Monte Carlo Methods for Particle Transport  book cover
1st Edition

Monte Carlo Methods for Particle Transport

ISBN 9781466592537
Published November 24, 2014 by CRC Press
272 Pages - 75 B/W Illustrations

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Book Description

The Monte Carlo method has become the de facto standard in radiation transport. Although powerful, if not understood and used appropriately, the method can give misleading results.

Monte Carlo Methods for Particle Transport teaches appropriate use of the Monte Carlo method, explaining the method’s fundamental concepts as well as its limitations. Concise yet comprehensive, this well-organized text:

  • Introduces the particle importance equation and its use for variance reduction
  • Describes general and particle-transport-specific variance reduction techniques
  • Presents particle transport eigenvalue issues and methodologies to address these issues
  • Explores advanced formulations based on the author’s research activities
  • Discusses parallel processing concepts and factors affecting parallel performance

Featuring illustrative examples, mathematical derivations, computer algorithms, and homework problems, Monte Carlo Methods for Particle Transport provides graduate students and nuclear engineers and scientists with a practical guide to the application of the Monte Carlo method.

Table of Contents


About the Author


History of Monte Carlo Simulation

Status of Monte Carlo Codes

Motivation for Writing This Book

Overview of the Book

Recommendations to Instructors

Author's Expectation


Random Variables and Sampling


Random Variables

Discrete Random Variable

Continuous Random Variable

Notes on pdf and cdf Characteristics

Random Numbers

Derivation of the Fundamental Formulation of Monte Carlo (FFMC)

Sampling One-Dimensional Density Functions

Analytical Inversion

Numerical Inversion

Probability Mixing Method

Rejection Technique

Numerical Evaluation

Table Lookup

Sampling Multidimensional Density Functions

Example Procedures for Sampling a Few Commonly Used Distributions

Normal Distribution

Watt Spectrum

Cosine and Sine Function Sampling




Random Number Generation (RNG)


Random Number Generation Approaches

Pseudorandom Number Generators (PRNGs)

Congruential Generators

Multiple Recursive Generator

Testing Randomness


Frequency Test

Serial Test

Gap Test

Poker Test

Moment Test

Serial Correlation Test

Serial Test via Plotting

Examples for PRNG Tests

Evaluation of PRNG Based on Period and Average

Serial Test via Plotting




Fundamentals of Probability and Statistics


Expectation Value

One-Dimensional Density Function

Multidimensional Density Function

Useful Theorems Associated with the "True Variance"

Definition of Sample Expectation Values Used in Statistics

Sample Mean

Expected Value of the Sample Variance

Precision and Accuracy of a Statistical Process

Uniform Distribution

Bernoulli and Binomial Distributions

Geometric Distribution

Poisson Distribution

Normal ("Gaussian") Distribution

Limit Theorems and Their Applications

Corollary to the de Moivre-Laplace Limit Theorem

Central Limit Theorem

Formulations of Uncertainty and Relative Error for a Random Process

General Random Process

Special Case of Bernoulli Process

Confidence Interval for Finite Sampling

Introduction to Student's t-Distribution

Determination of Confidence Interval and Application of the t-Distribution

Test of Normality of Distribution

Test of Skewness Coefficient

Shapiro-Wilk Test for Normality



Integrals and Associated Variance Reduction Techniques


Estimation of Integrals

Variance Reduction Techniques Associated with Integrals

Importance Sampling

Correlation Sampling Technique

Stratified Sampling Technique

Combined Sampling




Fixed-Source Monte Carlo Particle Transport


Introduction to the Linear Boltzmann Equation

Introduction the Monte Carlo Method

Determination of Free Flight, i.e., Path-Length

Selection of Interaction Type

Selection of Scattering Angle

A Monte Carlo Algorithm for Estimation of Transmitted Particles

Perturbation Calculations via Correlated Sampling

Analysis of Monte Carlo Results




Variance Reduction Techniques in Particle Transport


Effectiveness of Variance Reduction Algorithms

Biasing of Density Functions

Implicit Capture (or Survival Biasing)

Russian Roulette

Biasing the Path-Length to the Next Collision

Exponential Transformation

Forced Collision

Splitting Techniques

Geometric Splitting with Russian Roulette

Energy Splitting with Russian Roulette

Angular Splitting with Russian Roulette

Weight-Window Technique

Application of Combination of Importance Sampling, pdf biasing, and Splitting Technique in Particle Transport

Importance (Adjoint) Function Methodology in Deterministic Transport Theory

Determination of Detector Response

Use of Deterministic Importance (Adjoint) Function for Importance Sampling






Major Quantities in a Particle Transport Simulation

Tallying in a Steady-State System

Collision Estimator

Path-Length Estimator

Surface-Crossing Estimator

Analytical Estimator

Tallying in a Time-Dependent System

Tallies in Nonanalog Simulations

Estimation of Relative Error Associated Physical Quantities

Propagation of Error




Geometry and Particle Tracking


Discussion on a Combinatorial Geometry Approach

Definition of Surfaces

Definition of Cells


Description of Boundary Conditions

Particle Tracking




Eigenvalue or Criticality Monte Carlo Particle Transport


Theory of Power-Iteration for Eigenvalue Problems

Monte Carlo Eigenvalue Calculation

Random Variables Associated with a Fission Process

Direction of Fission Neutrons

Monte Carlo Simulation of a Criticality Problem

Estimators for Sampling Fission Neutrons

Issues Associated with the Standard Eigenvalue Calculation Procedure

Diagnostic Methods for Source Convergence

Fission Matrix (FM) Methodology

Issues Associated with the FM Method




Vector and Parallel Processing of Monte Carlo Methods


Vector Processing

Vector Performance

Parallel Processing

Parallel Performance

Vectorization of Monte Carlo Methods

Parallelization of the Monte Carlo Methods

Other Possible Parallel Monte Carlo Algorithms

Development of a Parallel Algorithm Using MPI




Appendices One to Six

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Alireza Haghighat is a professor at Virginia Tech. He has served as the director of the Nuclear Science and Engineering Lab in Arlington, Virginia, and led the Virginia Tech Theory Transport Group. He previously worked at Penn State and the University of Florida. He holds a Ph.D from the University of Washington. He has published numerous papers, received several best paper awards, and presented many invited workshops, seminars, and papers nationally and internationally. He is a recipient of the 2011 Radiation Protection Shielding Division’s Professional Excellence Award, and a recognition award from the Office of Global Threat Reduction. An ANS fellow, he has served in various ANS leadership positions.

Featured Author Profiles

Author - Alireza  Haghighat

Alireza Haghighat

Professor and Director, Virginia Tech/Nuclear Science and Engineering Lab
Arlington, VA, USA

Learn more about Alireza Haghighat »


"Dr. Haghighat’s Monte Carlo textbook is, in my opinion, the best textbook on Monte Carlo neutronics available. It does a great job covering the basics of Monte Carlo and of particle transport. I recommend it to my students in my Monte Carlo class and to everyone doing research in this field."
Leslie Kerby, PhD, MBA, Idaho State University, USA

"This is an outstanding reference and textbook on applied stochastic methods. It is a must-have for scientists, students, and practitioners interested in Monte Carlo methods for solving particle transport problems. This book provides an excellent description of the fundamentals through numerous example problems and a rich discussion of advantages and pitfalls of the Monte Carlo method. The chapter on solving eigenvalue problems is long overdue, where diagnosing convergence of the fission source in reactor physics problems with high dominance ratio is challenging and as a result has been a subject of much research."
Farzad Rahnema, Georgia Power Company Distinguished Professor and Chair of Nuclear and Radiological Engineering and Medical Physics, Georgia Institute of Technology, Atlanta, USA

"This is a very solid book for graduate students in nuclear engineering to learn how the Monte Carlo method can be used to solve reactor physics problems. It covers the fundamentals of Monte Carlo simulations before discussing how the technique can be used to solve fixed and fission sources neutron transport problems. Excellent examples are provided in the main text, in addition to a complete set of homework problems at the end of each chapter. This makes it an ideal textbook for those teaching a course on simulation methods in reactor physics."
G. Marleau, Professor of Nuclear Engineering and Engineering Physics, Director of the Institute of Nuclear Engineering, École Polytechnique de Montréal, Québec, Canada

"Professor Haghighat, based on his many years of experience in teaching the subject, has written a long-awaited book on Monte Carlo methods. The subject of the book based on particle transport has an old history in concept, but is becoming lately more important and enjoying heavy use with the advent of high-performance computers. Professor Haghighat has succeeded in writing a book that is concise, but also includes all ingredients in the Monte Carlo method. … [This book is] an excellent addition to the bookshelf of teachers, students, researchers, and practitioners in the field of nuclear reactor design and radiation shielding applications."
Nam Zin Cho, Professor of Nuclear and Quantum Engineering, Korea Advanced Institute of Science and Technology, Daejeon, South Korea

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