A Monte Carlo Primer: A Practical Approach to Radiation Transport

This text introduces the reader to the use of Monte Carlo methods for solving practical problems in radiation transport, and should also serve as a reference work for practitioners in the field. It assumes the reader has a general knowledge of calculus and radiation physics, and a knowledge of Fortran programming, but assumes no prior knowledge of stochastic methods or statistical physics. The subject is presented by a combination of theoretical development and practical calculations. Because Monte Carlo methods are closely linked to the use of computers, from the beginning the reader is taught to convert the theoretical constructs developed in the text into functional software for use on a personal computer. Example problems provide the reader with an in-depth understanding of the concepts presented and lead to the production of a unique learning tool, a probabilistic framework code that models in a simple manner the features of production of Monte Carlo transport codes. This framework code is developed in stages such that every function is understood, tested, and demonstrated - random sampling, generating random numbers, implementing geometric models, using variance reduction, tracking particles in a random walk, testing the thoroughness with which the problem phase space is sampled, scoring detectors, and obtaining estimates of uncertainty in results. Advanced topics covered include criticality, correlated sampling, adjoint transport, and neutron thermalization. Monte Carlo codes can produce highly precise wrong answers. The probability of this occurring is increased if production codes are run as opaque, "black boxes" of software. This text attempts to make Monte Carlo into a comprehensible, usable tool for solving practical transport problems.

Table of Contents:
1. Introduction.- 1.1 The Monte Carlo Method.- 1.2 The Evaluation of Random Processes>.- 1.3 Monte Carlo Evaluation of Definite Integrals.- Exercises.- 2. Monte Carlo Sampling Techniques.- 2.1 Probability Theory and Statistics.- 2.2 Sampling.- 2.3 Means and Variances.- 2.4 Estimations of Means and Variances.- 2.5 Introduction to Variance Reduction Techniques.- Exercises.- 3. Monte Carlo Modeling of Neutron Transport.- 3.1 Introduction.- 3.2 Neutron Interactions and Mean Free Path.- 3.3 Neutron Transport.- 3.4 A Mathematical Basis for Monte Carlo Neutron Transport.- 3.5 Monte Carlo Modeling of Neutron Motion.- 3.6 Particle Flight Path in Complex Geometries.- 3.7 Multi-Region Problems.- Exercises.- 4. Energy-Dependent Neutron Transport.- 4.1 Elastic Scattering of Neutrons.- 4.2 Transformation of Post-Collision Direction to Laboratory System.- 4.3 Energy-Dependent Cross Sections.- Exercises.- 5. A Probabilistic Framework Code.- 5.1 Introduction to PFC.- 5.2 Problem Definition in PFC.- 5.3 The Random Walk in PFC.- 5.4 Computing the Response.- Exercises.- 6. Variance Reduction Techniques.- 6.1 Introduction.- 6.2 Source Biasing.- 6.3 Survival Biasing.- 5.4 Russian Roulette.- 6.5 Splitting.- 6.6 Exponential Transform.- Exercises.- 7. Monte Carlo Detectors.- 7.1 Introduction.- 7.2 The Next-Event Estimator.- 7.3 Volumetric Flux Detectors.- 7.3 Surface-Crossing Flux Estimator.- 7.5 Expectation Surface-Crossing Flux Estimator.- Time-Dependent Detectors.- Exercises.- 8. Nuclear Criticality Calculations with Monte Carlo.- 8.1 Multiplying Assemblies.- 8.2 The Generation Method.- 8.3 The Matrix Method.- 8.4 Combination of Generation and Matrix Methods.- 8.5 Criticality Calculations Using Multigroup Cross Sections.- Exercises.- 9. Advanced Applications of Monte Carlo.- 9.1 Correlated Sampling.- 9.2 Adjoint Monte Carlo.- 9.3 Neutron Thermalization.- Exercises.- Appendix Random Number Generators.