score in the transformed game will be approximately independent of the position if Ati(P) = e '"M,(P) = Ae(x b)x cousy i.e., if b=X

(7.54)

Accordingly, a quasi-optimum stretched cross-section is 6-(13) = o'(P) — AIL

(7.55)

where A is the coefficient of x in the exponential spatial drop of the particle density in the analog game and kt is the cosine of the angle between the actual flight and the preferred direction. Note that in certain cases the stretched cross section in Equation (7.55) may become negative, which poses specific problems in the simulation. This matter will be discussed in Chapter 7.111. The optimal procedure outlined above was successfully applied in practical problems. 36 It will be seen in Chapter 7.111 that by defining a transformed game in which not only the transition kernel but also the collision kernel is biased, the direct statistical approach applied above will determine an approximation to a zero-variance game.

II. OPTIMIZATION OF GEOMETRICAL SPLITTING Geometrical splitting is one of the simplest variance-reducing and also efficiency-increasing techniques, and is used in almost all general and special-purpose Monte Carlo codes. It is especially favored in deep-penetration calculations, but is also efficient in enhancing the particle population in regions where the analog particle density is low. In spite of its conceptual simplicity, its use is still based mainly on intuition, practice, or "rules of thumb". Optimization of geometrical splitting has recently gained considerable attention, mainly because of the powerful mathematical tools provided by the Monte Carlo moment equations and the concept of the direct statistical approach. Early results in optimum splitting schemes''''' are based on very simple models, similar to those presented in previous Chapters. The variance of the score in the parallel use of splitting and exponential transformation was investigated by Sarkar and Prasad46 on the basis of the moment equations. Juzaitis21 extended the investigations to the efficiency of a game with splitting and proposed to solve the moment equations by a standard Sn code. The results so obtained for single-surface splitting in monoenergetic homogeneous simulation may serve as reference values for more sophisticated schemes. Dubi, Elperin, and Dudziak"° and later Dubi" presented a very detailed description of a general fixed-surface splitting game through the direct statistical approach. Practical applicability of their analysis is still hindered by the great number of bulk parameters to be determined in the model. Nevertheless, simplification of the model may yield feasible automatic optimization schemes.

463 An easy-to-use direct statistical approach to a deep-penetration splitting scheme,' based on considerations similar to those in Section 7.I.D, will be presented in Sections B and C, and will be generalized to a more rigorous model through the concept of virtual continuous splitting" in Section D. The continuous splitting model is optimized in Section E, and its practical applicability along with numerical results are demonstrated in Section F. A powerful in-code efficiency-increasing method, called the weight window technique,2.3•19 is outlined in Section G. A. GEOMETRICAL SPLITTING IN TERMS OF REGION IMPORTANCES As was seen in Section 5.IV.B, geometrical splitting in its most general form involves a large number of functions and quantities to be determined by the user. Recent theory and practice of optimization falls far short of covering the general case and addresses the following problem. For a given domain of simulation and a given reaction rate to be estimated, let us determine a system of splitting surfaces S, and a set of numbers n,. The surfaces and numbers define a splitting procedure in such a way that whenever a particle crosses S, from one direction, it is split into n, particles (the fragments all having 1/n, times the weight of the original particle), whereas if the particle crosses the surface Si from the opposite direction, Russian roulette is played with a survival probability 1/n1. The splitting surfaces divide the domain of simulation into distinct regions (also called cells), and the division makes it possible to define the splitting procedure in a more special way. Instead of assigning splitting ratios to each surface, let us define* the importance of every region and let the splitting procedure be modified so that a particle entering a region of importance I is split into a number of fragments, each having a weight W' =

(7.56)

with some constant W„. The procedure can be realized in practice in the following way: 1. Let W be the weight of a particle entering the region and let a = W/W' = WI/W0

(7.57)

2. If a is an integer, then the original particle is split into n = a fragments when entering the domain. 3. If a is not an integer but is greater than one, i.e., if a = n + v,

n>_1,

0 v < 1

then n + 1 fragments are started after the splitting with a probability v and n fragments with a probability 1 — v. The expected number of fragments then is (n + 1)v + n(1 — v) = a, and thus the expected total weight of the fragments is aW' = W, i.e., the weight of the incoming particle is preserved. 4. If a is less than unity, the particle survives a Russian roulette with a probability v = a and is killed with the complementary probability. (Note that this is a special case of Step 3 with n = 0.) * The importance function introduced here should not be confused with the function defined under the same name in Chapter 4. Use of identical names, although unfortunate, is well established in Monte Carlo theory. Importance here is assigned to a region, whereas the adjoint function in Chapter 4 defines the importance of a particle in future scoring.

464

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

In this special formulation of the splitting procedure, optimization means a choice of the regions and their importances such that the efficiency of the game is maximum. In practice, the structure of the domain of simulation more or less determines the surfaces and regions to be used in the splitting procedure, and thus the main emphasis should be placed on the optimum choice of the importances. Considerations to this effect are given in the next sections for essentially one-dimensional deep-penetration problems. B. A SIMPLE METHOD Let the system considered be composed of a sequence of parallel, infinite slab regions with faces perpendicular to the x axis. Let the surfaces of region i be located at x;_, and x, (x,_ < x). Assume that particles enter the system at x„ and let the quantity to be estimated be the number of particles that leave the outermost, m'th, region. In the derivations below, we assume that 1. The domain of simulation is nonmultiplying. 2. Apart from splitting and Russian roulette, the game is analog. 3. The probability that a particle crosses repeatedly any region surface is negligible. 4. The thicknesses of the regions are sufficiently large. The last two assumptions necessitate further explanation. Let t, denote the probability that a particle entering the region i at x; _, (i.e., in a positive direction) will eventually leave the region at x,. In brief, t, is the transmission probability through region i. Now, assumption 3 may be substituted by the premise that the number of particles entering region (i + 1) from region i is proportional to t,. Assumption 4 means that t, is considerably less than unity. The latter assumption is not essential, as will be clear from the derivation. Let I, denote the importance of region i. This value is to be optimized to yield a maximum efficiency. According to Equation (7.56) and also assumption 3, the weight of any particle in region i is W, = Wo/I,

(7.58)

Splitting or Russian roulette is played at the region boundaries, acid in view of Equation (7.57), a particle entering region i from region (i — 1) is split into a, = W, ,/W, =

(I0 = 1)

„

(7.59)

fragments (on an average). Then the splitting rules in 1 through 3 in the previous Section apply with a; =n +v; where n, and v, are the integer and fractional parts of a,. Let k,_, denote the number of particles that leave region (i — 1) in a positive direction (i.e., toward region i). Let us emphasize that k,_, is the number of simulation particles, i.e., progeny, of the analog original particles after a number of possible splittings. Accordingly, the number of particles continuing the random walk from the surface x,_, in region i is k: =

+ j,

0

j

(7.60)

465 with a probability —

IT(11k,-,) =

(7.61)

The probability that k, particles out of the k, that cross x,_, will reach the surface at x, follows from the simple binomial law as k p(kilk:) = (k t•y.

• ‘k!-ki

,

0

k:

(7.62)

Hence, the probability that k, particles leave region i in a positive direction provided k,_, particles reach the surface at x„ is k,_

=E j=0

Trolki_op(kilk)

(7.63)

Equations (7.60) through (7.63) describe the transmission of particles through a given region. Let P,(k,) denote the probability that k particles leave region i in a positive direction due to a single starter at the surface at xo. Then, obviously, the following recurrence holds: 13,(k) =

E

(7.64)

q(k,lk,_

k,_ , =0

where Po(ko) = and, since no splitting is assumed at the starting surface I,= 1,

n, = 1,v,=0

Equations (7.60) through (7.64) give the statistical description of the game, and as such, they substitute the score probability equation introduced in Chapter 5 for a rigorous treatment. What we are eventually interested in is the first and second moments of the number of particles leaving region m in a positive direction. These moments follow from the master Equation (7.64) in a recursive manner. If (10 denotes the r-th moment of the number of particles that leave region i in a positive direction, then (K)

=E

lq,(k)

It is shown in Appendix 7A that for r = 0, 1, and 2, we have (k?) = 1 (k,) = t,a, (kJ =

tia; J=1

(7.65)

466

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

and (kD = t?a (1q_ ) + lt1(1 — ti)ai +

v,(1 -

(7.66)

Now let T, denote the overall probability of transmission through the slab from x„ to xm. Then, by assumption 3 (7.67) and according to Equations (7.59) and (7.65) (10 = H ,=1

= T,I,

(7.68)

Since, by Equation (7.58), the weight of a particle leaving region i is w0/I„ the expected score at the m-th surface, i.e., the expected number of particles leaving the slab, is MI = Wm(km) = WoTm An exact model would obviously give the same expectation, and therefore it is apparent that the model investigated here is unbiased in spite of the seemingly rough assumption 3. It is easily seen from Equations (7.60) and (7.61) that the expected number of particles entering region i through the surface at x,_ is Wi > = ai(k, ,) = T1

(7.69)

as is expected from the definition of the splitting ratio a,. Before establishing a closed expression for the second moment of the score, let us make assumption 4 more specific. We shall assume that the following inequality holds for every

is ty,(1 — v,) 13")At(P")

(7.140)

where Fi,(P) = exp[b(P)]1,(P)

(7.141)

and I,(P) is the analog first-flight score in Equation (7.135). Let us now assume that the path-stretching function b(P) vanishes at the boundary of V for directions pointing outward, i.e., let b(P,,) = 0

for

con,

0

where n, is the outer normal of V at re. Then Equation (7.141) with Equation (7.135) reads .61 ,1(P) = exp[b(P)]exp[ — T(P,131)] = exp[ —7r(P,P')]

491 where T(P,P' ) = f dt&(r + tw,E) 0 the optical distance between P and P' in a medium where the total cross section is equal to the stretched cross section. Thus, the transformed moment Equation (7.140) becomes Pb

.M ,(P) = exp[ —;(P,P')] + J dP'T(P,P')IdP't(P',P")./lit i (P")

(7.142)

in perfect analogy to the analog Equations (7.134) and (7.135). Hence, the transformed game is to be played just like any analog game, but the coordinates of the events are to be selected from the kernels 'T and C in Equations (7.137) and (7.138), and the analog score is obtained from the transformed score according to Equation (7.136). Alternatively, the total score in the transformed game will be unbiased if the particles are started from the transformed (nonnormalized) source distribution Q(P) = Q(P)exp[ — b(P)]

(7.143)

as demonstrated in Section 5.V.D. The advantage and disadvantage of the transformed game vs. the equivalent nonanalog game were discussed at the end of Section 5.V.C. What is important here is, that the transformed game in its present form does not involve any statistical weight and hence can be treated as an analog game. Let us now investigate the second moment of the score in the transformed game. Thus far, we have not specified the estimator used in the simulation since all the partially unbiased estimators yield the same expected first-flight and final scores (cf. Chapter 5.VI). Different estimators, however, yield different variances. Here we consider two different types of estimators. First, a zero-variance scheme with a last-event estimator is derived; next, an equivalent scheme with the expectation estimator is established. Assume that only those flights contribute to the score that end outside the region V, i.e., the score assigned to a flight from P to P' is 11 if PEV,

P' (% V

F(P,P') =

(7.143) 0,

otherwise

This type of leakage estimator is the most obvious one, and it is often called the last-event estimator. The second moment of the transformed game with this contribution function follows from Equations (5.81) or (5.59) as Ph Alt2(P) =

exPl — ;(P,Pb)1 +

dP'T(P,P')IdrC(P',P'')/tt 2(P")

(7.144)

Note that Equations (7.142) and (7.144) are identical, i.e., they have the same solution ht2(P) = Ati(P) The equality is heuristically obvious since the final transformed score is either zero or one.

492

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

The variance of the score due to a particle that starts from P is V2(P) = Alt2(P)

MAP) = Ati(P)[ 1 — 3 Ut,(P)]

and evidently the variance is zero if /14,,(P) = 1. Taking note of Equation (7.136), this equality holds if expE — b(P)1 = M,(P) Hence, the optimum path-stretching function, which results in zero variance with the lastevent estimator, is b(P) = — logeM,(P)

(7.145)

where M,(P) is the expected score due to a particle started from P in the analog game. Note that M,(Pb) = 1 at the boundary of V for outgoing directions (the probability of the leakage is unity) and thus b(Pb) = 0, as required. In view of Equations (7.137) through (7.139), we have the following theorem. Theorem 7.1 — An exponential transformed game has zero variance if the last-event estimator is used and the path-stretching function satisfies Equation (7.145), i.e., if the stretched cross section is &(P) = cr(P) — coVlogeMI(P)

(7.146)

the transition kernel has the form T(P,P') = T(P,131)&(P')M,(PTcr(P')Mi(P)

(7.147)

and the collision kernel is (P' ,P") = C(131 ,P")o-(P')M,(P")/4;(131)Mi(P')

(7.148)

The source density in a zero-variance game follows from Equations (7.143) and (7.145) as Q(P) = Q(P)MI(P) At this point, the reader may have the impression that the scheme proposed here does not fit into the general form of the zero-variance, partially unbiased game derived in Section 5.VIII.A and may suspect that something went wrong in the derivation. Although the kernels in Equations (7.147) and (7.148) seem to be very different from those in Equations (5.288) through (5.290), in fact, they are special cases of the latter ones, as demonstrated below. Let us consider the following identity. ,;(171 ')Mi(P')/0-(13 ')

= [o(P') — oNlogeM,(P')]M,(P')/(r(P') = M,(P') — koVM,(P')Vcr(P') = e'('''''')(oV[e —T(P'P')MiOn]kr(P')

which holds for any point P with T(P,P') given in Equation (7.132). Let us now multiply

493 both sides of the equality by T(P,P') and integrate with respect to P' from P to infinity along w to obtain

j dP'T(P,P')[Ci(P')M,(P')/o-(P')] = M,(P) Comparing this relation with Equation (5.12), we conclude that (7.149)

&(13')MI(P')/0-(P') = J*(P')

where t15*(P) is the adjoint collision density. Hence, the optimum transformed kernels in Equations (7.147) and (7.148) can be rewritten as t(P,P') = T(P,P')41*(P')/M,(P) = T(P,P')f dP"C(P',P")M,(P")/M,(P) for P'eV

(7.150)

and C(P,P') = C(P,P')M,(P")/ir(P') = C(P',P")M,(P")/f dP"C(P',P")M,(P")

(7.151)

The second equalities follow from the adjoint integral transport equation (5.9): 111*(P) = f(P) + fdP'C(P,P')fdP"T(P',P")4,*(P") = f(P) + fdPV(P,P')M,(P') and from the fact that f(P), the weighting function in the escape rate, is zero inside V, i.e., LP* (P)

fdr C(P,P')M (P')

if

P'

E

V

(7.152)

Now, comparing the kernels in Equation (7.150) and (7.151) to the general zero-variance kernels derived in Section 5.VIII.A, the mutual correspondance is obvious. We note that the kernels in Equations (7.150) and (7.151) are equivalent to those in Equations (7.146) and (7.147) if the exact expected score, M,(P), is used in the formulas. In the case of approximations to the optimum formulas, however, they will not be equivalent any more, as will be seen in Section B. As always happens with a zero-variance scheme, its realization assumes knowledge of the quantity to be estimated. Therefore, a game with exactly zero variance is never feasible in practice; nevertheless, approximations to the optimum schemes may lead to substantial variance reduction. Practical realizations will be considered in Section C. The scheme above was derived under the assumption that the last-event estimator is applied. Zero-variance schemes with a general estimation procedure were derived in Section 5.VIII.A, and we mentioned there that although two ideal schemes with different estimators are equivalent from the point of view of efficiency, the efficiencies of their approximate

494

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

realizations may be different. Therefore, it may be useful to introduce zero-variance pathstretching schemes with estimators different from the last-event estimator. In what follows, we introduce a scheme with the expectation estimator. The contribution of a flight from a point P is then equal to the expected first flight score, L(P), in Equation (7.135), i.e., f(P,P') = L(P) A zero-variance scheme follows directly from the general form derived in Section 5.VIII.A. Inserting the expectation estimator above into the general Equation (5.288), the transformed transition kernel reads t(P,P') = T(P,P')[11(P) + f dP"C(P',P")Mi(P")]/MI(P)

(7.153)

Introducing again the adjoint collision density, two equivalent forms of the transition kernel follow from Equations (7.149), (7.150), and (7.152) as T(P,P') = T(P,P')[I,(P) + tfr*(P')]/M,(P) = T(P,P') o-(P')II

(P) +

g(r)Mi(P)

(7.154)

The transformed collision kernel in a general zero-variance scheme is given in Equation (5.290), and it is seen that the collision kernel is independent of the contribution function. Thus, in the zero-variance leakage estimation, this kernel is the same as the one given in Equation (7.148) or (7.151) for both the last-event and expectation estimators. There is one point to be emphasized here. The scheme with the expectation estimator as derived above is not an exponential transformed game. This is seen from Equations (7.153) and (7.154) since the terms multiplying the analog transition kernel on the RHS of the equations do not factorize to functions depending separately on P and P', respectively, and therefore T(P,P') in these equations does not conform to the exponential transformed (or importance-sampling) form of Equation (7.137). Therefore, realization of the scheme (if it were possible at all) would only be practicable by the use of a nonanalog simulation and statistical weights. Nevertheless, approximately optimum path stretching can also be defined with an expectation estimator, as will be discussed in the next section. B. DISCUSSION OF THE SCHEMES Before turning to practical realizations, we shall briefly discuss some specific properties of the schemes above. Let us first note that the probability of an absorption is zero in both schemes, i.e., the transformed collision kernel is normalized to unity:

f

dP"C(P',131') = 1

This is a common property of all partially unbiased zero-variance schemes, as pointed out in Section 5.VIII.A. Unit survival probability is reached by transformation of the collision kernel. This particular property of a nonanalog game can also be produced by survival biasing (cf. Section 5.VIII.D). In this case, the nonanalog collision kernel is

.

C(P',P") = C(P',P")/fdP"C(P',P")

495 Comparing this kernel to the transformed one in Equation (7.151), we see that when using survival biasing, the optimum kernel is approximated by putting [f dP"C(P',P")M,(Pl/MI (P") ~ fdP"C(P',P") Besides survival biasing, however, the transformed collision kernel also differs from the analog one in its direction-energy dependence, and it is this specific biasing that makes the simulation of a scattering very efficient from the point of view of the final estimate. Heuristically, the biasing emphasizes directions and energies where large contributions to the score are expected [i.e., where M,(P") is large]. Recall that path stretching in a strict sense means the alteration of the transition kernel only (with a possible survival biasing). The zero-variance scheme above suggests that parallel alteration of both kernels may result in a more efficient simulation than path stretching alone. 12,13,20,49,50 The transition kernel in the first scheme [Equations (7.147) and (7.150)] defines a pathstretching procedure with the stretched cross section cr''' in Equations (7.139) and (7.146). The simplest path-stretching schemes mentioned in Section 7.I.B and in Equation (7.129) are special cases of this general form. Also, special cases concerning transmission through slabs were introduced in Sections 5.V.D and E. It was shown that the general form of the stretched cross section in slab geometry is cr(x,R,E) = cr(x,E) +

a

— ax b(x,R,E)

(7.155)

In most practical path-stretching applications, the stretching function is chosen as

b(x,p,,E) = b(µ,E)

x

dx' — cr(x',E) = b(µ,E) T(x,X)

(7.156)

where X is the x coordinate of the boundary of the system to be traversed and T(x,X) is the optical distance from x to X along the x axis. Then the stretched cross section becomes Cr(x,R,E) = cr(x,E)[1 — b(µ,E)]

(7.157)

The simple examples mentioned in the introduction of this Chapter are special cases of this form. For example, the stepwise directionally independent scheme by Lewitt22'29'3° uses b b(µ,E) = t o

if if

>0 0

A linear direction dependence is obtained" by putting b(µ,E) = and the exponential dependence by Murthy' follows as b(µ,,E) = 1 — e -bPNow, comparing Equation (7.157) to Equation (7.146), it is seen that in a zero-variance

496

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

scheme, the path-stretching function and the expected score are related as

p.

ax

log M,(x,µ,E) = cr(x,µ)b(p,,E)

i.e., the expected score (leakage rate) due to a particle started from P = (x,µ,E) has the form = exp

b(µ,E)

T(x,X)}

(7.158)

In practice, Equation (7.158) never holds exactly, but sometimes the expected score can be approximated fairly well by a function of this form. As an example, we recall that in the straight-ahead scattering model, Equation (7.158) holds and therefore the specific form of path stretching in Equation (7.157) together with the survival biasing were seen to yield a zero-variance scheme (cf. Section 7.I.B). More realistic approximate schemes are introduced in the next section. The results above also justify the aproximate optimization discussed in Section 7.I.D. There we assumed that with some constant value X, the expected score could be written as Mi(x,P,,E) ~ Ae -mx -- x)

(7.159)

Cr(x,R,E) = o(x,E) — Xµ

(7.160)

and we chose

Obviously, this choice corresponds to the approximation 1 — b(µ,E)i-(x,X)

X(X — x)

or, equivalently o-(x,E)b(µ,E) Now, accepting the approximate form of the expected score in Equation (7.159), the optimum collision kernel follows from Equation (7.151) as ,P") = C(P'P")/1dP"C(P',P")

(7.161)

i.e., approximation (7.159) calls forth an optimized game where simple survival biasing represents the alteration of the collision kernel. The arguments above explain the success of schemes that apply pure path stretching with survival biasing, but no angular biasing, of the collision kernel. 22,29,30,43,47 Note, however, the little trick involved when introducing Equation (7.161) on the basis of Equations (7.159) and (7.151). Indeed, if the approximation in Equation (7.159) to the expected score is accepted, then the stretched cross section in Equation (7.160) does follow from Theorem 7.1. However, the corresponding collision kernel should be derived from Equation (7.148), not Equation (7.151), since Equations (7.148) and (7.151) are equivalent only if the exact

497 expected score is put into the formulas. Using Equation (7.148), we obtain (P' P") = C(P',131')cr(P')/[o-(P') —

(7.162)

Although this form of the biased collision kernel conforms with Theorem 7.1, it seems to be less favorable than the simple survival-biased kernel for two reasons. First, its dependence on the postcollision coordinates is not altered, compared to the analog kernel. Second, the number of secondaries per collision is not unity, while both these requirements are essential in a zero-variance scheme. Approximate collision kernels possessing these properties will be introduced in the next section. Recall that the approximate optimum path-stretching scheme in Equation (7.160) was derived in Section 7.I.D under the heuristically founded assumption that the path-stretching procedure is optimum if the transformed moment ht,(P) is independent of the position of the starter. As was seen in the derivation of the first scheme in the previous section, the heuristic assumption is justified; the zero-variance scheme defines a transformed game with A/11(P) = 1 . When we introduced the zero-variance scheme with the expectation estimator (second scheme in the previous section) we pointed out that this scheme does not define a real pathstretching procedure since the biasing factor in the transition kernel, Equation (7.153), has the form [I,(P) + fdP"C(P',P")M,(P")]/M,(P) which cannot be written in the general path-stretching biasing form &(P') exp[b(P) — b(P')1/cr(P') [cf. Equation (7.138)]. This fact might suggest that the expectation estimator, when used in approximate optimum path-stretching games, would be less efficient for leakage estimations than the last-event estimator. Although in certain cases it is indeed so, it has been shown that for medium and strong absorption, the expectation estimator is definitely competitive." Approximate optimum realizations of path stretching with expectation estimators are based on the observation that in the case of deep penetration, the first-flight score is considerably smaller than the total expected score, and therefore I,(P)

=

c 1 + b 1 2b log` 1 — b =

for b satisfies the Placzek Equation (7.165). If Q(x,µ) denotes the analog source density, then the transformed source density follows from Equation (7.143) as 0(x,µ) = eb'("-x)1:2(x,1-1)44)b(1-)

(7.170)

The scheme defined by Equations (7.168) through (7.170) can only be applied for homogeneous, monoenergetic isotropic transport. With heterogeneous and/or energy-dependent problems, the cross section o- and the survival probability c depend on the position and energy of the particle. Therefore, the asymptotic solution in Equation (7.167) of the moment equation does not apply directly to such cases. Approximate optimization of the pathstretching parameter can be obtained by this method e.g., by requiring that the leakage determined by Equation (7.167) with some homogenized material constants be equal to the estimated (i.e., real) leakage rate. Thus, in the case of the isotropic unit source at x = 0, i.e., if Q(x,R,E) = 8(x)QE(E);

p40,1]

a value (bo) is to be chosen such that the homogeneous expectation in Equation (7.167) satisfies the equality

where Mi is a preliminary estimate of the leakage rate in the real system. Accordingly, we require that e -b,srx

=

500

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

i.e., that 1 X

(bcr) = - - log M,

e

Then the stretched cross section becomes 6-(x,11,,E) = o-(x,E) +

logcM,

E

(7.171)

o-(x,E)ll -

and this cross section defines a biased transition kernel of the form t(x ->

= Cr(x,µ,E)exp[

dto-(t,R,E)/µ]

The transformed collision kernel can, for example, be written by analogy to that in Equation (7.169) as* , C(x',E') = CE(E -> Fix 40 2 (3-(x',µ1 ,E)

(7.172)

where CE(E ->

= fcip,V(µ,E ->

JdE'C(W , E ->

µ',E'lx') (7.173)

i.e., CE is the analog density function of the postcollision energy and 6'(x,E') is a normalization factor: 1 fl C(x,E) = [2

, cr(x,E) ]jr(x,R , ,E)

26(x,E)/loge[il + 6((x

(7.174)

with b(x,E) = -logeM,/[Xo-(x,E)]

(7.175)

According to Equation (7.170), the source density is also to be biased and its normalized

Notice that this form of the kernel is independent of the postcollision direction distribution of the analog kernel. This makes the normalization factor 5.(x',E) very simple, and the scattering properties of the analog kernel are then reflected by the statistical weight of the particle (cf. Theorem 7.2). An alternative form of the transformed collision kernel, which is more closely related to the analog scattering law, might be e(p,,E —>

') = C(p,,E —> p.' ,E' x')e(x ' )cr(x ' ,E')/Ci-(x '

,E')

with e(x') =

p.,',E'lx')o-(x',E')/ei(x',11',E')1

The calculation of the factor e(x'), however, may be rather troublesome in case of anisotropic scattering.' This makes the latter form of the kernel less attractive than the one in Equation (7.172).

501 form is 0(x, ,E) = 8(x)QE(E)

1 cr(x,E) q(x,E) Cr(x,I.L,E)

(7.176)

where — q(x,E) = f dp. cr(x,p,,E) o

1 1oge[l — Ii(x,E)] b( x,E)

The main advantage of the scheme in Equations (7.171) through (7.176) over conventional pure path-stretching strategies is that it also involves directional biasing of the scattering and source, thus enhancing the preferred direction not only in flights, but also in the selection of the flight direction. Let us emphasize that this scheme is only an approximation of the optimum transformed kernels derived in Section A and, as such, it does not define an exponential transformed game. Therefore, the game is necessarily played nonanalog with the use of statistical weights according to the weight-generation rules in Section 5.V.B. The statistical weights corresponding to this path-stretching game are detailed in Theorem 7.2 below. [For the sake of completeness, we note that by setting b(x,E) equal to a constant value, the scheme may be rewritten in the form of an exponential transformed game,' although usually less effectively than the one with the space-energy-dependent 6]. Two practical comments are proper here. First, it is seen that the collision kernel in Equation (7.172) and the source density in Equation (7.176) contain the factor acr/2cr- , which depends on the energy argument to be selected from the densities. Therefore, one might think that the selection of the postcollision energy would become more complicated because of the biasing factor. This, however, is not the case since this factor is normalized to unity in its angular variable µ ' . Thus, the postcollision energy is invariantly selected from the marginal density CE in Equation (7.173), and the postcollision direction is then drawn from the conditional density co-/2Q. Similarly, the starting energy is selected from QE and the direction from the biasing factor. Second, although the above scheme was derived assuming a fixed value of the escape rate M, obtained from a preliminary run, this value may, of course, be updated during a production run and the biasing factors may then be recalculated from more and more reliable estimates. Finally, we note that the above procedure is easily generalized to energy-dependent biasing. Indeed, if the approximate first moment M, is estimated as a function of the energy (i.e., in several energy groups), then the stretched cross section in Equation (7.171) becomes Cr(x,p.,E) = o(x,E) —

logeM,(E) = o(x,E)[1 — 116(x,E)]

whereas all the formulas following from it remain unchanged. Test calculations demonstrate that the angular biased scheme above may increase the efficiency of the estimation by as much as two orders of magnitude, compared to the efficiency of a pure path-stretching game."'" Two alternatives to the approximate method above are worth mentioning. In the first method, an approximate exponential drop of M,(x) again determines the biasing parameter. In particular, let us assume that the expected score due to a particle started from x is estimated and found to have the functional form IC41(x)

ex('-x)

502

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

Then we set the space-dependent parts of the expected scores in the homogeneous and real systems equal to each other (instead of requiring the equality of the leakage rates, as in the previous approach). Then

fi dRmi(x,R) = mi(x) ebo(x - X) =

eX(x - X)

which yields the biasing parameter as bo- = X Then the stretched cross section has the form Cr(x,R,E) = o(x,E) — X

(7.177)

instead of that in Equation (7.171), whereas the relation b(x,E) = X/cr(x,E)

(7.178)

replaces Equation (7.174). Otherwise, all the formulas in Equations (7.172) through (7.174) remain valid. In the second modification of the method, estimates of the collision rate N, absorption rate A, and flux integral F are required for the entire system. With these estimates, one defines the average quantities =1— and = N/F Then the path-stretching parameter is determined from the Placzek equation c 2b

1+b —1 1—b

(7.179)

o-(x,E) — bQµ

(7.180)

and the stretched cross section becomes Cr(x,R,E) Finally b(x,E) = bCr/o-(x,E)

(7.181)

is inserted into Equation (7.174). For an approximate solution of Equation (7.179), see Reference 6.

503 In all the approximations above, the stretched cross section is expressed as = o-(x,E)[1 — µ6(x,E)] where b(x,E) is one of the quantities in Equation (7.175), (7.178), or (7.181). The biased transition kernel then has the form 1 45-(x',R,E)expt — f dtcr(t,E)[1 — µ6(t,E)]/µ} T(x --> xlµ,E) = 71 (7.182)

= T(x —> x'lµ,E)[1 — µ13(x',E)]exp[i. dt6(t,E)o-(t,E)]

is the unbiased (analog) transition kernel. Assume that the analog where T(x --> collision kernel is written in the form = c(x',E)CE(E —> Elx1 ,µ)C,(µ --> tillx',E,E')

C(F.L,E

(7.183)

with

IdEVE(E —> E Ix'

= 11 cIWC,(11, ---* Fil ,E,E') = 1

This form of the collision kernel poses no limitation on the game since any collision kernel can be written in the form of Equation (7.183). Indeed, by setting diilc,(µ,E —>

c(x,E) = CE(E -->

--> µ',E'lx)/c(x,E)

= f

and

c(p. —>

=

—> til ,E1x)/f idp:C(µ,E

we have the factorization in Equation (7.183). The biased kernel is given by Equation (7.172) as --> li',E'lx') = C(x',E')CE(E —> El x,11)

= C(µ,E —>

-((x ,E) 2 cr(x',W,E')

CE(E --* Elx,10

c(x',E')

2[1 — p:b(x',E')]

C(',E') [c(x',E)Cp.(p.. —> 2[1 — 1.1:6(x',E')]

(7.184) Finally, let the analog source be written in the form Q(x,µ,E) = Q0(x,E)Q(µ,lx,E)

504

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

where dp..Qµ(µ1x,E) = f dxf dEQ,(x,E) = 1 Then the biased source density follows from Equation (7.176) as 0(x,i.L,E) = Q.(x,E){q(x,E)[1 — Q(x,E){q(x,E)[1 — 13(x,E)]Qp.(1-11x,E)}-1

(7.185)

We have argued that the biased game is to be played nonanalog with statistical weights. The weights to be applied follow from the weight generation rules in Section 5.V.B and are summarized in the following theorem. Theorem 7.2 — In the path-stretched game described by the densities in Equations (7.182), (7.184), and (7.185), the weight of a starter at P = (x,p„E) is Wµ(P) = Q(x,11,E)/0(x,R,E) = q(x,E)[1 — 6(x,E)1Q(µix,E) where q(x,E) =

— 6(x,E)]/6(x,E)

In a transition from P = (x,p„E) to P' = (x',µ,E), the particle weight is multiplied by w(P,P') = T(x —> x'1µ,E)/i(x ---> x'111,E) jx = exp[ — dt6(t,E)cr(t,E)]/[1 — whereas in a collision from P' to P" = (x1 4.1.,',E'), the factor multiplying the weight is wc(P',13") = C(µ,E —>

p,',E'lx')

= c(x',E)C,,(p, —> plx I ,E,E')2[1 — I.C6(x',E')]/C(x',E') where, according to Equation (7.174) c(x,E) = 26(x,E)/loge [

li(x,E)1 1 — li(x,E)j

and 6(x,E) is defined by one of the expressions in Equation (7.175), (7.178), or (7.181), depending on the approximation used.

Note that it is not necessary to calculate the exponential term in the weight factor w(P,P') in every flight separately, but it is more economical to sum up the exponents during the history and calculate the exponential only once at the time the particle escapes. In order to obtain an impression of the mechanism of variance reduction by the scheme,

505 let us determine explicitly the weight of a particle that leaves the domain of simulation. Let the particle be started at Po = (xo,µo,Eo) and let the pre- and postcollision coordinates of the i-th collision in the history be = (x„ p,, _ „E;_,) and P, = (x„p,„E,), respectively (i n + 1), where x„ < X and x, , X. Then the weight of the escaping particle = 1, is W=

1.1 w(P;_,,P:)wc(P:,P)w(P.,13'n+

= Q,,(µ„lx„,E„)q(x0,E„)[1 — µ06(x ,E )11-1 ""

[C(X, E , _ i

)

c(x,E,)

1 — p.,,fi(x„E) exp[— E ,6(x„E,_,) 1— i=i‘, -.

2C,p.„ (,lx„E, p., „Ed

— i-Ln6(xn+i,En/1-1

Now, in a monoenergetic, homogeneous isotropic transport e = c, and W=

q

e

-ba(xn+

xo)

where q=

441 —

= — loge(1 — 6)/b

The above result has two interesting consequences. First, in this simple case, one may completely forget about statistical weights during the simulation, and the weight is to be introduced in the last flight only. Thus, the homogeneous, monoenergetic, isotropic version of our scheme is analogous to an exponential transformed game. Second, since the medium at x > X is irrelevant from the point of view of the simulation, one may assume that it is filled with a black absorber of a very large cross section. Then xe+ = X, and W = qe

-"°)

(7.186)

i.e., every particle that is transmitted from x0 to X reaches X with the same weight W. It is tempting to attribute the low variance capability of the game to the fact that the contributions of all the leaking particles are nearly equal (sometimes it is so stated in the literature); however, recall that the analog game has the same property (as every transmitted particle scores unity) and the latter seldom results in zero variance. In fact, uniform scores yield low variance only if the number of transmitted particles due to a starter does not fluctuate heavily, i.e., if an approximately constant number of particles reach X in every history. Now, if 6 is chosen such that the probability M, of the transmission of an analog particle started from xo in a positive direction is mi

e - bo(X — .0)

then the weight in Equation (7.186) is related to this probability as

W = gm' On the other hand, if k denotes the expected number of particles transmitted in the biased

506

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

game (number of simulation particles), then the final score is M, = kW

k = 1/q The variance of the score in the biased game is 152 = W2k(1 — k) = wq2k(1 — k) = wi(q — I)

In contrast, the expected number of particles transmitted in the analog game is M„ each having a weight of unity, and therefore the variance of the analog game is D2 = M,(1 — M1) The ratio of the two variances is — 1 152/D2 = M,(q — 1)/(1 — M1) = e-b'(X-x°) lq— 1. There are two customary methods of overcoming these difficulties. The first method is based on the idea of delta scattering. In the second method, the stretched cross section is simply limited to acceptable values. If delta scattering is introduced (cf. Section 5.V.H), a delta-scattering cross section cro(x,E) is defined in such a way that (7.187)

16(x,E)I — cro(x,E)/ff(x,E) < 1 and the stretched cross section is modified to Cr,(x,R,E) = o(x,E)[1 — µ6(x,E)] + cro(x,E)

The free flight of a particle is selected from the transition kernel built up with the cross section /6-,0. The collision at the end of the flight is taken a delta scattering with a probability o-o/ Qs, and it will be a real collision with the complementary probability 45-4,. As was proven in Section 5.V.H, introduction of the delta scattering does not alter the distribution of real collisions. Therefore, with this artificial procedure, the problem of the negative stretched cross section in the transition is avoided without affecting the simulation of the real events. The problem of an undefined collision kernel, however, persists. The simplest thing one can do is to also insert the stretched cross section with delta scattering into the collision kernel. It is easily seen that Equation (7.187) is equivalent to the inequality 168(x,E)I =

o(x,E)b(x,E) < 1 cr(x,E) + cro(x,E)

(7.188)

Let us now use 6, = cr 6/(Q + cro) in place of 6 in Equation (7.184). Then the biased kernel becomes = CE(E —> Erix',µ) 2[1

11,68(x,,E,),

(7.189)

—

where C8(x,E) = 268(x,E)/logo[

1 + fis(x,E)1 l — 6,(x,E)]

(7.190)

508

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

Naturally, the weight factor multiplying the weight of a particle in a real collision is al, to be altered by inserting b, and e, in place of b and C in Theorem 7.2. The particle weigi is not changed in a delta scattering and is multiplied by w(P,P'), according to the theorem, in a flight between two real collisions at x and x'. Although delta scattering removes the problem of an undefined probability density, the statistical weights still may become negative and also arbitrarily large in modulus because of the weight factor 1/(1 — µb) associated with a transition. Also, delta scattering increases the number of collisions and, therefore, in unfortunate cases it may result in considerable loss of efficiency. The scheme may only be made safe from the above dangers by forcing the stretched cross section to be always positive. In this method, the modified stretching function 6,, in Equation (7.188) also replaces b in the transition kernel whenever 1411 1, and then the kernel reads T(x —> x'Iµ,E) = T(x —> x ' I µ ,E)[1 — µ68(x' ,E)]exp [ dt6,(t ,E)cr(t,E)] Using this kernel, every collision will be a real collision and will be played with the collision kernel in Equations (7.189) and (7.190). Proceeding this way, we certainly depart from the quasi-optimum game in every case when 16111 1, and therefore deterioration of the efficiency is expected. In practical problems, test calculations must determine the choice of one method or another.

APPENDIX 7A: APPROXIMATE MOMENTS OF THE NUMBER OF TRANSMITTED PARTICLES THROUGH MULTILAYER SLABS An approximate model of particle transport through slabs was introduced in Section 7.II.B and the geometrical splitting procedure is optimized in the model. For the optimization, the first two moments of the number of particles transmitted through slab regions are to be determined. According to Equations (7.63) and (7.64), the probability P,(k,) that k, particles leave region i in a positive direction satisfies the recurrence 13;(k) =

k,--

E1" E j= o

(A.1)

where the quantities appearing on the RHS are defined in Equations (7.60) through (7.62). Note that since the number of split fragments is bounded at any surface, the summation over k, in Equation (A.1) does not really extend to infinity (k, is certainly not greater than k:). Therefore, we can set the upper limit of the summation equal to k,'. The quantities to be determined are the r'th moment (r = 1, 2) of the number of particles: =

E

klPi(k)

ki

From Equation (A.1), we have

(kD = E E k;_ 1

j

i)

(A.2)

509 where, according to Equation (7.62) ki

ki

IVO =

E

klp000

=E

K-(t)

—

ti)kH

ki

Now, for the elementary properties of the binomial distribution po(k:) = 1,

p,(k:) = Kt, = (k,_,n, + j)t,

1)200 =

+

and —

where, from Equation (7.60), we have k: =

_.,n, + j

Inserting this expression, together with that of Tr(j1k, ,) in Equation (7.61), into Equation (A.2), the moments become (K) = 1 ki -

(1(,)

= E E (1,-)vx] -

v,)k'-- I --1 (k1 _ ,n, +

ki -1 j = 0

=E

t,(1c, ,n, + k1 _

(A.3)

IAA; -1)

ki_

and ki_

(10 =

EE -1j

9v1(1 -

Vi)ki

+ 2k,_ i n, j + I2./t; + = Eftfflq--1(n + 2ny, + k;_,

+

+ k_ 11);(1

li)lPi

i)

vi)

+ t,(1 — ti)a,k, 1113,_ 1 (k1 1) =

+ lt,(1 — ti)a, + t?v,(1 — 01(k,_,)

(A.4)

REFERENCES 1. Bending, R. C., Direction dependent exponential biassing, in Proc. NEACRP Meeting of a Monte Carlo Study Group, ANL-75-2 Report, Argonne National Laboratory, 1974, 295. 2. Booth, T. E., Automatic importance estimation in forward Monte Carlo, Trans. Am. Nucl. Soc., 41, 308, 1982. 3. Booth, T. E. and Hendricks, J. S., Deep penetration by Monte Carlo, Trans. Am. Nucl. Soc., 43, 609, 1982. 4. Booth, T. E. and Hendricks, J. S., Importance estimation in forward Monte Carlo calculations. Nucl. Technol. Fusion, 5, 90, 1984. 5. Burgart, C. E. and Stevens, P. N., A general method of importance sampling the angle of scattering in Monte Carlo calculations, Nucl. Sci. Eng., 42, 306, 1970.

510

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

6. Case, K. M. and Zweifel, P. F., Linear Transport Theory, Addison-Wesley, Reading, MA, 1967. 7. Clark, F. H., The Exponential Transform as an Importance-Sampling Device — A Review, ORNL-RSIC14 Report, Oak Ridge National Laboratory, 1966. 8. Dubi, A. and Dudziak, D. J., Optimal choice of parameters for exponential biasing in Monte Carlo, Nucl. Sci. Eng., 70, 1, 1979. 9. Dubi, A., et al., Geometrical splitting in Monte Carlo, Nucl. Sci. Eng., 80, 139, 1982. 10. Dubi, A. and Dudziak, D. J., Extended model of geometrical surface splitting in Monte Carlo, Nucl. Sci. Eng., 83, 487, 1983. 11. Dubi, A., General statistical model for geometrical splitting in Monte Carlo, Tr. Th. Stat. Phys., 14, 167, 195, 1985. 12. Dwivedi, S. R., Zero variance biasing schemes for Monte Carlo calculations of neutron and radiation problems, Nucl. Sci. Eng., 80, 172, 1982. 13. Dwivedi, S. R., A new importance biasing scheme for deep-penetration Monte Carlo, Ann. Nucl. Energy, 9, 359, 1982. 14. Ermakov, S. M. and Mikhailov, G. A., Course of Statistical Modelling, Nauka, Moscow, 1976 (in Russian). 15. Everett, C. J. and Cashwell, E. D., Cost of Splitting in Monte Carlo Transport, LA-7189-MS Report, Los Alamos Scientific Laboratory, 1978. 16. Goertzel, G. and Kalos, M. H., Monte Carlo methods in transport problems, in Progress in Nuclear Energy, Series I, Physics and Mathematics, Vol. 2, 1958, 315. 17. Gupta, H. C., A class of zero-variance biasing schemes for Monte Carlo reaction rate estimators, Nucl. Sci. Eng., 83, 187, 1983. 18. Gupta, H. C., Importance biasing scheme for expectation estimator in deep-penetration problems, Ann. Nucl. Energy, 11, 283, 1984. 19. Hendricks, J. S., A code-generated Monte Carlo importance fuction, Trans. Am. Nucl. Soc., 41, 307, 1982. 20. Hendricks, J. S. and Carter, L. L., Anisotropic angle biasing of photons, Nucl. Sci. Eng., 89, 118, 1985. 21. Juzaitis, R. J., Predicting the cost of splitting in Monte Carlo particle transport, Nucl. Sci. Eng., 80, 424, 1982. 22. Karcher, R. H., et al., The application of track-length distribution biasing in Monte Carlo deep-penetration calculations, Nucl. Sci. Eng., 31, 492, 1968. 23. Kahn, H., Random sampling (Monte Carlo) techniques in neutron attenuation problems, Nucleonics, 6, 27, 36, 60, 1950. 24. Kahn, H., Modification of the Monte Carlo Method, Rand Report P-132, Rand Corporation, Santa Monica, 1949. 25. Kalos, M. H., Importance sampling in Monte Carlo shielding calculations, Nucl. Sci. Eng., 16, 227, 1963. 26. LASL Group X-6, MCNP — A General Monte Carlo Code for Neutron and Photon Transport, LA-7396M Report, Los Alamos Scientific Laboratory, 1979. 27. Lanore, J. M., Theoretical Principles of a Weighting Method in the Monte Carlo Calculations, CEA Note N-817, Nuclear Research Center, Fontenay-aux-Roses, 1967. 28. Lanore, J. M., Weighting and biasing of a Monte Carlo calculation for very deep penetration of radiation, Nucl. Sci. Eng., 45, 66, 1971. 29. Lewitt, L., A simplified Monte Carlo approach to deep penetration problems, Trans. Am. Nucl. Soc., 7, 44, 1964. 30. Lewitt, L. B., The use of self-optimized exponential biasing in obtaining Monte Carlo estimates of transmission probabilities, Nucl. Sci. Eng., 31, 500, 1968. 31. Leimdtirfer, M., A Monte Carlo method for the analysis of gamma radiation transport from distributed sources in laminated shields, Nukleonik, 6, 58, 1964. 32. Leimdorfer, M., On the transformation of the transport equation for solving deep penetration problems by Monte Carlo method, Trans. Chalmers Univ. Technol., 286, 1964. 33. Leimdtirfer, M., On the use of Monte Carlo methods for calculating the deep penetration of neutrons in shields, Trans. Chalmers Univ. Technol., 287, 1964. 34. Lux, I., Variance and Efficiency in Transport Monte Carlo, KFKI-1979-35 Report, Central Research Institute for Physics, Budapest, 1979. 35. Lux, I., Variance versus efficiency in transport Monte Carlo, Nucl. Sci. Eng., 73, 66, 1980. 36. Lux, 1, HEXANN-EVALU — A Monte Carlo Program System for Pressure Vessel Neutron Irradiation Calculation, Research Report 210, Technical Research Centre of Finland, Helsinki, 1983. 37. Lux, I., A handy method for approximate optimization of splitting in Monte Carlo, Nucl. Sci. Eng., 83, 198, 1983. 38. Lux, I., A continuous model for the optimization of splitting in deep-penetration Monte Carlo, in Proc. 6th ICRS, Vol. 1, Tokyo, 1983, 219. 39. Lux, I., On zero variance Monte Carlo path-stretching schemes, Nucl. Sci. Eng., 84, 388, 1983.

511 40. Lux, I., On Geometrical Splitting in Nonanalog Games, KFKI-1985-04 Report, Central Research Institute for Physics, Budapest, 1985. 41. Murthy, K. P. N., Direction dependent exponential biasing in Monte Carlo simulation of radiation transport in thick shields, in Proc. 5th ICRS, Science Press, Princeton, 1977, 598. 42. Murthy, K. P. N., Tracklength biasing in Monte Carlo radiation transport, Atomkernenerg.-Kerntech., 34, 125, 1979. 43. Murthy, K. P. N., A comparative study of different tracklength biasing schemes, Ann. Nucl. Energy, 7, 389, 1980. 44. Ponti, C., Angular and tracklength distribution biasing in Monte Carlo deep penetration calculations, in Proc. Conf. New Developments in Reactor Mathetmatics and Applications, ORNL-RSIC-29 Report, Oak Ridge National Laboratory, 1971, 27. 45. Ragheb, M. M. H., Monte Carlo parametric importance sampling with particle tracks scaling, Atomkernenerg.-Kerntech., 39, 198, 1981. 46. Sarkar, P. K. and Prasad, M. A., Prediction of statistical error and optimization of biased Monte Carlo transport calculations, Nucl. Sci. Eng., 70, 243, 1979. 47. Spanier, J., An analytical approach to variance reduction, SIAM J. Appl. Math., 18, 172, 1970. 48. Spanier, J., A new multi-stage procedure for systematic variance reduction, in Proc. Conf. New Developments in Reactor Mathematics and Applications, ORNL-RSIC-29 Report, Oak Ridge National Laboratory, 1971, 760. 49. Tangk, J. S., et al., Monte Carlo shielding calculations using event-value path-length biasing, Nucl. Sci. Eng., 62, 617, 1977. 50. Tang, J. S., et al., Angular biasing of the collision process in multigroup Monte Carlo calculations, Nucl. Sci. Eng., 64, 837, 1977.

513

INDEX A Absorption defined, 25 photonuclear, 47 Adjoint Monte Carlo analysis, 126-138, 145-146 Analog games defined, 5 plausible modifications, 55-56 replacement of absorption and leakage by statistical weight reduction, 56-58 replacement of multiplication by increase of weight, 58 Russian roulette and splitting, 58-59 simulation of random walk collisions in general, 41-43 direction cosines of particle after scattering, 54 matter/neutron interactions, 47-54 mattter/photon interactions, 43-47 path length selection, 39-41 scoring, 54-55 selection of source parameters, 34-38 Anisotropic scattering, 73-75

B Bilinear forms estimation, 287-289

C Capture events, 48 Carlson method of energy selection, 68-69 Cartesian coordinates, 23 Central limit theorem, 63 Charged particle producing reactions, analog simulation, 53 Coherent (Rayleigh) scattering, 47 Collision densities defined, 27-29, 99-100 equations connecting, 101-103 transition and collision kernel definition, 100-103 Collision density and importance equations adjoint Monte Carlo analysis, 126-138, 145-146 collision density equations, 98-107 elementary variance—reducing techniques, 86-93 Fredholm-type integral equations, 93-98 integral calculation, 81-86 scoring, 108-119 special problems criticality studies, 125-126 path stretching (exponential transformation), 119-121 perturbation Monte Carlo, 121-125

variances, 138-141 Collision kernel alternative forms in moment analysis, 182-183 definition, 101-103 normalization, 106-107 of value equation in adjoint method, 131-134 Collisions, analog simulation, 41-43 Compton scattering, 44-46 Continuous splitting, 470-486 Correlated Monte Carlo analysis: perturbation analysis, 305-306 difference estimators, 314-315 examples and special techniques, 321-324 feasibility of, 311-314 moment equations, 307--311 parametric perturbations: integral Monte Carlo, 327-328 perturbation source method, 324-326 variance of the correlated score difference, 315-321 Correlated sampling, 92-93, 122-123 Correlation of estimators, 289-290 Coupled multiparticle simulation, 290-294 Criticality studies, 125-126, 346-347 one-step by acceleration of iteration, 364-367 parametric derivatives of kff, 376-377 practical realizations, 357-362 reactivity change due to perturbations, 367-376 source iteration convergence of, 355-357 methods, 347-355 unbiased estimation of reaction rates, 430-431 variance of estimated multiplication factor, 362-364

D Delta scattering, 222-226 Differential Monte Carlo analysis: sensitivity analysis, 123-125, 328-329 data adjustment with sensitivities, 336-338 discussion, 332-336 estimation of first-order derivatives, 329-332 estimation of higher-order derivatives, 338-340 example, 340-343 expansion to parameter-dependent estimators, 343-344 perturbation estimation: Taylor series approach, 344-346 Direction vector defined, 23 Direct simulation of physical processes analog simulation of the random walk, 34-55, see also Analog simulations angle selection for anisotropic scatterings, 73-78 energy selection from Klein-Nishina formula, 65-69

514

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

fission neutron energy selection, 71-73 plausible modifications of analog game, 55-62 statistical considerations, 62-65 thermal neutron energy selection, 69-71

E Elastic scattering, 48-50 ELP method, 263-265 Energy selection fission neutrons, 71-73 Klein-Nishina formula, 65-69 thermal neutron, 69-71 Expectation of matrix ARA, 434-435 Expectation ratio estimation, 419-425 Expected leakage probability method, 234-235 Expected values in collision density equations, 91-92 Exponential distribution sampling, 16 Exponential transformation, 119-121, 207-212

F First-moment equation analysis, 196-199 Fission, analog simulation, 52-53 Fission neutrons, energy selection, 71-73 Fluence rate, see Flux Flux point estimation, 60-61, 114-119 bounded-variance point estimators, 391-394 confidence limits for singular estimators, 382-386 estimators with first-order singularity, 386-391 next-event point estimator, 378-382 practical modifications of basic method, 394-399 reaction rate integral calculation, 111-112 Flux density, see Flux Flux-type quantities defined, 24 Forward-backward moment equations, 294-297 Fredholm-type integral equations, 93-98 Free flight, 184-186 Free paths and distances, 27

G Gamma distribution in thermal neutron energy selection, 70 Geometrical splitting, 186-192, 462-470

I Importance value used as importance function, 141 Importance sampling, 87-89, 96-98, 203-207 Inelastic scattering, 50-51 Initial directions sampling, 37-38 Initial energies sampling, 38 Integral calculation, 81-86 convergence of numerical methods, 85-86 domains of complicated shape, 83-85

Fredholm-type equations, 93-98 generalization to multidimensional, 83 one-dimensional, 81-83 Inverse distribution method, 8

K Kernel definition in collision density equations, 100-103 Kernel distortion, importance sampling, 96-98 Kernel normalization in collision density equations, 104-107 Klein-Nishina formula, 65-69 Klein-Nishina theory of Comptom scattering, 44-46

L Legendre expansion, 77-78

M Matter/particle interactions, definitions and notations, 25-27 Matter/photon interactions, analog simulation, 43-47 Maxwellian distribution in thermal neutron energy selection, 69-70 Mean estimation from rare sets, 411-417 MELP method, 263-265 Moment equations approximate solutions, 239-240 effect of surroundings, 246-249 quality of, 244-246 separation assumption, 241-244 simplified model, 240-241 empirical third moments, 435-436 extension to multiplying games, 169-170 collision kernel alternative forms, 182-183 equivalent nonmultiplying game, 173-178 expectation and second moment, 171-173 score probability equation, 170-171 splitting: nonmultiplying process played as multiplying, 178-182 first-moment equation analysis, 196-197 delta scattering, 222-226 generalized exponential transformation, 207-212 importance analysis, 203-207 nonanalog game feasibility, 216-221 nonanalog game without statistical weights, 203-207 path stretching, 212-213 time and number event per history, 213-215 unbiased estimators, 197-199 weight generation rules, 199-203 forward-backward model solutions, 294-297 further generalizations, 183-184 geometrical splitting, 186-192 inclusion of time dependence, 193-196 interruption and restart of free flight, 184-186

515 score probability in general time-independent game, 192-193 general considerations analog and nonanalog simulations, 149-151 conditions of existence and uniqueness, 146-149 definitions and notations, 151-155 heuristic interpretation, 155-158 relation of expected score to adjoint collision density, 145-146 miscellaneous specific bilinear forms estimation, 287-289 correlation of estimators, 289-290 coupled multiparticle simulation, 290-294 in nonmultiplying games analytical example, 166-169 moment of a general score function, 160-163 score probability equations, 158-160 special cases: expectation and second moment of the score, 163-166 partially unbiased estimators, 226-228 analysis of variances in straight-ahead scattering model, 236-239 commonly used estimators, 231-236 transformation theorems, 228-231 second-moment equation analysis, 249-250 boundedness of variance, 258-260 examples: survival biasing and ELP and MELP method, 263-265 of multiple convolutions, 297-300 optimization of source distribution, 284-286 relative merits of common estimators, 275-280 estimator, 280-283 sufficient conditions of variance reduction by nonanalog games, 260-263 variance and efficiency of equivalent nonmultiplying game, 265-271 variance versus efficiency in nonanalog game, 283-284 zero—variance partially unbiased estimators, 271-275 zero-variance schemes, 250-258 slab transmission of particles, 508-509 straight-ahead scattering model, 300-301 variance estimates by, 139-141 Moment-generating equation, 290 Monte Carlo method defined, 5-6 Multiparticle simulation, 290-294 Multiple-convolution second-moment analysis, 297-300 Multiplicative effects defined, 25 Multiplying games equivalent nonmultiplying, variance and efficiency of, 265-271 moment equations in, 169-170 expectation and second moment, 171-173 score probability equation, 170-171 nonmultiplying played as: splitting, 178-182

N Neumann series expansion of Fredholm—type equations, 94-95 Neutron/matter interactions, analog simulation, 47-54, 57-64 (n,2n) reactions, 53 (n,3n) reactions, 53 Nonanalog games defined, 5 feasability in moment equations, 216-222 importance sampling, 203-207 moment equations, 149-151 in second-moment equation analysis, 260-263 variance versus efficiency in, 283-284 Nonmultiplying games equivalent to multiplying, 173-178, 265-271 moment equations in analytical example, 166-169 expectation and second moment of the score, 163-166 moment of a general score function, 160-163 score probability equations, 158-160 played as multiplying: splitting, 178-182 Normal distribution sampling, 14-16

0 Optimization, 441-442 continuous splitting model, 470-486 by direct statistical approach, 458-462 of geometrical splitting, 462-470 of path stretching, 487-489 practical applications, 497-505 special problems, 506-508 in straight-ahead model, 447-455 zero-variance schemes, 489-497 of Russian roulette parameter, 455-458 of source distribution, 284-286 splitting schemes in straight—ahead model, 442-447 weight-window technique, 486-487

P Pair-production, 46-47 Partially unbiased estimators in first-moment equation analysis, 197-199 in moment equations, 226-237 self-improving, 280-283 Particle/matter interactions, definitions and notations, 25-27 Particle sources defined, 23 Particle transport, basic physical quantities, 22-30 Path length selection in analog simulation of random walk, 39-41 Path stretching, 119-121, 212-213 optimization, 487-489 practical applications, 497-505 special problems, 506-508

516

Monte Carlo Particle Transport Methods: Neutron and Photon Calculations

in straight-ahead model, 447-455 zero-variance schemes, 489-497 Pay-off functions in collision density equations, 110-119 Perturbation calculations, 121-125, 307-328, see also Correlated Monte Carlo analysis by differential games, 344-348 Taylor series approach, 344-348 Phase space defined, 23 Photoelectric effect, analog simulation, 43-44 Photon/matter interactions, analog simulation, 4347 Photonuclear absorption, 47 Point estimation of flux, 60-61, 378-399, see also Flux, point estimation of Power density function, first derivative sampling, 16-19 Power function selection of sampling probability distributions, 13-14 Probability mixing method, 9

Q Quantities and notations, 22-30 Quota sampling, 90-91

R Radiative capture, 48 Radioactive activity defined, 23 Random vector selection, 19-20 Random walk, analog simulation, see Analog games, simulation of random walk Rare events, 62 Rare sets estimation of combined variance, 417-419 estimation of common mean, 411-417 Rayleigh (coherent) scattering, 47 Rejection techniques, 9-12 in analog simulation of random walk, 36-37 in Klein-Nishina energy selection, 66-68 in Legendre expansion, 77-78 Russian roulette method, 58-59, 107 optimization of, 455-458 S Sample mean estimation, 399-406 Sampling of adjoint source, 130-131 correlated, 122-123 correlation, 92-93 importance, 87-89, 203-207, 96-98 initial directions, 37-38 initial energies, 38 mean and variance in straightforward, 86-87 quota, 90-91 small sets, 406-411, 431-434 space coordinates, 35-37 systematic, 89-90 Sampling probability distributions, 6-8

efficient selections from the exponential distribution, 16 first derivative of the probability density function, 16-19 inverse distribution method, 8 probability mixing method, 9 random vector selection, 19-20 rejection techniques, 9-12 sampling from normal distribution, 14-16 selection from power functions, 13-14 table look-up method, 12-13 two- and three—dimensional random orientations, 20-22 Scattering analog simulation, 43 angle selection for anisotropic, 73-75 Compton, 44-46 defined, 25 delta, 222-226 direction cosines of particle after, 54 elastic, 48-50 inelastic, 50-51 Rayleigh (coherent), 47 straight-ahead model, 236-239, 300-301 of thermal neutrons, 51-52 Scoring of adjoin[ Monte Carlo analysis, 134-137 in analog simulation of random walk, 54-55 in collision density equations, 108-119 expected values in analog modifications, 59-62 Second-moment equation analysis, 249-250 of multiple convolutions, 297-300 zero-variance schemes, 250-258, 271-275 Self-improving estimator, 280-283 Sensitivity analysis, 123-125, 328-346, see also Differential Monte Carlo analysis: sensitivity analysis Slab transmission, 59-60, 113-114, 508-509 Source density defined, 23 Source iteration, 347-357 Source parameter selection in analog games, 34-35 Space coordinate sampling, 35-37 Splitting, 58-59 continuous, 470-486, 186-192 optimization of, 462-470 in moment equations, 178-182 optimization in straight-ahead model, 442-447 Statistical considerations in analog games, 62-65 Statistical evaluation problems determination of theoretical variances, 426-429 estimation of common mean from rare sets, 411-417 estimation of ratio of expectations, 419-425 optimum combination of sample mean, 399-406 unbiased estimation of combined variants from small sample sets, 406-411 unbiased estimation of criticality reaction rates, 430-431 Straight-ahead model, 236-239, 300-301 optimization

517 path stretching, 447-455 splitting, 442-447 Survival biasing, 263-265 Systematic sampling, 89-90

T Table look-up method, 12-13, 75 Termination in collision density equations, 107 Thermal neutrons energy selection, 69-71 scattering, 51-52 Third moments, empirical, 435-436 Three-dimensional sampling, 20-22, 37 Time-dependent games, 193-196 Time-independent games, 192-193 Transformation theorems in moment equations, 228-231 Transition kernel definition, 100-101 normalization, 105-106 Trexpectation estimator, 235 Two-dimensional random orientation sampling, 20-22

V Variance reduction elementary in collision density/importance equations correlated sampling, 92-93 importance sampling, 87-89 mean and variance in straightforward sampling, 86-87 quota sampling, 90-91 systematic sampling, 89-90 use of expected values, 91-92 Variances in collision density equations, 138-141

w Weight generation rules for first-moment analysis, 199-203 Weight-window technique of optimatization, 486-487

Y Yield defined, 23

U Unbiased estimators in first-moment equation analysis, 197-199 Uncollided particles, 137-138 Unresolved resonance range cross-sections, 53-54

Zero-variance schemes of path stretching, 489-497 in second-moment equation analysis, 250-258, 271-275