Monte Carlo Methods for Radiation Transport Fundamentals and Advanced Topics

by Oleg N. Vassiliev

Monte Carlo Methods for Radiation Transport Fundamentals and Advanced Topics This book is a guide to the use of Monte Carlo techniques in radiation transport This topic is of great interest for medical physicists Praised as a gold standard for accurate radiotherapy dose calculations Monte Carlo has stimulated a high level of research activity that has produced thousands of papers within the past few years The book is designed primarily to address the needs of an academically inclined medical physicist who wishes to learn the technique as well as experienced users

Publisher : Springer International Publishing

Author : Oleg N. Vassiliev

ISBN : 9783319441405

Year : 2017

Language: en

File Size : 3.73 MB

Category : Used Textbooks

Biological and Medical Physics, Biomedical Engineering

Oleg N. Vassiliev

Monte Carlo
Methods for
Radiation
Transport
Fundamentals and Advanced Topics

BIOLOGICAL AND MEDICAL PHYSICS,
BIOMEDICAL ENGINEERING

BIOLOGICAL AND MEDICAL PHYSICS,
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Editor-in-Chief:
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Editorial Board:
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Tokyo Institute of Technology, Yokohama, Japan
Olaf S. Andersen, Department of Physiology,
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Robert H. Austin, Department of Physics,
Princeton University, Princeton, New Jersey, USA

Hans Frauenfelder,
Los Alamos National Laboratory,
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James Barber, Department of Biochemistry,
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and Medicine, London, England
Howard C. Berg, Department of Molecular
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More information about this series at http://www.springer.com/series/3740

Oleg N. Vassiliev

Monte Carlo Methods
for Radiation Transport
Fundamentals and Advanced Topics

123

Oleg N. Vassiliev
Department of Radiation Physics
The University of Texas
MD Anderson Cancer Center
Houston, TX, USA

ISSN 1618-7210
ISSN 2197-5647 (electronic)
Biological and Medical Physics, Biomedical Engineering
ISBN 978-3-319-44140-5
ISBN 978-3-319-44141-2 (eBook)
DOI 10.1007/978-3-319-44141-2
Library of Congress Control Number: 2016954643
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In memory of my parents

Then are you so certain that your roulette playing will get us
out of our difficulties?
F. Dostoevsky, “The Gambler”

Preface

This book is intended as an introductory graduate-level text on the application of
the Monte Carlo method to radiation transport problems. The target audience is
radiation medical physicists: students, faculty members, and researchers specializing in radiotherapy physics, medical imaging, or nuclear medicine. The book
should be of interest to clinicians as well, because Monte Carlo-based software,
no longer confined to the research environment, is gradually finding its way into
routine clinical practice.
The types of problems that are important in the field of medical physics
determined the material that was selected for the book. Rather than focusing on
the practical application of Monte Carlo techniques, however, the book focuses
on the fundamentals of the method: its mathematical foundations, the numerical
techniques on which it relies, its optimization strategies, and the statistical aspect
of its calculations. With this approach, most of the information is quite general,
and parts should be useful to a broad audience. More advanced topics are included
as well, such as the adjoint formulation of the transport problem, the transport
of charged particles in an external magnetic field, microdosimetry, elements of
stochastic transport theory, and grid-based solvers. Inclusion of these topics makes
the text more complete and extends the book into areas of recent significant
developments.
An important objective of this book is to introduce the basic concepts, terminology, and formalism of radiation transport theory. This material, of course, is
necessary to understand how transport problems are solved with the Monte Carlo
method. It is also of significant interest in its own right because it is the basis for
methods other than Monte Carlo, analytical and numerical, that have been used
extensively in radiation medical physics. Several such methods are covered in the
book.
Our didactic approach reflects the view expressed by N. Metropolis and S. Ulam
in their seminal paper “The Monte Carlo method” (1949) that Monte Carlo is
a “statistical approach to the study of differential equations, or more generally,
of integro-differential equations.” The equation that we study in this book is the

ix

x

Preface

Boltzmann transport equation. For this reason, we dedicate an entire chapter to the
equation and its various forms. Only after the equation is explained do we introduce
algorithms for solving it.
The chapters and appendix of the book can be summarized as follows:
• Chapters 1 and 2 present a general introduction to the Monte Carlo method
with an emphasis on sampling techniques, an essential element of any Monte
Carlo algorithm. Sampling techniques are used to generate random numbers and
vectors that have distributions required by the algorithm.
• Chapter 3 begins with definitions of the fundamental quantities of radiation
transport theory, such as cross sections, free path, and fluence. Next is a rather
elementary introduction to the Boltzmann equation followed by examples of its
various forms. We conclude the chapter with more advanced topics: a general
algorithm for solving the Boltzmann equation with the Monte Carlo method and
the related topic of biasing techniques, which together form the mathematical
basis for algorithm optimization.
• Chapter 4 discusses three main components of a Monte Carlo algorithm for
radiation transport problems: generation of a particle trajectory, tallying, and
variance reduction. Tallying is the process of deriving a numerical estimate of
a quantity of interest from information contained in particle trajectories. Here,
and throughout the book, the word estimate is used instead of calculate because
Monte Carlo is a statistical method. This by no means implies poor accuracy of
the result. Variance reduction is a broad term referring to a variety of optimization
methods that reduce statistical uncertainties without introducing systematic error
or bias.
• Chapter 5 is dedicated to the transport of charged particles such as electrons,
protons, and heavy ions. Most Monte Carlo algorithms for charged particles rely
on multiple scattering models. We cover all the classic models for energy loss
fluctuations (energy straggling), angular distribution, and transverse and longitudinal spatial displacements. This chapter also includes sections on transport in
magnetic fields and the charge exchange process, which is particularly important
near the end of a heavy ion track.
• In the last two chapters, Chaps. 6 and 7, we present two advanced topics: microdosimetry with elements of stochastic transport theory and grid-based solvers
of the Boltzmann equation. The calculation of microdosimetric characteristics
is a problem fundamentally different from more conventional problems, such
as the dose calculation, because the Boltzmann equation is not applicable in
this case. For this reason, in this chapter, we introduce another equation, the
stochastic transport equation, and discuss algorithms for solving it. Grid-based
Boltzmann equation solvers are deterministic algorithms that present a viable
alternative to Monte Carlo. The best-known algorithm of this type is Acuros
(Vassiliev et al. 2010), which was translated into the clinic almost instantly,
for treatment planning for radiotherapy of cancer. The grid-based Boltzmann
equation solver, however, remains a relatively new technology and has the
potential for improvement and for use in new applications. In Chap. 7, we explain
step-by-step how an algorithm of this type works.

Preface

xi

• Appendix A provides a summary of the concepts and methods of probability
theory and statistics to help the reader better understand the material presented
in the book and the statistical nature of the Monte Carlo method. In Appendix B,
some of the mathematics used in the book is clarified.

Acknowledgments
The author thanks I.A. Matveeva (M.Arch.) and E.A. Butovskaya (M.Arch.) who
created the first version of most of the illustrations in the book. Their work defined
the overall style and served as a template for further revisions.
Houston, TX, USA

Oleg N. Vassiliev

Reference
Vassiliev, O.N., Wareing, T.A., McGhee, J., Failla, G., Salehpour, M.R.: Validation of a new gridbased Boltzmann equation solver for dose calculation in radiotherapy with photon beams. Phys.
Med. Biol. 55(3), 581–598 (2010)

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1
The Monte Carlo Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2
The Monte Carlo Method in Radiation Medical Physics. . . . . . . . . . . .
1.3
Estimation of  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4
Calculation of Volumes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5
Calculation of Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
1
3
5
6
8
13

2

Sampling Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1
Sampling a Uniform Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2
The Inversion Method for Sampling Continuous Distributions . . . . .
2.3
The Inversion Method for Sampling Discrete Distributions . . . . . . . .
2.4
Simple Rejection Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5
Neumann’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6
Transformation of Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7
Sampling a Sum of Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.8
Compton Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.9
Superposition Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.10 Sampling a Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.11 Random Points and Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.12 Sampling a Joint Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.13 Simulating a Particle-Scattering Event . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.14 Algorithm Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.14.1 Histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.14.2 The 2 Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.14.3 The Likelihood Ratio Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15
15
17
20
21
24
26
27
29
31
33
37
40
42
43
43
45
47
48

3

The Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1
Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2
Introduction to the Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3
Adjoint Transport Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49
49
58
62
xiii

xiv

Contents

3.4
3.5
3.6
3.7
3.8

Overview of the Formalism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
The Lagrangian Form of the Boltzmann Equation . . . . . . . . . . . . . . . . . . 65
The Boltzmann Equation for Multiplying Systems . . . . . . . . . . . . . . . . . 67
Adjoint Transport Equation for Multiplying Systems . . . . . . . . . . . . . . . 68
The Boltzmann Equation in the Presence
of an External Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.9
Simplified Forms of the Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . . . 73
3.9.1
Unscattered Fluence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.9.2
The Boltzmann Equation in Planar Geometry . . . . . . . . . . . . . 76
3.9.3
Energy Degradation Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.9.4
Continuous Slowing Down Approximation . . . . . . . . . . . . . . . . 78
3.9.5
Continuous Slowing Down Approximation
for Soft Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.9.6
Fokker-Planck Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.9.7
PN Approximation in Planar Geometry . . . . . . . . . . . . . . . . . . . . 86
3.10 Fredholm Integral Equation of the Second Kind . . . . . . . . . . . . . . . . . . . . 90
3.11 The Boltzmann Equation in an Integral Form . . . . . . . . . . . . . . . . . . . . . . . 94
3.12 The Boltzmann Equation as a Basis for Biasing Techniques . . . . . . . 96
3.12.1 Source Biasing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
3.12.2 Trajectory Biasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4

Particle Trajectories, Tallies, and Variance Reduction . . . . . . . . . . . . . . . . . .
4.1
Planning Monte Carlo Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2
Neutral Particles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1
Starting a Trajectory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.2
Stepping to the Next Collision Point . . . . . . . . . . . . . . . . . . . . . . .
4.2.3
Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3
Tallies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1
Surface Crossing Tally . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.2
Boundary Crossing Tally . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.3
Collision Tally . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.4
Track End Tally. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.5
Path Length Tally . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.6
Adjoint Function Tally . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.7
Other Tallies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4
Variance Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.1
Algorithm Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.2
Particle Splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.3
Russian Roulette. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.4
Forced Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.5
Exponential Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.6
Using Symmetry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.7
Correlated Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5
Approximate Acceleration Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

105
105
109
109
110
114
115
116
116
117
118
118
120
122
125
125
127
128
128
131
132
136
138
139

Contents

5

xv

Transport of Charged Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1
Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2
Energy Loss Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.1
The Continuous Slowing Down Approximation . . . . . . . . . . .
5.2.2
The Gaussian Model of Energy Straggling . . . . . . . . . . . . . . . .
5.2.3
The Landau Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.4
The Vavilov Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3
Models for the Angular Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.1
The Fokker–Planck Approximation . . . . . . . . . . . . . . . . . . . . . . . .
5.3.2
The Molière Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.3
The Goudsmit–Saunderson Distribution . . . . . . . . . . . . . . . . . . .
5.4
Spatial Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.1
Longitudinal Displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.2
Transverse Displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.3
Moments of the Longitudinal Distribution . . . . . . . . . . . . . . . . .
5.4.4
Moments of the Transverse Distribution . . . . . . . . . . . . . . . . . . .
5.4.5
Estimation of Spatial Errors Due to the Finite
Step Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5
Heavy Charged Particles in Condensed Matter: Charge
Exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6
Transport of Charged Particles in Magnetic Fields . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

141
141
143
143
144
147
150
156
156
159
166
168
169
173
175
178

6

Microdosimetry. Elements of Stochastic Transport Theory. . . . . . . . . . . . .
6.1
Beyond the Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2
Event-by-Event Monte Carlo Algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3
Microdosimetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.1
Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.2
Application to Radiobiology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4
Intuitive Algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4.1
A Symmetry-Based Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4.2
Sampling Over Individual Transfers. . . . . . . . . . . . . . . . . . . . . . . .
6.4.3
Algorithm Based on Caswell’s Analytical Method . . . . . . . .
6.5
Fluctuation Detector Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5.1
Equation for the Distribution of Deposited Energy . . . . . . . .
6.5.2
Method FD-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5.3
Method FD-2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.6
The Effect of Energy Straggling on Microdosimetric Spectra . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

195
195
196
197
198
202
204
204
205
206
208
208
211
217
221
222

7

Grid Based Boltzmann Equation Solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.1
Beyond the Monte Carlo method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2
Discretize, Discretize, Discretize . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2.1
Discretization of Energy: The Multigroup
Approximation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2.2
Spherical Harmonics Series for the Collision Integral. . . . .

225
225
226

180
183
186
191

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Contents

7.2.3

Discretization of Angular Variables:
The Discrete Ordinates Method . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2.4
Discretization of Spatial Variables: The Finite
Elements Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3
The Galerkin Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4
Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4.1
The Loop Over Energy Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4.2
Source Iterations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.5
Electron-Photon Fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.6
Algorithm in an External Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.6.1
The Magnetic Force Term in Spherical Coordinates . . . . . .
7.6.2
Representation in Spherical Harmonics . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

233
234
239
239
240
241
243
243
245
249

A Probabilities and Statistics Refresher. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.1
Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.2
Convergence in Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.3
Conditional Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.4
Independent Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.5
The Total Probability Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.6
Probability Distribution of a Discrete Variable . . . . . . . . . . . . . . . . . . . . . .
A.7
Probability Distribution Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.8
Probability Density Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.9
Transformation of Random Variables and Their Distributions . . . . .
A.10 Distribution of a Sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.11 The Expectation Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.12 The Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.13 The Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.14 The Central Limit Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.15 The Binomial Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.16 The Poisson Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.17 The 2 Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.18 Other Characteristics of a Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.19 Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.20 Point Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.21 The Maximum Likelihood Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.22 The Method of Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.23 Order Statistics as Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.24 Interval Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.25 Markov Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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251
251
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252
252
252
252
253
253
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257
257
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258
259
259
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262
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