Radiation Shielding 0894484567, 9780894484568

Citation preview

RADIATION SHIELDING

by

J. Kenneth Shultis and

Richard E. Faw Department of Nuclear Engineering Kansas State University Manhattan, Kansas 66506

Published by the American Nuclear Society, Inc. La Grange Park, Illinois 60526 USA

Library of Congress Cataloging-in-Publication Data Shultis, J. Kenneth. Radiation shielding/ J. Kenneth Shultis, Richard E. Faw. p. cm. Originally published: Upper Saddle River, NJ : Prentice Hall PTR, c1996. Includes bibliographical references and index. ISBN 0-89448-456-7 1. Shielding (Radiation) 2. Ionizing radiation-Mathematical models. I. Faw, Richard E. II. Title. TK9210 .S58 2000 621.48'0289'9-dc21

00-022271

ISBN: 0-89448-456-7 Library of Congress Catalog Card Number: 00-022271 ANS Order Number: 350021 Copyright© 2000 by the American Nuclear Society, Inc. 555 North Kensington Avenue La Grange Park, Illinois 60526 USA This book was previously published by Prentice-Hall, Inc. All rights reserved. No part of this book may be reproduced in any form without written permission of the publisher. Printed in the United States of America

Preface

Tweuty yeaxs ago, with Art Chilton, we began the development of a textbook Principles of Radiation Shielding. In it, we tried to present the basic ideas supporting radiation shielding technology that had evolved in the previous half century. Since then, several major innovations have influenced this technology' First, the widespread availability of inexpensive, powerful microcomputers and workstations now permits the shielding analyst to perform complex calculations without recourse to the large mainframe computers which dominated the discipline when we wrote our first book. Second, there have been major changes not only in the conceptual framework for radiation protection but also in the units and quantities used to characterize radiation and its effects. Third, in the intervening years, many refinements have been made in our understanding of the interaction of radiation vrith matter and our knowledge of the funda,mental shielding data needed for accurate shield design and analysis. Because of these recent changes and the desire to present in a more unified fashion the principles and techniques of performing radiation shielding calculations, we have undertaken this present effort. We have attempted to present as much current data and information as possible, as well as to extend the scope of our treatment. Our goal has been to produce a volume that can serve both as a textbook for students in radiation shielding courses and as a reference work for shielding practitioners. Such a dual purpose has led us to emphasize the principles behind the many techniques used in various aspect of shield analysis. While this approach is certainly useful for the teaching of radiation shielding, it has prevented us from discussing the nuances of the large shielding codes that have come to dominate shielding design and analysis. Nonetheless, we trust that shielding experts will find this book useful to remind them of the underlying principles of these codes and, more important, to offer approximate methods of analysis, which often can be used to obtain radiation dose estimates at a small fraction of the cost incurred by using computationally intensive techniques. In writing this book we have intentionally omitted coverage of several topics

important in radiation shielding. The kindred disciplines of radiological assessment aaa

xtlt

xiv

Preface

and radiation dosimetry are addressed only as needed in defining and understand­ ing modern radiation dose units. Nor have we covered nuclear instrumentation and experimental methods, even though many of the simplified and specialized shielding analysis techniques are founded on experimental measurements. We have not cov­ ered operational radiation safety at all. We have omitted these topics, not because we minimize their importance, but for other reasons. First, there is a practical constraint on the size of this volume. Also, some of the topics are so specialized that few general principles can be derived. Finally, the neglected topics are covered very well by other excellent texts and references. In our presentation of the material in this book, we assumed that the reader has an understanding of mathematics through basic calculus and vector analysis. Chap­ ters 10 and 11 present the most advanced material on transport theory methods and require of the reader some knowledge of differential equations and statistics. We also assumed that the reader has a knowledge of the nuclear physics of radioac­ tive decay. For most chapters, problem sets are provided. Although students will develop proficiency while solving these problems, the shielding specialist will also find many useful results presented in these problems. We have also included suf­ ficient shielding data to enable the reader to perform a wide variety of shielding calculations without recourse to other sources of data. We have striven to present shielding principles in many different contexts, rather than in a single context, such as nuclear reactor shielding. For example, informa­ tion on charged-particle penetration is presented in a fashion that can be used for applications in space shielding or at a microscopic cellular level. We have also tried to unify as much as possible the treatment of the various types of radiation dis­ cussed in this book. A new treatment based on the point-pair distance distribution and the point-kernel concept is presented for use with neutrons, gamma-photons, electrons, and heavy charged particles. The concept of a detector response function is also used throughout the book as a unifying means of estimating different types of radiation dose. Use of the International System of Units (SI), generally followed throughout this book, has been leavened with the inclusion of other units in common usage, such as the curie, the roentgen, the electron-volt, and their multiples. Insofar as possible, nomenclature, symbols, abbreviations, and definitions are those prescribed by the International Commission on Radiation Units and Measurements and the American National Standards Institute, although a few nonstandard choices have been used to foster the unity of presentation we have emphasized.

Acknowledgments Little of the material presented in this book is original. Most has been gathered from the vast base of knowledge produced by radiation shielding researchers over the years. To these workers we extend our deepest appreciation for their leadership and their important technological contributions. Most of the material in this book has been accumulated by us over many years for use in courses we have taught in radiation shielding and radiological assess-

Acknowledgments

xv

ment. We have attempted to unify notation, to use modern radiological units, and to present the basic ideas in a fashion that we have found to be successful in the classroom. To our students who have struggled with earlier drafts of much of the material in this book and whose suggestions have been indispensable, we extend our gratitude. Although many colleagues and friends have encouraged us and helped with various aspects of this book, we would like to acknowledge especially direct as­ sistance from Jeffrey Ryman, Robert Stewart, Auu Gui, Sherrill Shue, and Ronald Brockhoff, and indirect help and inspiration from Lewis Spencer, Charles Eisen­ hauer, Martin Berger, and Stephen Seltzer. To David Trubey, Robert Roussin, and their colleagues at the Radiation Shielding Information Center, we extend our sin­ cere appreciation for their invaluable assistance. To Norman Schaeffer at Radiation Research Associates and Charles Negin at Grove Engineering we express apprecia­ tion for their encouragement of our own research program in radiation shielding at Kansas State University. Finally, we want to express our gratitude to Donald Knuth and Leslie Lamport for developing and making publicly available the TEX and H'-'IEX text preparation programs we used to typeset this book. Because of our use of this system, we have the bittersweet responsibility of acknowledging that all errors are ours alone. More important, this new methodology, by which authors can produce their own typeset pages ready for printing, allows publishing houses to undertake specialty projects like ours, which have limited sales potentials. For embracing this new publishing paradigm, we thank Prentice Hall. Manhattan, Kansas

J. Kenneth Shultis Richard E. Faw

Contents

Preface

xiii

Preface to the 2000 Printing

xvi

1

2

Introduction 1.1 Historical Roots 1.1.1 The Early Years 1.1.2 The Origins of Modern Shielding Practice 1.1.3 Modern Developments 1.1.4 Regulatory Development 1.2 Radiation Shielding Institutions 1.2.1 Professional Societies and Journals 1.3 Radiation Protection Institutions 1.3.1 United Nations Organizations 1.3.2 Governmental Organizations in the United States 1.4 Important Sources of Shielding Information 1.5 Final Remarks Characterization of Radiation Fields and Sources 2.1 Directions and Solid Angles 2.2 Fundamental Radiation Field Variables 2.2.1 Fluence and Fluence Rate 2.2.2 Net Flow and Net Flow Rate 2.2.3 Other Definitions 2.3 Directional Properties of the Radiation Field 2.3.1 Properties of the Fluence 2.3.2 Transformation of Variables 2.3.3 Angular Properties of the Flow and Flow Rate 2.4 Representations of Angular Dependence 2.5 General Specification of Radiation Sources 2.6 Distributed and Discrete Variables

1 2

2

4 4 6 7 8 8 9 9 10 12 15 16 17 17 18 19 19 19 20 21

22

23 25 V

vi

Contents

3 Interaction of Radiation with Matter

28

4 Common Radiation Sources Encountered in Shield Design

80

3.1 Interaction Coefficient 3.2 Microscopic Cross Section 3.3 Conservation Laws for Scattering Reactions 3.3.1 Conservation of Momentum 3.3.2 Conservation of Energy 3.3.3 Application of the Conservation Laws 3.3.4 Scattering of Photons by Free Electrons 3.3.5 Scattering of Neutrons by Atomic Nuclei 3.3.6 Limiting Cases in Classical Mechanics of Elastic Scattering 3.3.7 Elastic Scattering of Electrons and Heavy Charged Particles 3.4 Cross Sections for Photon Interactions 3.4.1 Thomson Cross Section for Incoherent Scattering 3.4.2 Klein-Nishina Cross Section for Incoherent Scattering 3.4.3 Incoherent Scattering Cross Sections for Bound Electrons 3.4.4 Coherent (Rayleigh) Scattering 3.4.5 Photoelectric Effect 3.4.6 Pair Production 3.4.7 Photon Attenuation Coefficients 3.4.8 Compton Absorption and Scattering Cross Sections 3.4.9 Photoelectric Absorption Cross Section 3.4.10 Absorption Cross Section for Pair Production 3.4.11 Corrections for Radiative Energy Loss 3.5 Neutron Interactions 3.5.1 Classification of Types of Interactions 3.5.2 Cross Sections for Neutron Scattering 3.5.3 Average Energy Transfer in Neutron Scattering 3.5.4 Radiative Capture of Neutrons 3.6 Charged-Particle Interactions 3.6.1 Collisional Energy Loss 3.6.2 Electron Radiative Energy Loss 3.6.3 Charged-Particle Range 3.6.4 Residual-Range Concept 3.6.5 Electron Radiation Yield 4.1 Neutron Sources 4.1.1 Fission Neutrons 4.1.2 Photoneutrons 4.1.3 Neutrons from (o:,n) Reactions 4.1.4 Activation Neutrons 4.1.5 Fusion Neutrons 4.2 Sources of Gamma Photons 4.2.1 Radioactive Sources

28 30 31 31 32 33 33 34 38 39 39 40 40 41 42 44 45 45 46 47 48 48 49 50 56 58 60 61 62 67 69 72 72 80 80 84 86 91 92 92 92

Contents

4.2.2 Prompt Fission Gamma Photons 4.2.3 Gamma Photons from Fission Products 4.2.4 Capture Gamma Photons 4.2.5 Gamma Photons from Inelastic Neutron Scattering 4.2.6 Activation Gamma Photons 4.2.7 Annihilation Radiation 4.3 Sources of X Rays 4.3.1 Characteristic X Rays 4.3.2 Bremsstrahlung 4.3.3 X-Ray Machines

vii 93 94 101 103 103 104 105 105 108 111

5 Photon and Neutron Response Functions 5.1 Dosimetric Quantities 5.1.1 Energy Imparted, Specific Energy, and Lineal Energy 5.1.2 Deterministic Quantities 5.1.3 Absorbed Dose 5.1.4 Kerma 5.1.5 Exposure 5.1.6 Linear Energy Transfer 5.2 Dose Equivalent Quantities 5.2.1 Quality Factor 5.3 Concept of Radiation Response Function 5.4 Local Response Functions for Point Targets 5.5 Charged-Particle Equilibrium 5.6 Local Response Functions for Neutrons 5.7 Local Response Functions for Photons 5.7.1 Photon Energy Deposition Coefficients 5.7.2 Photon Kerma, Absorbed Dose, and Dose Equivalent 5.7.3 Photon Exposure 5.7.4 Selection of Proper Mass Energy Deposition Coefficients 5.8 Response Functions for the Human as Target 5.8.1 Characterization of Ambient Radiation 5.8.2 Response Functions Based on Simple Geometric Phantoms 5.8.3 Response Functions Based on Anthropomorphic Phantoms 5.8.4 Comparison of Response Functions

121 121 123 123 124 124 124 125 126 126 129 130 131 134 137 137 139 141 141 144 144 145 147 149

6 Basic Methods for Radiation Dose Calculations 6.1 Uncollided Radiation 6.1.1 Exponential Attenuations 6.1.2 Mean-Free-Path Length 6.1.3 Uncollided Dose from a Point Source 6.1.4 Point Kernel for the Uncollided Dose 6.2 Uncollided Doses from Distributed Sources 6.2.1 Line Source

155 156 156 157 157 160 161 161

viii

Contents

6.2.2 Disk Source 6.2.3 Rectangular Area Source 6.2.4 Spherical Surface Source 6.2.5 Frustrum of a Cone 6.2.6 Infinite Slab Source 6.2.7 Cylindrical Volume Source Point-Kernel Concept for Total Dose 6.3.1 Dose in Terms of the Green's Function of Transport Theory 6.3.2 Point Kernel for the Total Dose 6.3.3 Isotropic Detector Without Spatial Dependence 6.3.4 Infinite Homogeneous Medium 6.3.5 Examples of Point Kernels Generalized Method for an Infinite Homogeneous Medium 6.4.1 Volumetric Sources 6.4.2 Absorbed Fraction and Reduction Factor 6.4.3 Advantages of the Generalized Approach 6.4.4 Limiting Source or Target Volumes 6.4.5 Reciprocity Theorem 6.4.6 Extension to Nonuniform and Surface Sources 6.4.7 Infinite Cylindrical Sources Calculation of Geometric Factors 6.5.1 Analytical Calculation of Geometry Factors 6.5.2 Examples of Geometry Factors and Point-Pair Distributions 6.5.3 Uncollided Dose Examples Using Geometry Factors 6.5.4 Monte Carlo Evaluation of Geometry Factors and Point-Pair Distance Distributions 6.5.5 Multiregion Geometries 6.5.6 Basic Geometry Factors for One-Dimensional Problems 6.5.7 Examples of Multiple Regions Effect of Density Variations 6.6.1 Theorems for Density Variations 6.6.2 Point Kernels in Media with Density Variations 6.6.3 Modified Point-Pair Distance Distributions 6.6.4 Modified Geometry Factors 6.6.5 Example of a Modified Point-Pair Distance Distribution 6.6.6 Example Problem Using Modified Geometry Factors Geometric Transformations 6.7.1 Circular Area (Disk)-to-Point Source Transformation 6.7.2 Volume-to-Surface Source Transformation

164 167 169 169 170 171 173 173 174 175 175 176 177 178 180 181 182 182 183 184 185 185 185 188

7 Special Techniques for Photons 7.1 Photon Buildup-Factor Concept 7.1.1 Isotropic, Monoenergetic Sources in Infinite Media 7.1.2 Comparison of Buildup Factors for Point and Plane Sources

214

6.3

6.4

6.5

6.6

6.7

189 191 192 193 195 196 198 199 199

200 200 203 203 204 215 215 218

ix

Contents

7.2 7.3

7.4

7.5

7.6

7.7

7.1.3 Empirical Approximations for Point-Source Buildup Factors 7.1.4 Point-Kernel Applications of Buildup Factors Buildup Factors for Heterogeneous Media 7.2.1 Boundary Effects in Finite Media 7.2.2 Treatment of Stratified Media Broad-Beam Attenuation of Photons 7.3.1 Attenuation Factors for Monoenergetic Photon Beams 7.3.2 Attenuation of Oblique Beams of Monoenergetic Photons 7.3.3 Attenuation Factors for X-Ray Beams 7.3.4 The Half-Value Thickness Photon Albedo Concept 7.4.1 Differential Number Albedo 7.4.2 Integrals of Albedo Functions 7.4.3 Application of the Albedo Method 7.4.4 Single-Scatter Albedo 7.4.5 Approximation for the Single-Scatter Dose Albedo 7.4.6 Chilton-Huddleston Formula 7.4.7 Photon Albedo Data Photon Streaming Through Ducts 7.5.1 Characterization of Incident Radiation 7.5.2 Line-of-Sight Component for Straight Ducts 7.5.3 Wall-Penetration Component for Straight Ducts 7.5.4 Single-Scatter Wall-Reflection Component 7.5.5 Transmission of Gamma Rays Through Two-Legged Rectangular Ducts Shield Heterogeneities 7.6.1 Limiting Case for Small Discontinuities 7.6.2 Small Randomly Distributed Discontinuities 7.6.3 Large Well-Defined Heterogeneities Gamma-Ray Skyshine 7.7.1 Open Silo Example 7.7.2 Shielded Skyshine Sources

8 Special Techniques for Neutrons

8.1 Differences Between Neutron and Photon Calculations 8.1.1 Buildup Factors 8.1.2 Neutron Dose Units 8.2 F ission Neutron Attenuation by Hydrogen 8.3 Removal Cross Sections 8.3.1 Extensions of the Removal-Cross-Section Model 8.4 Fast-Neutron Attenuation Without Hydrogen 8.5 Calculation of the Intermediate and Thermal F luences 8.5.1 Diffusion Theory for Thermal Neutron Calculations

220 223 225 225 227 229 229 231 231 234 235 236 237 237 239 240 240 241 242 244 245 248 248 250 252 253 253 255 256 257 259 269

271 271 272 273 279 282 288 291 291

Contents

X

Fermi Age Treatment for Thermal and Intermediate-Energy Neutrons 8.5.3 Removal-Diffusion Techniques Capture-Gamma-Photon Attenuation 8.6.1 Response from Uncollided Photons 8.6.2 Response from Scattered Photons Neutron Shielding with Concrete 8.7.1 Concrete Slab Shields Neutron Albedo 8.8.1 Fast Neutron Albedo 8.8.2 Intermediate-Energy Neutron Albedo 8.8.3 Thermal Neutron Albedo 8.8.4 Emission of Secondary Photons During Neutron Reflection Duct Streaming for Neutrons 8.9.1 Straight Ducts 8.9.2 Ducts with Bends 8.9.3 Empirical and Experimental Results Neutron Skyshine 8.5.2

8.6 8.7 8.8

8.9

8.10 9

Special Techniques for Charged Particles

9.1 Introduction 9.2 Alpha and Beta Decay 9.2.1 Alpha Decay 9.2.2 Beta Decay 9.3 Spatial Distribution of the Absorbed Dose 9.3.1 Point and Plane Kernels Defined 9.3.2 Electron and Beta-Particle Dose Distributions 9.4 Applications of the Point-Kernel 9.4.1 Line Source of Electrons 9.4.2 Plane Isotropic Source of Electrons 9.4.3 Volume Source of Electrons 9.5 Energy Spectrum of the Fluence 9.5.1 CSDA Approximation 9.5.2 Fluence Energy Spectra for Electron Sources

10 Deterministic Transport T heory

10.1 Transport Equation 10.1.1 Explicit Form for the Three Basic Geometries 10.1.2 Integral Form of the Transport Equation 10.1.3 Transport Equation for Photons 10.1.4 Transport Equation for Neutrons 10.2 Implications of the Transport Equation 10.2.1 Existence and Uniqueness 10.2.2 Spatially Uniform Flux Density

293 295 300 303 304 304 307 312 312 315 317 317 318 318 320 322 322 333

333 333 333 334 336 336 338 342 342 344 347 348 349 349

355

355 359 361 365 368 371 371 371

xi

Contents

10.2.3 Plane-Density Variations 10.2.4 Scaling of Radiation Fields 10.2.5 Volume-to-Surface Source Transformation Approximations to the Transport Equation 10.3.1 Exponential Attenuation 10.3.2 Diffusion Approximation 10.3.3 Multigroup Approximation Method of Moments Discrete-Ordinates Method Integral Transport Method 10.6.1 Direct Integration of the Scattering Source Term 10.6.2 Implementation

372 374 375 377 378 380 384 386 392 398 400 402

11 Monte Carlo Methods for Radiation Transport Calculations 11.1 Random and Pseudorandom Numbers 11.2 Selection Techniques for Stochastic Variables 11.2.1 Selection of Discrete Variables 11.2.2 Cumulative Distribution Method for Continuous Variables 11.2.3 Rejection Method 11.2.4 Composition Method 11.2.5 Composition-Rejection Method 11.3 Simple Analog Monte Carlo Calculation 11.3.1 Geometric Transformations 11.3.2 Particle Tracking 11.3.3 Scoring 11.4 Variance Reduction and Nonanalog Methods 11.4.1 Central-Limit Theorem 11.4.2 Importance Sampling 11.4.3 Truncation Methods 11.4.4 Splitting and Russian Roulette 11.4.5 Interaction Forcing 11.4.6 Exponential Transformation 11.4.7 Deterministic Scoring Methods

408 409 411 411 412 413 414 415 415 415 416 419 420 421 421 423 424 424 424 425

Appendix A: Constants and Conversion Factors

431

Appendix B: Mathematical Tidbits B.1 Elliptic Integrals B.2 Sievert or Secant Integral B.3 Exponential Integral Function B.4 Legendre Polynomials B.5 Chandrasekhar's H Function B.6 Dirac Delta Function

433 433 433 435 440 442 443

10.3

10.4 10.5 10.6

xii

Contents

Appendix C: C ross Sections and Related Data

446

Appendix D: Photon and Neutron Response Functions

468

Appendix E: Photon Buildup and Neutron Attenuation Factors

478

Appendix F: Skyshine Response Functions

498

Appendix G: Fission-Product Source Parameters

506

Appendix H: Photons Emitted by Selected Radionuclides

514

Index

523

Chapter 1

Introduction The twentieth century has seen radiation and radioactivity evolve from laboratory curiosities to essential features of a modern technological age. Along with under­ standing of the characteristics of the different types of radiation and the potential benefits of their use came awareness of the potential harm that could arise from ra­ diation exposure. Thus from the need for protection was the discipline of radiation shielding analysis and design born. This discipline has evolved during this century from one based on empirical rules of thumb to one based on sophisticated analysis techniques requiring, on occasion, the fastest computers and the most extensive data libraries. The term shielding analysis can be used in a narrow sense to refer to the quan­ tification of how a system of shields near a source of radiation affects the radiation field at some point of interest. In a more general sense-and the one used in this text-shielding analysis is the study of how radiation is created, how it migrates from its source, how it interacts with matter, how it creates microscopic changes in the medium it traverses, and how these changes affect the medium. To under­ take such analysis, one must have a thorough understanding of technical subjects such as radiation source characteristics, radiation protection criteria and the special dosimetric units involved, radiation reflection properties of media, and the funda­ mental physics of the interaction of radiation with matter. Diverse areas of physics, mathematics, nuclear technology, radiation protection practice (health physics), and material science also are important in shielding analysis. These topics are the subjects of this book. With this technical background, the shielding analyst or designer can proceed to the study of shields themselves. This study, which is the principal purpose of this book, has many aspects: radiation transport within shields, radiation levels outside shields, deposition of heat in shields, radioactivation of materials in shields and their surroundings, radiation penetration through holes in shields, radiation scattering around shields, selection of shield materials, and optimization of shield design subject to economic and other constraints. The shield analyst is concerned with the type of radiation that causes ionization of the media with which it interacts. Such radiations may be energetic particles carrying an electric charge, such as beta particles, alpha particles, protons, and other recoil nuclei. These radiations are called directly ionizing, since they cause 1

2

Introduction

Chap. 1

ionization by direct action on electrons in atoms of the media through which they pass. Other types of radiation, such as neutrons and x-ray or gamma-ray photons, are not charged and therefore cause ionization through a more complicated mech­ anism involving the emission of energetic secondary charged particles which cause most of the ionization. Consequently, radiation such as photons and neutrons is called indirectly ionizing. The classification of ionizing radiations into two principal types has impor­ tant implications in the study of shields. Directly ionizing radiation interacts very strongly with shielding media and is therefore easily stopped. Indirectly ionizing radiation, by contrast, may be quite penetrating and the shielding required may be quite massive and expensive. For these reasons, the reader will find that much attention in this book is paid to the shielding of neutrons and photons, the two types of indirectly ionizing radiation most frequently encountered. The ionizing ability of these types of radiation is the reason for the importance of the study of shields, since the significant biological effects resulting from radia­ tion exposure are intimately related to the ionization of atoms in the human body. This book does not undertake an examination of such biological effects, although a shield designer should be aware of them and the various regulatory standards that are imposed for protection purposes. The interested reader can obtain a detailed summary of the biological effects of radiation in a companion book, Radiological Assessment [Faw and Shultis 1993]. Other reviews can be obtained from publi­ cations by the International Commission on Radiological Protection [ICRP 1991], the United Nations Scientific Committee on the Effects of Atomic Radiation [UN­ SCEAR 1988a,b], the National Council on Radiation Protection and Measurements [NCRP 1993a, 1993b], and the Advisory Committee on the Biological Effects of Ionizing Radiation of the National Academy of Science [NAS 1990].

1.1 HISTORICAL ROOTS 1.1.1 The Early Years A brief comment on the historical developments that have led to the present state of knowledge about shielding against ionizing radiation may be useful. The origins go back to the science of optics, before ionizing radiation was even discovered. The exponential attenuation behavior of light in its transmission through semitranspar­ ent media has long been recognized; and the use of the "ray" approach in geometric optics has been carried over and is still useful in many shielding situations involving photons and neutrons. The culmination of decades of study of cathode rays and luminescence phe­ nomena was W.C. Roentgen's 1895 discovery of x rays and A.H. Becquerel's 1896 discovery of what Pierre and Marie Curie in 1898 called radioactivity. The beginning of the twentieth century saw the discovery of gamma rays and the identification of the unique properties of alpha and beta particles. These radiations were discovered and characterized in part by the extent to which they were attenuated by various media-marking the first shielding analyses. Polonium, then radium and radon,

Sec. 1.1.

Historical Roots

3

were identified among the radioactive decay products of uranium and thorium and were characterized by the radiations they emitted. The importance of x rays in medical diagnosis was immediately apparent and, within months of their discovery, their bactericidal action and ability to destroy tumors were revealed. The high concentrations of radium and radon associated with the waters of many mineral spas led to the mistaken belief that these radionuclides possessed some subtle, broadly curative powers. The genuine effectiveness of radium and radon in treatment of certain tumors was also discovered and put to use in medical practice. What were only later understood to be ill effects of exposure to ionizing radiation had been observed long before the discovery of radioactivity. Fatal lung disease, later diagnosed as cancer, was the fate of many miners exposed to the airborne daughter products of radon gas, itself a daughter product of the decay of uranium or thorium. Both uranium and thorium have long been used in commerce, thorium in gas-light mantles, and both in ceramics and glassware, with unknown health consequences. The medical quackery and commercial exploitation associated with the supposed curative powers of radium and radon may well have led to needless cancer suffering in later years. Certain ill effects of radiation exposure, such as skin burns, were observed shortly after the discovery of x rays and radioactivity. Other effects, such as cancer, were not soon discovered because of the long latency period, often of many years' duration, between radiation exposure and overt cancer expression. Quantification of the degree of radiation exposure, and indeed the standardiza­ tion of x-ray equipment, were for many years major challenges in the evaluation of radiation risks and the establishment of standards for radiation protection. Fluores­ cence, darkening of photographic plates, and threshold erythema (skin reddening as though by first-degree burn) were among the measures early used to quantify radiation exposure. However, variability in equipment design and applied voltages greatly complicated dosimetry. What we now identify as the technical unit of ex­ posure, measurable as ionization in air, was proposed in 1908 by Villard but was not adopted for some years because of instrumentation difficulties. Until 1928, the threshold erythema dose (TED) was the primary measure of x-ray exposure. Unfor­ tunately, this measure depends on many variables, such as exposure rate, site and area of exposure, age and complexion of the person exposed, and energy spectrum of the incident radiation. The first quarter of the twentieth century was thus the period in which the hazards from ionizing radiation gradually became known, and the use of heavy materials such as lead in the form of sheets and bricks became commonplace for shielding use against x rays and gamma rays. The uses of such radiation were largely confined to medical applications or to fundamental scientific studies. A body of shielding knowledge of elementary character began to be accumulated, largely summarized in the form of half-thickness values or simple exponential factors involving absorption coefficients.

4

1.1.2

Introduction

Chap. 1

The Origins of Modern Shielding Practice

The second quarter of the twentieth century was marked by a rapid increase in the knowledge of the fundamentals of the interaction between radiation and matter, made possible by the development of quantum mechanics as a theoretical tool. The discovery of the neutron in 1932 began the series of events that led eventually to the use of nuclear reactors and nuclear explosives, both intense sources of neutrons. This new type of particle was soon recognized to be a most significant hazard, protection against which might require massive shielding. Until recent decades, most information on radiation shielding was developed on an ad hoc basis, with little concern being felt that the subject might deserve atten­ tion as an abstract discipline in its own right. Medical radiologists, supported by radiological physicists, have developed, largely on an experimental basis, shielding data for radiation from x-ray machines and from the more commonly used radioiso­ topes. These data are expressed in either graphical or tabular form (NCRP 1976]. The advent of nuclear weapons required new programs for radiological defense in the military establishment and its counterpart in the civil defense establishment. As a result of extensive experimental work in nuclear weapons testing during the 1950s, supplemented by some profound theoretical work on gamma radiation transport (Spencer and Fano 1951; Fano, Spencer, and Berger 1959], a technology evolved for weapons radiation shielding. In the early 1960s, particular emphasis was placed on radiation shielding of gamma rays from nuclear-weapon fallout, and a well-developed system of shielding analysis resulted from a combination of theoretical calculations, experimental laboratory work, and extensive engineering methodology development (Kimel 1966; Spencer 1962; OCD 1974; Spencer, Chilton and Eisenhauer 1980]. The development of nuclear power reactors introduced new ideas and data on shielding, especially against neutrons. The specific programs that emphasized these shielding advances included the submarine nuclear propulsion program, the nuclear­ powered aircraft program, and the development of central-station nuclear power plants. In the 1950s the approach to neutron shielding was largely empirical, in­ volving variants of the old exponential attenuation approach. Since that time, these empirical methods have been supplemented by more accurate and more powerful techniques for solving the basic mathematical equations governing radiation trans­ port. In the 1960s, manned space vehicle development brought about an increase in knowledge of shielding against the very high energy charged particles encountered in outer space. In the 1970s, the rapid increase in fusion reactor studies brought about an emphasis on such specialized shielding problems as heat deposition, thermoelastic stress analysis, and tritium production within the reaction blanket (which also serves as the shield).

1.1.3

Modern Developments

It is important to recognize how important the development of computer technol­ ogy has been in the evolution of radiation shielding design and analysis methods.

Sec. 1.1. Historical Roots

5

Although the basic physical principles governing the transport of radiation, as ex­ pressed by the Boltzmann transport equation were known many decades previously, it was only through the use of computers that these principles could be applied to photon and neutron shielding with any hope of acceptable accuracy. Since the 1960s, much effort has gone into the development of large general-purpose radiation trans­ port codes based on the numerical solution of the underlying transport equation or based on Monte Carlo transport simulations. At the same time small computers with the power of large mainframe computers of only a decade earlier have become affordable and readily available to the shielding specialist, and it is common to find today large multidimensional transport calculations being performed routinely on desktop computer systems. Although many of the ideas and techniques still used in shielding analysis do not require direct access to computers, much of the data on which they depend was developed by computerized calculations; and if reasonably accurate answers to complex shielding situations are needed, it is almost mandatory to use computerized techniques. In many ways the science and technology of radiation shielding is mature. The principles and data upon which shielding design and analysis are based have been well established and accepted by the shielding community. Indeed, these principles form the basis of this book. On the other hand, the discipline is still undergo­ ing rapid changes driven by two recent developments: (1) recent changes in the dosimetric concepts and definitions used by regulatory agencies to assess radiation fields, and (2) the tremendous advances in computing power available to the shield­ ing analyst. The result of the first makes many data bases obsolete, requiring new data bases expressed in modern dosimetry units to be created, while the second permits computational analyses not previously feasible. A wide array of special­ and general-purpose shielding codes has become available in the past 20 years, and enhancement of old codes and development of new codes continues at a rapid pace. Modern computing capability affords completely new approaches to shield analysis. For example, visualization techniques are just beginning to be used to better un­ derstand the results of analysis and to construct more detailed three-dimensional models used in shielding design. As is broadly true in engineering design and analysis, recent advances in comput­ ing, as applied to radiation shielding, require very disciplined application. Modern shielding analyses often unnecessarily tend to be exercises in number crunching us­ ing large general-purpose radiation transport programs whose results, because of the pedigree of the program, tend to assume an aura of accuracy that often is not justified. It is very easy with these "black-box" computer codes to obtain a very precise result that is quite inaccurate because, for example, (1) a poor model of the problem is used, (2) inappropriate assumptions are used to generate the calcu­ lational model, and (3) the program user is unaware of program limitations. The disciplined application of these powerful new shielding programs still requires the user to have judgment, experience, and a sound understanding of the programs' methodologies in order to obtain accurate results.

6

Introduction

Chap. 1

In many cases accurate shielding design can be accomplished much more eco­ nomically by using many of the simplified techniques discussed in this book. Results of such calculations can also be used as a rapid check to ensure that results of a large computer calculation are reasonable or to estimate the effect of some small component whose neglect would greatly simplify a calculational model. Thus, the development of the modern computerized shielding tools and new radiological con­ cepts requires the shielding_ analyst to have even greater skill and knowledge.

1.1.4

Regulatory Development

Apace with the development of radiation shielding technology was the introduction of many standards and regulations for radiation protection. Promulgated by various institutions, they continue to evolve. Such standards and regulatory activities have a direct impact on shielding analysis since, in many instances, shield designs are dictated by these standards. Thus it is also useful to have an understanding of the various institutions involved in setting these standards. The first organized efforts to promote radiation protection took place in Europe and America in the period 1913 to 1916 under the auspices of various national advi­ sory committees. After the 1914-1918 world war, more detailed recommendations were made, such as those of the British X-Ray and Radium Protection Commit­ tee in 1921. Signal events in the history of radiation protection were the 1925 London and 1928 Stockholm International Congresses on Radiology. The first led to the establishment of the International Commission on Radiological Units and Measurements (ICRU). The 1928 Congress led to the establishment of the Inter­ national Commission on Radiological Protection (ICRP). Both these commissions continue to foster the interchange of scientific and technical information and to pro­ vide scientific support and guidance in the establishment of standards for radiation protection. In 1929, upon the recommendation of the ICRP, the Advisory Committee on X-Ray and Radium Protection was formed in the United States under the aus­ pices of the National Bureau of Standards. In 1946, the name of the committee was changed to the National Committee on Radiation Protection. In 1964, upon receipt of a congressional charter, the name was changed to the National Coun­ cil on Radiation Protection and Measurements (NCRP). It and similar national advisory organizations in other countries maintain affiliations with the ICRP, and committees commonly overlap substantially in membership. A typical pattern of operation is the issuance of recommendations and guidance on radiation protection by both the NCRP and the ICRP, with those of the former being more closely re­ lated to national needs and institutional structure. The recommendations of these organizations are in no way mandatory upon government institutions charged with promulgation and enforcement of laws and regulations pertaining to radiation pro­ tection. In the United States, for example, the Environmental Protection Agency (EPA) has the responsibility for establishing radiation protection standards. Other federal agencies, such as the Nuclear Regulatory Commission, or state agencies have the responsibility for issuance and enforcement of such laws and regulations.

Sec. 1.2. Radiation Shielding Institutions

1.2

7

RADIATION SHIELDING INSTITUTIONS

An institution of enormous benefit to the radiation shielding community is the Ra­ diation Safety Information Computational Center (RSICC) at Oak Ridge National Laboratory. Until a few years ago, this Center was named the Radiation Shielding Information Center (RSIC). This center maintains a comprehensive collection of literature, computer programs, and data libraries contributed by shielding special­ ists from around the world. Throughout this book, there are references to many codes and data libraries available through RSIC (or now RSICC), and this is the first place a shielding analyst should contact to obtain a particular code or data library. RSICC may be addressed at P.O. Box 2008, Oak Ridge National Labo­ ratory, Oak Ridge, TN 37831-6362, and can be reached through the internet at http://www-rsicc.ornl.gov/rsic.html. Other important institutions providing shielding information are: • National Nuclear Data Center, Bldg. 197D, Brookhaven National Laboratory, Upton, NY 11973-5000. Provides basic cross-section and nuclear data through the internet at http://www.nndc.bnl.gov. • International Atomic Energy Agency, P.O. Box 100, A-1400 Vienna, Austria. Provides many publications and, through its Nuclear Data Section, provides basic nuclear data and reports. The Nuclear Data Section may be reached through the internet at http://www. iaea.or.at. • OECD Nuclear Energy Agency Data Bank, 12, boulevard des Iles, 92130, Issy-les-Moulineaux, France. Provides nuclear data and other shielding infor­ mation through the Data Bank. It may be reached through the internet at http://www.nea.fr/html/databank/. • National Institute of Standards and Technology, US Dept. of Commerce, Gaithersburg, MD 20899-0001, through its Physics Laboratory issues reports, programs and nuclear and atomic data. Reachable through the internet at URL http:/ !physics .nist .gov. • National Council on Radiation Protection and Measurements (NCRP), at 7910 Woodmont Avenue, Suite 800, Bethesda, MD 20814-3095, issues various guidelines and recommendations for the United States. It may be reached through the internet at http://www. ncrp.com. • International Commission on Radiation Units and Measurements (ICRU), at 7910 Woodmont Avenue, Suite 800, Bethesda, MD 20814-3095, issues inter­ national shielding and dosimetry guidelines and recommendations. It may be reached through the internet http://users.erols.com/icru/index.htm). • The German Institute of Radiation Protection, an institute of the Forschungs­ zentrum fiir Umvelt und Gesundheit (GSF), provides shielding and dosimetry reports. It may be reached at Ingolstadter LandstraBe 1, D-85764 Neuherberg, Germany, or at the www URL http://www.gsf .de/englisch/index. html.

8

Introduction

Chap. 1

• National Radiological Protection Board, Chilton, Didcot, Oxon OXll 0 RQ, England. Provides shielding and dosimetry reports for the United Kingdom. It may be reached through the internet at http://www. nrpb. uk. • Radiation Protection and Shielding Division of the American Nuclear So­ ciety, 555 North Kensington Avenue, La Grange Park, IL 6052 6 (internet http://www. ans. org/main. html). This Division of the ANS provides shield­ ing standards, conference proceedings, and journals. It may be reached at the internet address http://www-rsic. ornl. gov/rspd. html. • International Commission on Radiological Protection (ICRP), SE-17116 Stock­ holm, Sweden. Publishes international shielding and radiological guidelines and recommendations. Internet address http://www. icrp. org. • Health Physics Society, 1313 Dolley Madison Boulevard, Suite 402 , McLean, VA 22101. It issues dosimetry standards and the Health Physics journal, and may be reached through the internet at http://www. hps. org.

1.2.1

Professional Societies and Journals

Professional societies serving the radiation shielding and protection community in­ clude the American Nuclear Society, the Health Physics Society, and the American Association of Physicists in Medicine in the United States, and counterpart soci­ eties in other countries. Professional journals of special interest to the shielding community include Health Physics, Journal of Nuclear Medicine, Medical Physics, Nuclear Science and Engineering, Nuclear Technology, Progress in Nuclear Energy, Radiation Protection Dosimetry, and Radiology.

1.3

RADIATION PROTECTION INSTITUTIONS

Persons with a professional interest in shielding and radiation protection are con­ fronted with an array of statutes, regulations, codes of practice, regulatory guides, recommendations, guidelines, and procedures, not to mention sanctions which may be imposed if bad advice is mistaken for good or requirements are interpreted as recommendations. All this well-intentioned guidance issues from national and inter­ national institutions with missions and authorities as confusing as their acronyms. The following is an attempt to help the reader understand the roles of the various institutions. Emphasis is placed on U.S. institutions, not out of national pride but because the authors are more familiar with government bureaucracy in the United States. Much attention has been given in this chapter to two international and one U.S. organization: the International Commission on Radiation Units (ICRU), the Inter­ national Commission on Radiological Protection (ICRP), and the National Coun­ cil on Radiation Protection and Measurement (NCRP). These organizations have enormous influence on radiation-protection standards, not because of any power or authority, but because of their independence and their long-recognized scientific preeminence.

Sec. 1.3. Radiation Protection Institutions

9

With equal scientific credibility but with slightly different roles are organiza­ tions such as the National Research Council (NRC) of the National Academy of Sciences (NAS) in the United States. Of special note is the NAS/NRC Committee on the Biological Effects of Ionizing Radiation (BEIR), which periodically issues comprehensive reports. While NCRP and ICRP activities are more closely related to standards for protection, those of the BEIR Committee are more closely associ­ ated with evaluation and interpretation of scientific research on the biological effects of ionizing radiation. Independent international and national industrial standards organizations be­ come involved in radiation protection through their publication of specific codes of practice and procedures. Activities of the standards organizations are typically integrated with those of professional and trade organizations. 1.3.1 United Nations Organizations One UN organization has had far-reaching influence on the understanding of radi­ ation hazards. The United Nations Scientific Committee on the Effects of Atomic Radiation (UNSCEAR), established in 1955, reports yearly to the General Assembly and periodically issues comprehensive reports on radiation hazards. Several other UN organizations have influence on the practice of radiation pro­ tection. Among these are the International Atomic Energy Agency (IAEA), the World Health Organization (WHO), and the Food and Agriculture Organization (FAO). Established in 1957, the IAEA has as its principal objective "to acceler­ ate and enlarge the contribution of atomic energy to peace, health, and prosperity throughout the world." The WHO, created in 1948, has among other goals the promotion of international cooperation in collection and dissemination of health statistics and establishing international health standards. Along with the FAO, and the Joint FAO/WHO Codex Alimentarius Commission, the WHO is involved with establishing codes of practice for the operation of radiation facilities for the preservation of foods. 1.3.2 Governmental Organizations in the United States The Atomic Energy Act of 1954 gave exclusive authority to an Atomic Energy Com­ mission (AEC) to exercise federal regulation on the use, transportation, and disposal of radioactive materials used in or produced by the nuclear fission process. The au­ thority did not extend to naturally occurring radionuclides, except for uranium and thorium as source materials for nuclear fuels. The first AEC radiation-protection regulations (10CFR20) were published in 1955 and became effective in 1957. The Energy Reorganization Act of 1974 gave rise to a Nuclear Regulatory Commission (NRC) to which were transferred the regulatory functions of the AEC. Other func­ tions of the AEC devolved to the Energy Research and Development Administration (ERDA), later given cabinet status as the Department of Energy (DOE), which has responsibility for radiation protection and safety at national laboratories.

10

Introduction

Chap. 1

The AEC, then NRC, has been permitted since 1959 to delegate to the individual states the responsibility and licensing authority for all regulatory activity save that for nuclear reactors. These agreement states, which number about half those in the union, are required to maintain radiation protection standards at least as strict and comprehensive as the federal standards. During the period 1959 to 1970, the responsibility for establishing standards for radiation protection was assigned to an interagency Federal Radiation Coun­ cil (FRC). Implementation of the standards through regulations and enforcement practices was the responsibility of the competent federal agency. The Department of Health, Education and Welfare (HEW) was assigned the primary responsibility for collation, analysis, and interpretation of environmental radiation levels, includ­ ing medical use of isotopes and x rays. In 1970, the authority of the FRC, along with certain surveillance responsibilities of the Public Health Service (PHS), were transferred to the Environmental Protection Agency (EPA). The Atomic Energy Commission retained the responsibility for implementation and enforcement of ra­ diation standards through its licensing authority. The regulation of radiation from consumer products remained with HEW, now HHS, the Department of Health and Human Services. However, certain functions of the Food and Drug Administration (FDA) Bureau of Radiological Health were transferred to the EPA. As a result of the Orphan Drug Act of 1983 (Public Law 97-414), the National Institutes of Health (NIH) of HHS has had the responsibility for preparation of probability-of-causation tables for adjudication of claims of radiation carcinogenesis.

1.4 IMPORTANT SOURCES OF SHIELDING INFORMATION There are many sources of shielding information, some of which are referenced at the end of each chapter. In particular, the authors have found the following publications and data libraries to be particularly useful and significant.

Textbooks and Monographs on Radiation Shielding Blizard, E.P. (ed.), Reactor Handbook, Vol. III, Part B: Shielding, 2nd ed., Inter­ science, New York, 1962. Chilton, A.B, J.K. Shultis, and R.E. Faw, Principles of Radiation Shielding, Prentice Hall, Englewood Cliffs, NJ, 1984. Goldstein, H., Fundamental Aspects of Reactor Shielding, Addison-Wesley, Reading, MA, 1959. Jaeger, R.G. (ed.), Engineering Compendium on Radiation Shielding, Vol. I, Shielding Fundamentals and Methods; Vol. II, Shielding Materials and Design; Vol. III, Shield Design and Engineering, Springer-Verlag, New York, 1968-1975. Price, B.T., C.C. Horton, and K. T. Spinney, Radiation Shielding, Pergamon Press, Elmsford, NY, 1957. Profio, A.E., Radiation Shielding and Dosimetry, Wiley, New York, 1979. Rockwell, T. III, (ed.), Reactor Shielding Design Manual, D. Van Nostrand, Princeton, NJ, 1956.

Sec. 1.4. Important Sources of Shielding Information

11

Schaeffer, N.M. (ed.), Reactor Shielding for Nuclear Engineers, TID-25951, National Technical Information Service, Springfield, VA, 1973.

Textbooks and Monographs on Radiation Protection Attix, F.H., and W.C. Roesch (eds.), Radiation Dosimetry, 2nd ed., Vols. I-III, Aca­ demic Press, New York, 1966. Cember, H., Introduction to Health Physics, 2nd ed., Pergamon Press, Elmsford, NY, 1983. Eisenbud, M., Environmental Radioactivity, 3rd ed., Academic Press, Orlando, FL, 1987. Faw, R.E., and J.K. Shultis, Radiological Assessment: Sources and Exposures, Pren­ tice Hall, Englewood Cliffs, NJ, 1993. Shapiro, J., Radiation Protection, 3rd ed., Harvard University Press, Cambridge, MA, 1990. Turner, J.E., Atoms, Radiation, and Radiation Protection, 2nd ed., Wiley, New York, 1995.

Reference Books with Emphasis on Nuclear Data Browne, E., and R.B. Firestone, Table of Radioactive Isotopes, Wiley, New York, 1986. Courtney, J.C. (ed.), A Handbook of Radiation Shielding Data, A Publication of the Shielding and Dosimetry Division of the American Nuclear Society, La Grange Park, IL, 1975. ICRP, Radionuclide Transformations: Energy and Intensity of Emissions, Publication 38, Annals of the ICRP, Vols. 11-13, Pergamon Press, Elmsford, NY, 1983. Kocher, D.C., Radioactive Decay Tables, Report DOE/TIC-11026, National Technical Information Service, Springfield, VA, 1981. Nuclear Data Tables,

Academic Press, Orlando, FL.

Weber, D.A., K.F. Eckerman, L.T. Dillman, and J.C. Ryman, MIRD: Radionuclide Data and Decay Schemes, Society of Nuclear Medicine, New York, 1989.

Standards and Guidelines American National Standards Institute, published by various professional societies, for example, the American Nuclear Society, 555 North Kensington Avenue, La Grange Park, IL 60525, and the Health Physics Society, 1313 Dolley Madison Boulevard, Suite 402, McLean, VA 22011. Annals of the International Commission on Radiological Protection, available from Pergamon Press ( U.S. address: Elsevier Science, Inc., 660 White Plains Rd., Tarry­ town, NY, 10591-5153). NM/MIRD Pamphlets, published by the Medical Internal Radiation Dose Committee of the Society of Nuclear Medicine, 136 Madison Avenue, New York, NY 10016. Reports of the International Commission on Radiological Units, available in the United States from ICRU Publications, 7910 Woodmont Avenue, Suite 800, Bethesda, MD 20814-3095.

12

Introduction

Chap. 1

Reports of the National Council on Radiation Protection and Measurements, available in the United States from NCRP Publications, 7910 Woodmont Avenue, Suite 800, Bethesda, MD 20814-3095. UNSCEAR, IAEA, FAO, and WHO publications, available from United Nations Pub­ lications, Room DC2-853, Department 701, New York, NY 10017.

Electronic Data Libraries Capture Gamma Photons: THERMGAM: Prompt Gamma Rays from Thermal­ Neutron Capture, available from Radiation Shielding Information Center (RSIC) as Data Library Collection DLC-140. Charged Particle Stopping Powers: STAR CODES: Code System for Calculating Stopping-Power and Range Tables for Electrons, Protons, and Helium Ions, available from Radiation Shielding Information Center (RSIC) as Code Package PSR-330 Electron Cross Sections: Perkins, S.T., D.E. Cullen, and S.M. Seltzer, Tables and Graphs of Electron-Interaction Cross Sections from 10-eV to 100 GeV Derived from the LLNL Evaluated Electron Data Library (EEDL), Z = 1-100, Report UCRL-50400 Vol. 31, Lawrence Livermore National Laboratory, Livermore, CA, 1991. Neutron Cross Sections: MCNPDAT: MCNP, Version 4, Standard Neutron Cross Section Data Library Based on ENDF/B-V, available from Radiation Shielding In­ formation Center (RSIC) as Data Library Collection DLC-105C. Nuclear Decay Data: NUCDECAY, ICRP and MIRD Nuclear Decay Data, avail­ able from Radiation Shielding Information Center (RSIC) as Data Library Collection DLC-172. Photon Buildup Factors: ANS 6.4.3: Geometric Progression Gamma-Ray Buildup Factor Coefficients, available from Radiation Shielding Information Center (RSIC) as Data Library Collection DLC-129. Photon Cross Sections: PHOTX: Photon Interaction Cross Section Library, avail­ able from Radiation Shielding Information Center (RSIC) as Data Library Collection DLC-136. Photon Cross Sections: Cullen, D.E., M.H. Chen, J.H. Hubbell, S.T. Perkins, E.F. Plechaty, J.A. Rathkopf, and J.H. Scofield, Tables and Graphs of Photon Interaction Cross Sections from 10 eV to 100 GeV Derived from the LLNL Evaluated Photon Data Library (EPDL), Report UCRL-50400, Vol. 6, Rev. 4, Part A: Z = 1-50, Part B: Z = 51-100, Lawrence Livermore National Laboratory, Livermore, CA, 1989.

1.5

FINAL REMARKS

No personal credit has been given to this point to the many individuals who have contributed to the development of shielding technology. Their names will become known to the reader as he or she reads this book, especially the bibliographic in­ formation provided as references at the end of the chapters. Significant textbooks and handbooks providing broad coverage on shielding have been listed in Section 1.4. Similarly, many reports, monographs, and journal articles have been published on various particular aspects of nuclear radiation shielding.

13

References

One thing is noteworthy about the reference works. To a great extent, they were written within the context of the needs of a single research program-sometimes a single source type. The basic ideas are all present in one or another of these books and reports, but there is a great diversity in what they emphasize, the nomenclature they employ, and the form in which the information is cast. It is the purpose of the present book to emphasize those principles which are common to all shielding applications and thus establish shielding as an integrated field of knowledge. It is hoped that the student who uses this work will thereby learn the fundamentals of shielding in a way that will permit their application to a wide variety of practical problems.

REFERENCES FANO, U., L.V. SPENCER, AND M.J. BERGER, "Penetration and Diffusion of X-Rays," in Handbuch der Physik, Vol. 38, No. 2, S. Fliigge (ed.), Springer-Verlag, Berlin, 1959. FAW, R.E. AND J.K. SHULTIS, Radiological Assessment: Sources and Exposures, Prentice Hall, Englewood Cliffs, NJ, 1993. ICRP, 1990 Recommendations of the International Commission on Radiological Protection, Publication 60, Annals of the ICRP, 21 No. 1-3, Pergamon Press, Oxford, 1991. KIMEL, W.R. (ed.), Radiation Shielding: Analysis and Design Principles as Applied to Nuclear Defense Planning, OCD/DSU Doc. TR 40, U.S. Government Printing Office, Washington, DC, 1966. NAS, Health Effects of Exposure to Low Levels of Ionizing Radiation, Report of the BEIR V Committee, National Academy of Sciences/National Research Council, Washington, DC, 1990. NCRP, Structural Shielding Design and Evaluation for Medical Use of X Rays and Gamma Rays of Energies up to 10 Me V, Report 49, National Council on Radiation Protection and Measurements, Washington, DC, 1976. NCRP, Risk Estimates for Radiation Protection, Report 115, National Council on Radia­ tion Protection and Measurements, Bethesda, MD, 1993a. NCRP Limitation of Exposure to Ionizing Radiation, Report 116, National Council on Radiation Protection and Measurements, Bethesda, MA, 1993b. OCD, Shelter Design and Analysis, Vol. 1: Fallout Radiation Shielding, Doc. TR-20(Vol. 1), Office of Civil Defense (July 1967), as amended by Change 1 (July 1969), Change 2 (January 1970), and Change 3 (November 1974). SPENCER, L.V. Structure Shielding against Fallout Radiation from Nuclear Weapons, NBS Monograph 42, U.S. Government Printing Office, Washington, DC, 1962. SPENCER, L.V., A.B. CHILTON, AND C.M. EISENHAUER, Structure Shielding Against Fall­ out Gamma Rays from Nuclear Detonations, NBS Special Publication 570, U.S. Gov­ ernment Printing Office, Washington, DC, 1980. SPENCER, L.V., AND U. FANO, "Penetration and Diffusion of X-Rays: Calculation of Spatial Distributions by Polynomial Expansion," J. Res. Natl. Bur. Stand., 46, 446 (1951).

14

Introduction

Chap. 1

UNSCEAR, Sources, Effects and Risks of Ionizing Radiation, United Nations Scientific Committee on the Effects of Atomic Radiation, United Nations, New York, 1988. UNSCEAR, Sources and Effects of Ionizing Radiation, United Nations Scientific Commit­ tee on the Effects of Atomic Radiation, United Nations, New York, 1993.

Chapter 2

Characterization of Radiation Fields and Sources

This book is concerned with the transmission of directly and indirectly ionizing radiation through matter and its interaction with matter. The radiation is concep­ tualized as particles-photons, electrons, neutrons, and so on. The term radiation field refers collectively to the particles and their trajectories in some region of space or through some boundary, either instantaneously or accumulated over some period of time. Characterization of the radiation field, for any one type of particle, requires accounting for the spatial variation of the joint distribution of particle energy and direction. In certain cases, such as those encountered in neutron scattering ex­ periments, properties such as spin may be required for full characterization. Such infrequent and specialized cases are not considered in this book. The sections to follow describe how to characterize the radiation field in a re­ gion of space in terms of the particle fluence and how to characterize the radiation field at a boundary in terms of the particle flow. The fluence and flow are called radiometric quantities, as distinguished from dosimetric quantities, which are dis­ cussed in Chapter 5. It will be seen that the fluence and flow concepts apply both to measurement and to calculation. Measured properties are inherently stochastic, in that they involve enumeration of individual particle trajectories. Measurement, too, requires finite volumes or boundary areas. The same is true for fluence or flow calculated by Monte Carlo methods, as described in Chapter 11, since such calculations are in large part computer simulations of experimental determinations. In the methods of analysis discussed in this book, the fluence or flow is treated as a deterministic point function and should be interpreted as the expected value, in a statistical sense, of a stochastic variable. It is perfectly proper to refer to the fluence, flow, or related dosimetric quantity at a point in space. But it must be recognized that any measurement is only a single estimate of the expected value. 15

16

2.1

Radiation Fields and Sources

Chap. 2

DIRECTIONS AND SOLID ANGLES

The directional properties of radiation fields are almost universally described using spherical polar coordinates as illustrated in Fig. 2.1. The direction vector is a unit z w

X

Figure 2.1. Spherical polar coordinate system for specification of the unit direction vector 0, polar angle iJ, azimuthal angle 1/J, and associated direction cosines ( u, v, w ).

vector, given in terms of the orthogonal Cartesian unit vectors i, j, and k by 0

= iu + jv +kw= i siil'Ocos� + j siil'Osin� + kcos-0.

(2.1)

Increase in '!9 by d-0 and � by d� sweeps out area dA = sin '!9 d-0 d� on a unit sphere. The solid angle encompassed by a range of directions is defined as the area cut on the surface of a sphere divided by the square of the radius of the sphere. Thus, the differential solid angle associated with differential area dA is dfl

= sin-Od-Od�.

(2.2)

The solid angle is a dimensionless quantity. Nevertheless, to avoid confusion when referring to a directional distribution function, units of steradians, abbreviated sr, are attributed to the solid angle. A substantial simplification in notation can be achieved by making use of w = cos '!9 as an independent variable instead of the angle '!9, so that sin '!9 d-0 -dw.

=

17

Sec. 2.2. Fundamental Radiation Field Variables

A derivative benefit is a more straightforward integration over direction and the avoidance of a common error. 1 The benefit is evident when one computes the solid angle subtended by "all possible directions," namely, n

2.2

=

1

0 11"

d'I? sin 19

12,r d'I/J = 11 dw 12,r d'I/J = 41r. 0

-1

(2.3)

0

FUNDAMENTAL RADIATION FIELD VARIABLES

The evolution of radiation shielding and dosimetry has benefited from contributions made in many areas of the physical and biological sciences. Consequently, there exist several different systems of nomenclature. To the extent possible, this book adopts the nomenclature prescribed by the International Commission on Radiation Units and Measurements [ICRU 1993].

2.2.1

Fluence and Fluence Rate

A fundamental way of characterizing the intensity of a radiation field is in terms of the number of particles that enter a specified volume. To make this characterization, the radiometric concept of fluence is introduced. The particle fl.uence, or simply \

(a)

\

\

\

\

I

I

I

I

I

(b)

Figure 2.2. Element of volume d V in the form of a sphere with cross-sectional area dA. In (a) the attention is on the number of particles passing through the surface into the sphere. In (b) the attention is on the paths traveled within the sphere by particles passing through the sphere.

fl.uence, at any point in a radiation field may be thought of in terms of the number of particles A.Np that, during some period of time, penetrate a hypothetical sphere of cross section AA centered on the point, as illustrated in Fig. 2.2(a). The fl.uence is defined as (2.4) 1 The error is to state the differential solid angle incorrectly as d,{) d'ljJ instead of sin,{) d,{) d'ljJ.

18

Radiation Fields and Sources

Chap. 2

An alternative and arguably more useful definition of the fluence is in terms of the sum Li Si of path-length segments within the sphere, as illustrated in Fig. 2.2(b). The fluence can also be defined as = lim [ �V-+O

Li Si . ]

AV

(2.5)

In Problem 1 at the end of this chapter, the reader is asked to verify the equivalence of these two definitions of the fluence. While the difference quotients of Eqs. (2.4) and (2.5) are useful conceptually, beginning in 1971 the ICRU prescribed that the fluence should be given in terms of differential quotients, in recognition that ANp is the expectation value of the number of particles entering the sphere. Thus, _ dNP = dA '

(2.6)

where dNp is the number of particles which penetrate into a sphere of cross-sectional area dA. In the words of ICRU Report 19 [1971], "a detailed analysis of the char­ acteristics of such expectation values has to be based on a consideration of the cor­ responding stochastic quantities and their probability distributions. In many appli­ cations these considerations can be disregarded and all definitions of non-stochastic quantities can be adequately understood without consideration of random fluctua­ tions." The definition of Eq. (2.6) has been retained in subsequent reports of the ICRU [1980, 1993]. The fluence rate, or flux density, is expressed in terms of the number of particles entering a sphere, or the sum of path segments traversed within a sphere, per unit time, namely, d _ d2 NP (2.7) = - dt - dAdt·

2.2.2

Net Flow and Net Flow Rate

Another radiometric measure of a radiation field is the net number of particles crossing a surface with a well-defined orientation, as illustrated in Fig. 2.3. In this text, net particle flow (or simply net flow) at a point on a surface is the net number of particles in some specified time interval that flow across a unit differential area on the surface, in the direction specified as positive. As shown in the figure, one side of the surface is characterized as the positive side and is identified by a unit vector n normal to the area AA. If the number of particles crossing AA from the negative to the positive side is AMt and the number from the positive to the negative side AM;, then the net number crossing toward the positive side is AMp = AMt - AMP-. The net flow at the given point is designated as Jn, with the subscript denoting the unit normal n from the surface, and is defined as (2.8)

19

Sec. 2.3. Directional Properties of the Radiation Field

or in the limit,

_ dMp Jn = dA. The flows of particles in the positive and negative directions, J,: and J;;, are defined in terms of l:lM: and l:lM;; in a similar manner. The relation between the net flow and the positive and negative flows is (2.10)

The net flow rate is expressed in terms of the net number of particles crossing an area perpendicular to unit vector n, per unit area per unit time, namely,

(2.9)

n

(2.11) Flow rates can also be defined, in analogy with Eq. (2.10), so that . - ·+ ·- . (2.12) Jn = Jn - Jn ·

Surface Figure 2.3. Ele­ ment of area �A in a surface. Particles cross the area from either side.

The concepts of fluence and particle flow appear to be very similar, both being defined in terms of a number of particles per unit area. However, for the concept of the fluence, the area presented to incoming particles is independent of the direction of the particles, whereas for the particle flow concept, the orientation of the area is well defined.

2.2.3

Other Definitions

Fluence and its time rate, flux density, or fluence rate, can be defined in way s other than through the spherical detector approach. See Problem 1, for example. One alternative approach is to identify the fluence rate of monoenergetic particles at a point as the product of the particle number density at the point and the speed of the particles. For particles distributed in velocity, the total fluence rate is obtained by integrating the product of the number density and the velocity, that is,

= fo 00 dvvn(v).

(2.13)

2.3 DIRECTIONAL PROPERTIES OF THE RADIATION FIELD 2.3.1 Properties of the Fluence The computed fluence is a point function of position r. Measurement of the fluence requires a radiation detector of finite volume; therefore, there is not only uncertainty due to experimental error but also ambiguity in identification of the "point" at which to attribute the measurement. The nature of the particles is implicit, and

20

Radiation Fields and Sources

Chap. 2

the argument r in cf>(r) is sometimes implicit. With no other arguments, cf> or cf>(r), represents the total fluence, irrespective of particle energy or particle direction, that is, integrated over all particle energies and directions. In many circumstances, it is necessary to broaden the scope of the fluence to include information about the energies and directions of particles. To do so requires the use of distribution functions. In the same way that ages or heights of persons in a large population have continuous distributions, particle energies and direc­ tions require, in general, fluences expressed as distribution functions. For example, cf>(r, E) dE lim�E=O cf>(r, E) tl.E is, at point r, the fiuence energy spectrum-the fluence of particles with energies between E and E + dE. 2 The angular dependence of the fluence is a bit more complicated to write. The angular variable itself is the vector direction n. The direction is a function of the polar and azimuthal angles {) and 'lj;. Similarly, the differential element of solid angle is a function of the same two variables, namely dO. = sin{) d{) d'lj; = dw d'lj;. Thus, cf>(r, 0) dO. or cf>(r, w, 'lj;) dw d'lj; is, at point r, the angular fluence-the fluence of particles with solid angles in dO. about n. The joint energy and angular distribution of the fluence is defined in such a way that cf>(r, E, 0) dE dO. is the fluence of particles with energies in dE about E and with directions in dO. about n. These distribution functions are related by

=

cf>(r) = / dO. cf>(r, 0) = f')O dE cf>(r, E) = }47r lo

/

lo

co

dE { dO. cf>(r, E, 0).

141r

(2.14)

Occasionally finding use in radiation shielding and dosimetry is a quantity called the energy fiuence. It is given by w(r) =

/

lo

co

dE { dO.Ecf>(r,E,O).

}47r

(2.15)

In the system of notation adopted, it is necessary that the energy and angular variables appear specifically as arguments of cf> to identify the fluence as a distribu­ tion function in these variables. 3

2.3.2

Transformation of Variables

When making changes in the independent variable in a distribution function, special care must be taken. For example, in Chapter 3 it is convenient to convert from distributions using the energy variable E to distributions using the wave number variable ,\ = m e c2 / E, in which m e c2 is the rest-mass energy of the electron. Since there is a unique relationship between the energy and the wave number, it must be true that the fluence of particles within a certain range of energies dE must be 2Or, in other words, with energies in dE about E. 3ICRU Report 33 [1980] refers to the energy distribution as the spectral distribution and to the angular distribution as the radiance. The symbol p is used for ¢,(r, n) and the symbol PE for ,j,(r, E, 0).

Sec. 2.3.

21

Directional Properties of the Radiation Field

equal to the fluence of particles within the corresponding range4 of wave numbers -d)... Thus, (2.16)

2.3.3

Angular Properties of the Flow and Flow Rate

Definition of the Angular Flow Just as it is very often necessary to account for the variation of the fluence with particle energy and direction, the same is true for the flow and flow rate. Treatment of the energy dependence is no different from the treatment for the fluence, so here we need only examine the angular dependence. With an element of area and its orientation as illustrated in Fig. 2.3, it is perfectly proper to define the angular flow in such a way that Jn (r, fl) df! is the flow of particles through a unit area with directions in dn about n. The angular flow rate is written as in (r, fl). Angular Flow vs Angular Fluence Figure 2.4 illustrates particles within a differential element of direction df! about direction n crossing a surface perpendicular to unit vector n. Also shown in the figure is a sphere whose surface just intercepts all the particles. It is ap­ parent that if �A is the cross-sectional area of the sphere, then the corre­ sponding area in the surface is �A sec'!?, where cos'!? = n•O. Thus, since the same number of particles pass through the sphere and through the area in the surface, Jn (r, 0) �A= cos'!? �A(r,w)

= L bn(r)wn. n=O

(2.23)

The coefficients bn can be computed from the coefficients ae. When either the power series or the Legendre polynomial series is truncated to a finite number of terms, it is possible that the approximate series yields negative values of cf>(r,w) for some values of w. Various schemes may be used to accommodate such anomalies, the easiest of which simply is to ignore negative values. For example, at the outer surface of a reactor core, the fluences of neutrons or gamma rays are well approximated as being isotropic for outward directions. If n is an outward normal to the surface, then the angular fluence is well approximated as being proportional tow, but only for positive w.

2.5

GENERAL SPECIFICATION OF RADIATION SOURCES

Solution of a shielding problem requires a thorough understanding of the charac­ teristics of the source of radiation. In this chapter, sources are discussed from an abstract point of view, but the properties described are those one must identify and quantify in practical cases. Detailed and specific sourced descriptions are provided in Chapter 4. The most fundamental type of source is a point source. A real source can be approximated as a point source provided (1) that the volume is sufficiently small, that is, with dimensions much smaller than the dimensions of the attenuating medium between source and detector, and (2) that there is negligible interaction of radiation with the matter in the source volume. Requirement 2 may be relaxed if source characteristics are modified to account for source self-absorption.

24

Radiation Fields and Sources

Chap. 2

In general, a point source may be characterized as depending on energy, direc­ tion, and time. In this text and, indeed, in almost all shielding practice, time is not treated as an independent variable. Because of the negligible time between radiation emission from the source and energy deposition in a target receptor, ra­ diation problems may be treated in a steady-state approximation. In other words, the radiation dose rate varies with time in direct proportion to the source variation with time. Therefore, the most general characterization of a point source used in this text is in terms of energy and direction, so that Sp (E, 0) dE dn is the number of particles emitted with energies in dE about E and in dn about n. If it is un­ derstood that one is interested in the dose rate or some other response rate, then Sp is the number of particles emitted per unit time. Common practical units for Sp (E, 0) are Mev- 1 sr- 1 or MeV - 1 sr- 1 s- 1. Most radiation sources treated in shielding practice are isotropic, so that source characterization requires only knowledge of Sp(E) dE, which is the number of parti­ cles emitted with energies in dE about E (per unit time), and has common practical units of Mev- 1 (or Mev- 1 s- 1 ). Radioisotope sources are certainly isotropic, as are fission sources and capture gamma-ray sources. A careful distinction must be made between the activity of a radioisotope and its source strength. Activity is precisely defined as the expected number of atoms undergoing radioactive transformation per unit time. It is not defined as the number of particles emitted per unit time. Decay of two very common laboratory radioiso­ topes illustrate this point. Each transformation of 6 °Co, for example, results in the emission of two gamma rays, one at 1.173 MeV and the other at 1.333 MeV. Each transformation of 137 Cs, accompanied by a transformation of its decay product 137 m Ba, results in emission of a 0.662-MeV gamma ray with probability 0.85. The traditional unit of activity is the curie (Ci), equivalent to 3.7x10 10 transfor­ mations per second. The SI unit is the becquerel (Bq), equivalent to 1 transforma­ tion per second. In medical physics and health physics, radiation source strengths are commonly figured on the basis of accumulated activity, such as mCi-hours in traditional units or Bq-seconds in SI units. Such time-integrated activities account for the cumulative number of transformations in some biological entity during the transient presence of a radionuclide in the entity. Of interest in such circumstances is not the time-dependent dose rate to that entity or some other nearby region, but rather the total dose accumulated during the transient. Similar practices are followed in dose evaluation for reactor transients, solar flares, nuclear weapons, and so on. Radiation sources may be distributed along a line, over an area, or within a vol­ ume. Source characterization requires, in general, spatial and energy dependence, with S1(r, E) dE, S a (r, E) dE, and S v (r, E) dE representing, respectively, the num­ ber of particles emitted in dE about E per unit length, per unit area, and per unit volume. Occasionally it is necessary to include angular dependence. This is especially true for effective area sources associated with computed angular flows across certain planes. Clearly, for a fixed surface, S a (r, E, 0) and Jn (r, E, 0) are equivalent specifications.

25

Sec. 2.6. Distributed and Discrete Variables

2.6

DISTRIBUTED AND DISCRETE VARIABLES

Radiation shielding and radiation transport methods are very often formulated with sources and fl.uences treated as continuous functions of position, energy, and direc­ tion. However, sources may be monoenergetic and unidirectional. Indeed, a volume or area-source formulation may have to accommodate a point source. Such accom­ modations may be made using the Dirac delta function, which is discussed in some detail in Appendix B.6. The function 8(x - x0 ) has the property of being zero for all x except x 0 , and of rendering (2.24) The integral equals to zero if x 0 is out of the range of integration. Two examples are given below to illustrate the use of the Dirac delta function. 1. Suppose that a problem formulation requires the fl.uence to be a distributed function of position, energy, and direction. However, the actual fl.uence is monoenergetic and unidirectional, with spatial variation given as f(x). Then, (r, E, !l)

= f(x)8(E - E0 )8(!l - !l0 ), = f(x)8(E - E0 )8(w - w0 )8(1/) -1/)0

).

(2.25)

2. Suppose that a problem formulation requires the volume source strength to be a distributed function of position, energy, and direction. However, the actual source is a plane source at x = 0. It is monoenergetic and isotropic, with total source strength Sa . Then, (2.26)

REFERENCES ICRU, Radiation Quantities and Units, Report 19, International Commission on Radiation Units and Measurements, Washington, DC, 1971. ICRU, Radiation Quantities and Units, Report 33, International Commission on Radiation Units and Measurements, Washington, DC, 1980. ICRU, Quantities and Units in Radiation Protection Dosimetry, Report 51, International Commission on Radiation Units and Measurements, Bethesda, MD, 1993.

26

Radiation Fields and Sources

Chap. 2

PROBLEMS 1. Equations (2.4) and (2.5) are two defini­ tions of the fluence. Consider the sphere illustrated to the right, with a uniform parallel beam of particles incident on its surface, so that !:l.Np particles enter the sphere. Verify that the mean chord length is (£) = 4R/3 and use this iden­ tity to demonstrate the equivalence of the two definitions. 2. Show that if the angular flux density (r, f!) is isotropic, the current density j(r) is equal to zero. 3. Show that the angular fluence (r)8(w - w0 )8('1/J - 1/J0 ), in which 8 represents the Dirac delta function and w cos{). Use Eq. (2.19) to determine j(r), the current density. Show that the number of particles per unit time per unit area crossing the surface is w0 ¢>(r), that is, that the z component of j is the number of particles per unit time crossing a unit area perpendicular to the z-axis.

=

5. Show that the solid angle subtended by the disk shown to the right at the point P which is a dis­ tance z from the disk on a line perpendicular to the center of the disk is

p

,, z

n � 21r(l - cos{)). 6. Show that the solid angle subtended by a rectan­ gular area W x L, at a point distance z from the center of the area on a line perpendicular to the center of the area is given by !1 in which

f

= 4 tan-1 ( ✓1 +€ 2 + 2) ,,, ,,, €

= W/L and 'f/ = 2z/L.

p

w L

27

Problems

7. At a certain plane in space, the particle flux density is isotropic for 0 ≤ ω ≤ 1 and zero for negative ω, where ω is the cosine of the angle made by the particle direction and a normal to the plane of reference. In that plane, what is (a) jn (ω), the angular distribution of the flow rate, and (b) jn+ , the flow rate for all positive ω? 8. Neutrons emerge from a plane surface into a vacuum with an angular flow rate proportional to the square of the cosine of the angle with respect to an outward normal n ˆ to the surface. Let jn be the net flow rate at the surface. What is the total (not angular) flux density in terms of jn ? 9. At a hypothetical plane surface normal to the unit vector n ˆ there is a plane isotropic source of radiation of strength Sa (cm−2 s−1 ). In terms of Sa , what is (a) the net flow rate jn and (b) the flux density φ at the plane? 10. At some location r, the angular flux density of monoenergetic particles is

φ(r, Ω) =

(

ω 7/4 ,

0 ≤ ω ≤ 1,

0,

ω < 0,

where ω ≡cos ϑ. The polar angle of direction is ϑ; the azimuthal angle of direction is ψ. The Cartesian representation of Ω is i sin ϑ cos ψ + j sin ϑ sin ψ + k cos ϑ. (a) Use Eq. (2.14) to derive an expression for the (total) flux density φ(r). (b) Use Eq. (2.19) to derive an expression for the vector current density j(r). (c) Use Eq. (2.17) to derive an expression for the (scalar) angular flow rate at r through an area perpendicular to the z-axis, namely jk (r, Ω). (d) Use Eq. (2.18) and the result of part (c) to derive an expression for the net flow rate across an area perpendicular to the z-axis, namely jk (r). Verify the result using Eq. (2.20) and the result of part (b). (e) Determine the net flow rates through the x-z and y-z planes. (f ) Use Legendre polynomials in ω, of order 0 through 4, to approximate the angular flux density. Compute and plot the flux density in all approximations. (g) Explain and give examples of the distinctions between the angular flux density and the angular flow rate, and between the total flux density and the net flow rate.

Copyrighted Materials Copyright© 2000 ANS (American Nuclear Society) Retrieved from app.knovel.com

Chapter 3

Interaction of Radiation with Matter

3.1

INTERACTION COEFFICIENT

The interaction of a given type of radiation with matter may be classified according to the type of interaction and the matter with which the interaction takes place. The interaction may take place with an electron, and in many cases the electron behaves as though it were free. Similarly, the interaction may take place with an atomic nucleus, which in many cases behaves as though it were not bound in a molecule or crystal lattice. However, in some cases, particularly for radiation particles of comparatively low energy, molecular or lattice binding must be taken into account. Despite these differences, for consistency in this book, all cross sections are expressed on an atomic basis. The interaction may be a scattering of the incident radiation accompanied by energy change. A scattering interaction may be elastic or inelastic. Consider, for example, the interaction of a photon with an electron in what is called Compton scattering. In the sense that the interaction is with the entire atom within which the electron is bound, the interaction must be considered as inelastic, since some of the incident photon's energy must compensate for the binding energy of the electron in the atom. However, in most practical cases electron binding energies are orders of magnitude lower than gamma-photon energies, and the interaction may be treated as purely elastic scattering of the photon by a free electron. Neutron scattering by an atomic nucleus may be elastic, in which case the incident neutron's kinetic energy is shared by that of the scattered neutron and that of the recoil nucleus, or it may be inelastic, in which case some of the incident neutron's kinetic energy is transformed to internal energy of the nucleus and thence to a gamma ray emitted from the excited nucleus. It is important to note that for both elastic and inelastic scattering unique relationships between energy exchanges and angles of scattering arise from conservation of energy and linear momentum. Other types of interactions are absorptive in nature. The identity of the incident particle is lost, and total relativistic momentum and energy are conserved, some of the energy appearing as nuclear excitation energy, some as translational, vibrational, 28

Sec. 3.1. Interaction Coefficient

29

and rotational energy. The ultimate result may be the em1ss1on of particulate radiation as occurs in the photoelectric effect and in neutron radiative capture. The interaction of radiation with matter is always statistical in nature, and therefore must be described in probabilistic terms. Consider a particle traversing a homogeneous material and let P denote the probability that this particle interacts in some specified manner, say absorption, while traveling a distance �x. It is found that the quantity µ lim6. x -+o(P/ �x) is a property of the material for a given interaction. In the limit of small path lengths, µ is seen to be the probability per unit differential path length that a particle undergoes a specified interaction. That µ is constant for a given material and for a given type of interaction implies that the probability of interaction per unit path length is independent of the path length traveled prior to the interaction. In this book, when the interaction coefficient is referred to as the probability per unit path length of an interaction, it is understood that this is true only in the limit of very small path length. The interaction probability per unit path length is fundamental in describing how radiation interacts with matter and is usually called the linear attenuation coefficient [ICRU 1980]. It is perhaps more appropriate to use the words linear interaction coefficient since many interactions do not "attenuate" the particle in the sense of an absorption interaction. Although this nomenclature is widely used to describe photon interactions, µ is often referred to as the macroscopic cross section and given the symbol � when describing neutron interactions. The utility of the linear interaction coefficient in describing interaction of radi­ ation with matter becomes apparent when one interprets the fluence as the path length traveled by particles in a unit volume, in the limit of a very small volume. The product µ is thus seen to be the number of interactions per unit volume. Division by the material density yields(µ/ p), the number of interactions per unit mass. The linear interaction coefficient is a function of the energy of the particle. De­ pending on the nature of the interaction, it may also be a function of (1) the energy of the particle after scattering,(2) the energy of the recoil atom or electron,(3) the angles of deflection of the scattered radiation and recoil atom or electron,1 and(4) the angles of emission of secondary particles. For example, the doubly differential scattering interaction coefficient is defined in such a way that µ( E, E', .is ) dE' dD, is the probability per unit path length for an interaction in which the incident particle of energy E emerges from the interaction with energy between E' and E' + dE' and with scattering angle .is , measured with respect to the incident direction, within the differential solid angle dD.. In this form, µ has units such as cm-1 Mev-1 sr-1. Alternatively, the interaction coefficient could be expressed in terms of energies and angles for recoil electrons or atoms, or secondary radiations. Often it is of interest to deal only with, say, the energy dependence or the angular distribution of scattered radiation. In this case, one or the other of the following

=

1This dependence is true for isotropic media, which is assumed to be the case unless otherwise noted specifically. For crystalline and other anisotropic media, µ 8 generally depends on the incident radiation direction O and the exit radiation direction !1 1.

30

Interaction of Radiation with Matter

Chap. 3

forms of the linear interaction coefficient may be used:

= }47r{ dflµ(E,E','19.) = µ(E,'19.) = 1 dE' µ(E,E','19.). µ(E,E')

or

(3.1) (3.2)

Here µ(E,E') dE' is the probability per unit path length for scattering into dE' about E' without regard to scattering angle, and µ(E,'19.) dfl is the like probability for scattering into direction range dfl without regard to the energy of the exit radiation. Also of interest is µ(E)

= 1 = dE'µ(E,E'),

(3.3)

which is just the total linear interaction coefficient for the scattering of incident radiation of energy E, without regard to energy loss or angle of scattering. When necessary to avoid ambiguity, a specific type of interaction may be designated by subscript, the symbolµ., for example, to designate scattering.

3.2

MICROSCOPIC CROSS SECTION

The linear interaction coefficient depends on the type and energy of the incident particle, the type of interaction, and the composition and density of the interacting medium. One of the more important quantities that determine µ is the density of target atoms or electrons in the material. It seems reasonable to expect that µ should be proportional to the "target" atom or electron density N in the material, that is, µ = uN, where u is a constant of proportionality independent of N.2 The quantity u is called the microscopic cross section and is seen to have dimensions of area. It is often interpreted as being the effective cross-sectional area presented by the target atom or electron to the incident particle for a given interaction. Indeed, in many cases u has dimensions comparable to those expected from the physical size of the nucleus. However, this simplistic interpretation of the microscopic cross section, although conceptually easy to grasp, leads to philosophical difficulties when it is observed that u generally varies with the energy of the incident particle and, for a crystalline material, the particle direction. The view that u is the interac­ tion probability per unit differential path length, normalized to one target atom or electron per unit volume, avoids such conceptual difficulties while emphasizing the statistical nature of the interaction process. Cross sections are usually expressed in units of cm2 . A widely used special unit is the barn, equal to 10- 2 4 cm2 . Just as the linear interaction coefficient for scattering may be differential in na­ ture, so also may be the corresponding microscopic cross section. In this book, these 2This proportionality is not strictly true for coherent interactions in which the incident particle interacts collectively with multiple target atoms or electrons, as in interactions with orbital electrons of a single atom or with atoms of a crystal lattice. Even for such cases, though, the microscopic cross section is usually defined in this manner.

31

Sec. 3.3. Conservation Laws for Scattering Reactions

differential microscopic cross sections will be denoted by O"(E,E',�.), O"(E,E'), and O"(E,�.), which are related by integral expressions analogous to those of Eqs. (3.1) to (3.3). The reader should be aware of alternative forms of notation; for ex­ ample, O"(E,E',�.) d 2 0"jdE'dn, and the latter form is used in some texts. Sometimes, to emphasize the transition from one energy E to another E', or from one direction !l to another !l', the notation E --+ E' or !l --+ !l' is used. The following are samples of identical notations: O"(E --+ E') O"(E,E') and O"(E--+ E',!l--+ !l') O"(E,E',!l•!l') O"(E,E',w.), where w. !l•!l' =cos�•. All these alternative forms are used in this book. Data on cross sections and linear interaction coefficients, especially for photons, are frequently expressed as the ratio ofµ to the density p, called the mass interaction coefficient. Since the atomic density N for a medium composed of a single element is p (3.4) N= ANa,

=

=

=

= =

in which Na is Avogadro's number (mo1-1) and A is the atomic weight, µ NO" Na -=-=-(}". A p p

(3.5)

Thus µ/ p is an intrinsic property of the interacting medium-independent of its density. This method of data presentation is used much more for photons than for neutrons, in part because, for a wide variety of materials and a wide range of photon energies, µ/ p is only weakly dependent on the nature of the interacting medium. For compounds or homogeneous mixtures, the linear and mass interaction coef­ ficients are, respectively, (3.6) and

�=

�Wi (�},

(3.7)

in which w; is the weight fraction of component i. In Eq. (3.6), the atomic density N; and the linear interaction coefficient µi are values for the ith material after mixing.

3.3 CONSERVATION LAWS FOR SCATTERING REACTIONS 3.3.1 Conservation of Momentum Consider the nuclear reaction illustrated in Fig. 3.1. Reactants are a stationary target of rest mass M and an incident particle of rest mass m, kinetic energy E, and momentum p. Products are one particle of rest mass m', kinetic energy E', and momentum p', and a second particle of rest mass M', kinetic energy T, and momentum P r · Energies E, E', and Tare kinetic energies in the laboratory system, except for photons for which the adjective kinetic is not used.

32

Interaction of Radiation with Matter

Chap. 3

E'

E p

T

(b)

Figure 3.1. Nuclear reaction with one reactant initially stationary and with products emerging at angles {J, and {Jr in the laboratory system. (a) Energies and scattering angles. (b) Conservation of linear momentum.

Energy and momentum of the incident particle are related by the equation E=

Jp2 c2 + m2 c4 - mc2 ,

(3.8)

in which scalar p is the magnitude of the momentum p and c is the speed of light in vacuum, 2.9979 x 108 m/s. Similar equations relate E' and p', and T and P r · Conservation of linear momentum is depicted in the vector diagram Fig. 3.l(b), in which the initial momentum vector p must equal the sum of the product momentum vectors p' and P r . From the law of cosines, (3.9) and

2 2 p; = p + (p') - 2pp' cos.is . It is also clear from the figure that from the law of sines,

p' I sin -Or

3.3.2

= Pr I sin -Os .

(3.10) (3.11)

Conservation of Energy

The fact that total energy, the sum of kinetic and rest-mass energy, is conserved in the nuclear reaction is expressed by the equation E= E' +T-Q,

(3.12)

in which Q is the Q value of the reaction. If it is assumed that the reactants are initially in the ground state, i.e., m = m 0 and M = M0 , the Q value is given by (3.13) in which !::,.M0 c2 is energy equivalent of the changes in the ground-state rest masses of the reactants, and W is the sum of the nuclear excitation energies Eex of the reaction products.

33

Sec. 3.3. Conservation Laws for Scattering Reactions

3.3.3 Application of the Conservation Laws If an incident particle is a photon, the rest mass is zero and, from Eq. (3.8), p = E /c. If the incident particle is a neutron, classical mechanics may be assumed to apply for energies less than about 15 MeV, and p = J2mE. If the incident particle is an electron, then relativistic conservation laws generally should be applied and the following equations are useful. The ratio of the particle speed to the speed of light in vacuum is identified as /3, and if E E /mc2, then

=

(3.14) and (3.15) Equations (3.8) through (3.15) provide the framework for treatment of the kine­ matics of a large class of nuclear reactions. They will be drawn upon throughout this book. The topics now to be addressed are scattering reactions, for which m' = m and M' = M. Most reactions addressed are elastic scattering, for which the Q value is zero. The exception is inelastic scattering of a neutron by an atomic nucleus. That reaction is endothermic, and the Q value is negative and equal in magnitude to the excitation energy of the final state of the nucleus with respect to its ground state.

3.3.4 Scattering of Photons by Free Electrons This scattering process is known as the Compton effect, to recognize its 1923 dis­ covery by A. H. Compton. Since the photon has zero rest mass, p = E /c and p' = E' /c. Because the electron is free, the scattering is elastic and E = E' + T. Thus, Eqs. (3.9) and (3.10) reduce to (3.16) and

T=

2m c2 E 2 cos2 {) e r , (E + me c2 ) 2 - E2 cos2 {Jr

0'.5_{)r '.5_1r/2.

(3.17)

Here me c2 is the rest-mass energy of the electron, 0.51100 MeV or 8.1871 x 10- 14 J. These equations are very much simplified if expressed in terms of the dimensionless variables3 A= me c2 / E and 7 = T/me c2 , namely, (3.18) 3In this book, the dimensionless variable >. is called the wavelength. Clearly it does not have units of length. It might better be called the wave number. It may be construed more properly as the ratio of the actual wavelength he/ E to the Compton wavelength >. c = hc/m.c2 = 2.4263 X 10-12 m, in which h is Planck's constant, 6.6261 x 10-34 J s. More correctly, then, >. is the wavelength in units of the Compton wavelength.

34

and

Interaction of Radiation with Matter

2

2 cos {)r T=------(1 + ..\) 2

-

cos2 {)r ·

Chap. 3

(3.19)

Note that ,\ S ,\' S ,\ + 2. Note too that the maximum energy transfer to initial kinetic energy of a recoil electron is given by

(3.20) which approaches unity for incident photons of very high energy. Similarly, E' E

(3.21)

and (3.22)

which approaches unity for incident photons of very low energy. A relationship between scattering angles is given by (3.23)

3.3.5

Scattering of Neutrons by Atomic Nuclei

The scattering interaction is the most probable interaction of fast neutrons and is the mechanism relied upon to slow these neutrons to thermal energies, at which they can be absorbed through (n, 'Y) reactions. There are two distinct types of scattering processes, both of importance in fast neutron attenuation. In capture scattering the incident neutron is absorbed by the scattering nucleus to form a compound nucleus which subsequently decays by the emission of a neutron. If the residual nucleus is left in the ground state, the scattering is called elastic. If the residual nucleus is left in an excited state, the scattering is called inelastic. The other type of scattering is referred to as potential scattering. In this process, which is always elastic, the incident neutron is scattered by the nucleus as a whole-analogous to the diffraction of the incident neutron wave by the entire nuclear potential. Capture-scattering cross sections generally exhibit resonance behavior, while potential scattering cross sections usually vary slowly with energy. In all scattering processes the total energy and momentum must be conserved. Except for thermal neutron scattering, for which the thermal motion of the target atoms may be comparable to the neutron speed, one can properly neglect the initial kinetic energy of the scattering nucleus in the laboratory coordinate system. Fur­ thermore, these scattering interactions may be treated by classical mechanics, so that energy and momentum are related by p2 = 2mE. The ratio M/m of the mass of the target nucleus to the mass of the neutron is so nearly equal to the atomic

35

Sec. 3.3. Conservation Laws for Scattering Reactions

mass A of the target nucleus that in this book, the difference is neglected. In the general case of inelastic scattering, Eqs. (3.8) to (3.10) reduce to (see Problem 12) w (E E ) s ' /

in which

W8

1 (A+ 1) 2 [

=-

= cos-O , -1 S s

W8

Ii'

- - (A - 1) E

{-:,

- - -Q E' ./EE' A

S +1. It follows that

l

'

(3.24)

(3.25) and (3.26) in which

� = Q(l + A) AE

(3.27)

and OS -Or S 1r/2. Since ...fEi physically must be non-negative, only the plus sign in Eqs. (3.25) and (3.26) gives meaningful results for elastic scattering ( = 0) and for most inelastic Q slightly greater than scattering. However, when a neutron with energy only I Q I is inelastically scattered, both signs may lead to physically realistic results. This so-called "double value" region is discussed below. Threshold Energies for Neutron Inelastic Scattering.

From Eq. (3.25), it is clear that for E' to be real it is necessary that (3.28) The least value of E allowing inelastic scatter, the threshold energy Et, corresponds to W8 = 1, namely, (3.29) In Table 3.1 the energy levels of the first two excited states are listed for selected nuclides. There it is seen that the threshold for inelastic scattering tends to decrease as the atomic mass of the scatterer increases. Notice that the level spacings of the light elements and the magic number4 nuclides are comparatively large and hence inelastic scattering is generally less significant for these nuclides. Moreover, the odd-even and even-odd nuclides tend to have smaller thresholds than the even-even nuclides. 4 A magic number nucleus is one in which the number of neutrons or protons equals 2, 8, 20, 50, 82, or 126. When the nucleus is magic, a particularly stable configuration of the nucleons in the nucleus is achieved analogous to closed electron shells in atomic physics.

36

Interaction of Radiation with Matter

Chap. 3

Table 3.1. Energies of the first and second excited states in MeV for selected nuclides.

Nuclide

�H, �He

�Li

lLi iiB 11B iic iic

1ic b 1io a

110 b

1io b b fiK

f8 K

a �gca b a �ijC

��Sc g�Mn

a

F irst excited state

Second excited state

none

none

b giFe

0.478

2.185

Nuclide

4.445

1.454

6.590

b 2i1Pb a �Pb 2i 2i\Bi

3.684 6.130

0.871

3.055

2.523

2.814

3.352

3.736

0.030

3.555

0.800

1.943

2.001

0.126

0.984

0.012

0.137

b g�Fe b i g�N

Ni b �g

1.982

0.014

g�Fe

7.654

6.094

2.538

4.63

4.439

6.049

1.408

0.847

1.740

3.088

Second excited state

g�Fe

3.562

0.717

2.125

First excited state

0.376

2iipb b

2i�Bi b

2tiTh 2t�u 2ttu 2iiu

2iiu 2t�u

0.811

1.332

0.803

0.570

2.085

1.675

2.460 2.159

1.167

0.898

2.615

3.198

0.897

1.609

0.063

0.510

0.049

0.162

0.043

0.143

0.045

0.150

0.040

75 eV

0.045

0.092

0.013 0.148

a Indicates the numbers both of neutrons and protons are magic. b lndicates a magic number of neutrons or protons. Source: Lederer and Shirley (1978].

It is also clear from Eq. (3.25) that for values of E just greater than Et , the scat­ tered neutron can appear in the forward direction, W s > 0, with either of two distinct positive energies. The largest value of E for which this is possible-the cutoff energy Ee-is the largest value of E for which w8 ../E-JE(w� + A 2 -1) + A(A + l)Q 2:'.: 0, namely, AQ (3.30) Ee= ---. A-1 For E between Et and Ee, there is a maximum scattering angle, or minimum W 8 , permitting real values of E', namely, Ws ,mi n = J1-A 2 -QA(A + 1)/E. Discus­ sion of the implications of dual values of E' may be found in the works of Evans [1955] and Amaldi [1959]. Neutron Scattering in the Center-of-Mass System

Particle kinematics may be described and understood completely in the laboratory coordinate system used implicitly to this point. It may therefore seem a needless

37

Sec. 3.3. Conservation Laws for Scattering Reactions

V

(a)

(b)

(c)

Figure 3.2. Conversion between laboratory and center-of-mass coordinate systems. (a) Velocities in the laboratory system. (b) Velocities in the center-of-mass system. (c) Relationship between scattering angles in the two systems.

complication to introduce a different coordinate system-the center-of-mass sys­ tem. Nevertheless, there are certain aspects of particle kinematics that are greatly simplified when described in the new system. This is especially true for angular characteristics of scattering cross sections and angular distributions of neutrons af­ ter scattering. Such distributions are more easily and more precisely described in the center-of-mass system. For example, the angular distribution of neutrons scat­ tered from hydrogen atoms or, indeed, the angular distribution of billiard balls after elastic collision is very anisotropic in the laboratory system but completely isotropic in the center-of-mass system. Consider the scattering reaction illustrated in Fig. 3.2(a). The target nucleus is initially at rest. Prior to scatter, the center of mass is moving to the right with velocity v defined in such a way that the total linear momentum mv is equal to (m + M)v 0 • In the center-of-mass system, Fig. 3.2(b), the center of mass is stationary. Prior to scatter, the target atom is moving to the left with velocity v 0 and the incident neutron is moving to the right with velocity Ve = v - v 0 • The total momentum before and after scattering is zero. Thus, products of the scattering must move in opposite directions, the neutron with velocity v� and the recoil nucleus with velocity V�. Figure 3.2(c) illustrates the relationship between the scattering angles in the two systems and, as addressed in Problem 13, reveals that in terms of their cosines, O

(3.31) and5 (3.32) 5For elastic scattering, only the positive sign in Eq. (3.32) applies, with one exception. For A= 1, Ws 2'. 0, and We = -1 + 2w;. For inelastic scattering, the positive sign applies except in the region of dual values of E', and then W8 > 0 and We is dual valued.

38

Interaction of Radiation with Matter

Chap. 3

in which (see Problem 14) 'Y

=v

0

v�

= [A 2 +

A(A

+ l)Q -l/2

(3.33)

]

E

Energies after scattering are given by (see Problem 15) A E' = �E(l+a)+ �(1-a)Ewc✓l + � + AQ 2 2 +1

and

(3.34)

✓

T = E- E' + Q = ½E(l-a) [1-wc l+�]+ � , A l

(3.35)

in which

O!=[���r Note that E' /E always lies between the limits E:nin

= [ E

2 A✓f+K-1 ] and l+A

E:nax

E

=

(3.36)

[A✓f+K+1

l+A

2 ]

(3.37)

3.3.6 Limiting Cases in Classical Mechanics of Elastic Scattering By setting Q = 0 and E' = E-T in Eqs. (3.25) and (3.34) and using Eq. (3.11), one finds that T 2A 4A 2 w , (3.38) (1-wc) = E = (1 + A)2 (1 + A)2 r where A= M/m. Thus, Wr

=

v{1=-z:;: � = ·('2''le)

(3.39)

Slll

By substituting We = 1-2w; from this equation into Eq. (3.31), one finds that for the classical mechanics of elastic scattering, 1-2Aw;/(A + 1)

Ws = -;:======;=====� Jl - 4Aw�/(A + 1) 2

(3.40)

Listed as follows are limiting values of scattering parameters for three cases. Case 1: m = M. This case describes neutron-proton scattering and the scatter­ ing by stationary free electrons of low-energy electrons or positrons. All scattering angles are possible in the center-of-mass system, but in the laboratory system, fJ8 and f}r are limited to the range O to 1r /2. f}c

f}s

0 1r/2

0 1r/4 1r /2

1r

f}r

E'/E

T/E

/2 1r/4 0

1 1/2 0

0 1/2 1

1r

39

Sec. 3.4. Photon Cross Sections

Case 2: m < M. This case describes electron scattering by nuclei and neutron scattering from heavier nuclei. Scattering is possible for values of {)5 and {Jc between 0 and 1r but for {)r only between 0 and 1r/2. As A increases, {)5 approaches {Jc · T/E

E'/E 1 (1 + a)/2

0 7r

/2

0

(1- a)/2 1-et

Ct

7r

Case 3: m > M. This case approximately describes scattering of heavy charged particles by electrons. Scattering is possible for 0 :S sin {)5 :S M/m. As M/m-+ 0, {)8 -+ 0.

3.3. 7

Elastic Scattering of Electrons and Heavy Charged Particles

The basic working equation follows directly from the conservation laws, Eqs. (3.8) through (3.13), namely, 2Mc2 p2 c2 w; ---'-. 2�-. T=-[ -=--=---=2 2 2 2c + J c + M m c4 ] - p2 c2 w;. p

=

The maximum energy is transferred to the recoil target when (3 v/c, and in terms of Eqs. (3.14) and (3.15),

(3.41) Wr

= 1.

In terms of

(3.42) Electron-Electron Scattering Here M = m = m e . The scattered and recoil electrons are indistinguishable, so the convention is adopted that the one with the lesser energy is identified as the recoil:

and T,max

3.4

(32 = __ 2 1- (3

[

l

m e c2 = E. 1 + 1/ �

(3.43)

(3.44)

CROSS SECTIONS FOR PHOTON INTERACTIONS

For details of the mechanisms of photon interactions, the reader is referred to the standard reference works of Reitler [1954] and Evans [1955]. For comprehensive data tabulations, the reader is referred to ANSI [1991], Biggs and Lighthill [1972], Cullen [1994], Cullen et al. [1989], Hubbell [1969, 1982], Hubbell and Seltzer [1995],

40

Interaction of Radiation with Matter

Chap. 3

Hubbell et al. (1975, 1979, 1990], Plechaty et al. [1975], Seltzer (1993], and Trubey et al. (1989]. Photon energies between 10 eV and 10 MeV are important in radiation shielding design and analysis. For this energy range, only the photoelectric effect, pair production, and Compton scattering mechanisms of interaction are significant. Of these three, the photoelectric effect predominates at the lower photon energies. Pair production is important only for higher-energy photons. Compton scattering predominates at intermediate energies. In rare instances one may need to account also for coherent scattering.

3.4.1

Thomson Cross Section for Incoherent Scattering

Incoherent scattering refers to the interaction of a photon with an individual elec­ tron, as distinguished from the coherent interaction of a photon with all electrons of an atom. It is assumed in the discussion that follows that the incident radiation is not polarized. In the limit of zero photon energy, scattering of the photon by a free electron may be treated by the classical theory of radiation. The electron in the electromag­ netic field of the incident radiation vibrates with the same frequency as that of the incident radiation, thereby giving rise to the emission of secondary electromagnetic radiation of the same frequency. Named for discoverer of the electron, J.J. Thomson (1856-1940), the total cross section per electron for such scattering is

(TT

=

8

2

(3.45)

:f/lTe,

in which r e is the classical electron radius. The value of re is given by

re

=

e2

----, 2 47l"fomec

(3.46)

where e is the electronic charge, 1.6022 x 10-19 C, and € 0 is the permittivity of 1 2.8179 x 10- 13 cm and CTT free space, 8.8542 x 10- 14 F cm- • Thus, r e 2 25 6.6525 x 10- cm • For unpolarized incident radiation, the Thomson electronic cross section per steradian for scattering at angle fJ8 , as in Fig. 3.1, is

=

CTT(fJs)

= 21 r

2

e (l

+ COS2 fJ8).

=

(3.47)

Knowledge of the Thomson cross section is important for two reasons. It is the low­ energy limit for the incoherent (Compton) scattering cross section. It is also, as is evident from Eq. (3.55), the basis for computing Ray leigh-scattering cross sections for interactions of photons coherently with atomic electrons.

3.4.2

Klein-Nishina Cross Section for Incoherent Scattering

The atomic Compton cross section for incoherent scattering may be approximated as the atomic number times the electronic cross section for scattering by a free

41

Sec. 3.4. Photon Cross Sections

electron, namely σc = Z σKN .

(3.48)

The differential scattering cross section per electron σKN (λ, ϑs ) is given by the KleinNishina [1929] differential scattering cross section: σKN (λ, ϑs ) =

1 2 λ2 re 2 (1 + λ − cos ϑs )2   1 + λ − cos ϑs λ 2 − sin ϑs . × + 1 + λ − cos ϑs λ

(3.49)

A convenient alternative form is σKN (E, ϑs ) =

1 2 r q[1 + q 2 − q(1 − cos2 ϑs )], 2 e

(3.50)

in which q ≡ E 0 /E = λ/λ0 is given by Eq. (3.21). This cross section is illustrated in Fig. 3.3. Note that as λ approaches infinity, that is, E approaches zero, q approaches unity and Eq. (3.50) reduces to the Thomson formula, Eq. (3.47). A related quantity is the energy scattering differential cross section, σce (E, ϑs ) ≡

E0 Z σc (E, ϑs ) ' re2 q 2 [1 + q 2 − q(1 − cos2 ϑs )]. E 2

(3.51)

The total cross section per atom, based on the free-electron approximation, is obtained from Eq. (3.49) by integration over all directions. σc (λ) ' ZσKN (λ) = 2Zπ

Z

+1

d(cos ϑs ) σKN (λ, ϑs )

−1

=

3.4.3

πZre2 λ

   2(1 + 9λ + 8λ2 + 2λ3 ) 2 + . (3.52) (1 − 2λ − 2λ ) ln 1 + λ (λ + 2)2



2

Incoherent Scattering Cross Sections for Bound Electrons

The equations for scattering from free electrons break down when the kinetic energy of the recoil electron would be comparable to its binding energy in the atom. Thus, binding effects might be thought to be an important consideration for the attenuation of low-energy photons in media of high atomic number. For example, the binding energy of K-shell electrons in lead is 88 keV. However, under these same circumstances, cross sections for the photoelectric interaction of photons greatly exceed incoherent scattering cross sections. Radiation attenuation in this energy region is dominated by photoelectric interactions, and in most attenuation calculations, the neglect of electron binding effects on incoherent scattering causes negligible error. Corrections for electron binding and related data are available in the literature [e.g., Storm and Israel 1967; Biggs and Lighthill 1972; Plechaty, Cullen, and Howerton

42

Interaction of Radiation with Matter

Chap. 3

....... ,._

104�----___:!!:!!....----

;0

sr-r"�CD

10·2����-����-���-����-���-���

-1.

0.0

-0.5

0.5

1.

cos,Jg Figure 3.3. Klein-Nishina cross section for incoherent scatter of a photon by a free electron.

1975; Hubbell 1982; Trubey, Berger, and Hubbell 1989]. The total atomic cross section for incoherent scattering by bound electrons is given by crc(-X)

1

= 21r _ d(cos'!9.) S(x, Z)o-KN(.\, '19.), +1 1

(3.53)

in which S(x, Z) is the incoherent scattering function [Hubbell et al. 1975] and xis the momentum-transfer parameter, given approximately by

(3.54) Figure 3.4 shows the relative importance, in lead, of electron binding effects by comparing photoelectric cross sections with those for incoherent scattering from both free and bound electrons. As is apparent, S(x, Z) is very small for low-energy photons and approaches Z as photon energy increases.

3.4.4

Coherent (Rayleigh) Scattering

In competition with the incoherent scattering of photons by individual electrons is coherent scattering by the electrons of an atom collectively. Since the recoil mo-

43

Sec. 3.4. Photon Cross Sections

4 ---- 10

C 0 QJ VJ

Coherent scattering

--V;---......__'vr,",

Photoelectric effect

VJ VJ

b

102

Pair production

\ ,

Free erI�coh�rent-·-·-·-•-.:...

\ 100 10-3

\' 10-2

10-1

100

101

102

Energy (MeV) Figure 3.4. Comparison of scattering, photoelectric-effect, and pair-production cross sections for photon interactions in lead.

mentum in the Rayleigh interaction is taken up by the atom as a whole, the energy loss of the gamma photon is slight and the scattering angle small. For example, for 1-MeV photons scattering coherently from iron atoms, 75% of the photons are scattered within a cone of less than 4 ° half-angle [Hubbell 1969]. As is appar­ ent from Fig. 3.4, coherent scattering cross sections may greatly exceed incoherent scattering cross sections, especially for low-energy photons and high-Z materials. However, because of the minimal effect on photon energy and direction, and be­ cause the coherent scattering cross section is so much less than the cross section for the photoelectric effect, it is common practice to ignore Rayleigh scattering in radiation shielding calculations, especially when electron binding effects mentioned in the preceding paragraph are ignored. Named for the 4th Lord Rayleigh, R.J. Strutt (1856-1940), the scattering cross section per atom is (3.55)

44

Interaction of Radiation with Matter

Chap. 3

in which F(x, Z) is the atomic form factor, and the momentum-transfer parameter xis given by Eq. (3.54). Form factors are tabulated by Hubbell and Overb0 [1979]. As E approaches zero, F(x, Z) approaches Z and UR varies as Z2• Effects of coherent scattering are addressed in detail by Trubey and Harima [1987] and data are available in an ANSI standard [1991].

3.4.5 Photoelectric Effect In the photoelectric effect, a photon interacts with an entire atom, resulting in the emission of a photoelectron, usually from the K shell of the atom. Although the difference between the photon energy E and the electron binding energy Eb is distributed between the electron and the recoil atom, virtually all of that energy is carried as kinetic energy of the photoelectron because of the comparatively small electron mass. Thus, T = E - Eb. K-shell binding energies Ek vary from 13.6 eV for hydrogen to 7.11 keV for iron, 88 keV for lead, and 116 keV for uranium. As the photon energy drops below Ek, the cross section drops discontinuously. As E decreases further, the cross section increases until the first L edge is reached, at which energy the cross section drops again, then rises once more, and so on for the remaining edges. These "edges" for lead are readily apparent in Fig. 3.4. The cross section varies as E-n, where n '.::='. 3 for energies less than about 150 keV and n '.::='. 1 for energies greater than about 5 MeV. The atomic cross section varies as z m , where m varies from about 4 at E = 100 keV to 4.6 at E = 3 MeV. As a very crude approximation in the energy region for which the photoelectric effect is dominant, (3.56) While it is true that for light nuclei, K-shell electrons are responsible for almost all photoelectric interactions, such interactions are normally much less important than incoherent scattering. As a general rule, about 80% of photoelectric interactions with heavy nuclei result in ejection of a K-shell electron. Consequently, the approx­ imation is often made for heavy nuclei that the total photoelectric cross section is 1.25 times the cross section for K-shell electrons. As the vacancy left by the photoelectron is filled by an electron from an outer shell, either fluorescence x rays or Auger electrons6 may be emitted. The probability of x-ray emission is given by the fluorescent yield. For the K shell, fluorescent yields vary from 0.005 for Z = 8 to 0.965 for Z = 90. Although x rays of various energies 6If an electron in an outer shell, say Y, makes a transition to a vacancy in an inner shell, say X, an x ray may be emitted with energy equal to the difference in binding energy between the two shells. Alternatively, an electron in some other shell, say Y', which may be the same as Y, may be emitted with energy equal to the binding energy of the electron in shell X less the sum of the binding energies of electrons in shells Y and Y'. This electron is called an Auger electron. If an electron makes a transition from one subshell to a vacancy in another subshell of the same shell, the small difference in binding energies may be transferred to an outer-shell electron, in this case called a Coster-Kronig electron.

Sec. 3.4. Photon Cross Sections

45

may be emitted, the approximation is often made that only one x ray or Auger electron is emitted, with energy equal to the binding energy of the photoelectron.

3.4.6

Pair Production

In this process, the incident photon is completely absorbed and in its place appears a positron-electron pair. The phenomenon is induced by the strong electric field in the vicinity of the nucleus and has a photon threshold energy of 2me c2 (= 1.02 MeV). It is possible but much less likely that the phenomenon is induced by the electric field of an electron (triplet production), for which case the threshold energy is 4me c2 . The discussion that follows is limited to the nuclear pair production process. In this process, the nucleus acquires indeterminate momentum but negligible kinetic energy. Thus, T+ + T− = E − 2me c2 , (3.57) in which T+ and T− are the kinetic energies of the positron and electron, respectively. To a first approximation, the total atomic pair production cross section varies as Z 2 . The cross section increases with photon energy, approaching a constant value at high energy. As illustrated in Fig. 3.5, the resulting electron and positron are widely distributed in energy. Both have directions not far from the original direction of the photon but separated by π radians in azimuth about the photon direction. As an approximation, the angles ϑ with respect to the photon direction are me c2 /E radians. The fate of the positron is annihilation in an interaction with an electron, generally after slowing to practically zero kinetic energy. The annihilation process usually results in the creation of two photons moving in opposite directions, each with energy me c2 .

3.4.7

Photon Attenuation Coefficients

The photon linear attenuation coefficient µ is, in the limit of small path lengths, the probability per unit distance of travel that a gamma photon undergoes any significant interaction. Thus, for a specified medium, µ(E) = N [σc (E) + σph (E) + σpp (E)] ,

(3.58)

in which N = ρNa /A is the atom density. Note that Rayleigh scattering and other minor effects are specifically excluded from this definition.7 More common in data presentation is the mass interaction coefficient Na µph µpp µc µ = [σc (E) + σph (E) + σpp (E)] = + + , (3.59) ρ A ρ ρ ρ in which Na is Avogadro’s number. It is important to note that the mass interaction coefficients are independent of the mass density ρ of the material, and it is for this reason that µ/ρ rather than µ values are usually tabulated. 7 When

referring to data tables in other publications, the reader should be aware that occasionally Rayleigh scattering is included.

46

Interaction of Radiation with Matter

Chap. 3

0.050 E=100MeV 0.040

-!2'

1:

£

50MeV 25MeV

0.030

10MeV

::1.1a. 0.020

0.010

0.2

0.4

0.6

0.8

1.

2

Energy variable: (T + m 0 c )/E

Figure 3.5. Differential pair-production cross section in iron, in the form of the mass interaction coefficient. The function µ(E, T) dT is the probability per unit distance of travel that a photon of energy E experiences a pair-production interaction in which a product electron has kinetic energy between T and T + dT. Data derived using the PEGS4 Code [Nelson, Hirayama, and Rogers 1985].

3.4.8 Compton Absorption and Scattering Cross Sections The cross section per unit wavelength for scattering of a photon into wavelength >.' without regard to angle is, from Eq. (3.51),8 lTc(,\,>.') � z(TKN (,\,,\')

= Z1r r; ( ;,

y [ ( ;, ) + ( �) + (,\' - ,\)(,\' - ,\ -

2)] .

(3.60) A related quantity is the cross section per unit electron energy for creating a recoil electron with energy T. Here it is convenient to use the ratio T T/m e c2• Since 1/,\ - 1/>.' = T, d>.'/dT = ,\ 2 (1- AT)- 2 . Thus, lTKN (,\,T) = lTKN (,\,>.')d,\' jdT, or

=

lTc(A,T) � 1rZ r;,\2 (1- ,\T) + (l - AT)-1 + [

+ 2,\T- 2) ] (l - ,\T)2

(,\ 2 T)(,\ 2 T

(3.61)

The mean fraction of the photon energy transferred to the recoil electron is 8Note that a(.\!?,)= -a(>.,>.') x d>.' /dO., and that d>.' /dO., = (1/2-n-) d>.' /d!.,;, = -1/21r.

Sec. 3.4. Photon Cross Sections

47

designated as fc and the Compton energy-absorption cross section9 per electron is defined as (3.62) O" ca (A) fc O"c(A).

=

The factor fc is the weighted average of T/ E = AT, namely, fc

=

1

O"c(A)

2/[>.(,\+2)] 1 0

dr AT .+2 .X. d.X.' O"c (A, .X.'). 1 N (A) O"c >.

(3.65)

Associated with the energy-absorption and energy-scattering cross sections are the mass energy-absorption and energy-scattering coefficients

and

µ ca _ - Na O" ca (E)

(3.66)

µ ce _ Na O"ce (E) A p

(3.67)

P

A

3.4.9 Photoelectric Absorption Cross Section The photoelectric absorption cross section

Cl>

2 10

Q..

......_ ...J

.... Cl>

"' VJ C:

1 0'

,._

�

Cl> C: Cl>

"'

100

Cl> C:

I 1 02 1 0-

o ,o

I 10-

1 0'

10

2

Energy (MeV) Figure 3.12. Linear energy transfer L/ p in mass units (MeV cm 2 /g) for protons and electrons. The dashed line for aluminum is the collisional LET for positrons. Electron data from ICRU [1984] and proton data from Janni (1982].

A values of Ecut on the order of 200 eV is often chosen. The restricted stopping power L c 0 u(E,Ecut) is defined in analogy with Eq. (3.90), but with Tmax expressed in terms of Ecut• The result is pZC [

L coll(E,Ecut)= in which 77

=E

cu t/ E.

2Aj3 2

2 ln(E/I)+ln(l+E/2)+G ± (E,77) -8 ],

The factor

(3.100)

c- for electrons is given by

c- (E,77) = (1- /32 ) [E2 772 /2+(2E+1) ln(l - 77)] - 1- /32+In [477(1 - 77)]+(1- 77)- 1•

(3.101)

The factor c + for positrons is given by c + (E,77)

= ln 477 - /32 [1 + (2 - e)77 - (3+e)(tE/2)772 3 2 + (1+ Eo(eE /3)77

in which

t = (E+2)- 1.

3.6.2

Electron Radiative Energy Loss

-

4 (eE3 /4)77 J,

(3.102)

Classical electromagnetic theory requires that the rate at which electromagnetic en­ ergy is radiated from an accelerating charged particle be proportional to the square

68

Chap. 3

Interaction of Radiation with Matter 1.

- E = 1 MeV 2 MeV - 5 MeV 100 MeV 0.4 ,-

1 � � W

0.2

[

0.0 r- - -- - ___[_ ____j___ 10-2

L _L___L__ l

_J____J__j=�-�--�--:_��""";....,_"::""===""""'..........._j 10-

1

10

°

E'/E Figure 3.13. Energy spectrum of bremsstrahlung photons released in lead by radiative energy losses of electrons with initial energy E.

of the acceleration. In the electric field of a nucleus with atomic number Z, the force on a charged particle with z elemental charges is proportional to the product Zz. The resulting acceleration of the charged particle is proportional to this force and inversely proportional to the particle's mass m. Thus, the rate of radiative en­ ergy loss by the charged particle would be expected to be proportional to ( Z z / m)2• This inverse-square dependence on particle mass explains why bremsstrahlung from protons or heavier charged particles is negligible in most circumstances compared to that from electrons or positrons. The dependence on the square of Z also explains why, except for the lightest elements, bremsstrahlung in the field of the nucleus far exceeds bremsstrahlung in the fields of atomic electrons. One may define a differential interaction coefficient for radiative energy loss in such a way that µ ra d(E, E') dE' is the probability per unit differential distance of travel that deceleration of an electron of energy E results in emission of a photon with energy between E' and E' + dE'. However, it is not possible to write a simple expression for µ ra d(E, E') or for the associated microscopic cross section O"ra d(E, E'). The interaction coefficient for radiative energy loss by an electron is illustrated in Fig. 3.13, which is based on calculations performed by the PEGS4 computer program [Nelson, Hirayama, and Rogers 1985]. Electron Radiative LET No simple formula can describe the stopping power or LET for electron radiative energy losses (bremsstrahlung). The total stopping power can be written as the sum of stopping powers associated with radiative losses in the force fields of nuclei and

Sec. 3.6. Charged-Particle Interactions

69

2=1

30

10

i i

----

- --· ·

o�-�����-�-,����-������-����� 10- 2

10- 1

100

10 1

102

E (MeV} Figure 3.14. Scaled dimensionless radiative energy loss cross sections 'P rad (E, Z) for elec­ trons in various media. Based on data from Seltzer and Berger [1986].

electrons, namely, L ra a(E) = L ra d,n(E)+L ra d,e(E) = L ra d,n(E)(l+(/z), in which ( is the "correction term" L ra d,e/L ra d,n· According to ICRU [1984], ( depends hardly at all on the medium and is less than 1.2 in magnitude for all E. It is about 0.5 at E = 700 keV and approaches zero at low E. The radiative stopping power may be written as [ICRU 1984] (3.103) in which N is the atomic density, a -:= 1/137 is the fine-structure constant, and 9'!(E, Z) is the dimensionless scaled radiative energy-loss cross section given by (3.104) The function 9'!, too, is not strongly dependent on Z, as is illustrated in Fig. 3.14 for several values of Z. Tables of data are given by Seltzer and Berger [1985, 1986].

3.6.3

Charged-Particle Range

If it were true that a charged particle traveled along a straight path and lost energy continuously along its path at the rate Ltot (E), which is the sum of the collisional and radiative LETs, then the path length traveled while slowing through any dif­ ferential energy range dE would be dE/Ltot ( E). The total path length traveled as

70

Interaction of Radiation with Matter

Chap. 3

10 2 10 1

�

1 00

"'

10-1 10-2 10-3 10-4 10-2

100 Energy (MeV)

Figure 3.15. Range or path length pA, in mass units (g/cm 2 ), in the continuous slowing­ down approximation. Electron data from ICRU [1984] and proton data from Janni [1982].

the particle slowed from initial energy A(Eo) =

E0 to a full stop would thus be Eo dE {

Jo

Ltot(E) ·

(3.105)

As charged particles, especially electrons, slow, they experience deflections along their paths and statistical fluctuations in energy loss. Paths are not straight, and for a group of particles all starting at the same energy, there are fluctuations in total path length traveled. Thus, one cannot identify an unambiguous "range." However, as a measure of range one may identify precisely an effective path length based on the continuous slowing-down approximation (CSDA). This CSDA range, very often identified as r 0 , is given by A from Eq. (3.105). Note that r 0 is a function of the particle's initial energy E0 • Under the CSDA approximation it is assumed that a particle slows continuously, with no energy-loss fluctuations, and with a mean energy loss per unit differential path length given by the total LET evaluated at the local value of the particle's energy. There are practical difficulties in evaluating the integrand of Eq. (3.105) as E approaches zero because LETs are not well known at low energy. It is usually assumed [ICRU 1984] that the integrand is zero at E = 0 and increases linearly with E to 1 keV or to the known value at the least energy below 1 keV. CSDA ranges are illustrated in Fig. 3.15 and an approximation is provided in Table 3.6. Extensive tables of CSDA ranges are given by the ICRU [1984] for electrons, by Janni [1982]

71

Sec. 3.6. Charged-Particle Interactions

Table 3.6. Constants for the empirical formula y = a + bx+ cx2 relating charged-particle energy and CSDA range, in which y is log10 of pA (g/cm 2 ) and xis log10 of the initial particle energy E0 (MeV). Valid for energies between 0.01 and 100 MeV. Electrons

Protons Material Aluminum Iron Gold Air Water Tissuea Boneb

a

b

C

a

b

C

-2.3829 -2.2262 -1.8769 -2.5207 -2.5814 -2.5839 -2.5154

1.3494 1.2467 1.1664 1.3729 1.3767 1.3851 1.3775

0.19670 0.22281 0.20658 0.21045 0.20954 0.20710 0.20466

-0.27957 -0.23199 -0.13552 -0.33545 -0.38240 -0.37829 -0.33563

1.2492 1.2165 1.1292 1.2615 1.2799 1.2803 1.2661

-0.18247 -0.19504 -0.20889 -0.18124 -0.17378 -0.17374 -0.17924

a striated

muscle (ICRU). Cortical bone (ICRP). Source: Based on calculations using the ESTAR code [Berger 1992].

b

for protons, by Hubert, Bimbot, and Gauvin [1990] for heavier charged particles, and by the ICRU [1993] for protons and helium nuclei. It may be inferred from data given by Cross, Freedman, and Wong [1992] that for a beam of electrons ranging in energy from 0.025 to 4 MeV normally incident on water, about 80% of the electron energy is deposited within a depth of about 60% of the CSDA range, 90% of the energy within about 70% of the range, and 95% of the energy within about 80% of the range. For a point isotropic source of monoenergetic electrons of the same energies in water, 90% of the energy is deposited within about 80% of the range, and 95% of the energy within about 85% of the range. In either geometry, all of the energy is deposited within about 110% of the CSDA range. For protons in aluminum, for example, pA is 0.26 mg/cm 2 at 100 keV. However, for such protons normally incident on a sheet of aluminum, the likely penetration depth, 1 2 in mass thickness, is only 0.22 mg/cm2 . For 10-keV protons incident on gold, the average penetration depth is only about 20% of the CSDA range. If radiation losses and charge fluctuation may be neglected, interpolation of range data for high-energy, heavy charged particles is facilitated by the following rules, which may be deduced from Eqs. (3.96) and (3.105): 1. For particles of the same initial speed in a given medium, pA is approximately proportional to m/ z 2 , in which m is the particle mass and z is the particle charge number. 1 2 If a particle starts its trajectory along the z-axis, the penetration depth is the final z-coordinate when the particle has come to rest. Berger [1992] defines the detour factor as the ratio of the average penetration depth to the average path length (practically, the CSDA range).

72

Interaction of Radiation with Matter

Chap. 3

2. For particles of the same initial speed, in different media, pA is approximately proportional to (m/ z 2 )(Z/A), where Z is the atomic number of the stopping medium and A is its atomic weight. Thus, from rule 1, in a given medium a 4-MeV alpha particle has about the same range as a 1-MeV proton. However, these rules fail for particle energies less than about 1 MeV per atomic mass unit. For example, the CSDA range of a 0.4-MeV alpha particle in aluminum is about twice that of a 0.1-MeV proton.

3.6.4 Residual-Range Concept Suppose that a beam of charged particles is normally incident on a shield of thick­ ness x. The incident particles are assumed to have a flow-rate energy spectrum j;(E). Often of interest is the energy spectrum of the flow rate Jt (E') of charged particles after transmission through the shield. Under the continuous slowing-down approximation, one may express the energy of a charged particle as a function of its path length. Except as pointed out previously, one may also treat the path length as a reasonable approximation of the penetration depth of a particle in matter. For charged particles energetic enough to penetrate the thickness x, those emerg­ ing with energy E' and residual range x = A(E') must have entered with energy E whose range corresponds to A(E') + x. In other words,

= j;(E(A(E') + x) dE,

(3.106)

dE dA(E')/dE' Ltot (E) . . dE' - dA(E)/dE - Ltot (E')

(3.107)

Jt (E') dE' where

3.6.5 Electron Radiation Yield The radiation yield for an electron of initial energy E0 is the fraction of that energy released as bremsstrahlung along the path of the electron as it is slowed. Yields as reported in Table 3.2 are computed on the basis of the continuous slowing-down approximation. In that approximation the path length traveled by an electron that slows through the energy range dE is dE/L tot (E), and the energy lost radiatively is just this length multiplied by L ra d(E). The radiation yield is thus (3.108) In carrying out the integration, it is assumed that there is an energy E; < E0 , say 1 keV or some lesser energy at which LETs are known. Then, for E < E;, the approximation may be made that the integrand is given by E L ra d(E;) L ra d(E) . Ltot (E) - E; Ltot (E;)

(3.109)

73

References

REFERENCES

AMALDI, E., "The Production and Slowing Down of Neutrons," Handbuch der Physik, Vol. 38, Part 2, S. Fliigge (ed.), Springer-Verlag, Berlin, 1959. ANSI, Gamma-Ray Attenuation Coefficients and Buildup Factors for Engineering Mate­ rials, ANSI/ ANS-6.4.3, American National Standards Institute, New York, 1991. BARKAS, W.H., AND M.J. BERGER, Tables of Energy Losses and Ranges of Heavy Charged Particles, Report NASA SP-3013, National Aeronautics and Space Administration, Washington, DC, 1964. BERGER, M.J., ESTAR, PSTAR, and ASTAR: Computer Programs for Calculating Stop­ ping Power and Range Tables for Electrons, Protons, and Helium Ions, Report NISTIR 4999, National Institute of Standards and Technology, Gaithersburg, MD, 1992. [Dis­ tributed as Peripheral Shielding Routine PSR-330 by Radiation Shielding Information Center, Oak Ridge National Laboratory, Oak Ridge, TN.] BETHE, H.A., Annalen d. Physik, 5, 325 (1930). BETHE, H.A., Zeits. f. Physik, 76, 293 (1932). BHABA, H.J., "Scattering of Positrons by Electrons with Exchange on Dirac's Theory of the Positron," Proc. R. Soc. London, A154, 195 (1936). BHABA, H.J., Proc. R. Soc. London, A164, 257 (1938). BIGGS, F., AND R. LIGHTHILL, Analytical Approximations for Photon-Atom Differential Scattering Cross Sections Including Electron Binding Effects, Report SC-RR-72 0659, Sandia Laboratories, Albuquerque, NM, 1972. CROSS, W.G., N.0. FREEDMAN, AND P.Y. WONG, Tables of Beta-Ray Dose Distributions in Water, Report AECL-10521, Chalk River Laboratories, Atomic Energy of Canada, Ltd., Chalk River, Ontario, 1992. CULLEN, D.E., M.H. CHEN, J.H. HUBBELL,, S.T. PERKINS, E.F. PLECHATY, J.A. RATH­ KOPF, AND J.H. SCOFIELD,, Tables and Graphs of Photon Interaction Cross Sections from 10 eV to 100 GeV Derived from the LLNL Evaluated Photon Data Library {EPDL),

Report UCRL-50400, Vol. 6, Rev. 4, Part A: Z = 1-50, Part B: Z = 51-100, Lawrence Livermore National Laboratory, Livermore CA, 1989.

CULLEN, D.E., Program EPICSHOW: A Computer Program to Allow Interactive Viewing of the EPIC Data Libraries, Report UCRL-ID-116819, Lawrence Livermore National Laboratory, Livermore CA, 1994. EVANS, R.D., T he Atomic Nucleus, McGraw-Hill, New York, 1955. REITLER, W., T he Quantum Theory of Radiation, 3rd ed., Oxford University Press, Ox­ ford, 1954. HUBBELL, J.H., Photon Cross Sections, Attenuation Coefficients, and Energy Absorption Coefficients from 10 keV to 100 Ge V, Report NSRDS-NBS 29, National Bureau of Standards, Washington, DC, 1969. HUBBELL, J.H., AND I. 0VERB0, "Relativistic Atomic Form Factors and Photon Coherent Scattering Cross Sections," J. Phys. Chem. Ref. Data, 9, 69 (1979). HUBBELL, J.H., "Photon Mass Attenuation and Energy Absorption Coefficients," Int. J. Appl. Radiat. !sot., 33, 1269-1290 (1982).

74

Interaction of Radiation with Matter

Chap. 3

HUBBELL, J.H., AND SELTZER, S.M., Tables of X-Ray Attenuation Coefficients 1 keV to 20 MeV for Elements Z = 1 to 92 and 48 Additional Substances of Dosimetric Interest, Report NISTIR 5632, National Institute of Standards and Technology, Gaithersburg, MD, 1995. HUBBELL, J.H., W.J. VEIGELE, E.A. BRIGGS, R.T. BROWN, D.T. CROMER, AND R.J. HOWERTON, "Atomic Form Factors, Incoherent Scattering Functions, and Photon Scat­ tering Cross Sections," J. Phys. Chem. Ref. Data, 4, 471-616 (1975). HUBERT, F., R. BIMBOT, AND H. GAUVIN, "Range and Stopping-Power Tables for 2.5500 MeV /Nucleon Heavy Ions in Solids," At. Data and Nucl. Data Tables, 46, 11-213 (1990). ICRU, Radiation Quantities and Units, Report 33, International Commission on Radiation Units and Measurements, Washington, DC, 1980. ICRU, Stopping Powers for Electrons and Positrons, Report 37, International Commission on Radiation Units and Measurements, Washington, DC, 1984. ICRU, Stopping Powers and Ranges for Protons and Alpha Particles, Report 49, Interna­ tional Commission on Radiation Units and Measurements, Washington, DC, 1993. JANNI, J.F., "Proton Range-Energy Tables, 1 keV-10 GeV," At. Data and Nucl. Data Tables, 27, 147-529 (1982). KINSEY, R., ENDF/B-V Summary Documentation, Report BNL-NCS-17541 (ENDF-201), 3rd ed., Brookhaven National Laboratory, Upton, NY, 1979. LEDERER, C.M., AND V.S. SHIRLEY (eds.), Table of Isotopes, 7th ed., Wiley-Interscience, New York, 1978. M0LLER, C., "Passage of Hard Beta Rays Through Matter," Ann. Phys., 14, 531 (1932). NELSON, W.R., H. HIRAYAMA, AND D.W.0. ROGERS, The EGS4 Code System, Report SLAC-265, Stanford Linear Accelerator Center, Stanford, CA, 1985. PERKINS, S.T., D.E. CULLEN, AND S.M. SELTZER, Tables and Graphs of Electron-Interac­ tion Cross Sections from 10 eV to 100 GeV Derived from the LLNL Evaluated Electron Data Library (EEDL), Z = 1-100, Report UCRL-50400, Vol. 31, Lawrence Livermore National Laboratory, Livermore, CA, 1991. PLECHATY, E.F., D.E. CULLEN, AND R.J. HOWERTON, Report UCRL-50400, Vol. 6, Rev. 1, National Technical Information Service, Springfield, VA, 1975. [Data are available as the DLC-139 code package from the Radiation Shielding Information Center, Oak Ridge National Laboratory, Oak Ridge, TN.] ROSE, P.F., AND C.L. DUNFORD (eds.), ENDF-102, Data Formats and Procedures for the Evaluated Nuclear Data File ENDF-6, Report BNL-NCS 44945 (Rev.), Brookhaven National Laboratory, Upton, NY, 1991. Rossi, B., High-Energy Particles, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1952. SELTZER, S.M., "Calculation of Photon Mass Energy-Transfer and Mass Energy Absorp­ tion Coefficients," Rad. Res., 136, 147-179 (1993). SELTZER, S.M., AND M.J. BERGER, "Bremsstrahlung Spectra from Electron Interactions with Screened Atomic Nuclei and Orbital Electrons," Nucl. Instrum. Methods in Phys. Res., B12, 95-134 (1985).

75

Problems

SELTZER, S.M., AND M.J. BERGER, "Bremsstrahlung Energy Spectra from Electrons with Kinetic Energy 1 keV-10 GeV Incident on Screened Nuclei and Orbital Electrons of Neutral Atoms with Z = 1-100," At. Data and Nucl. Data Tables, 35, 345-418 (1986). STORM, E., AND H.I. ISRAEL, Photon Cross Sections from 0.001 to 100 Me V for Elements 1 through 100, Report LA-3753, Los Alamos Scientific Laboratory, Los Alamos, NM, 1967. TRUBEY, D.K., M.J. BERGER, AND J.H. HUBBELL, "Photon Cross Sections for ENDF/B­ VI," American Nuclear Society Topical Meeting, Advances in Nuclear Engineering Com­ putation and Radiation Shielding, Santa Fe, NM, 1989. [Data are available as the DLC136/PHOTX code package from the Radiation Shielding Information Center, Oak Ridge National Laboratory, Oak Ridge, TN.] TRUBEY, D.K., AND Y. HARIMA, "New Buildup Factor Data for Point Kernel Calcula­ tions," in Proceedings of a Topical Conference on Theory and Practices in Radiation Protection and Shielding, Knoxville, TN, April 1987, Vol. 2, p. 503, American Nuclear Society, La Grange Park, IL, 1987. [Data are available as the DLC-129 code package from the Radiation Shielding Information Center, Oak Ridge National Laboratory, Oak Ridge, TN.]

PROBLEMS 1. A small homogeneous sample of mass m (g) with atomic mass A is irradiated uniformly by a constant flux density (cm- 2 s-1). If the total atomic cross section for the sample material with the irradiating particles is denoted by CTt (cm2 ), derive an expression for the fraction of the atoms in the sample that interact during a 1-h irradiation. State any assumptions made. 2. Calculate the linear interaction coefficients in pure air at 20 ° C and 1 atm pressure for a 1-MeV photon and a thermal neutron (2200 m s-1). Assume that air has the composition 78% nitrogen, 21% oxygen, and 1% argon by volume. Use the following data: Photon

Neutron

Element

µ/p (cm g-1)

CTtot (b)

Nitrogen Oxygen Argon

0.0636 0.0636 0.0574

11.9 4.2 2.2

2

3. Assume that dry air has the composition 78% nitrogen, 21% oxygen, and 1% argon by volume. For argon, use data from Problem 2. ( a) If the air behaves as an ideal gas, what is the effective molecular weight?

76

Interaction of Radiation with Matter

Chap. 3

(b) Calculate the density of dry air at atmospheric pressure (101.3 kPa) and 273.2 K and the linear interaction coefficient for 1-MeV gamma rays in that air.

( c) Calculate the density of humid air with 45% relative humidity at atmo­ spheric pressure (101.3 kPa) and 293.2 K and the linear interaction coef­ ficient for 1-MeV gamma rays in that air. 4. Derive the Compton formula, Eq. (3.16), given that the momentum of a photon of energy E is equal to E/c and the momentum of an electron of kinetic energy Tis equal to (1/c)JT(T + 2mec2). 5. Verify Eqs. (3.18) to (3.23). 6. Verify Eq. (3.45); that is, given Eq. (3.47), carry out the integration

7. Suppose that 20-MeV gamma rays interact in the atmosphere by Compton scattering.

(a) Derive an expression for, compute, and plot the distribution function f(w s ), defined in such a way that f(w8 photons within µ� r > µtr > µen , the corresponding response functions are ordered as (5.40)

One may then argue that by "compensating errors," the four cases give approxi­ mately the same dose, that is, (5.41)

5.8 RESPONSE FUNCTIONS FOR THE HUMAN AS TARGET Two types of dose quantities have been developed for radiation-protection purposes in occupational and public health. Both types are phantom related, in that they are determined on the basis of physical or mathematical models of the human body. The more precise quantities, called limiting dose quantities, are used by organi­ zations such as the ICRP and NCRP in their recommendations of dose limits in radiation protection. They are based on anthropomorphic phantoms and represent weighted averages of organ doses in human subjects. For radiation protection pur­ poses at doses well below limits for public or occupational exposure there is a need for operational dose quantities. These quantities are based on simple geometric phantoms. They are measurable, in principle, and instruments may be calibrated in their terms.

5.8.1

Characterization of Ambient Radiation

A problem very often encountered in radiation shielding is as follows. At a given reference point representing a location accessible to the human body, the radiation field has been characterized in terms of the flux densities or fluences of radiations of various types computed in the free field, that is, in the absence of the body. Suppose for the moment that only a single type of radiation is involved, say either photons or neutrons, and the energy spectrum (E) of the fluence is known at the reference point. What is needed is the ability to define and to calculate, at that point and for that single type of radiation, a dose quantity R for a phantom representation of the human subject, which can be calculated using an appropriate response function as

R=

fo 00 dE '.R(E)(E),

(5.42)

analogous to Eq. (5.7). Here '.Ris a phantom-related response function and is the fluence energy spectrum, not perturbed by the presence of the phantom. Generation

Sec. 5.8. Human as Target

145

of the response function, of course, requires determination of the absorbed dose and accounting for the radiation transport inside a phantom resulting from radiation incident in a carefully defined angular distribution (usually, a parallel beam). Suppose one knows the angular and energy distributions of the fl.uence of ion­ izing radiation at a point in space, that is, the radiation field at the point. Both operational and limiting dose quantities are evaluated as radiation doses in phan­ toms irradiated by uniform radiation fields derived from the actual radiation field at the point. In the expanded field, the phantom is irradiated over its entire surface by radiation whose energy and angular distributions are the same as those in the actual field at the point of interest. In the expanded and aligned field, the phantom is irradiated by unidirectional radiation whose energy spectrum is the same as that in the actual field at the point.

5.8.2

Response Functions Based on Simple Geometric Phantoms

Of the geometrically simple mathematical phantoms, the more commonly used is the ICRU sphere of 30-cm diameter with density 1.0 g/cm3 and of tissue-equivalent composition, by weight, 76.2% oxygen, 11.1% carbon, 10.1% hydrogen, and 2.6% nitrogen. The dose quantity may be the maximum dose within the phantom or the dose at some appropriate depth. Response functions for the phantoms are computed for a number of irradia­ tion conditions, for example a broad parallel beam of monoenergetic photons or neutrons. At selected points or regions within the phantom, absorbed-dose values, often approximated by kerma values, are determined. In this determination, contri­ butions by all secondary charged particles at that position are taken into account; and for each type of charged particle of a given energy the £ 00 value in water and, therefore, Q are obtained. These are then applied to the absorbed-dose contribu­ tion from each charged particle to obtain the dose equivalent contribution at the given location in the phantom. The resulting distributions of absorbed dose and dose equivalent throughout the phantom are then examined to obtain the maxi­ mum value, or the value otherwise considered to be in the most significant location, say at 10 mm depth. The prescribed response function is then that value of either absorbed dose or dose equivalent divided by the fl.uence of the incident beam. These response functions are intended for operational dose quantities and are designed to provide data for radiation protection purposes at doses well below lim­ its for public exposure. The dose quantities may be treated as point functions, determined exclusively by the radiation field in the vicinity of a point in space. Application of the response functions for these dose quantities is explained in depth by the ICRU [1988]. Dose quantities are defined as follows, and tabulations are provided in Appendix D. Spherical Phantoms Deep Dose Equivalent Index. For this dose quantity, H1,d, the radiation field is assumed to have the same fl.uence and energy distribution as those at a reference point but expanded to a broad parallel beam striking the phantom. The dose is the

146

Response Functions

Chap. 5

maximum dose equivalent within the 14-cm-radius central core of the ICRU sphere. There are difficulties in using this dose quantity when the incident radiation is polyenergetic or consists of both neutrons and gamma rays. The reason is that the depth at which the dose is maximum varies from one type of radiation to another or from one energy to another. Thus, this quantity is nonadditive. Shallow Dose Equivalent Index. This dose quantity, H1,s, is very similar to the deep dose equivalent index, except that the dose equivalent is the maximum value between depths 0.007 and 0.010 cm from the surface of the ICRU sphere (corre­ sponding to the depths of radiosensitive cells of the skin). Ambient Dose Equivalent. For this dose, H* ( d), the radiation field is assumed to have the same fluence and energy distribution as those at a reference point but expanded to a broad parallel beam striking the phantom. The dose equivalent is evaluated at depth d, on a radius opposing the beam direction. This calculated dose quantity is associated with the measured personal dose equivalent Hp (d), the dose equivalent in soft tissue below a specified point on the body, at depth d. For weakly penetrating radiation, depths of 0.07 mm for the skin and 3 mm for the lens of the eye are employed. For strongly penetrating radiation, a depth of 10 mm is employed. Directional Dose Equivalent. For this dose quantity, H'(d, fl), the angular and energy distributions of the fluence at a point of reference are assumed to apply over the entire phantom surface. The depths at which the dose equivalent is evaluated are the same as those for the ambient dose equivalent. The specification of the angular distribution, denoted symbolically by argument n, requires specification of a reference system of coordinates in which directions are expressed. In the particular case of a unidirectional field, H'(d, fl) may be written as H'(d) and is equivalent to H*(d). Irradiation Geometries for Spherical Phantoms. Photon and neutron response functions for deep and shallow indices and for directional dose equivalents at depths of 0.07, 3, and 10 mm have been calculated for radiation protection purposes and have been tabulated by the ICRP [1987] for the following irradiation geometries: • PAR: a single plane parallel beam • OPP: two opposed plane parallel beams • ROT: a rotating plane parallel beam (i.e., a plane parallel beam with the sphere rotating about an axis normal to the beam) • ISO: an isotropic radiation field For a single plane parallel beam, the more conservative of the irradiation ge­ ometries, response functions for H1,d and H*(d) at 10 mm are virtually identical for photons. For neutrons, the two differ only at low energies, with the deep dose equivalent index being greater and thus more conservative.

Sec. 5.8. Human as Target

147

Slab and Cylinder Phantoms

Response functions are also available for plane parallel beams incident on slabs and on cylinders with axes normal to the beam. Slab-phantom deep-dose response functions are tabulated by the ICRP [1987] for high-energy photons and neutrons. Cylinder-phantom deep-dose response functions reported by the NCRP [1971] are of special interest in that they are employed in U.S. federal radiation protection regulations [USNRC 1991].

5.8.3

Response Functions Based on Anthropomorphic Phantoms

The effective dose equivalent HE and the effec­ tive dose £ are limiting doses based on an an­ thropomorphic phantom for which doses to in­ dividual organs and tissues may be determined. Averaging the individual doses with weight fac­ tors related to radiosensitivity leads to the effec­ tive dose or effective dose equivalent. In many calculations, a single phantom represents the adult male or female. In other calculations, separate male and female phantoms are used. These dose quantities have been developed for radiation-protection purposes in occupational and public health and, to some extent, in in­ ternal dosimetry as applied to nuclear-medicine procedures. The dose quantities apply, on aver­ age, to large and diverse populations, at doses well below annual limits. Their use in assess­ ment of health effects for an individual subject Figure 5.5. Sectional view of the male anthropomorphic phantom used requires very careful judgment. The male phan­ in calculation of the effective dose. tom, Adam, is illustrated in Fig. 5.5. Adam has a companion female phantom, named Eva [Kramer et al. 1982]. In yet other calcula­ tions [Cristy and Eckerman 1987], a suite of phantoms is available for representation of the human at various ages from the newborn to the adult. The many phantoms used for measurements or calculations are described in ICRU Report 48 [1992]. Anthropomorphic phantoms are mathematical descriptions of the organs and tissues of the human body, formulated in such a way as to permit calculation or numerical simulation of the transport of radiation throughout the body. In cal­ culations leading to response functions, monoenergetic radiation is incident on the phantom in fixed geometry. One geometry leading to conservative values of response functions is anteroposterior (AP), irradiation from the front to the back with the beam at right angles to the long axis of the body. Other geometries, posteroante­ rior (PA), lateral (LAT), rotational (ROT), and isotropic (ISO) are illustrated in Fig. 5.6. The ROT case is thought to be an appropriate choice for the irradiation pattern experienced by a person moving unsystematically relative to the location

148

Response Functions

LAT

Chap. 5

ROT

Figure 5.6. Irradiation geometries for the anthropomorphic phantom. From ICRP [1987].

of a radiation source. However, the AP case, being most conservative, is the choice in the absence of particular information on the irradiation circumstances.

Effective Dose Equivalent. In 1977 the ICRP introduced the effective dose equiv­ alent HE, defined as a weighted average of mean dose equivalents in the tissues and organs of the human body, namely,

{5.43)

in which Dr is the mean absorbed dose in the organ and Qr is the mean quality factor. If mr is the mass of organ or tissue T, then Qr

=

1 - f dm QD mr

1 f dmf dLoo D(Loo)Q(Loo), =mr

{5.44)

in which D{L 00 ) dL 00 is that portion of the absorbed dose attributable to charged particles with LETs in the range dL 00 about L 00 • Tissue weight factors to be used with the effective dose equivalent are listed in Table 5.4. They are determined by the relative sensitivities for stochastic radiation effects such as cancer and first­ generation hereditary illness. The 1977 values are still of importance because of their implicit use in federal radiation protection regulations [USNRC 1991] in the United States. Effective Dose. In 1991 the ICRP recommended replacement of the effective dose equivalent by the effective dose. This recommendation was endorsed in 1993 by the NCRP in the United States. The effective dose £ is defined as follows. Suppose

Sec. 5.8.

149

Human as Target

Table 5.4. Tissue weight factors adopted by the ICRP [1977] for use in determination of the effective dose equivalent. Tissue 0.25 0.15 0.12 0.12 0.03 0.03 0.30

Gonads Breast Red bone marrow Lung Thyroid Bone surface Remaindera

a A weight of 0.06 is applied to each of the five organs or tissues of the remainder receiving the highest dose equivalents, the compo­ nents of the GI system being treated as separate organs.

that the body is irradiated externally by a mixture of particles of different type and different energy, the different radiations being identified by the subscript R. The effective dose may then be determined as £

=L T

wr Hr

=L T

wr

L WR Dr,R,

(5.45)

R

in which Hr is the equivalent dose in organ or tissue T, Dr,R is the mean absorbed dose in organ or tissue T from radiation R, wR is a radiation weighting factor for radiation R, as determined from Table 5.5, and wr is a tissue weight factor given in Table 5.6. Note that in this formulation, WR is independent of the organ or tissue and wr is independent of the radiation. In computing the response function for the effective dose, one assumes that the phantom is irradiated by unit fluence of monoenergetic particles of energy E. Neither local values nor tissue-average dose equivalents but only tissue-average ab­ sorbed doses are calculated. The tissue-average absorbed doses are multiplied by quality factors determined not by the LET distributions in the tissues and organs but by quality factors characteristic of the incident radiation. This is a fundamental departure from the methodology used in determination of the response functions for the effective dose equivalent.

5.8.4

Comparison of Response Functions

Figures 5. 7 and 5.8 compare response functions for photons and neutrons, respec­ tively. At energies above about 0.1 MeV, the various photon response functions are very nearly equal. This is a fortunate situation for radiation dosimetry and surveil­ lance purposes. Instruments such as ion chambers respond essentially in proportion to absorbed dose in air. Personnel dosimeters are usually calibrated to give re­ sponses proportional to the ambient dose. Both the ambient dose and the absorbed dose in air closely approximate the effective dose equivalent.

150

Response Functions

Chap. 5

Table 5.5. Mean quality factors (radiation weighting factors) adopted by the ICRP [1991] and NCRP [1993]. They apply to the radiation incident on the body or, for internal sources, emitted from the source. Radiation Gamma and x rays of all energies Electrons and muons of all energies Protons (> 1 MeV, other than recoil) Neutrons b < 10 keV 10-100 keV 100 keV-2 MeV 2-20 MeV > 20 MeV Alpha particles, fission fragments, heavy nuclei

5 10 20 10 5 20

a weight factor of 2 according to NCRP. The neutron quality factor may be approximated as (with the neutron energy E in MeV) WR= 5 + 17 exp{-[ln(2E)]2 /6}.

b

Table 5.6. Tissue weight factors adopted by the ICRP [1991] and the NCRP [1993] for use in determination of the effective dose. 0.01

0.05

0.12

0.20

Bone surface Skin

Bladder Breast Liver Esophagus Thyroid Remaindera

Bone marrow Colon Lung Stomach

Gonads

The remainder is composed of the following additional organs and tissues: adrenals, brain, small intestine, large intestine, kidney, muscle, pancreas, spleen, thymus, uterus, and others selectively irradiated. The weight factor of 0.05 is applied to the average dose in the remainder tissues and organs except as follows. If one of the remainder tissues or organs receives an equivalent dose in excess of the highest in any of the 12 organs or tissues for which factors are specified, a weight factor of 0.025 should be applied to that tissue or organ and a weight factor of 0.025 to the average dose in the other remainder tissues or organs. a

The comparison of response functions for neutrons is not so straightforward. The tissue kerma in air always has the smallest response function, largely because no quality factor is applied to the kinetic energy of a secondary charged particle. Fortunately, the ambient dose and the deep dose equivalent index have response functions that are very similar, indeed equal at energies above about 5 keV. There­ fore, historic dosimetry records based on personnel dosimeters calibrated in terms of the deep dose equivalent index do not diverge significantly from those that would

151

Sec. 5.8. Human as Target 102 c,-----,--�-r--r---T--r-,--r-r-----.------;r-r-,-r-,-,-,r--,----,---,---,--.-,--ri:, Photon response functions

_,'ii? � :,�

..

,.. .. :::7 -1/

-

/,/

_ _....--

��::...._..___ ----_.--_.::z_�"--Ambient dose (PAR, 10 mm)

� Effective dose equivalent (AP) Absorbed dose in air

10·2 '----'--'--.........-'--'-'-'-'-----'--'--'--'-'-'-"LI..-L--���-'-�� 10·2 Photon energy (MeV) Figure 5.7. Comparison of photon response functions. Data are from [ICRP 1987].

_,.,-

"' >

101 (!;"

'?'�

10°

Ji

10_1

£

10·2

,g

I

Deep dose equivalent index (PAR) ·-·- -

,,,,-

/

, ,,___

--------·-·-·=�: _____,,..,-_____,,-

Ambient dose (PAR, 10 mm)7 ffective dose equiv. (AP) Tissue kerma in air �-/

----..... -----------____,,__,,--

__,,,/

_ /

,,,,_,,,,____,,,,, ... -- ....-

_....--

__.,,,

Neutron response functions

10·3 ������������������������ 10·6 Neutron energy (MeV) Figure 5.8. Comparison of neutron response functions. Data are from [ICRP 1987].

have been recorded using more modern dose standards. Furthermore, the ambi­ ent dose always exceeds the effective dose equivalent index. Thus, calibration of personnel dosimeters in terms of ambient dose is a conservative practice.

152

Response Functions

Chap. 5

REFERENCES

ATTIX F.H., Introduction to Radiological Physics and Radiation Dosimetry, W iley, New York, 1986. CRISTY, M., AND K.F. ECKERMAN, Specific Absorbed Fractions of Energy at Various Ages from Internal Photon Sources, Report ORNL/TM-8381 (6 vols.), Oak Ridge National Laboratory, Oak Ridge, TN, 1987. FODERARO, A., L.J. HOOVER, AND J.H. MARABLE, "Heat Generation by Neutrons," in Engineering Compendium on Radiation Shielding, Vol. 1, R.G. Jaeger (ed.), Springer­ Verlag, New York, 1968, Sec. 7.2. GOLDSTEIN, H., AND J.E. W ILKINS, JR., Calculations of the Penetrations of Gamma Rays, NDA/ AEC Report NYO-3075, U.S. Government Printing Office, Washington, DC, 1954. HUBBELL, J.H., AND M.J. BERGER, "Attenuation Coefficients, Energy Absorption Coef­ ficients, and Related Quantities," in Engineering Compendium on Radiation Shielding, Vol. 1, R.G. Jaeger (ed.), Springer-Verlag, New York, 1968, Sec. 4.1. ICRP, Radiation Protection: Recommendations of the International Commission on Ra­ diological Protection, Adopted Sept. 17, 1965, Publication 9, International Commission on Radiological Protection, Elmsford, NY, 1966. ICRP, Recommendations of the International Commission on Radiological Protection, Pub­ lication 26, International Commission on Radiological Protection, Annals of the ICRP, Vol. 1, No. 3, Pergamon Press, Oxford, 1977. ICRP, Statement from the 1985 Paris Meeting of the International Commission on Radio­ logical Protection, Publication 45, International Commission on Radiological Protection, Pergamon Press, Oxford, 1985. ICRP, Data for Use in Protection Against External Radiation, Publication 51, Interna­ tional Commission on Radiological Protection, Annals of the ICRP, Vol. 17, No. 2/3, Pergamon Press, Oxford, 1987. ICRP, 1990 Recommendations of the International Commission on Radiological Protection, Publication 60, International Commission on Radiological Protection, Annals of the ICRP, Vol. 21, No. 1-3, Pergamon Press, Oxford, 1991. ICRU, Radiation Quantities and Units, Report 19, International Commission on Radiation Units and Measurements, Washington, DC, 1971. ICRU, Neutron Dosimetry for Biology and Medicine, Report 26, International Commission on Radiation Units and Measurements, Washington DC, 1977. ICRU, Average Energy Required to Produce an Ion Pair, Report 31, International Com­ mission on Radiation Units and Measurements, Washington DC, 1979. ICRU, Radiation Quantities and Units, Report 33, International Commission on Radiation Units and Measurements, Washington, DC, 1980. ICRU, Determination of Dose Equivalents from External Radiation Sources - Part 2, Report 43, International Commission on Radiation Units and Measurements, Bethesda, MD, 1988. ICRU, Phantoms and Computational Models in Therapy, Diagnosis and Protection, Report 48, International Commission on Radiation Units and Measurements, Bethesda, MD, 1992.

153

Problems

ICRU, Quantities and Units in Radiation Protection Dosimetry, Report 51, International Commission on Radiation Units and Measurements, Bethesda, MD, 1993. KRAMER, R., M. ZANKL, G. WILLIAMS, AND G. DREXLER, The Calculation of Dose from External Photon Exposures Using Reference Human Phantoms and Monte Carlo Methods, Part I: The Male (Adam) and Female (Eva) Adult Mathematical Phantoms, Report GSF-Bericht S-885, Gesellschaft f¨ ur Umweltforschung, Munich, 1982 (Reprinted 1986). NCRP, Protection Against Neutron Radiation, Recommendations of the National Council on Radiation Protection and Measurements, Report 38, National Council on Radiation Protection and Measurements, Washington, DC, 1971. NCRP, Recommendations on Limits for Exposure to Ionizing Radiation, Report 91, National Council on Radiation Protection and Measurements, Bethesda, MD, 1987. NCRP, Recommendations on Limits for Exposure to Ionizing Radiation, Report 116, National Council on Radiation Protection and Measurements, Bethesda, MD, 1993. USNRC, Standards for Protection Against Radiation, Title 10, Code of Federal Regulations, Part 20, 56FR23360, U.S. Nuclear Regulatory Commission, Washington, DC, 1991. WHYTE, G.N., Principles of Radiation Dosimetry, Wiley, New York, 1959.

PROBLEMS 1. Consider the bombardment of a slab of material (bounded by vacuum) by a beam of photons or neutrons. Make a sketch showing qualitatively the ratio of the absorbed dose to the kerma as a function of depth of penetration into the slab along the direction of the beam. Explain the features of the sketch. 2. On the assumption of elastic isotropic scattering in the center-of-mass system, estimate the water-kerma response function for 0.1-MeV neutrons. How does this compare with the corresponding value for tissue given in Appendix D? Take scattering cross sections for hydrogen at this energy to be 12.8 b and for oxygen to be 3.5 b. 3. A small volume of tissue acting as a detector is located in a beam of 1-MeV neutrons which provides a fluence of 1013 cm−2 . Compute the kerma using the tissue composition of Section 5.8.2 and Table D.8 of Appendix D. Then compute the kerma using the response function of the same table. It may be assumed that scattering is isotropic in the center-of-mass system and that the scattering cross sections for 1-MeV neutrons are 4.3 b for hydrogen, 2.6 b for carbon, 2.0 b for nitrogen, and 8 b for oxygen. 4. Consider Fig. 5.7. Explain why the three response functions diverge at lower energies. Explain why, for 20-keV photons, the response function for tissue dose in air exceeds those for the effective dose and the deep dose index.

154

Response Functions

Chap. 5

5. Consider Fig. 5.8. Explain why the response function for tissue kerma in air is less than the other response functions at all energies. Explain why the response function for the deep dose equivalent index exceeds those for the ambient dose (at certain energies) and the effective dose. 6. Listed in the accompanying table are various coefficients (cm2 g−1 ) for gammaray interactions in water. (a) Using only the data given in the table, fill in the blanks, making an internally self-consistent data set. Refer to Sections 3.4.8–3.4.10 and 5.7.1. (b) Explain why µen /ρ < µtr /ρ < µa /ρ. Coefficient

Energy (MeV)

(cm2 /g)

0.01

µc /ρ µph /ρ µpp /ρ µca /ρ µpha /ρ µppa /ρ µ/ρ µa /ρ µtr /ρ µen /ρ

4.720 0.0 0.0029 4.720 0.0 4.875 4.723 4.723 4.723

0.1 0.163 0.0024 0.0 0.0024 0.0 0.165 0.0248 0.0248 0.0248

1 0.0 0.0 0.0311 0.0 0.0 0.0707 0.0311 0.0311 0.0311

10 0.0171 0.0

0.0 0.0221 0.0167 0.0162 0.0158

100 0.0 0.01454 0.0022 0.0 0.0143 0.0172

0.0119

7. A rule of thumb for exposure from point sources of photons in air at distances over which exponential attenuation is negligible is as follows: 6CEN X˙ = , r2 where C is the source strength (Ci), E is the photon energy (MeV), N is the number of photons per disintegration, r is the distance in feet from the source, and X˙ is the exposure rate (R h−1 ). (a) Reexpress this rule in units of Bq for the source strength and meters for the distance. (b) Over what ranges of energies is this rule accurate within 10% and within 25%?

Chapter 6

Basic Methods for Radiation Dose Calculations

In this chapter, simplified methods for estimating the dose under specialized source and geometric conditions are reviewed. For the most part, isotropic radiation sources are considered and the methods developed here are confined to infinite media or to those situations that can be approximated well by an infinite medium. For more complex problems not well modeled by the methods given in this chapter, it is necessary either to use specialized simplified techniques suitable to the partic­ ular type of radiation (see Chapters 7 to 9) or to use transport models to estimate the radiation fields (see Chapters 10 and 11). First, doses from uncollided radiation are considered. Uncollided radiation doses are calculated for several basic source configurations using the traditional point­ source superposition technique. The concept of the point kernel is introduced for uncollided dose and total dose calculations. A generalized method based on the geometry factor or point-pair distance distribution is then introduced for calculating doses resulting from quite general source and detector configurations. This powerful generalized method for dose calculations is also shown to be applicable to media with material heterogeneities. Finally, the use of geometric transformations that relate doses between different source geometries is discussed. It should be noted that in these and in all other dose calculations in this book, the calculated doses are mechanistic and distinct from stochastic measured doses. The calculations are based on the average or expected interaction behavior of ra­ diation, whereas a measured dose represents but a single statistical sample of the radiation field. Different measurements under the same conditions yield different values, sometimes greatly different if only a small number of radiation particles are measured. The mechanistically calculated dose represents the statistical average that would be obtained if a sufficiently large number of measurements were used. In some cases, for example, measurement of the small neutrino fluence on earth produced by a distant supernova explosion, measurements cannot be repeated and 155

156

Basic Methods for Radiation Dose Calculations

Chap. 6

the calculated fl.uence from various models may be quite different. In any event, it is important to appreciate the fundamental difference between the nature of measured doses and of calculated doses. In this chapter, the many results derived for the dose D are for radiation sources emitting S particles which in turn produce a fl.uence cI> at the detector location. These results may also be used for steady source emission rates S, in which case the dose D becomes the dose rate b. Although only radiation doses are considered in this chapter, the same results can be applied to any detector response.

6.1

UNCOLLIDED RADIATION

The fl.uence or dose at some point of interest is in many situations determined primarily by the uncollided radiation that has streamed directly from the source without interaction in the surrounding medium. For example, if only air separates a gamma-ray or neutron source from a detector, interactions in the intervening air or in nearby solid objects, such as the ground or building walls, are often negligible, and the radiation field at the detector is due almost entirely to uncollided radiation coming directly from the source. Scattered and other secondary radiation in such situations is of minor importance. In this section, some basic properties of the uncollided radiation field are presented, and methods for estimating the dose from this radiation are derived.

6.1.1

Exponential Attenuations

Consider radiation emitted at x = 0 along the x-axis in a homogeneous medium characterized by a total interaction coefficientµ for the radiation. Let P(x) be the probability that a radiation particle emitted at x = 0 travels at least a distance x without making any interaction. Also let P(x) be the probability that a radiation particle interacts within a travel distance x. These two probabilities are related by P(x) = 1 - P(x). To determine P(x) it is sufficient to recognize that a radi­ ation particle's probability of interacting in its future travel through a medium is independent of the particle's past history, that is,

P(x + �x) = P(x)P(�x).

(6.1)

From the complementary relation between P and P, this equation can be rewritten as

x) P(x + �x) = P(x)[l - P(�x)] = P(x) [1 - p�� �x],

(6.2)

which can be rearranged to give

(6.3)

157

Sec. 6.1. Uncollided Radiation

In the limit as Ax - 0 and from the definition of the interaction coefficient, namely, µ = lima x -+O P(Ax)/Ax (see Section 3.1), Eq. (6.3) becomes

dP(x) dx Since P(O)

= -µP(x).

(6.4)

= 1, the solution of this differential equation is (6.5)

In words, the uncollided radiation is exponentially attenuated as it streams from the source in a homogeneous medium. From this result, the half-value thickness X2 that is required to reduce the uncol­ lided radiation to one-half of its initial value [i.e., P(x2) = e-µa: 2 = 1/2] can readily be found, namely X2 = In 2/µ. Similarly, the tenth-value thickness x 10, which is the distance the uncollided radiation must travel to be reduced to 10% of its initial value, is found to be X10 = In 10/µ. The concepts of half-value and tenth-value thicknesses, although derived here for uncollided radiation, are also often used to describe the attenuation of the total radiation dose (see, e.g., Section 7.3.4). 6.1.2 Mean-Free-Path Length The average distance A that a particle streams from the point of its birth to the point at which it makes its first interaction is called the mean-free-path length. Let p(x) Ax be the probability that a radiation particle travels a distance x without interacting and then interacts between x and x + Ax. This probability in terms of P and Pis p(x) Ax= P(x)P(Ax) (6.6) or

P(Ax) _ px ( ) - P(x) Ax ·

(6.7)

In the limit as Ax - 0, P(Ax)/Ax - µ, so that the probability density function (PDF) p(x) is p(x) = µP(x) = µe-µ"'. (6.8) With this PDF the average distance a particle travels before interacting with the medium is found to be

A=

1 = xp(x) dx = 1 = x µe-µ"' dx = 1/µ.

(6.9)

6.1.3 Uncollided Dose from a Point Source In the following subsections, basic expressions are derived for the dose from uncol­ lided radiation produced by isotropic point sources.

158

Basic Methods for Radiation Dose Calculations

t�

\

r

$ )I( p

\

,¢ p I

(a)

Chap. 6

I

I

SP

p

p

r

(b)

(C)

Figure 6.1. Point isotropic source (a) in a vacuum, (b) with a slab shield, and (c) with a spherical-shell shield. Point Pis the location of the point detector.

Point Source in Vacuum Consider a point isotropic source that emits Sp particles into an infinite vacuum as in Fig. 6.l(a). All particles move radially outward without interaction, and because of the source isotropy, each unit area on an imaginary spherical shell of radius r has the same number of particles crossing it, namely, Sp /(41rr 2 ). It then follows from the definition of the fluence that the fluence 0 of uncollided particles a distance r from the source is s_ O(r) = _P (6.10) 41rr 2 _ If the source particles all have the same energy E, the response of a point detector a distance r from the source is obtained by multiplying the uncollided fluence by the appropriate detector response function '.R, which usually depends on the particle energy E, namely, Do( ) = Sp '.R (6.11) r 47rT 2. Notice that the dose and fluence decrease as 1/r 2 as the distance from the source is increased. This decreasing dose with increasing distance is sometimes referred to as geometric attenuation. Point Source in a Homogeneous Attenuating Medium Now consider the same point monoenergetic isotropic source embedded in an in­ finite homogeneous medium characterized by a total interaction coefficient µ. As the source particles stream radially outward, some interact before they reach the imaginary sphere of radius r and do not contribute to the uncollided fluence. The number of source particles that travel at least a distance r without interaction is Sp e-µr, so that the uncollided dose is D o(r) = Sp '.R e -µr. 47rT 2

(6.12)

The term e-µr is referred to as the material attenuation term to distinguish it from the 1/r 2 geometric attenuation term.

Sec. 6.1. Uncollided Radiation

159

Point Source with a Shield Now suppose that the only attenuating material separating the source and the detector is a slab of material with attenuation coefficient µ and thickness t as shown in Fig. 6.1(b). In this case, the probability that a source particle reaches the detector without interaction is e-µt, so that the uncollided dose is (6.13) This same result holds if the attenuating medium has any shape [e.g., a spherical shell of thickness t as shown in Fig. 6.l(c)] provided that a ray drawn from the source to the detector passes through a thickness t of the attenuating material. If the interposing shield is composed of a series of different materials such that an uncollided particle must penetrate a thickness t i of a material with attenuation coefficient µi before reaching the detector, the uncollided dose is (6.14) Here L i µi t i is the total number of mean-free-path lengths of attenuating material that an uncollided particle must traverse without interaction, and exp(- L i µi t i ) is the probability that a source particle traverses this number of mean-free-path lengths without interaction.

Point Source in a Heterogeneous Medium The foregoing treatment for a series of different attenuating materials can be ex­ tended to an arbitrary heterogeneous medium in which the interaction coefficient µ(r) is a function of position in the medium. In this case the probability that a source particle reaches the detector a distance r away is e-L, where £ is the total number of mean-free-path lengths of material that a particle must traverse with­ out interaction to reach the detector. This dimensionless distance can be written formally as (6.15) where the integration is along the ray from source to detector. 1 The uncollided dose then is given by (6.16)

Polyenergetic Point Source So far it has been assumed that the source emits particles of a single energy. If the source emits particles at several discrete energies, the total uncollided dose is found 1The distance £ is sometimes referred to as the optical depth or optical thickness, in analogy to the equivalent quantity used to describe the exponential attenuation of light. This nomenclature is used throughout this chapter to distinguish it from the geometric distance r.

160

Basic Methods for Radiation Dose Calculations

Chap. 6

simply by summing the uncollided doses for each monoenergetic group of source particles. Thus if a fraction Ji of the Sp source particles are emitted with energy Ei, the total uncollided dose is

(6.17) For a source emitting a continuum of particle energies such that N(E) dE is the probability that a source particle is emitted with energy in dE about E, the total uncollided dose is D 0 (r)

6.1.4

=

1 o

00

dE

Sp N(E) (E) � exp[-1 ds µ(s,E)] 41rr o r

(6.18)

Point Kernel for the Uncollided Dose

Consider the case for which the isotropic point source is placed at position rs and the isotropic point detector (or target) is placed at rt in a homogeneous medium. The detector response in this case depends not on rs and rt separately, but only on the distance lr s - rt ! between the source and detector. If, in addition, the source emission Sp is unity, the detector response is [cf. Eq. (6.12)] go(rs, rt, E)

=

'.R(E)

47rr s -rt 12

I

e-µ(E)lr,-r,I

(6.19)

Here g 0 (r s ,rt ,E) is the uncollided dose point kernel and equals the dose at rt per particle of energy E emitted isotropically at r s . This result holds for any geometry or medium provided that the material through which a ray from rs to rt passes has a constant interaction coefficient µ. If the medium is heterogeneous between r s and rt , then, in the expression above, exp[-µ(E)lrs -rt l] is replaced by exp(-£), where £ is the distance in mean-free-path lengths between rs and rt given by Eq. (6.15). With this point kernel, the uncollided dose due to distributed sources is readily expressed. For example, consider a monoenergetic isotropic volume source Sv (r s ) distributed throughout some region S with volume Vs of space. Here Sv (r s ) dVs is the number of particles emitted isotropically in a differential volume dVs at rs . These source particles from dVs then produce an uncollided dose at rt of

(6.20) The total uncollided dose at rt from all particles emitted in Vs is obtained by integrating this differential dose over all Vs , namely,

(6.21) This result can be generalized to give the average dose in some target region

T with volume ½ that is produced by an energy-dependent source distributed in

Sec. 6.2.

161

Uncollided Doses from Distributed Sources

region S. If Sv (r.,E)dV.dEis the number of particles emitted with energies in dE about Efrom sources in dV. at r., the average uncollided dose in ½ is2 Do(T .- S)

=

_.!._

{

½JE

dE { dV. { d½

lv.

Jv,

Sv (rs,E)J Ymax where Ymax is the maximum ray length from source to target within the plane of the cylindrical cross sections. Kellerer [1981] shows this construction of the three-dimensional geometry factor from the two-dimensional geometry factor for infinite cylinders of arbitrary cross section can also be extended to the case of cylinders of finite length.

6.5 CALCULATION OF GEOMETRIC FACTORS The key to using the generalized method for computing doses with the point kernel method of the previous section is to be able to evaluate the geometry factor (or, equivalently, the point-pair distance distribution) appropriate for a specified source and target geometry. 6.5.1 Analytical Calculation of Geometry Factors For source and target volumes with simple geometric shapes, the geometry factor g(r,T - S) or the point-pair distance distribution p(r,T - S) can often be calcu­ lated analytically. In this section, geometric factors are derived for several simple cases. Additional results are given in the problems at the end of this chapter. The determination of these distributions is a standard problem in geometric probability theory [Weil 1983], and many results can be found in the literature [Mader 1980; Kellerer 1981, 1984; NCRP 1991; Faw and Shultis 1992, 1993] Considerable atten­ tion has been given to the autologous case (Sand T the same region) for which the chord-length distribution and the geometry factor are closely related in a convex region [Kellerer 1971]. 6.5.2

Examples of Geometry Factors and Point-Pair Distributions

Line Source and Point Target

For an isotropic line source {region S) of length L and a point detector (region T) opposite one end of the source [see Fig. 6.lO{a)], the probability that a ray from the source to the detector has a length between r and r + dr is p(r,T- S)dr

= dx/L.

{6.104)

186

Basic Methods for Radiation Dose Calculations

Chap. 6

T L x+dx X

r+dr S

r

dP

0 ___h ___ T

(a)

(c)

(b)

Figure 6.10. Geometries for evaluating the point-pair distance distribution for (a) a line source and point detector, (b) a disk source and point detector, and (c) a spherical source and detector.

From the geometry shown in Fig. 6.lO(a), x 2 = r 2 - h2 , from which dx/dr r/Jr2 - h2 . Thus the point-pair distance distribution for this geometry is

p(r,T +- S) =

2 { LJr; - h '

0,

=

(6.105) otherwise.

Because the detector is a point, the geometry factor for this case is zero. Isotropic Disk Source and Point Target Consider the isotropic disk source shown in Fig. 6.lO(b). The probability that a line from the detector to a random point on the disk will have a length between r and r + dr is equal to the fraction of the disk's area that is in the annulus between p and p + dp, namely,

p(r,T +- S)dr =

21rpdp 1rR 2 .

(6.106)

From the geometry it is seen that p2 = r 2 - h2 , from which pdp = rdr. Thus the point-pair distance distribution is

p(r,T +- S) =

{ 2r

R2 '

0,

(6.107) otherwise.

Again, the geometry factor is zero for this point-detector problem. Spherical Source and Target Figure 6.lO(c) shows a spherical source region of radius R containing a uniform volume source. The target volume is the same sphere. To evaluate the geometry factor for this problem, consider rays of length r that originate at a distance u from

187

Sec. 6.5. Calculation of Geometric Factors

Jt _____ T I

I

\

\

\

\

\

\

Figure 6.11. Geometry for evaluating the geometry fac­ tor for an infinite slab source S with a uniform isotropic source. The point target T and the slab are in an infinite homogeneous medium with the same material properties as the source slab.

the center of the sphere and end on the spherical surface illustrated by the dashed sphere in Fig. 6.lO(c). The fraction of these rays ending within the source sphere, that is, O, t f 4rr in Eq. (6.93), is unity for r ::; R - u and is otherwise just the solid angle fraction subtended by the half conical angle (3, namely, (1 - cos (3)/2. By making use of the law of cosines and by averaging over the volume of the sphere as prescribed by Eq. (6.93), one arrives at the following geometry factor g(R,T +- S), which is the fraction of all rays of length r originating within the source sphere and terminating in the same sphere:

g(r,T

+-

3 R3

[1

=1-

3r 4R

S) =

R

-r

O

duu2

+ -21 1

R

R

+

r3 3' 16R

-r

(

duu2 1-

u2 + r 2 - R 2 ) 2ur

l

(6.108)

0 ::; r ::; 2R.

The geometry factor is zero if r > 2R. For an application of this result, see Berger [1973). Finally, from Eq. (6.93) the point-pair distance distribution is found to be

p(r,T

+-

4rrr 2 S) = ½g(r,T

+-

S) =

3r 2 g(r,T R3

+-

S).

(6.109)

Infinite Slab Source and Point Target

As a final example of the analytical determination of the geometry factor, consider the uniform slab source S of thickness H shown in Fig. 6.11. A point detector T is placed a distance t from one face of the source slab. The target and source are in an infinite homogeneous medium with the same interaction coefficients and density as the material of the slab source. Because the detector is a point, g(r, T +- S) = O; however, g(r, S +- T) is not.

188

Basic Methods for Radiation Dose Calculations

Chap. 6

Rays of length r emitted isotropically from T terminate on the surface of a sphere represented by the dashed circle in Fig. 6.11. The fraction of these rays, g(r, S ← T ), that end in the slab S is simply 1/4π times the solid angle subtended at T by that portion of the sphere’s surface inside the slab (i.e., with angles between θ1 and θ2 ). Consider the case shown in the figure, for which r ≥ t + H. The solid angle subtended by the spherical surface 0 ≤ θ ≤ θi , i = 1 or 2, is just 2π(1 − cos θi ). Thus the geometry factor is g(r, S ← T ) =

2π(1 − cos θ2 ) − 2π(1 − cos θ1 ) 1H = , 4π 2 r

r > t + H.

For the case t < r ≤ t + H, no ray passes through the source slab and   2π(1 − cos θ2 ) 1 t g(r, S ← T ) = = 1− , t < r ≤ t + H. 4π 2 r

(6.110)

(6.111)

And if r ≤ t, no ray reaches the slab and g(r, S ← T ) = 0. For this example, because the source region is infinite in extent, the point-pair distance distribution collapses to zero.

6.5.3

Uncollided Dose Examples Using Geometry Factors

To illustrate the use of the geometry factor or point-pair distance distribution for determining doses, the uncollided dose kernel G o (r) of Eq. (6.79) is used in the generalized dose method to determine uncollided doses for some of the examples analyzed in Section 6.2 with the source superposition technique. In all these examples the source is assumed to be monoenergetic. Line Source in an Attenuating Medium This problem is illustrated in Fig. 6.2(a), where the source region S becomes the line source and the target region T the point detector at P . For a line source, the total source emission Stot = Sv Vs in Eq. (6.90) becomes Sl L, to give Z ∞ dr G o (r)p(r, T ← S). (6.112) Do (T ← S) = Stot 0

Substitution of G o (r) = Re−µr /(4πr2 ) and the point-pair distance distribution of Eq. (6.105) into this result yields Z √h2 +L2 Sl R dr e−µr √ D (T ← S) = , (6.113) 4π h r r2 − h2 √ which with the geometric relations r2 − h2 = h tan θ and r = h sec θ reduces to Z Sl R θo Sl R F (θo , µh). Do (T ← S) = dθ e−µh sec θ = (6.114) 4πh 0 4πh o

This is the same result as that obtained in Eq. (6.31). As µ → 0, this result reduces to the nonattenuating result of Eq. (6.26).

Sec. 6.5. Calculation of Geometric Factors

189

Disk Source in a Homogeneous Attenuating Medium In Fig. 6.4 the total emission from the disk source (region S) is Stot = Sa πa2 . The detector (region T ) is located a distance h above the center of the disk in a homogeneous medium with interaction coefficient µ. Substitution of the point-pair distance distribution for this geometry as given by Eq. (6.107) (with R = a) and the uncollided dose kernel into Eq. (6.112) yields Sa R D (T ← S) = 2 o

Z

√ h2 +a2 h

Sa R dr −µr e [E1 (µh) − E1 (µh sec θo )], = r 2

(6.115)

which is the same result as Eq. (6.42). Shielded Infinite Slab Source The geometry for this problem shown in Fig. 6.11, in which the source slab S and the target T are embedded in an infinite homogeneous medium with the same material properties as the slab. The uncollided dose at T is found immediately by substituting the geometry factor for this case [Eqs. (6.110) and (6.111)] and the uncollided dose kernel G o (r) into Eq. (6.96). The result is (Z )  Z ∞  t+H t Sv R o −µr −µr H + 1− dr e dr e D (T ← S) = 2 r r t t+H =

   Sv R  −µt e − µtE1 (µt) − e−µt−µH − µ(t + H)E1 (µt + µH) 2µ

=

Sv R [E2 (µt) − E2 (µt + µH)] , 2µ

(6.116)

where the relation (see Appendix B) E2 (x) = e−x − xE1 (x) has been used. This result is the same as that of Eq. (6.58).

6.5.4

Monte Carlo Evaluation of Geometry Factors and Point-Pair Distance Distributions

For target and source regions with geometric shapes that are sufficiently complex to preclude analytical evaluation of the geometry factor or point-pair distance distribution, Monte Carlo techniques can be used to determine numerically either of these functions. When one of the functions is determined, then the other is obtained easily from Eq. (6.93). If the target and source regions are both finite in size (including point and other zero-volume regions), it is far more efficient computationally to evaluate the pointpair distance distribution than the geometry factor. To determine p(r, T ← S) numerically, select N pairs of random points, one of each pair in S and the other in T , as illustrated in Fig. 6.12(a), and calculate the distance r between each pair. Create a histogram of the number of lengths that fall between ri and ri + ∆r and

190

Basic Methods for Radiation Dose Calculations

Chap. 6

(a) Figure 6.12. Monte Carlo determination of (a) the point-pair distance dis­ tribution p(r, T +- S), and (b) the geometry factor g(r, T +- S).

normalize the histogram to unit area by dividing by N �r. After a sufficiently large number of pairs have been used, the normalized histogram will approximate quite closely p(r, T +- S), the approximation becoming better the smaller �r becomes. It is critical when choosing random points in S and T that points are selected uniformly in the volumes. Such sampling techniques are described in Chapter 11. The geometry factor g(r, T +- S) can also be determined by a similar Monte Carlo procedure. Random points uniformly distributed in Sare generated, and from each point a ray of length r oriented isotropically is drawn as shown in Fig. 6.12(b). In the limit of a large number of rays, the fraction of the rays that terminate in T equals the geometry factor g(r, T +- S). This procedure is then repeated for different values of r to obtain g at a set of appropriate discrete r values. Clearly, this procedure is computationally very inefficient because so many rays are "wasted" (i.e., do not contribute to g). By contrast, all random points contribute to the determination of the point-pair distance distribution. Although it is almost always better to determine first the point-pair distance distribution through Monte Carlo calculations and then to determine the geometry factor from Eq. (6.93), there is one important class of problems for which the geom­ etry factor must be determined directly by Monte Carlo methods. When either S or T becomes infinite, p(r, T +- S) and p(r, S +- T) go to zero while g(r, S +- T) or g(r, T +- S), respectively, not being a probability density function, remains nonzero over an infinite range of r values. Once a histogram approximation for p(r, T +- S) or g(r, T +- S) has been obtained, the absorbed dose D(T +- S) is readily evalu­ ated from Eq. (6.90) or Eq. (6.96) by replacing the integral over r by a sum over all histogram bins. Examples of Monte Carlo Calculated Geometry Factors

Two examples of p(r, T +- S) determined by the Monte Carlo method are shown in Fig. 6.13. On the left, Monte Carlo results for the autologous case (S = T) of a sphere with unit radius is compared to the analytic results of Eq. (6.108). The right-hand example is for a more complex case of a uniform source in the liver producing a dose in the spleen.

191

Sec. 6.5. Calculation of Geometric Factors

0.20

en .sa.

0.8

0

0.6

I

c

�

en .sa. I

V

c

:::,

0

..c

EIll 'o � ·;;;

a. I

1=

·o a.

:.::;

Age 5 0.15

0.10

·.::: .....

0.4

Ill

'o

� ·;;;

0.2

a. l,

·o 0.50

1.5

lO

2.0

0.050

0.0

0

5

10

15

20

25

30

Distance, r (cm)

Distance, r

Figure 6.13. Two examples of the point-pair distance distribution determined by Monte Carlo techniques. The left example is for a sphere of unit radius as both source and target. The Monte Carlo results are shown as a histogram and the analytical result of Eq. (6.108) as a continuous curve. The example to the right shows the point-pair distance distribution for the liver as the source and the spleen as target for an adult and child. The organ geometries are those specified by Cristy and Eckerman [1987].

6.5.5

Multiregion Geometries

Many dosimetric problems can be modeled by a few regions with simple shapes embedded in an infinite medium. For example, a model might consist of several concentric spherical regions for which the absorbed dose in one spherical shell arising from a source in another is sought. In this case, some simple properties of the geometry factor can be used to express the needed geometry factor in terms of geometry factors of a simple problem: for example, that for a spherical source to a concentric shell. From physical arguments, if A, B, and C represent any three regions with volumes Va, Vi, and Ve in an infinite homogeneous medium, one has the multiple-target property

g(r, B+ C+-A)= g(r, B+-A)+ g(r, C+-A).

(6.117)

From the reciprocity relation for the geometry factor, that is, Va g(r, B+-A)=

Vi g(r, A+-B),

(6.118)

and the multiple-target property, one can derive the following multiple-source prop­ erty: (Va +

Vi)g(r, C+-A+ B)=

Va g(r, C+-A)+

Vi g(r, C+-B).

(6.119)

The multiple-source and multiple-target properties can be generalized to any num­ ber of regions. In particular, if the infinite homogeneous medium is decomposed

192

Basic Methods for Radiation Dose Calculations

Chap. 6

into N distinct contiguous regions ¼, one has

L g(r, ½ N

+-

¼) = i.

(6.120)

j=l

The above relations can often be used to obtain analytical results for a multire­ gion problem in terms of results for simpler problems and thereby to avoid a direct, and often complex, analytic evaluation in the multiregion geometry. Examples of the use of these relations are given later in Section 6.5. 7. 6.5.6 Basic Geometry Factors for One-Dimensional Problems Certain geometry factors for simple geometries are fundamental in the sense that they may easily be used, in conjunction with the multiple-source and multiple-target relationships ofEqs. (6.118) through (6.120), to derive geometry factors for far more complicated geometries. These basic geometry factors are illustrated in Fig. 6.14 and derived below. Cartesian Coordinates

Consider a half-space target region contiguous with a slab source region of finite thickness L. It can be shown [Faw and Shultis 1991] that geometry factor ge(r, L ), defined as the fraction of rays of length r originating in the source region and ending in the target region, is given by r r � L, 4£' (6.121) ge(r, L) = { !:_ �_ r ?:. L. 2 4r' From Eq. (6.120) it can be shown that the equally important geometry factor for self-absorption in the same slab region of thickness L is given by g(r,S +- S) = 1- 2ge(r, L ) = Spherical Coordinates

{ L

2' � 1 _!__ ' 2£

r ?:. L,

(6.122)

Consider a spherical source region of radius R 1 and a concentric spherical target of radius R2 ?:. R1 . It can be shown [Faw and Shultis 1993] that the geometry factor g 8 (r, R1 , R2 ), defined as the fraction of rays of length r originating in the source region and ending in the target region, is given by gs(r, R1 R2 ) _ - 2_ ' Rf

[Hr +6 � _ r(Rr 8+ R�) + 48r _ (R�-l6rRt) ] · 3

2

(6.123)

The geometry factor is zero for r ?:. R1 + R2 and unity for r � R2 - R1 . Note that Eq. (6.108), for the autologous spherical source and target regions, is just a special case of Eq. (6.123), with R1 = R2 = R.

193

Sec. 6.5. Calculation of Geometric Factors

c

1.

0.8

1-2g,(c,L)

0

J!! 0

m

l

'

gc(r,R,R)

0.6

9 5 (r,R,R)

0.4

0.2

0.0

0.0

0.40

0.80

1.2

1.6

2.0

r/L or r/R Figure 6.14. Geometry factors for self-absorption in a slab region of thickness L and in spherical and cylindrical regions of radius R.

Cylindrical Coordinates Consider a source region consisting of an infinitely long circular cylinder of radius R1 and a target region consisting of a concentric cylinder of radius R2 2: R 1 . Let r now represent, not lengths of rays from one point to another, but projections of such rays in a plane normal to the cylinder axis (see Section 6.4.7). It can be shown [Faw and Shultis 1992, 1993] that the geometry factor 9c (r,R1 ,R2), defined as the fraction of ray projections of length r in the plane that originate within the source region and end within the target region, is given by _ -1 {cos _ 1 9c ( r, R 1, R2 ) 1r

(r2 + Ri - R�) + R�2 cos _ (r2 2 rR 1

r - [4

6.5. 7

2 R�

R

1

Ri + R� ) 2 r R2

-

1

- (Ri - R� - r 2 ) 2 ] 1 / 2 } 2Ri

(6.124)

Examples of Multiple Regions

As an example of the use of the multiple-source and multiple-target properties to construct geometry factors for complicated geometries from basic geometry factors, consider the spherical shells illustrated in Fig. 6.15. This geometry was chosen in an approximate calculation of the radiation dose to nuclei of thyroid follicular cells resulting from radioiodine present in follicles [Faw, Eckerman, and Ryman 1991]. The target is a spherical shell identified in the figure as region C, with r 2 � r � r 3•

194

Basic Methods for Radiation Dose Calculations

Chap. 6

Figure 6.15. Spherical shell target region separated from two source regions, one a central sphere, the other all space outside a spherical core.

One source region, S1, is a sphere identified as region A, with r < r 1. A second source region, S2 or region E, is all space beyond r = r 4• Regions B and D are "inert" shells separating sources and target. From the multiple-target property, Eq. (6.117), g(r, T +- S1)

= g(r, C +-A)= g(r,A+ B + C +- A) - g(r,A+ B +- A).

(6.125)

From Eq. (6.123), g(r,A+ B +-A)= gs (r, r1, r 2 ),

(6.126)

g(r,A + B + C +-A)= gs (r,r1,r3).

(6.127)

By direct substitution from Eq. (6.123), so long as r 2 - r 1 g(r, T +- S1 )

=

3 r� - r� r 2 - - - (r 3 3 { -6 8 Tl

-

r 22 )

1 4 [r 16r 3

- -

-

< r < r 1 + r 3,

4 2 2 r 2 - 2r 1 ( r 3

-

2 )] }

r2

(6.128) In determination of the dose in the target region resulting from sources in region S2, which is infinite in volume, it is necessary to make use of Eq. (6.96), that is, to make use not of g(r, T +- S2 ) but of g(r, S2 +- T) g(r, E +- C). From Eq. (6.120),

=

g(r, E +- C)

= l - g(r,A+ B +- C) - g(r, C +- C) - g(r, D +- C).

(6.129)

With the use of the reciprocity relation of Eq. (6.118) and the multiple-target prop­ erty of Eq. (6.119), the three geometry factors on the right-hand side of Eq. (6.129)

195

Sec. 6.6. Effect of Density Variations

can be expressed as follows.

g(r,A + B +-C)

=

=

b V;: g(r,C +-A+B)

b V;: {g(r,A+B+C..-A+B)-g(r,A+B..-A+B)}.

(6.130)

Similarly,

g(r,C +-C) =

1

{Va+b+e g(r,A+B + C +-A+B + C)

Ve

-2Va+b g(r,A+B + C +-A+B) + Va+b g(r,A+B +-A+B)}] (6.131) and

g(r,D +-C)

=

1

Ve {Va+b+e [g(r,A + B + C + D +-A+B + C)

- g(r,A+B + C +-A+B + C)] - Va+b [g(r,A+B + C + D +-A+B) - g(r,A+B + C +-A+B)]}.

(6.132)

Substitution of these results into Eq. (6.129) gives, for r

g(r, S2 +- T) = g(r, E +-C) = 1 - Va;:

+e

> r4 - r3,

g(r,A+B + C + D +-A+B + C)

+ V;:b g(r,A + B + C + D +-A+B) (6.133)

6.6

EFFECT OF DENSITY VARIATIONS

The methods for dose evaluation discussed in the preceding section apply rigorously only when the target and source regions have the same homogeneous composition and density as the infinite medium in which they are embedded. Of practical im­ portance is the situation in which a single region has a density different from that of the surrounding medium. For example, bone and lung have different densities from that of the surrounding soft tissue. Nevertheless, the previously discussed methods for an infinite uniformly homogeneous medium can be adapted to account for den­ sity variations in subregions provided that the interaction properties of the different regions are similar, as is often the case in many internal dosimetry problems. For example, soft tissue, bone, and air have very similar microscopic cross sections or mass interaction coefficients for charged particles and for photons because of the similar average atomic numbers of the elemental constituents.

196

Basic Methods for Radiation Dose Calculations

Chap. 6

6.6.1 Theorems for Density Variations There are some very general theorems for media with density variations that can be derived from the transport equation governing the penetration of radiation through matter. Three of these results are presented here without proof. Fano's Theorem One of the most basic theorems involving density variations is known as Fano's theorem [Fano 1954; Spencer Chilton, and Eisenhower 1980]. This theorem, which has important implications in the design of instrumentation for dosimetry, can be stated as follows. In an infinite medium with arbitrary density variations, the radiation field and hence the absorbed dose rate are everywhere the same provided that (1) the atomic interaction properties are constant, (2) the energy spectrum and angular distribution of the sources are independent of position, and (3) the volumetric source strengths are everywhere proportional to the local density. Theorem on Plane-Density Variations For the special case in which material properties are everywhere the same except for plane-density variations [i.e., the density p(x) varies only with one spatial coordinate x] and in which the strength of the radiation sources also varies only with the distance x, the radiation field at any depth x equals that at a corresponding depth x' in a medium of any constant density p' with the same mass interaction coefficients. Corresponding distances x and x' are at the same mass thickness depth, that is, x x' p' = fo p(x) dx, from corresponding reference planes here taken as x = x' = 0. The source strength per unit mass in the constant-density medium must be the same as that in the variable-density medium at corresponding points. This plane­ density theorem also holds for semi-infinite media if the boundary conditions of the variable- and constant-density media are the same. Figure 6.16 illustrates the attenuation of a beam of 1-MeV electrons normally incident on the plane surface of a half-space of water. What is shown is the absorbed dose as a function of the depth of penetration x into the water measured in units of mass thickness px. In one case, a water layer between depths of 0.218 and 0.262 g cm- 2 is replaced by a bone layer of the same mass thickness but of higher density. According to the plane-density variation theorem, if bone and water had the same, not just similar, mass interaction coefficients, then the absorbed dose distribution for both cases should be identical when plotted as a function of mass thickness. As can be seen, the bone produces only a minor distortion of the absorbed-dose distribution, namely a slight enhancement in reflection from the bone layer. If the bone layer were replaced by a metallic layer of the same mass thickness but whose mass interaction coefficients were quite different from those of water, the distortion would be much more pronounced. Scaling Theorem A very useful theorem for media with density variations, known as the scaling theo­ rem [Spencer, Chilton, and Eisenhower 1980], applies even to a finite heterogeneous medium. Consider two media A and B, one of which is a scaled version of the other,

197

Sec. 6.6. Effect of Density Variations

N I

I

Dose with H20 layer

0.06

Dose with bone layer

cu L

cu

Q.

cu

" 1/)

0

cu

0.04

0.02

·---H20 or bone

0.0L_.__JL___[___[_____L__.J..___L___l___,_---=i.....___, 0.0 0.1 0.2 0.3 0.4 0.5 2 Mass thickness depth (g cm- ) Figure 6.16. Absorbed dose versus depth for 1-MeV electrons normally inci­ dent on a water medium. Also shown is the dose profile for the case in which a layer of water is replaced by a layer of bone of the same mass thickness.

that is, the coordinates ra and rb of corresponding points in A and B, respectively, are related by rb = (ra , where ( is a constant scaling parameter. The two media are said to be similar if (1) all material interfaces occur at corresponding points, (2) the material at corresponding points is the same except that the density at a point in A is ( times that at the corresponding point in B, and (3) radiation sources at corresponding points are identical in energy and angular distributions and have the same strength per unit mass. Equivalently, the strength per unit (unscaled) volume at a point in A is ( times that at the corresponding point in B. Corresponding subregions in similar media have identical shapes but different volumes and masses. The scaling theorem then states that if two media are similar, then the radiation fields at corresponding points are identical. Under the conditions required by the scaling theorem, the absorbed doses at corresponding points in similar media are also equal.

198

Basic Methods for Radiation Dose Calculations

Chap. 6

As an application of this scaling theorem, consider an important dosimetry prob­ lem in which the density p' of a single region with volume V' and mass m' = p'V' is different from the density p of the surrounding infinite medium (e.g., a bone in a large mass of soft tissue). The volume V', which may have an arbitrary distribution of sources, is also the target region of interest. It is supposed that the mass interac­ tion coefficients are the same for both V' and the surrounding medium. Then from the arguments above, the volume V' can be scaled to an identically shaped volume V = V'(p' / p) 3 and mass m = m'(p' / p) 2 with density p so that the absorbed fraction ' for volume V' is approximately the same as the absorbed fraction in a volume V [ICRU 1979]. Because the masses contained in the two volumes are different, the specific absorbed fractions for the two problems scale as = (p/ p')2 if,'. If in the scaling of V' into V the density of the surrounding medium had also been scaled, the equivalence of the absorbed fractions would be exact. However, by keeping the density of the medium outside V at p so as to obtain a completely homogeneous medium, one has neglected the slightly different amounts of energy that would escape from and reenter the two regions V and V' either from backscatter or because the surfaces wrap around on themselves with areas of local concavity.

6.6.2

Point Kernels in Media with Density Variations

For infinite media problems in which there are material discontinuities between or in the source and target regions, the infinite-medium point-kernel techniques can still be used to good approximation provided that all materials have nearly the same electron range measured in terms of mass thickness pr or nearly the same photon mass interaction and energy absorption coefficients. Such is the case for constituents of the human body. The medium in and around the regions of interest can thus be homogenized conceptually and a point kernel used for the homogenized medium. Alternatively, the point kernel can be written with an additional variable, either f, the mass thickness, or £, the optical thickness, defined as O

f or

1r,-r,1 r ds p(s)

= lo r

e = lo

1r,-r.1

ds µ(s),

(6.134)

(6.135)

where the integral is along the line between the points rt and r s . For such a medium, the point kernel can be decomposed rigorously as (6.136) or

Q(rt,r s )

1 = Q(r,£) = 9'(£), 41rr 2

in which the energy dependence of the kernel is implicit.

(6.137)

199

Sec. 6.6. Effect of Density Variations

Other methods have also been used to account for discontinuities in material composition in some geometries. An important example is found in the determina­ tion of the dose to the skin arising from radioactive gases in the atmosphere [Berger 1974], a problem addressed later in this chapter and in Chapter 9.

6.6.3

Modified Point-Pair Distance Distributions

Use of the point kernel of Eq. (6.137) in dose evaluation requires a dual point-pair distance distribution defined in such a way that p(r,f,T +- S) dr df represents the joint probability that random points in T and S are separated by geometrical paths in dr about r and by mass thicknesses in the range of df about f. The analog of Eq. (6.90) is clearly

D(T +- S)

= Sv Vs

100 100

= Sv Vs

1

df

00

lo

Integration over r yields

D(T +- S)

= Sv

dr Q(r,f)p(r,f,T +- S)

1

df Q'(f)

v.100

00

lo

dr �p(r,f,T +- S). 41rr

df Q'(f)p'(f,T +- S),

(6.138)

(6.139)

in which the modified point-pair distance distribution is given by

p'(f,T +- S)

=

1

00

lo

dr

1 41rr 2

p(r,f, T +- S),

(6.140)

a weighted-average distribution, with the geometrical attenuation factor 1/41rr 2 as weight factor. Note also that the point-pair distance distribution for a homogeneous medium p(r,T +- S) is given by

p(r,T +- S)

6.6.4

=

100

df p(r,f,T +- S).

(6.141)

Modified Geometry Factors

As an alternative to the dual point-pair distance distribution, there is a dual ge­ ometry factor defined in such a way that g(r, f, T +- S)df represents the fraction of rays of length r that originate in S and reach T while passing through a mass thicknesses in the range of df about f. In analogy with Eq. (6.93),

g(r,f,T +- S)

=

½ p(r,f,T +- S). 41rr 2

(6.142)

200

Basic Methods for Radiation Dose Calculations

It follows from Eq. (6.139) that6 D(T +- S)

= Sv �

100

Integration over r yields D(T +- S)

= Sv �

df (}'(f)

100 100

100

dr g(r,f,T +- S).

df (}'(f)g'(f,T +- S),

Chap. 6

(6.143)

(6.144)

in which the modified geometry factor is given by g'(f,T +- S)

=

dr g(r,f,T +- S).

(6.145)

Note that whereas the geometry factor g(r,T +- S) is dimensionless, the modified geometry factor g'(f,T +- S) = ½p'(f,T +- S) has dimensions of cm 3 /g.

6.6.5

Example of a Modified Point-Pair Distance Distribution

Consider two touching spheres of unit radius and unit density p in a vacuum, one sphere representing a source region, the other a target. Let f be the mass thickness separating points rt and r s as defined by Eq. (6.134). Illustrated in Figs. 6.17 and 6.18 are results of a Monte Carlo calculation of the dual point-pair distance distribution p(r,f,T +- S) as well as the individual distributions p(r,T +- S) and p'(f,T +- S).

6.6.6

Example Problem Using Modified Geometry Factors

A very important shielding and dosimetry problem involves one half-space as a source region and a target point within an adjoining half-space, as illustrated in Fig. 6.19. For example, evaluation of the dose within the skin arising from beta­ particle sources in the atmosphere is commonly accomplished on the basis of this geometric model [Berger 1974]. This is possible because, per unit mass, the at­ tenuation properties of air are not very different from those of skin. Similarly, the same geometric model may be applied to the approximate determination of dose to the lining of the gastrointestinal system or the urinary bladder as a result of beta­ particle sources within the contents of these bodies. The same geometric model may be applied to determine the dose in air above soil or water containing a distributed source of beta particles or gamma rays. In the system illustrated in Fig. 6.19, the composition is assumed to be uniform. In the half-space containing the target, density p is assumed to be constant, as is density P s in the source region. The source strength Sv is assumed to be constant. 7 6 An alternative derivation of the analog of Eq. (6. 94) requires recognition that the energy absorbed in target region T as a result of source points within region S, separated from target points in region T by mass thicknesses between r and f + di' and geometric paths between r and r + dr, is given by the product of Sv V841rr 2 dr and [Q'(r)/41rr 2 ]g(r, f, T H, the two slabs separated by distance a. Show that the geometry factor is given by [NCRP 1991) 1

g(r,T +- S)=

a 4 - 4r' r-a-H/2 2r h (h+h+a-r)2 4Hr 2r h 2r'

a:::; r:::; a+H, a+H:::; r:::; a+h, a + h :::; r r

:::;

a + H + h,

> a+h+H.

213

Problems

27. Consider a source region consisting of a spherical surface of radius R s and a target region consisting of a concentric sphere of radius Rt. Show that [NCRP 1991] g(r,T +-S)

=

1

{

2-

r

4R s +

Jtf-R2s 4R s r

'

I Rs

- Rt

I< r < Rs + Rt,

1,

28. Consider a source region consisting of a spherical surface of radius R s and a target region consisting of a concentric spherical shell of inner and outer radii R1 and R2• Use the results of the previous problem to determine g(r,T +-S) for all r [NCRP 1991]. 29. Derive Eq. (6.123) and show that it reduces to Eq. (6.123) for the special case of R1 = R2. 30. Derive Eq. (6.124). 31. Verify the following equations associated with modified point-pair distance dis­ tributions and geometry factors: g(r,f,S +-T)

=

Vs p(r,f,S +-T), 41rr 2

= Vtg'(f,S +-T), g (f,S +-T) = Vs p'(f,S +-T),

¼g'(f,T +-S) 1

D(T +-S)

= Sv

fo

00

dfQ'(f) g'(f,S +-T).

Chapter 7

Special Techniques for Photons

This chapter describes the engineering methodology that has evolved for the design and analysis of shielding for gamma and x rays with energies from about 1 keV to about 20 MeV. To support this methodology, very precise radiation transport calculations have been applied to a wide range of carefully prescribed situations. The methods used for the transport calculations are discussed in Chapters 10 and 11. The results are in the form of buildup factors, attenuation factors, albedos or reflection factors, and line-beam response functions. Buildup factors relate total doses to doses from uncollided photons alone and are most applicable to point monoenergetic radiation sources with shielding well distributed between the source point and points of interest. Attenuation factors apply equally well to monoenergetic sources and to polyenergetic sources such as x-ray machines and are most applicable when a shield wall separates the source and points of interest, the wall being sufficiently far from the source that radiation strikes it as a nearly parallel beam. There are many common features of buildup and attenuation factors and it is possible to represent one factor in terms of the other. Albedos relate flows of radiation into and reflected from a shield and are applicable in ways similar to attenuation factors, except that points of interest are on the same side of the reflecting surface as is the source. Line-beam response functions are applicable to problems in which sources are collimated so that no direct beam can reach a point of interest and so that radiation reaches a point of interest as a result of scattering throughout a volume of material, not as a result of reflection from a fixed surface. Discussed first in this chapter are buildup factors for point isotropic and mono­ energetic sources in infinite media. Incorporation of these buildup factors into the point kernels introduced in Chapter 6 is treated next. Then addressed are three topics associated with the use of buildup factors. The first is the use of empirical buildup-factor approximations designed to simplify engineering design and analysis. The second is the use of buildup factors with point kernels to treat spatially distributed radiation sources. The third is the application of approximate methods 214

215

Sec. 7.1. Photon Buildup

to permit the use ofbuildup factors in media with variations in composition. Next addressed are albedos and their application to the analysis ofstreaming ofradiation along ducts through shields. Last addressed are line-beam response functions and their application in evaluation ofradiation doses from so-called skyshine radiation.

7.1

PHOTON BUILDUP-FACTOR CONCEPT

Whatever the photon source and the attenuating medium, the energy spectrum of the total photon fluence �(r, E) at some point ofinterest r may be divided into two components. The unscattered component �0 (r, E) consists of just those photons that have reached r from the source without having experienced any interactions in the attenuating medium. The scattered component �8 (r, E) consists of source photons scattered one or more times, as well as secondary photons such as x rays and annihilation gamma rays. Accordingly, the dose or detector response D(r) at point of interest r may be divided into unscattered (primary) and scattered (secondary) components D 0 (r) and D 8 (r). The buildup factor B(r) is defined as the ratio ofthe total dose to the unscattered dose, the latter usually being calculated with relative ease according to the methods of Chapter 6. Thus _ D(r) D 8 (r) B(r) = no 1+ o . = D (r) (r)

(7.1)

The doses may be evaluated using response functions described in Chapter 5, so that f dE '.R(E)�8 (r, E) B(r) = 1 + (7.2) J dE '.R(E)�0 (r, E) · in which the integrations are over all possible E. It is very important to recognize that in Eq. (7.2), the fluence terms depend only on the source and medium, not on the type ofdose or response. The response functions depend only on the type of dose, not on the attenuating medium. For these reasons it is imperative to associate with buildup factors the nature of the source, the nature of the attenuating medium, and the nature of the response. When the source is monoenergetic, with energy E0 , then �0 (r, E) = �0 (r)8(E­ E0 ), so that (7.3) In this case, the response nature is fully accounted for in the ratio '.R(E)/'.R(E0 ).

7.1.1

Isotropic, Monoenergetic Sources in Infinite Media

By far the largest body of buildup-factor data is for point, isotropic, and mo­ noenergetic sources of photons in infinite homogeneous media. The earliest data [Goldstein and Wilkins 1954; Fano, Spencer, and Berger 1959; Goldstein 1959] were based on moments-method calculations (see Chapter 10) and accounted only for buildup of Compton-scattered photons. Subsequent moments-method calculations

216

Special Techniques for Photons

10-MeV photons in lead

1-MeV photons in lead

10-MeV photons in water

1-MeV photons in water

Chap. 7

Figure 7.1. Comparison of photon transport in lead and water. Each box is 10 mean free paths on a side. Each depicts the projection in a plane of primary and secondary photon tracks arising from 10 primary photons originating at the box center, moving to the right in the plane of the paper. Tracks computed using the EGS4 code, courtesy of Robert Stewart, Kansas State University.

217

Sec. 7.1. Photon Buildup

[Eisenhauer and Simmons 1975; Chilton, Eisenhauer, and Simmons 1980] accounted for buildup of annihilation photons as well. Buildup-factor calculations using the discrete-ordinates ASFIT code [Subbaiah et al. 1982] and the integral-transport PALLAS code [Takeuchi, Tanaka, and Kinna 1981] account for not only Compton­ scattered and annihilation photons, but also for fluorescence and bremsstrahlung. These calculations supplement the later moments-method calculations, together supplying the data prescribed in the American National Standard for buildup fac­ tors [ANSI/ANS 1991]. Calculation of buildup factors for high-energy photons requires consideration of the paths traveled by positrons from their creation un­ til their annihilation. Such calculations have been performed by Hirayama [1987] and by Faw and Shultis [1993a] for photon energies as great as 100 MeV. Point­ source buildup factors described or used in this text exclude coherently scattered photons and treat Compton scattering according to the Klein-Nishina cross section, Eq. (3.49), for photon scattering with free electrons. This is also true for the buildup factors in the standard. Thus, in computing the dose or response from unscattered photons, coherent scattering should be excluded and the total Klein-Nishina cross section, Eq. (3.52), should be used. Total mass interaction coefficients, excluding coherent scattering, are identified in Appendix C as µ/ p. Correction for coherent scattering, significant for only low energy photons at deep penetration, is discussed in ANSI/ANS [1991]. 0.30 Figure 7.1 gives a qualitative im­ 2 pression of the buildup of secondary �., photons during the attenuation of pri­ Reflected >-. 0.20 mary photons. For 10-MeV photons in e' .,C lead, there is considerable buildup of ! ., annihilation photons, which are emit­ 0.10 ted isotropically, and bremsstrahlung, which deviates little in direction from >-. the path of the decelerating electron or .,e' C L&.J positron. For 1-MeV photons in lead, 0.0 0.0 1.0 2.0 0.50 1.5 there is very little buildup of secondary photons, owing to the strong photoelec­ Energy (MeV) tric absorption of the primary photons. Figure 7.2. Transmitted and reflected energy In water, both 1- and 10-MeV photons fluences for 2-MeV photons normally incident experience Compton scattering princi­ on a concrete slab of optical thickness 2. pally. However, for the higher-energy primary photons, the scattering leads to relatively small change in direction, as is evident from Fig. 3.3. Figure 7.2 illustrates the energy spectrum of the energy flu­ ence Eip(E) of reflected and transmitted photons produced by 2-MeV primary pho­ tons normally incident on a concrete slab two mean free paths in thickness. These fluences are normalized to unit incident flow and, thus, are dimensionless. Note that transmitted photons have energies up to the energy of the primary photons. However, the reflected photons, mostly single scattered, are much more restricted in energy, as is apparent from the Compton formula, Eq. (3.16). a. VJ

r·

,.

218

Special Techniques for Photons

Chap. 7

A limited set of buildup factors is provided in Appendix E. The data are more broadly applicable than might be thought at first glance. As indicated in Eq. (7.3), it is the ratio '.R(E)/'.R(E0 ) that defines the dependence of the buildup factor on the type of dose or response. For responses such as kerma or absorbed dose in air or water, exposure, or dose equivalent, the ratio is not very sensitive to the type of response. Thus, buildup factors for air kerma may be used with little error for exposure or dose equivalent. It can be shown that for a point isotropic source of monoenergetic photons in an infinite homogeneous medium, the buildup factor depends spatially only on the optical thickness, that is, the number of mean free paths C µr separating the source and the point of interest. Here, µ is the total interaction coefficient (excluding coherent scattering) in the attenuating medium at the source energy, namely µ(E0 ). Thus, as in Chapter 6, we write the buildup factor as B(µr), but it must be recognized that there is an implicit dependence on the source energy, the nature of the attenuating medium, and the nature of the response. Where necessary in this chapter, some of these parameters are displayed explicitly. For example, B''Jc(E0 ,µr) represents the exposure buildup factor for photons of energy E0 at distance r from a point isotropic source in medium m. Figure 7.3 illustrates the buildup factor for concrete, plotted with the photon energy as the independent variable and the number of mean free paths as a pa­ rameter. That there are maxima in the curves is due to the relative importance of the photoelectric effect, as compared to Compton scattering, in the attenuation of lower-energy photons and to the very low fluorescence yields exhibited by the low-Z constituents of concrete. Figure 7.4 illustrates the buildup factor for lead, plotted with the number of mean free paths as the independent variable and the photon energy as a parameter. For high-energy photons, pair production is the dominant attenuation mechanism in lead, the cross section exceeding that for Compton scat­ tering at energies above about 5 MeV, as may be seen from Fig. 3.4. The buildup is relatively large because of the production of 0.511-MeV annihilation gamma rays. As may also be seen from the figure, the attenuation factor increases greatly at pho­ ton energies just above the 0.088-MeV K-edge for photoelectric absorption, each absorption resulting in a cascade of x rays. For these reasons, buildup factors may be extraordinarily large, as evidenced by the line for 0.089-MeV photons in Fig. 7.4 At energies below the K-edge the buildup factors are very small, as indicated in Table E.4. The importance of fluorescence in the buildup of low-energy photons is addressed by Tanaka and Takeuchi [1986] and by Subbaiah and Natarajan [1987].

=

7.1.2 Comparison of Buildup Factors for Point and Plane Sources Many so-called "point-kernel codes" finding wide use in radiation shielding design and analysis make exclusive use of buildup factors for point isotropic sources in infinite media. This is so even when the source and shield configuration is quite different from that of an infinite medium. A good example is that of a point source and point receptor, each some distance in air from an intervening shielding wall. Is the use of infinite-medium buildup factors a conservative approximation? That

219

Sec. 7.1. Photon Buildup

,o

........u 0

�

,

3

,o

....

.su

o2

�

C.

J2

.0

.0

:,

·5

Ill

0 C.

>

0,

= 1 - WOm = 1 - ( 1 + ,82)-m/2

7 ( .67)

= 1, that is, for isotropic incident fl,uence, ( ) D0P D 0 (0)

=

11 21r'

7 ( .68)

in which n is the solid angle subtended by the duct entrance at the center of the duct exit, given in Problem 5, Chapter 2. The line-of-sight component of the dose at the exit of a cylindrical duct is il­ lustrated in Fig.7.19 and, in Fig. 7.21 of Section 7.5.4. As is quite evident from these figures, for a < < Z, D 0 (P)/ D 0 (0) '.::: (m/2),8 2 , simply an expression of the inverse-square law for attenuation of radiation from a point source. Line-of-Sight Component for the Rectangular Duct

Evaluation of the line-of-sight component for rectangular ducts is illustrated in Fig. 6.5 for one quadrant of the rectangular entry plane. Let the total rectangular dimensions be WxL, that is, b = L/2 and a= W/2. For Jn0 ( w) = m ( +l)wm J;t /21r, and from Eq. ( 6.50), D 0 (P)

/2

= -1T2 �(m + l)J;; 1L O

dy 1 0

W/ 2

dx z m (x 2 + y 2 + Z 2)-(m+2)/ 2•

(7.69)

248

For the special case of m

Special Techniques for Photons

Chap. 7

= 1, that is, for isotropic incident fluence, D 0 (P) n = 21r' D 0 (0)

(7.70)

in which n is the solid angle subtended by the duct entrance at the center of the duct exit, given in Problem 6, Chapter 2.

7.5.3 Wall-Penetration Component for Straight Ducts Consider the cylindrical duct illustrated in Fig. 7.18. Of interest is the radiation penetrating the wall through the lip of the duct entrance. This component D w (P) may be evaluated in a way very similar to that for the line-of-sight component, Eqs. (7.65) through (7.67), except that p� a and attenuation in the wall material must be accounted for, as is illustrated in the figure. Suppose that the effective linear interaction coefficient for the wall material is µ. Then, the attenuation factor for a ray toward P from radius pis exp[-µ(Z sec'!? -acsc'I?)]. The analog of Eq. (7.67) is D w (P) m-l [ ( 1 /3 = m l dww exp -µZ � - ✓ (7.71) )] . D o ( o) 1 -w2 o This ratio, which depends on both f3 and the mean free paths µZ of wall thickness, is illustrated in Fig. 7.19. Obviously, for thinner walls and narrower ducts, the wall-penetration component can dominate the dose at the duct exit.

r0

7.5.4 Single-Scatter Wall-Reflection Component This development is only for straight cylindrical ducts, although numerical results are likely to be similar for rectangular ducts with the same cross-sectional area. A necessary approximation for this development is that the particles entering the duct at its entrance may be treated as though coming from a point source on the duct axis. Only singly reflected particles are taken into account. While this is a reason­ able approximation for gamma rays, which experience relatively very low albedos peaked in directions along the duct axis, it is not reasonable for thermal neutrons, which experience relatively very high albedos with more nearly isotropic reflection. Neutron albedos, albedos for capture gamma rays, and neutron streaming through ducts are addressed in Chapter 8. Because of the neglect of multiple reflections along the duct, the methods illustrated here underestimate transmitted doses and thus must be used with caution. The geometry and notation for duct-wall reflection are illustrated in Fig. 7.20. The equivalent point source on the axis is located at point PO at the duct entrance and dose is evaluated at point P on the axis at the other end of the duct. The source emits S(-0) monoenergetic particles per steradian, with azimuthal symmetry about the duct axis. If Jn(-0) = [(m + 1)/21r]J;; cosm '19 is the fluence at the duct entry plane, then S(-0) = 1ra2 Jn ('19) = [(m + 1)/2]a2 J;; cosm '19 = [(m + 1)/2]a2 J;;wm . In accord with Eq. (7.43), the incident flow Jno at reflecting area dA = 21radz is given by cos '!9 1 S('!9)/41rri, and since cos -0 1 = a/r1, it follows that the portion of the dose

249

Sec. 7.5. Photon Streaming

z

Figure 7.20. Geometry for evaluation of single wall reflection in a straight cylindrical duct.

m

=

1

concrete ,o -

Line-of-sight component --.,__

1

,.... 0

',....

'-'

0

,0

-2

Q. '-'

0

,o-3

--

"'

,

•

,,,-­

,,,._,,, .,,,--

-­

, ,,,,,,,,,,,, '

,/

, ,.,,,,, '

,,,,

•

------

/,, o.,

Wall-reflection component

✓

,,,--

,,,-·,,,.

,_,,,

_,,, ,

·· ,,,..-• · .--

··

,o-4 .__...c..__,____.__..CL,.L-.,__.L...J_.__,_____,___.,____._-'---'--'--'--'-' 2 ,0-

1 ,0-

Aspect ratio (a/Z) Figure 7.21. Line-of-sight and single wall-reflection component for photons inci­ dent with isotropic fluence on a straight cylindrical duct in a concrete wall. The independent variable is /3, the aspect ratio, and the parameter is the photon en­ ergy. The data points represent the multiple-reflection component for a 0.1-MeV equivalent point source at the entry of a cylindrical duct in a 2-m thick concrete wall, computed using the MCNP Monte Carlo radiation-transport computer code.

250

Special Techniques for Photons

Chap. 7

at P due to reflection from area dA is given by dD1 =

2πa2 dz RS(ϑ) αD (Eo , ϑ1 ; ϑ2 , 0). r13 r22

(7.72)

Note that all reflections leading to the dose point require zero change in azimuthal angle ψ. The total reflected dose is given by Z Z S(ϑ) D1 (P ) = 2πa2 R dz 3 2 αD (Eo , ϑ1 ; ϑ2 , 0). (7.73) r1 r2 0 A simplification is made if the variable of integration is changed from z to u ≡ z/Z and the aspect ratio a/Z is called β. The result is D1 (P ) = π(m + 1)Jn+ Rβ 4

Z 0

1

du um

(β 2

αD (Eo , ϑ1 ; ϑ2 , 0) . + u2 )(m+3)/2 [β 2 + (1 − u)2 ]

(7.74)

For m > 0, this can be expressed in terms of the dose at the source plane Do (0). + From Eq. (7.60), Do (0) = [(m + p the broadly illuminated duct en√1)/m]RJn for 2 2 trance, and since ω = cos ϑ = z/ z + a = 1/ 1 + β 2 /u2 , the ratio of the singlereflection dose at the duct exit to the dose at the center of the broadly illuminated duct entrance is, for m > 0, Z 1 D1 (P ) αD (Eo , ϑ1 ; ϑ2 , 0) 4 = πmβ , (7.75) du um 2 o D (0) (β + u2 )(m+3)/2 [β 2 + (1 − u)2 ] 0 in which ϑ1 = cot−1 [β/u] and ϑ2 = cot−1 [β/(1 − u)]. The reader will note that, for specified photon energy and wall material, the reflection component of the exit dose is a function of only the aspect ratio. Representative results are illustrated in Fig. 7.21. Even for concrete, which has higher albedos than iron or lead, the wall-reflection component of the dose is generally much less than the line-of sight component.

7.5.5

Transmission of Gamma Rays Through Two-Legged Rectangular Ducts

The treatment of radiation transmission through multiple-legged ducts of arbitrary cross section is beyond the scope of this book. Approximation techniques that might be employed in such a general case are illustrated here in the analysis of radiation transmission through two-legged rectangular ducts. Details of the analysis and refinements to account for lip and corner penetration have been described by LeDoux and Chilton [1959]. The geometry for the analysis is illustrated in Fig. 7.22. The method of analysis is applicable for ducts whose lengths are appreciably greater than widths and heights and whose walls are of uniform composition and at least one mean free path thick.

251

Sec. 7.5. Photon Streaming

s( )

P A

1

A

3

A

2

s( )

r1 '

s( )

P

r2 '

r1 '

A

P

'

1

A

2

2

r2

1

S( )

P " 2

" 2

A

3

Figure 7.22. Prime scattering areas in radiation transmission through two-legged rectangular 00 00 ducts. In the bottom figure the leg from S has length r1 with an angle ϑ1 . The leg to P has 00 length r2 .

The dose D(P ) is evaluated at the center of the duct exit. Radiation incident on the duct is approximated as though it emerged from an anisotropic source S(ϑ) at the center of the duct entrance. For example, if the axisymmetric angular flow at the duct entrance plane is Jn (ϑ) and the cross-sectional area of the duct entrance is A, S(ϑ) = AJn (ϑ). A monoenergetic source is assumed, although generalization of the method to polyenergetic sources is straightforward.

252

Special Techniques for Photons

Chap. 7

The method of LeDoux and Chilton is based on the approximation that D(P) consists principally of responses to radiation reflected from prime scattering areas, that is, areas on the duct walls visible to both source and detector and from which radiation may reach the detector after only a single reflection. Four prime scattering areas may be identified in Fig. 7.22: areas A1 and A 2 on the walls, and areas A3 and A4 on the floor and ceiling (considering the figure to be a plan view). Reflection from each area is treated as though it occurred from a single point at or near the centroid of the area. Thus, the detector response may be expressed as

(7.76) According to Eq. (7.43), Di(P) = '.RA1S(7r /2- -OU cos '!9 �0:v(E0 , '!9 �; 0 , 0) , (r1r2)2

(7.77) (7.78)

D (P) = '.RA3 S(7r /2- -ancos '!9 �0:v(E0 , '!9 �; '!9 �, 7r /2) (7.79) 3 ' (r111 r2")2 in which the various arguments of the albedo function o:v are as illustrated in Fig. 7.14.

7.6

SHIELD HETEROGENEITIES

Occasionally, an analyst encounters a shield that includes regions of composition different from the bulk shield material. These regions may be large with a well-defined geometry, such as embedded pipes or instrumentation channels transverse to the direction of radiation penetration. By contrast, there may be incorporated into the shield material small irregularly shaped and randomly distributed voids or lumps of other material, such as pieces of scrap iron used to increase the effectiveness of a concrete shield against gamma radiation. Rigorous calculation of the effect of such heterogeneous re­ gions in a shield usually requires Monte Carlo tech­ niques (see Chapter 11). In this section, some sim­ plified techniques, based on ray theory, illustrate the effect of shield heterogeneities. Consider two rays through a shield (see Fig. 7.23) containing ran­ domly distributed small voids in a continuous phase which has an effective linear attenuation coefficient

0 0

-----101-c--

0 2

0

◊ ◊ 0 0 □ ◊ ◊ 0 0

□

oo □

Figure 7.23. Shield containing randomly distributed voids.

253

Sec. 7.6. Shield Heterogeneities

µ. Ray 1 travels a distance t with transmission probability T1 = e-µ,t . Ray 2 trav­ els a distance t - 8 with transmission probability T2 = e-µ,(t-6). The average path length for these two rays is f = t - 8/2, and the average transmission probability is

(7.80) Thus, it is seen that use of the average path length of a ray through the shield ma­ terial underpredicts the average transmission probability. If the void is replaced by a material with an attenuation coefficient different from that of the shield material, a similar analysis shows that use of the average path length (in mean free paths) also underestimates the average transmission probability. This effect is known as channeling and its neglect leads to an overestimation of the effectiveness of a shield. Channeling is seen in shields with randomly included heterogeneities as well as in shields with well-defined placement of voids and heterogeneous regions.

7.6.1

Limiting Case for Small Discontinuities

Suppose in Eq. (7.80) that 8 < < µ-1 , that is, the lumps or voids are much smaller in size than the radiation relaxation length in the continuous phase. In this case the channeling effect is negligible and shield transmission factors may be estimated using an average mass attenuation coefficient µ/p. Suppose that µ, p, and w are the effective linear attenuation coefficient, density, and weight fraction of the continuous phase, and µ', p', and w' are the same quantities for the discontinuous phase. If v is the volume fraction of the voids or lumps in the shield and pis the average shield density, the average mass attenuation coefficient is (7.81) and the transmission probability is thus T(t)

= e-(µ,/ p)pt = e-[(1-v)µ,+vµ,'] t = e-µ,te-(µ,'-µ,)vt.

(7.82)

This transmission probability is the same as if the shield material (continuous and discontinuous components) were conceptually homogenized and the average atten­ uation coefficient for the homogeneous mixture were used.

7.6.2

Small Randomly Distributed Discontinuities

Channeling effects have been treated by a statistical technique attributed to Cov­ eyou [Burrus 1968a]. Consider an infinite slab shield of thickness t uniformly and normally illuminated on one side by radiation. The shield contains randomly dis­ tributed lumps with a different linear attenuation coefficient (µ') from that of the shield material (µ). Based on ray theory, the transmission probability is T(t)

t

= 1 dx P(x, t)e-µ,'xe-µ,(t-x),

(7.83)

254

Special Techniques for Photons

Chap. 7

where P(x,t) dx is the probability that an incident ray passing through the shield of thickness t will also pass though a thickness between x and x + dx of the discon­ tinuous material. The probability distribution function P(x,t) can be calculated rigorously for special cases. For example, consider cubic lumps of size 8 all oriented with a face parallel to the shield surfaces, so that a ray hitting a lump must pass through µ'8 mean free paths of the lump material. Further assume that the lumps occupy a volume fraction v in the shield. Now conceptually divide the shield into N contiguous subslabs each of thickness 8 (i.e., t = N8). Let P(n, N) denote the probability that a penetrating ray passes through n lumps while traversing the N subslabs. This probability function is unchanged if the lumps are shifted left or right slightly so that each lies totally within the subslab in which it mostly lies. Then, since v is the probability that a ray encounters a lump while traversing any one subslab, P(n,N) is given by the binomial distribution P(n,N) = Binomial(n,N) =

!

� ' vn (l-v) N-n _ (N n).n.1

(7.84)

The transmission probability of Eq. (7.83) becomes for this discrete case

T (t )

= L P(n,N) e-µ' n8 e-µ( N-n)8 N

n=O

..f-.,

N! -v- -(µ'-µ)o] _ (l-v) N e-µ N8 L..-(N-n) ! n! [l-v e n=O

n

(7.85) This result can be written in the form of the limiting homogeneous case of Eq. (7.82), namely (7.86) Here, c0 is the cross-section effectiveness ratio (see Fig. 7.24) and is obtained by equating Eq. (7.86) to Eq. (7.85), namely,

= Co

(µ' - µ)v8 ln [(l-v) + ve-(µ'-µ)8] ·

(7.87)

Equation (7.86) can also be used for small inclusions that present a varying thickness 8 to the incident radiation by interpreting 8 to be the mean chord length

255

Sec. 7.6. Shield Heterogeneities

through the lump. For an arbitrary convex lump the mean chord length is just four times the volume/surface area ratio. For concave and irregularly shaped lumps, the value of δ must be determined by specific calculation or by measurement of slices of the shield material. Other methods for using the Coveyou approach for irregularly shaped lumps are discussed by Burrus [1968a].

µ µ δ

Figure 7.24. Cross-section effectiveness ratio co for various volume fractions v of a heterogeneous phase each lump of which presents a thickness δ to incident radiation.

7.6.3

Large Well-Defined Heterogeneities

For a large well-defined heterogeneity in the shield (most often a void or region of low interaction coefficient), ray theory can again be used effectively. In general, if Sv (r0 ) is the volumetric source strength of radiation at r0 , the dose at some other point r in the shield is given by Z D(r) = dV 0 Sv (r0 )G(r0 , r). (7.88) V

If ray-theory is applicable, the dose kernel G depends only on the material properties along a straight line between r and r0 . For monoenergetic photons of energy Eo , for example, one often uses [see Eq. (6.81)] G(r0 , r) =

R B(`)e−` . 4π|r − r0 |2

(7.89)

Here ` is the optical thickness between r and r0 , given by the path integral Z |r−r0 | ds µ(Eo , s), (7.90) `= 0

256

Special Techniques for Photons

Chap. 7

in which s is measured along a line from r to r'. For simple geometry (e.g., a spherical or cylindrical void in a slab shield) Eq. (7.88) can be evaluated analyti­ cally. Otherwise, numerical integration techniques are used. Examples are given by Rockwell [1956], Burrus [1968b], and Chilton, Shultis, and Faw [1984].

7.7

GAMMA-RAY SKYSHINE

In many facilities with intense localized sources of radiation, the shielding against radiation that is directed skyward is usually far less than that for the radiation emitted laterally. However, the radiation emitted vertically into the air undergoes scattering interactions and some radiation is reflected back to the ground, often at distances far from the original source. This atmospherically reflected radiation, referred to as skyshine, is of concern both to workers at a facility and to the general population outside the facility site. A rigorous treatment of the skyshine problem requires the use of computa­ tionally expensive methods based on multidimensional transport theory. Al­ ternatively, several approximate proce­ dures have been developed for both gamma-photon and neutron skyshine sources. See Shultis, Faw, and Bas­ set [1991] for a review. This section summarizes one approximate method, 0 which has been found useful for bare or shielded gamma-ray skyshine sources. 1 ao · 2500 This method, termed the integral linebeam skyshine method, is based on the availability of a line-beam response function �(E, 1(0) exp(-k1x '), the result above reduces to St1i(x )

=

k

cf> ) i /O [err(��+ k1 ✓,;;:-)

- erf(k1 � -

2�)]

where the error function, erf(x), is defined by erf(x)

=

2

r

l V7f o

e-Y dy. 2

e-k,(x-k,r,h ), (8.49) (8.50)

The error function approaches +1 (or -1) as its argument becomes a large positive (or negative) number. If the shield thickness T :» -fi;,:, then for x far removed from either surface of the shield, the argument of the first error function in Eq. (8.49) is positive and large, while the argument of the second error function is large and negative. Thus, (8.51)

If the thermal neutrons are absorbed near the point at which they reach thermal energies, then under steady conditions the number absorbed, µ acf>th(x), must equal the number thermalized, St1i(x ). Thus, from Eq. (8.51), cf>t1i(x )

k '.'.:::'. J cf>1(0) exp[-k1(x - k gh )] µa (8.52)

Sec. 8.5. Calculation of the Intermediate and Thermal Fluences

295

This result implies that inside the shield the thermal neutron fluence is proportional to the fast-neutron fluence displaced toward the source by a displacement distance krr,h . The thermal neutron fluence inside a shield can thus be expected to parallel the fast-neutron fluence-a result usually observed (see, e.g., Fig. 8.1). Age theory can also be used to estimate the energy-dependent fluence (x, E) of neutrons with intermediate energies [Price, Horton, and Spinney 1957]. The volumetric rate at which fast neutrons slow down past energy E [or equivalently, reach age r(E)] deep in a shield is q (x, E )

=

(' [ d f ( x')] ( ' ( ) ) ' q x, x , r E dx . -

Jo

dx '

(8.53)

If the fast-neutron fluence is again assumed to be attenuated exponentially in the then following the same shield, that is, 1 (x ') = 1 (0) exp(-k1x '), and if T reasoning used to obtain Eq. (8.51), one finds

» .Ji,

(8.54) In an infinite, nonabsorbing medium, the fluence (E) is related to the slowing­ down density q(E) by [Lamarsh 1966]

(8.55) so that for a thick neutron shield (with negligible neutron absorption above thermal energies), the intermediate-energy-neutron fluence can be described approximately by k1 1 (0) exp[-k1 (x - k1r(E))]. (x , E) = (8.56) �(E)µs(E) E This result can be expressed in terms of the fast-neutron fluence as (8.57) where B(E) is an effective buildup factor for the intermediate-energy neutrons, given by

(8.58) It should be noted that this intermediate-energy-neutron buildup factor is indepen­ dent of position.

8.5.3

Removal-Diffusion Techniques

Although the results of the preceding section are adequate for initial estimates, more accurate techniques are often needed without the effort and expense of a full-scale multigroup transport calculation. Multigroup diffusion theory, which is considerably less expensive and complex to use than transport theory, is remarkably

296

Special Techniques for Neutrons

Chap. 8

successful at describing the slowing down and thermalization of neutrons in a reac­ tor core. However, for describing neutrons deep within a shield, it met with only limited success [Taylor 1951], although better accuracy was obtained by introducing extraneous renormalization techniques to describe the penetration of the fast neu­ trons [Haffner 1968; Anderson and Shure 1960]. That strict diffusion models should be of limited use to describe fast-neutron penetration and subsequent thermaliza­ tion is not surprising since diffusion theory requires both the differential scattering cross sections and the angular fluence to be well described by first-order Legendre expansions (i.e., at most, to vary linearly with w). Such conditions usually hold in a reactor core where the neutron fluence is approximately isotropic; however, the flu­ ence deep within a shield is determined by those very energetic neutrons which are highly penetrating and whose angular distribution is therefore highly anisotropic. The penetrating fast neutrons are described very successfully by removal the­ ory. The migration of the neutrons, once they are removed from the anisotropic fast group and begin to thermalize, is small compared to the distance traveled by the unremoved neutrons. Further, during thermalization, the fluence becomes more isotropic as more scatters occur. Consequently, one would expect multigroup dif­ fusion theory to be applicable for the description of the slowing-down process and the subsequent diffusion at thermal energies. Thus, one approach to compute the buildup of low-energy neutrons inside a shield is to combine removal theory (to describe the penetration of fast neutrons) with multigroup diffusion theory (to de­ scribe the subsequent thermalization and thermal diffusion). This combination of removal and diffusion theory, in many formulations, has proved very successful. With the recent advent of inexpensive and ubiquitous computing power un­ dreamed of a decade ago, the use of removal-diffusion theory has waned and trans­ port theory codes are now almost universally used in place of removal-diffusion codes. However, we include this section, not only for historical completeness, but for the insight it affords the analyst on how fast neutrons migrate through a shield.

Original Spinney Method The first wedding of removal theory to diffusion theory was introduced by Spinney in 1958 [Avery et al. 1960]. In the original formulation the fast source region, 0 to 18 MeV, is divided into 18 equal-width energy bands. The source neutrons in each band penetrate the shield in accordance with removal theory. The density of removal collisions from all bands is then used as the source of neutrons in the first diffusion group. Explicitly, this diffusion source density at r is given by Sd(r)

=

tr i =l

lv

Sv(r')x i µr, i exp(-µr, i lr - r'I) d V(r'). 41rlr' - rl 2

(8.59)

where Sv (r') is the production of source neutrons per unit volume at r' in the source region, X i is the fraction of source neutrons in the ith removal band, and µr, i (r) is the removal coefficient for the ith band at position r. The term µr, i lr - r'I is the total number of removal relaxation lengths between r and r' for a fast neutron in the ith band.

297

Sec. 8.5. Calculation of the Intermediate and Thermal Fluences

These removal neutrons are inserted as source neutrons into the top energy group of five energy groups, with the fifth group representing the thermal neutrons. The transfer of neutrons from diffusion group to diffusion group is determined by Fermi age theory [Lamarsh 1 966], a continuous slowing-down model, and consequently neutrons can be transferred only to the energy group directly below. Thus, the diffusion group equations are written as (8.60) where �i is the fl.uence for group i, µa ,i is the linear absorption coefficient for group i, Di is the ith group diffusion coefficient, and Ti is the square of the slowing-down length from group i to the next lower group i + 1, or equivalently, the Fermi age of neutrons starting from group i and slowing down to group i + 1 (for the thermal group, Ti- 1 = 0). The source term for the ith diffusion group is then given by S·(x) •

=

{

Sd(r) from Eq. (8.59), i 1

= 1,

i>l.

Di - 1 Ti-=.. 1 �i -1(r),

(8.61 )

The group constants are obtained by averaging over the energy range of each group (Ei + 1 ,Ei ) with a 1 /E fl.uence weighting for the slowing-down groups (i = 1,..., 4) and with a Maxwellian distribution for the thermal group (i = 5). Explicitly, for the slowing-down groups, the group constants are given by

. _ jE;

µa ,,-

/JE;

µa (E) --dE E,+1 E

dE E;+1 E

(8.62)

I

[ E; dE [ E; dE (8.63) }E,=l 3[µin (E) + µes(E)(l - Wo )]E }E;+i E and Ti is the Fermi age from Ei to Ei +l, namely, [ E; dE Ti = (8.64) E E µi µ ( ( )(l - Wo )]µi(E),(E)E ' + ) es }E,=i 3[ n where w 0 is the mean cosine of the laboratory scattering angle for elastic scattering, and µa , µin , µes, and µt are, respectively, absorption, inelastic scattering, elastic scattering, and total interaction coefficients. With the assumptions that elastic scattering isotropic and that inelastic scattering takes place from only a single energy level, E*, the mean logarithmic energy decrement per scatter, ,(E), is given by Di

=

(8.65) where 'es=

{ 2

A+ 2/ 3' 1,

A>l, A= 1.

(8.66)

298

Special Techniques for Neutrons

Chap. 8

Improved Removal-Diffusion Models The original Spinney method, just described, was quite successful in predicting the low-energy neutron fluences in the concrete shields around early graphite reactors. However, to obtain better accuracy for a wider range of shield configurations, several obvious improvements could be made. First, more diffusion groups could be used to describe better the continuous slowing-down model implied by Fermi age theory. Second, neutrons should be allowed to transfer past intermediate diffusion groups in a single step to account for the possibility of large energy losses in inelastic scattering or elastic scattering from light nuclei. Third, more detail should be given for the removal of fast neutrons from the removal bands to the diffusion groups. Fast­ neutron diffusion cannot be neglected altogether, and hence, the upper diffusion groups should overlap the same energy region spanned by the lower-energy removal bands. Further, when neutrons suffer a removal interaction, they should be allowed to enter any one of several diffusion groups, depending on the severity of the removal interaction. This improved description of the removed neutrons would give more information about the fast-neutron fluence, an important consideration for radiation damage studies. Shortly after the introduction of the Spinney method, several variations of it were introduced which implement some or all of the improvements described above [Bendall 1962; Peterson 1962; Hjarne 1964]. The removal band and diffusion group structure as well as the allowed neutron transfers in three of these improved schemes are illustrated in Fig. 8.10. Perhaps the most refined of the removal-diffusion models in the variation used is the NRN code [Hjarne 1964]. This version has the most energy detail (30 removal bands and 24 diffusion groups) as well as the most flexibility in describing the transfer of neutrons between energy bands and groups. In this formulation, the ith-group diffusion equation is D/v 2 0) [Chilton, Shultis, and Faw 1984]. The result above holds for cylindrical ducts with a small aspect ratio a/Z. For larger ratios the importance of the infinite number of internal reflections implied by Eq. (8.101) becomes less. Artigas and Hungerford [1969] have produced a more complicated version of Eq. (8.102), which gives better results for a/Z > 0.3 [Selph 1973].

8.9.2

Ducts with Bends

To decrease the amount of radiation reaching the duct exit, shield designers often put one or more bends in the duct, and consequently, calculating the effect of bends is an important task for the designer. This is a difficult task to do accurately. How­ ever, a few simplified techniques are available for estimating transmitted neutron doses through bent ducts. Albedo methods are widely used for treating neutron streaming through bent ducts. These range from simple analytical models, such as those presented in this section, to elaborate Monte Carlo methods that use albe­ dos to reflect neutrons from duct walls and thereby allow them to penetrate large distances along the duct.

Figure 8.17. Geometry for the two-legged duct model.

Two-Legged Ducts Consider first the two-legged cylindrical duct of radius a shown in Fig. 8.17. The two legs are bent at an angle {J such that neutrons emitted from the source plane across the duct entrance at A cannot stream directly to the duct exit at C. Both legs are assumed to have small aspect ratios a/ £1 < < 1 and a/ L2 < < 1. The uniform source plane emits neutrons into the duct with a general cosine fl.ow distribution

321

Sec. 8.9. Duct Streaming for Neutrons

J;Jt/J) = (m+ l)J;;cosm 'l/;/(27r), where 'I/; is the angle with respect to the normal to the source plane. The uncollided dose on the duct centerline at the duct bend B arising from the disk source at the duct entrance is given by Eq. (6.36) for m= 0, or by Eq. (6.47) for m > 0. For a/ L1 0. The slab thickness tis also very large, so that for all practical purposes the slab can be considered to be infinitely thick. Derive a formula for the absorbed dose at the front surface of the shield (x = 0) due to the secondary gamma photons produced in the shield. 17. In a 40-cm-thick iron slab shield, the thermal neutron fluence (cm- 2) varies approximately as

Problems

331

where x is the distance in centimeters from the front face of the shield. Calculate at the exit surface of the slab the exposure arising from the production of capture gamma photons in the shield. Perform your calculations first for the uncollided photon component and then for the total detector response. 18. A 1-mg source of 252 Cf is placed in a large water-filled tank. Estimate, as a function of distance from this source, the absorbed dose rate in tissue due to (a) fast neutrons, (b) thermal neutrons, (c) primary gamma photons, (d) capture gamma photons. For photons, the absorbed dose in tissue can be approximated by the absorbed dose in water. 19. A 100-keV neutron beam uniformly and normally illuminates one side of a 100cm-thick concrete shield composed of NBS Type 04 ordinary concrete with 5.5% water content. Estimate the percentage change in the neutron plus secondary gamma-ray dose on the other surface of the shield for each of the following changes: (a) the water content is decreased to 3%, (b) the shield thickness increases to 150 cm, (c) the neutron beam is incident at 45◦ , (d) the limestone aggregate is replaced by a quartz-sand and aggregate mixture, and (e) the neutron energy is increased to 10 MeV. 20. Consider the occluded source problem shown in Fig. 7.27 and specified in Problem 14, but with the gamma-ray source replaced by the following two neutron sources. Specifically, estimate the ambient dose equivalent at the detector for a point isotropic source emitting 109 neutrons with energy of (a) 14 MeV, and (b) 2 MeV. 21. An infinitely-broad parallel beam of neutrons is normally incident on a halfspace of concrete. The incident fluence is denoted by Φo and differential and total number albedos by αN and AN , respectively. The reflected fluence Φr at a normal distance h outside the concrete. (a) Prove that an assumption of an isotropic albedo (i.e., αN = AD /2π) leads to an infinite result for the reflected fluence. (This leads to a suspicion that a completely isotropic differential albedo is unrealistic.) (b) Assume a cosine distribution for αN (i.e., αN = AN cos ϑ/π) and find the reflected fluence at the detector. Show that this result is independent of the detector distance h. 22. Consider the general result for wall reflection in a straight cylindrical duct given by Eq. (8.98). Different portions of the wall contribute to the dose at the duct exit in varying degrees. (a) Show that for an isotropic incident flow (m = 0) and an isotropic albedo (γ = 1) the single-scatter contribution comes from portions of the duct wall near the source plane.

332

Special Techniques for Neutrons

Chap. 8

= 0) and a cosine albedo ('y = 0) the single-scatter contribution comes equally from portions of the duct wall near the source plane and near the exit.

(b) Show that for an isotropic incident fl.ow (m

(c) Show that for an isotropic incident :fl.uence (m = 1) and a cosine albedo ('y = 0) the single-scatter contribution comes predominantly from portions of the duct wall near the exit. 23. A vertical cylindrical channel above an experimental reactor core is unplugged, allowing an intense beam of gamma and neutron radiation to stream into the atmosphere. Estimate the resulting skyshine dose as a function of distance from the reactor (normalized to a single emitted particle) for (a) a 0.1-MeV photon, (b) a 1-MeV photon, (c) a 0.1-MeV neutron, and (d) a 1-MeV neutron.

Chapter 9

Special Techniques for Charged Particles

9.1

INTRODUCTION

This chapter deals with evaluation of the spatial distribution of absorbed dose aris­ ing from electron, beta-particle, and other charged-particle sources in various ge­ ometries. The analytical treatment of electron attenuation is distinctly different from that for neutrons and gamma photons. The indirectly ionizing neutrons and gamma photons travel along paths composed of straight-line segments punctuated by distinct interactions with atoms or electrons in matter, these interactions being governed by short-range forces. Consequently, the particles have no identifiable "ranges" and their trajectories are best described in terms of concepts such as the interaction coefficient or the mean free path. By contrast, directly ionizing charged particles are attenuated by an almost continuous slowing-down process governed largely by long-range Coulombic forces. While the electron range is not defined precisely, the spatial distribution of the energy deposition around a source of elec­ trons may be described precisely in terms of a measure of range called the CSDA range, that is, the range in the continuous slowing-down approximation. The point-kernel concept introduced for neutrons and gamma rays is equally valid for charged particles. Moreover, for many electron dosimetry problems, the effects of boundaries or material heterogeneities can be neglected, and the use of infinite-medium point-kernel techniques to compute the dose received in one re­ gion from an electron source in another results in the dose being expressible as a one-dimensional integral involving an appropriate geometry factor as described in Chapter 6.

9.2 ALPHA AND BETA DECAY 9.2.1 Alpha Decay Radioactive decay by alpha-particle emission is possible only in the heavier el­ ements. One radionuclide may have several transitions involving alpha-particle emission; but for each transition, the particles are monoenergetic and usually of 333

334

Special Techniques for Charged Particles

Chap. 9

Table 9.1. Half-lives of alpha-particle emitters; energies and frequencies of particles emitted, based on the EDISTR code. Nuclide

Half-life

E (MeV)

Percent

2 1 4p0 2 1 8p0 2 1 2Bi

164.3 µs 3.05 m 60.55 m

2 10 p0 226Ra

138.4 d 1600 y

23su

4.5x109 y

232Th

1.4x10 10 y

7.687 6.002 6.051 6.090 5.305 4.602 4.785 4.147 4.196 3.953 4.010

100 100 25.2 9.6 100 5.5 94.4 23 77 23 77

Source: Dillman [1980].

several MeV energy. The Geiger-Nuttall rule [Evans 1955] accounts for the strong inverse relationship between the half-life of a radionuclide and the energy of its alpha particle. Table 9.1, which lists several important alpha-particle emitters, illustrates this inverse relationship.

9.2.2

Beta Decay

Beta decay of a radionuclide, in effect, involves either (1) a transformation within the nucleus of a neutron into a proton accompanied by the emission of an electron, or beta particle, and an antineutrino, or (2) a nuclear transformation of a proton in the decaying nucleus to a neutron accompanied by emission of a positron and a neutrino. The energy released in the process-the Q value-is shared by the electron and the antineutrino or by the positron and the neutrino. Beta decay of a nucleus may occur by any of several different nuclear transitions. For each there is a unique endpoint energy, the maximum kinetic energy of the electron or positron. Also, for each transition, there is a unique statistical distribution of energy between the electron and antineutrino or positron and neutrino, and to a very minor extent the residual nucleus. For the balance of this discussion, we deal with the more common beta-particle emission, with the recognition that positron emission may be treated very similarly. While the energy spectrum of the beta particles for any one transition certainly depends on the endpoint energy, the spectrum also depends in a very complicated way on the nuclear spin and parity quantum numbers of the nucleus before and after beta-particle emission. These quantum numbers determine whether the transition is referred to as allowed or forbidden, and if forbidden, whether the transition is unique (U) or nonunique (non-U). In a transition, the spin quantum number J may remain

335

Sec. 9.2. Alpha and Beta Decay

constant or it may change by one or more units, that is, l:l.J = 0, 1, 2, .... The parity quantum number II may or may not change, that is, l:l.II = -1 (no change) or +1 (change). To a first approximation, a single classification index n may be used to characterize the energy spectrum. For l:l.J 2'.: 2, n = l:l.J - 1. Otherwise, the various combinations of l:l.II and l:l.J establish the following selection rules, which determine the shape factor, which, in part, governs the beta-particle energy spectrum for a particular transition.

l:l.J

l:l.II

Transition classification

n

0,1 0,1 2 2 3

-1 +1 +1 -1 -1

Allowed Nonunique, first-forbidden Unique, first-forbidden Nonunique, second-forbidden Unique, second-forbidden

0 0 1 1 2

As an example, consider the decay of 38Cl to 38 Ar. Three beta transitions are possible, with the following characteristics: Frequency

(%)

32.5 11.5 56.0

Emax (keV)

Eavg (keV)

l:l.J

l:l.II

n

Classification

1107 2749 4917

420 1182 2244

1 0 2

-1 +1 +1

0 0 1

Allowed Non-U, first forbidden Unique, first forbidden

The energy spectrum of beta particles arising from a transition is conveniently expressed in terms of the electron total energy W (including rest-mass energy), that is, W = E +m e c2, and the momentum p = JE2 + 2m e c2E / c . The energy spectrum Ni(E) for transition i is defined in such a way that Ni(E)dE is the probability that, in a transformation of the radionuclide via transition i, a beta particle is emitted with kinetic energy in the range dE about E. This spectrum can be expressed as

in which Ema:x ,i represents the maximum (or endpoint) energy, Z and A are the charge and mass numbers of the nucleus after the transition, F is the Fermi function, Sn is a shape factor determined by the selection-rule classification index n, and Ci is a normalization constant. The product p W [Ema:x ,i - E]2 is a statistical factor and is a major determinant of the energy spectrum. The shape factor Sn is unity for allowed and nonunique first-forbidden transitions; for other transitions it is a complicated function. Suffice to say here that it is an extremely tedious task to evaluate the Fermi function and the shape factor. Procedures are clearly described by Dillman and Von der Lage [197 5] and by Dillman [1980], as are procedures required to account for screening by

336

Special Techniques for Charged Particles

0.6

.c'> OJ

e w

'z' e

.bu

(' 0.4 II \\ I

OJ

� >. OJ C:

w

0.2

Composite

'z'

2.0

u

1.5

�

1.0

>.

OJ C:

// I

/.,.. .,..-1--..... I

0

2.5

OJ

..----

3.0

e OJ

,I,/

0.1 0.0

7

Emax = 2.749 {11.5%) Emax = 4.917 {56.0%)

I I I I I I

0.3

3.5

38CI Beta Particles Emax = 1.107 {32.5%)

0.5

Chap. 9

90sr

0.50

', ..... 2

3

5

4

1.0

0.50

Energy (MeV)

1.5

2.0

2.5

Energy {MeV)

Figure 9.1. Energy spectra of 38CI beta particles.

Figure 9.2. Energy spectra of selected beta­ particle sources.

atomic electrons, an effect that is especially important for positron decay. Figure 9.1 illustrates individual and composite spectra for the three transitions involved in the beta decay of 38 Cl. In Fig. 9.2 the spectra for four important beta-emitting radionuclides are given. If Ii is the frequency for the ith transition, then the spectrum normalization constant Ci is determined by the requirement that (9.2) The average beta-particle energy for a particular transition is (Ei )

=

1 Ii

/

lo

Emax, i

dE ENi(E).

(9.3)

The average beta-particle energy released per transformation of the radionuclide is given by :E i Ii (Ei )-

9.3

SPATIAL DISTRIBUTION OF THE ABSORBED DOSE

This section deals with point and uniform-beam sources of monoenergetic charged particles in infinite homogeneous media. In particular, spatial distributions of the absorbed dose are expressed in terms of point and plane kernels and associated scaled dimensionless dose distributions.

9.3.1

Point and Plane Kernels Defined

The expected absorbed dose at distance r from a unit point isotropic source of monoenergetic charged particles of energy E is denoted as Q(r, E). Typical units

337

Sec. 9.3. Spatial Distribution of the Absorbed Dose

Table 9.2. Electron CSDA mass range f0 water.

E (MeV) 0.0005 0.0006 0.0008 0.0010 0.0015 0.002 0.003 0.004 0.005 0.006 0.008 0.010 0.015 0.02 0.03 0.04 0.05 0.06

=

E (MeV)

fo (g/cm 2 ) 2.27 X 10-6 2.90 X 10-6 4.33 X 10-6 5.98 X 10-6 1.09 X 10-5 1.71 X 10-5 3.28 X 10-5 5.27 X 10-5 7.65 X 10-5 1.04 X 10-4 1.69 X 10-4 2.52 X 10-4 5.15 X 10-4 8.57 X 10-4 1.76 X 10-3 2.92 X 10-3 4.32 X 10-3 5.94 X 10-3

pr 0 in liquid

fo (g/cm 2 ) 9.77 X 10-3

0.08 0.10 0.15 0.2 0.3 0.4 0.5 0.6 0.8 1.0 1.5 2 3 4 5 6 8 10

1.43 2.82 4.49 8.42 1.29 1.77 2.27 3.30 4.37 7.08 9.79 1.51 2.04 2.55 3.05 4.03 4.98

X X X X X X X X X X X

10- 2 10- 2 10- 2 10- 2 10-l 10-l 10-l 10-l 10-l 10-l 10-l

Source: Berger [1992] for E � 0.01 MeV, and Berger

[1973] for E < 0.01 MeV.

for g are MeV / g. The point kernel g is very conveniently expressed in terms of the dimensionless dose distribution F(r / r O, E), with radial distance r scaled by the CSDA range r O, namely, Q(r,E)

=

!

47r:T

pr 0

F(r/ro,E)

=

47r:T

�A

r0

F(f/f 0 ,E),

(9.4)

=

in which f pr and f0 is the CSDA range of Eq. (3.105), in mass units, as given in Table 9.2. A useful interpretation of F, which is illustrated in Fig. 9.3, is that the fraction of E deposited between radii r and r + dr is (dr/r 0 )F (r/r 0 , E). Note that, without a subscript, F and g refer to a point isotropic source. Similarly, for an infinite plane perpendicular 1 source of monoenergetic particles in an infinite homogeneous medium, the expected absorbed dose at distance r along the beam is denoted as 9;z (r,E). This is conveniently expressed in terms of a scaled dimensionless dose distribution as .L 9p1 (r,E)

.L

E F 1 (r/r =pr p o

0

,E),

{9.5)

which is illustrated in Fig. 9.4. 1The source is an area source with all particles released in a parallel beam normal to the source area.

338

Special Techniques for Charged Particles

Figure 9.3. Scaled dimensionless dose distributions, F (r/ro , E), for point isotropic electron sources in water. From Cross, Freedman, and Wong [1992].

Chap. 9

Figure 9.4. Comparison of F (r/ro , E) ⊥ (r/r , E) parallel-beampoint-source, Fpl o source, and CSDA scaled dimensionless dose distributions for 2-MeV electron sources in water. Data for the point and parallel-beam sources from Cross, Freedman, and Wong [1992].

Dose Distribution in the CSDA Approximation In the continuous slowing-down approximation, all particle tracks are straight and of length ro . It is easily shown that in this approximation, 

⊥ Fpl (r, E)



CSDA

= [F(r, E)]CSDA =

L(ro − r) , E/ro

r < ro .

(9.6)

Note that the linear energy transfer L is evaluated at the energy of the particle after passage through distance r, that is, for a particle with residual range ro − r. The residual range2 determines both the energy and the LET of a particle. The denominator of Eq. (3.105) is just the average LET along the entire path of the particle. Point and plane dose distributions are compared with the distribution in the CSDA approximation in Fig. 9.4.

9.3.2

Electron and Beta-Particle Dose Distributions

The pioneering moments-method electron-transport calculations of Spencer [1959] were used by Berger [1971] and Cross et al. [1982] to construct scaled dimensionless dose distributions for monoenergetic electrons and beta-particle sources. These were revised by Berger in 1973, based on Monte Carlo calculations. Cross, Freedman, and Wong [1992] also used Monte Carlo methods to prepare comprehensive tables of dose distributions for point, plane parallel, and plane isotropic sources of electrons and 2 Evaluation

of the LET and application of the residual range concept are discussed in Chapter 3.

Sec. 9.3. Spatial Distribution of the Absorbed Dose

50keV Figure 9.5. Tracks of 30 electrons from a 50keV point isotropic source in water, shown as orthographic projections of tracks into a sin­ gle plane. The box has dimensions 2r0 X 2r0 • Calculations performed using the EGS4 code, courtesy of Robert Stewart, Kansas State University.

339

1 MeV Figure 9.6. Tracks of 30 electrons from a 1MeV point isotropic source in water, shown as orthographic projections of tracks into a sin­ gle plane. The box has dimensions 2ro X 2ro. Calculations performed using the EGS4 code, courtesy of Robert Stewart, Kansas State University.

beta particles in water. Example dose distributions for point sources are reproduced in Table 9.3 and illustrated in Fig. 9.4. Figure 9.4 compares kernels for 2-MeV point and plane parallel electron sources in water. In contrast, the point kernel based on the CSDA approximation is shown to be a very poor approximation for electrons. The reason is apparent from exami­ nation of Figs. 9.5 and 9.6, which show the spatial distributions of energy deposition from point isotropic sources of monoenergetic electrons. Not only is much of the energy deposited well within distance r0 , but the spatial distribution of the energy distribution scaled by r0 is insensitive to E. The scaled patterns of Figs. 9.5 and 9.6 for 50-keV and 1-MeV electrons reinforce the logic for the similarity in shapes of the curves in Fig. 9.3. Kernels in Non-Aqueous Media It has long been observed [Cross 1967; Berger 1971) that when distance is measured in mass thickness f = pr, the point kernel in medium mat distance f is proportional to that in water, medium w, at a scaled distance 'f/T. The scale factor 'f/ was found to depend on medium m but only very weakly on the energy, so that the same scale factor could be applied to beta-particle sources as well as monoenergetic electron sources. Conservation of energy for a point source requires that

340

Special Techniques for Charged Particles

Chap. 9

Table 9.3. Dimensionless dose distribution :F(r/r0 , E) at distance r from a point isotropic source of monoenergetic electrons of energy E in water. Distances are given relative to r0 , the CSDA range. E (MeV) r/ro

0.025

0.050

0.100

0.200

0.400

0.700

1.000

2.000

4.000

0.000 0.025 0.050 0.075 0.100 0.125 0.150 0.175 0.200 0.225 0.250 0.275 0.300 0.325 0.350 0.375 0.400 0.425 0.450 0.475 0.500 0.525 0.550 0.575 0.600 0.625 0.650 0.675 0.700 0.725 0.750 0.775 0.800 0.825 0.850 0.875 0.900 0.925 0.950 0.975 1.000 1.025 1.050 1.075 1.100 1.125 1.150

0.564 0.580 0.599 0.620 0.643 0.670 0.703 0.740 0.780 0.824 0.870 0.921 0.980 1.043 1.107 1.175 1.251 1.329 1.403 1.471 1.530 1.578 1.612 1.632 1.639 1.625 1.578 1.505 1.404 1.289 1.173 1.050 0.916 0.780 0.648 0.523 0.409 0.308 0.227 0.162 0.114 0.079 0.054 0.035 0.020 0.004 0.000

0.570 0.587 0.604 0.623 0.645 0.670 0.701 0.734 0.767 0.805 0.853 0.900 0.952 1.005 1.068 1.138 1.212 1.289 1.373 1.453 1.517 1.569 1.612 1.642 1.661 1.659 1.625 1.565 1.483 1.378 1.246 1.102 0.958 0.812 0.661 0.518 0.398 0.296 0.211 0.144 0.094 0.059 0.035 0.020 0.011 0.005 0.000

0.589 0.600 0.613 0.632 0.656 0.683 0.712 0.744 0.777 0.815 0.861 0.906 0.957 1.007 1.068 1.134 1.202 1.271 1.342 1.411 1.478 1.540 1.594 1.635 1.662 1.668 1.641 1.588 1.516 1.420 1.291 1.143 0.984 0.820 0.660 0.510 0.377 0.263 0.176 0.111 0.066 0.036 0.019 0.010 0.005 0.002 0.000

0.627 0.642 0.658 0.676 0.697 0.718 0.744 0.768 0.801 0.832 0.872 0.917 0.966 1.020 1.077 1.138 1.207 1.278 1.348 1.416 1.481 1.541 1.596 1.640 1.667 1.670 1.638 1.578 1.489 1.379 1.257 1.119 0.959 0.792 0.629 0.477 0.345 0.235 0.155 0.096 0.053 0.024 0.009 0.005 0.003 0.001 0.000

0.692 0.700 0.714 0.732 0.751 0.772 0.797 0.819 0.844 0.872 0.905 0.942 0.984 1.031 1.088 1.149 1.210 1.273 1.341 1.407 1.468 1.520 1.561 1.590 1.606 1.606 1.587 1.542 1.461 1.355 1.232 1.093 0.940 0.780 0.615 0.457 0.320 0.207 0.128 0.074 0.039 0.019 0.009 0.003 0.000 0.000 0.000

0.761 0.773 0.787 0.800 0.816 0.832 0.847 0.864 0.885 0.907 0.936 0.964 0.998 1.038 1.087 1.140 1.195 1.251 1.308 1.365 1.423 1.477 1.523 1.557 1.574 1.571 1.544 1.497 1.434 1.348 1.226 1.083 0.931 0.771 0.599 0.438 0.312 0.210 0.128 0.068 0.034 0.016 0.008 0.002 0.000 0.000 0.000

0.807 0.814 0.821 0.833 0.847 0.862 0.877 0.894 0.914 0.935 0.956 0.979 1.010 1.046 1.085 1.135 1.193 1.252 1.308 1.360 1.405 1.444 1.481 1.507 1.514 1.507 1.489 1.453 1.396 1.316 1.206 1.075 0.924 0.764 0.605 0.455 0.324 0.213 0.128 0.067 0.034 0.017 0.009 0.002 0.000 0.000 0.000

0.892 0.902 0.910 0.920 0.930 0.939 0.950 0.961 0.974 0.988 1.003 1.020 1.040 1.057 1.082 1.113 1.148 1.185 1.221 1.259 1.299 1.340 1.383 1.419 1.436 1.437 1.420 1.387 1.345 1.280 1.175 1.050 0.920 0.784 0.642 0.499 0.358 0.233 0.142 0.077 0.038 0.016 0.006 0.002 0.001 0.001 0.000

0.952 0.960 0.965 0.972 0.978 0.985 0.992 0.998 1.005 1.012 1.018 1.027 1.036 1.048 1.063 1.079 1.096 1.115 1.141 1.170 1.198 1.226 1.252 1.275 1.295 1.307 1.308 1.297 1.276 1.236 1.166 1.077 0.974 0.857 0.723 0.583 0.443 0.313 0.205 0.120 0.066 0.032 0.014 0.005 0.004 0.002 0.000

Source: Cross, Freedman, and Wong [1992].

Sec. 9.3. Spatial Distribution of the Absorbed Dose

341

It therefore follows that (9.8) Similarly,

(9.9)

In terms of the dimensional dose distribution :F, as given in Table 9.3, the point kernel for medium m is (9.10) in which the mass CSDA range fm (E) of electrons in medium m, in terms of that in water, is given by fm (E) = f0 (E)/rJ. Thus, :F may be interpreted as a universal function of the ratio f/fm = r/rm , the distance from the point source in units of the CSDA range. Cross, Freedman, and Wong [1992] recommend that 'T/ be computed using the following formula, which is based on experiments and calculations for beta-particle attenuation in various media relative to that in water: -2 (L/p)m , rJ = 0.77(1 + 0.0491Z - 0.0009Z ) (9.11) (L/P)w in which the mean atomic number is given as follows, based on mass fractions of the atomic constituents in the medium:

z = :Ei WiZt /Ai . :Ei

WiZi/Ai

(9.12)

The ratio of stopping powers is insensitive to energy, and Berger [1971] recommends that they be evaluated at 0.2 MeV. Cross, Freedman, and Wong [1992] recommend that, instead, (L/P)m /(L/P)w be replaced by fw/fm evaluated at 500 keV. These formulas predict that 'T/ = 0.89 for air. Note that the ratio of the mass CSDA range in water to that in air varies from 0.89 for 0.01-MeV electrons to 0.91 for 3-MeV electrons. For the human body, rJ is 0.98 for soft tissue and 0.97 for compact bone. Kernels for Radionuclide Sources

Consider a point radionuclide source with combined beta-particle, Auger-electron, and conversion-electron energy spectrum N(E) per MeV per transformation. The expected absorbed dose (MeV /g) at distance r in water, per transformation, is given by the integral over all source energies: E N( E ) r 1 E . 913(r) = / dE N( E)g(r, E) = dE :F fo(E) ( ro , ) 41rr2 /

(9.13)

For absorbed doses in media other than water, the same procedures may be applied as are applied to monoenergetic sources. Selected dose distributions for radionu­ clides (excluding gamma and x rays) are presented in Fig. 9.7.

342

Special Techniques for Charged Particles

� �

102

CL>

'\

.£: C:

� CL>

Chap. 9

101

90Sr-

\

90 y

\ \

0 -0

\

-0

-� 0 C:

"'�

100

10-1�--�-�������--�-������ 10-1 10-2 r (cm)

Figure 9. 7. Beta-particle and electron absorbed dose versus distance from ra­ dionuclide point sources in liquid water. Based on data from Cross, Freedman, and Wong [1992].

9.4 APPLICATIONS OF THE POINT KERNEL 9.4.1 Line Source of Electrons Monoenergetic Electrons in Water

Consider an infinite water medium containing an infinite line source of unit strength emitting electrons of energy E. The line kernel 91ine (r,E) is the absorbed dose at distance r from the line. It can be shown (Problem 3) that the relationship between point and line kernels is given by 91i ne (r,E)

= 2r

1

2 O -n:/

d1J sec 1J Q(r sec1J,E). 2

(9.14)

For the electron source, the absorbed dose is zero for r > l.2r 0• Therefore, the upper limit on the integral can be replaced by � ax= cos-1(r/l.2r 0 ). In terms of

343

Sec. 9.4. Point-Kernel Applications

C 0

B � ...."' '6 "'.,

:::, ..Cl

....·.:::"' '6

.,"' ., � "' ., � ., -.;

1.2

0 "'Cl

.,

0 "'Cl

0.80

�"' ., �

:::, 0

0.8

I

I

"'Cl

2.

C

1.6

:;:::;

C

0.40

0.4

]l "'

u

u

(/]

(/]

0.20

0.40

0.60

0.80

1.0

1.2

0.0 '--'---'---'---'-------''-----''-----''-----'-"""'-...0.--'-------' 0.0 0.20 0.40 0.60 0.80 1.0 1.2

Dimensionless distance r/r 0

Dimensionless distance, z/r 0

Figure 9.8. Scaled dimensionless dose distributions, Fline (r, E), for line isotropic electron sources in water. Computed on the basis of data from Berger [1973, as corrected in 1990].

Figure 9.9. Scaled dimensionless dose dis­ tributions, Fp 1;(z/r 0 , E), for plane isotropic electron sources in water. Computed on the basis of data from Berger [1973, as corrected in 1990].

scaled dimensionless dose distributions,

E gline (r,E} = --21rrpr o

111.nax

O

E d{)F(r sec{)/ro,E) = --:Fii ne (r/ro,E) 21rrpr o

-

(9.15}

The line kernel, given by Fli ne (r fro, E)

/11.nax

= o J

d{)F(r sec{)/ro, E),

(9.16}

is interpreted in such a way that (dr/r 0 }Fli ne (r,E) is the fraction of the energy released from the line source that is absorbed between radial distances r and r + dr from the line. ( see Fig. 9.8). Extension to Polyenergetic Sources and Non-Aqueous Media

It follows from Eqs. (9.10} and (9.15} that for medium m with associated scale factor r,, (9.17} gm,li ne (f ,E} = � Fli ne( f ,E) , 21rrro/ 'f/ To/ 'f/ Now consider a beta-particle line source of activity per unit length given by Qli ne (Bq/ cm) and energy spectrum N(E} (Mev-1 per transformation). In water, the absorbed dose rate D(r) (MeV g- 1 s- 1) at distance r is A

D(r ) = Qii ne

Qli ne dE N(E) gli ne (r,E) = - 21rr /

J

dE

r EN(E} F1i ne ( -,E) , (9.18} r0 r0 A

344

Special Techniques for Charged Particles

Chap. 9

Application of the Line Kernel As an example of the use of the line kernel, consider a uniform cylindrical source of volumetric strength Sv . The source, which is of radius R, emits monoenergetic elec­ trons of energy E. The source and its sur­ dA roundings are the same medium, here taken as water. This example is representative of the problem of determining the absorbed u dose in the human body in the vicinity of R blood vessels carrying radiopharmaceuticals y [Faw and Shultis 1992]. Figure 9.10 illus­ trates a plane cut through the source cylin­ der. Differential area dA is the cross section of a line source of effective strength Sv dA and the absorbed dose at radial distance y from the center of the cylinder is at distance Figure 9.10. Sectional view cutting a uni­ T from the effective line source. Therefore, form cylindrical source of radius R. The by Eq. (9.15), the absorbed dose at distance source is effectively infinite in length and is embedded in a medium of the same compo­ y from the radius R source of electrons of energy E is given by sition as the source medium. D(y,R, E)

S

E

v -= 27rpT 0

J

1 T dA-Fline ( -, E ) . T T0

(9.19)

Note that y may be greater or less than R. From the figure it is apparent that u2 + y2 - 2uy cos 'l9. The absorbed dose may be dA udud'l9 and that T 2 expressed in the form of a reduction factor cp, the ratio of the local absorbed dose rate, namely, D(y,R, E), to the rate of energy release per unit mass in the source, namely, Sv E /p. The reduction factor may be determined by the integration of

=

=

cp(y,R, E )

1

=T 'lr o

1 R 1,r o

du u

o

1 T d'l9 -Fzine ( -, E T T0

)

.

(9.20)

The reduction factor, illustrated in Fig. 9.11, is seen to be very insensitive to the electron energy. If the radius of the cylinder is greater than 2T O, then there is within the cylinder a central core of radius R- 2T 0 for which the reduction factor is unity.

9.4.2

Plane Isotropic Source of Electrons

Monoenergetic Electrons in Water Consider an infinite water medium containing a plane isotropic source of unit strength emitting electrons of energy E. The plane isotropic kernel 9p zi(z, E) is the absorbed dose at a distance z from the plane. It can be shown (Problem 3) that the relationship between point and plane source kernels is given by 9p zi(z, E)

= 21r

100

dT T Q(T, E).

(9.21)

345

Sec. 9.4. Point-Kernel Applications

0.8

�

----0.6

0 (.)

0.4 (.) 0

0.2

0.0

Solid line: E = 0.01 MeV Broken line: E = 1.0 MeV 0.0

0.40

0.80

1.6

1.2

2.0

Dimensionless radial distance, y /r 0 Figure 9.11. Local reduction factor cp(y, R, E) for the uniform cylindrical volume source in a uniform medium, as a function of the radial distance in units of the CSDA range, namely y/ro .

For the electron source, the absorbed dose is zero for z > 1.2T0• Therefore, the upper limit on the integral can be replaced by Zmax = 1.2T In terms of dimensionless scaled dose distributions (see Fig. 9.9), O•

(9.22) The plane kernel is nonzero only for z


6

Without reflection

7 C,

"'E

C,

3-5 mg cm-2

4

E

u

>

Q)

� "'0

� "'0

4

Q)

Q)

Q)

"'C

"'C Q) ,t, "iij

Chap. 9

"'C

2

"'C

.�

2

"iij

E0

E0

z

z

0

,o

-2

O -1 ,o ,o Electron energy (MeV)

,0

With reflection 0

1

O -1 ,o ,o Electron energy (MeV)

2 ,o-

Figure 9.12. Absorbed dose versus elec­ tron energy for three layers of skin for plane isotropic and monoenergetic electron sources at the surface of the skin, excluding elec­ tron reflection from the atmosphere. The absorbed dose (MeV /g) is normalized to one electron per cm 2 of skin surface. Calcu­ lations performed using the TIGER Monte Carlo code, as reported in Faw [1992].

1 ,0

Figure 9.13. Absorbed dose versus elec­ tron energy for three layers of skin for plane isotropic and monoenergetic electron sources at the skin-atmosphere interface. The ab­ sorbed dose (MeV /g) is normalized to one electron per cm 2 of skin surface. Calcu­ lations performed using the TIGER Monte Carlo code, as reported in Faw [1992].

Now consider a beta-particle plane source of activity per unit area given by Q a (Bq cm- 2) and energy spectrum N(E) (Mev- 1 per transformation). In water, the absorbed dose rate D(z) (MeV g- 1 s- 1) at a distance z from the plane is

D(z) = Q a

J

Qa dE N(E)gpli(z, E) = 2

J

dE

EN(E) z ,T Fpli( -, E) . 0

To

(9.25)

A Direct Application of the Plane Kernel

Consider an expanse of bare skin covered by a plane isotropic source of electrons and the problem of evaluating the expected dose to the radiosensitive epithelial cells located about 0.05 to 0.10 mm deep, or nominally 0.07 mm deep. The approximation is sometimes made that the air and the skin are sufficiently alike that mass CSDA ranges f in the two media are the same. If that were so, and if the expanse of skin were sufficiently great, then the theorem on plane density variations discussed in Section 6.6.1 would allow the source to be treated as an infinite plane isotropic source in an infinite homogeneous medium. Equation (9.25) would then apply. This result would also be a good approximation if the skin were covered by layers of clothing. Then, backscattering of electrons from the clothing would be similar to that from an air layer if the layers were infinite in lateral extent. However, electrons released from bare skin to the atmosphere, because their ranges in air are significant in comparison with dimensions of the body, are likely lost to the skin. Therefore,

347

Sec. 9.4. Point-Kernel Applications

methods using kernels for the infinite plane isotropic source are not suitable, and more exact transport calculations must be undertaken. Skin dose calculations are compared in Figs. 9.12 and 9.13. In one case, Fig. 9.12, a plane isotropic source is on the interface between a vacuum and a half-space of skin tissue. This situation is representative of a source on bare skin, for which, if electrons are released away from the skin, they are unlikely to be reflected. In the other case, Fig. 9.13, a plane isotropic source is on the interface between two half-spaces, one of air, the other of skin tissue. It is quite apparent that the contribution of reflected electrons to skin dose can be significant if contaminated skin is covered with clothing or some other medium.

9.4.3 Volume Source of Electrons The problem of computing the dose to the bare skin resulting from beta­ particle emitters in the atmosphere is chosen to illustrate the use of electron kernels for volumetric sources. The skin dosimetry problem is represented ap­ proximately as a half-space of air con­ taining a volumetric source of strength Sv (cm- 3 ) that is adjacent to a half­ space of skin tissue. Both the skin and Skin the air are assumed to have the atten­ uation properties of water for beta par­ ticles. The air is assumed to be of Figure 9.14. Geometry for evaluating skin uniform density 3 Pa and the electron dose from electron sources in the atmosphere. source is assumed to be monoenergetic, at energy E. Figure 9.14 illustrates the geometry. Consider first the effective plane isotropic source represented by the region of the atmosphere between planes at u and u + du. Its strength is Sv du. From Eq. (9.22), that part of the dose (MeV/g) at depth z in the skin is given by Sv du 9pti(u, E). To allow for differences between air and skin density, it is necessary to make use of mass thicknesses such as du = Pa du and z = pz. Then the absorbed dose at mass thickness depth z due to the differential area source is Sv (du/Pa )9pti(u, E). A convenient way of representing the absorbed dose in the skin is in the form of a dimensionless reduction factor 1.p(z, E), the ratio of the absorbed dose in the skin to the energy released per unit mass in the atmosphere, namely ESv /Pa • Based on 9pli from Eq. (9.22), (E')µ.(E', E) + S,, li(E - E0 ).

352

Special Techniques for Charged Particles

Chap. 9

Source energy spectrum N(E) (MeV-1) x 0.1

E

:, ....,...

"'

X :, ;;:::

Conversion and Auger electrons

,o-3 .______.___,_____.__..__.,__'-'-_.___.____�.____.____.__.,__.,__L.....L-'---' ,o-2

,o-1

Energy (MeV) Figure 9.17. Energy spectra of the normalized electron and beta-particle fluences arising in air from a uniform concentration of 133Xe. Also shown is the energy spectrum of the source beta particles.

Figure 9.17 illustrates the results of a series of calculations involving a uni­ formly distributed source of 133 Xe in the atmosphere. One line in the figure shows the energy spectrum N(E) of the beta particles, a normalized probability density function. The source strength Sv (E)/p is the product of N(E) and the specific activity (Bq/g). The energy spectrum of the beta-particle :fluence shown in the figure is normalized to unit specific activity. Shown for comparison in the figure is the normalized energy spectrum of the electron :fluence arising from conversion and Auger electrons released in the decay of 133 Xe.

REFERENCES BERGER, M.J., Distribution of Absorbed Dose Around Point Sources of Electrons and Beta Particles in Water and Other Media, NM/MIRD Pamphlet 7, J. Nucl. Med., 12, Suppl. 5 (1971}.

BERGER, M.J., Improved Point Kernels for Electron and Beta-Ray Dosimetry, Report NBSIR 73-107, National Bureau of Standards, Washington, DC, 1973. BERGER, M.J., "Beta-Ray Dose in Tissue-Equivalent Material Immersed in a Radioactive Cloud," Health Phys., 26, 1-12 (1974). BERGER, M.J., ESTAR, PSTAR, and ASTAR: Computer Programs for Calculating Stop­ ping Power and Range Tables for Electrons, Protons, and Helium Ions, Report NISTIR

353

Problems

4999, National Institute of Standards and Technology, Gaithersburg, MD, 1992. [Dis­ tributed as Peripheral Shielding Routine PSR-330 by Radiation Shielding Information Center, Oak Ridge National Laboratory, Oak Ridge, TN.] CROSS, W.G., "The Distribution of Absorbed Energy from a Point Beta Source," Can. J. Phys., 45, 2121-2040 (1967). CROSS, W.G., H. ING, N.O. FREEDMAN, AND J. MAINVILLE, Tables of Beta-Ray Dose Distributions in Water, Air, and Other Media, Report AECL-7617, Atomic Energy of Canada Limited, Chalk River, Ontario, 1982. CROSS, W.G., N.O. FREEDMAN, AND P.Y. WONG, Tables of Beta-Ray Dose Distribu­ tions in Water, Report AECL-10521, Atomic Energy of Canada Limited, Chalk River, Ontario, 1992. DILLMAN, L.T., AND F.C. VON DER LAGE, Radionuclide Decay Schemes and Nuclear Parameters for Use in Radiation Dose Estimates, NM/MIRD Pamphlet 10, Society of Nuclear Medicine, New York, 1975. DILLMAN, L.T., EDISTR-A Computer Program to Obtain a Nuclear Decay Data Base for Radiation Dosimetry, Report ORNL/TM-6689, Oak Ridge National Laboratory, Oak Ridge, TN, 1980. EVANS, R.D., The Atomic Nucleus, McGraw-Hill, New York, 1955. FAW, R.E., "Absorbed Doses to Skin from Radionuclide Sources on the Body Surface," Health Phys., 63, 443-448 (1992). FAW, R.E., AND J.K. SHULTIS, "Dosimetry Calculations for Concentric Cylindrical Source and Target Regions with Application to Blood Vessels," Health Phys., 62, 334-350 (1992). MCGINNIES, R.T., Energy Spectrum Resulting from Electron Slowing Down, Circular 597, National Bureau of Standards, Washington, DC, 1959. SPENCER, L.V., AND U. FANO, "Energy Spectrum Resulting from Electron Slowing Down," Phys. Rev., 93, 1172 (1954) SPENCER, L. V., Energy Dissipation by Fast Electrons, NBS Monograph 1, National Bu­ reau of Standards, Washington, DC, 1959.

PROBLEMS 1. Consider a point source of monoenergetic electrons. Let x = r / rO ( E) be the scaled distance from the source. Verify that dx F(x, E) is the expected fraction of E deposited between x and x + dx. 2. Verify that under the continuous slowing-down approximation for charged par­ ticles, L(r0 - r) FP.L1 (r,E)=F(r,E)= E/ro , in which L(r0 - r) is the LET evaluated at the energy of the particle with residual CSDA range r0 - r.

354

Special Techniques for Charged Particles

Chap. 9

3. Verify Eqs. (9.14) to (9.17) and (9.21) to (9.24) for line and plane sources in nonaqueous media. 4. Suppose that water, as a surrogate for tissue, is uniformly irradiated by mono­ energetic neutrons of energy up to En . Consider only recoil protons and suppose that the (n,p) scattering cross section is isotropic in the center-of-mass system. Use the following approximation for the LET (MeV/cm) as a function of proton energy (MeV): Y

- 0.08969x = 1 + 2.4207 0.26323x + 0.13025x 2 '

in which y = log 10 L and x = log 10 E.

(a) Show that if one recoil proton is created per cm3, then the energy spectrum of the proton source strength is S(E) the proton fluence is

(E)

En

= L(E)-1 }{

E

= E;; 1

dE S(E)

and the energy spectrum of

= (1- E/En )/L(E).

Show that under the continuous slowing-down approximation, for a proton of initial energy E, the mean LET as the proton slows in given by L(E) =

�1

E

dE L(E).

(b) Assume that quality factor vs. LET is given by the nonlinear relationship of Table 5.2. Compute the mean quality factor as a function of neutron energy up to En = 10 MeV. Compare results with the tabulated quality factor data given in Table 5.5. ( c) Assume that quality factor vs. LET is given by a piecewise linear relation­ ship between the data of Table 5.1. Compute the mean LET of secondary protons as a function of neutron energy up to En = 10 MeV and corre­ sponding mean quality factors based on the mean LET. Compare results with the tabulated quality factor data given in Table 5.5. ( d) Assume that quality factor vs. LET is given by a piecewise linear rela­ tionship between the data of Table 5.1. Show that in terms of the mean LET L(E) of protons of initial energy E, the mean quality factor may be computed as (2/E;) f0En dE EQ(L(E)). Compare the results with the tabulated quality factor data given in Table 5.5.

Chapter 10

Deterministic Transport Theory

For many shielding situations in which particle scattering or production of secondary radiation (e.g., fission neutrons, annihilation photons, secondary electrons, etc.) is important, the approximate methods of the earlier chapters may not be capable of describing the radiation field with sufficient accuracy. The exact description of the radiation particles that takes into account all the possible particle-medium interactions and the effects of medium composition and geometry is the realm of transport theory. In its most general form, transport theory is a special branch of statistical mechanics which deals with the interaction of one species of particles (here, the radiation field) with another species (the shielding medium). There are two basic approaches used for rigorous transport calculations. One is to perform a simulation using Monte Carlo techniques of how radiation particles migrate through a medium. This powerful technique is the subject of the next chapter. The other basic approach is to use an equation which rigorously describes the radiation field and whose solution gives the expected fluence or flux density of the radiation particles throughout the shielding medium. This is the approach followed in the present chapter. In this and the next chapter, the transport of only electrically neutral photons and neutrons is considered. Charged-particle transport is considerably more involved. Fortunately, in many practical shielding situations, the transport of charged particles can be treated by the approximations described in earlier chapters.

10.1

TRANSPORT EQUATION

The shielding analyst ideally would like to know the distribution of photons or neutrons everywhere in the shielding medium. For most applications the steady­ state spatial and energy distribution of the particle density n(r, E) is all that is

355

356

Deterministic Transport Theory

Chap. 10

needed. 1 Unfortunately, there is no equation for this quantity that holds rigorously in all situations. The simplest equation which accurately describes the particle distribution in a medium is for the differential energy and directional flux density /ox and 8¢>/oy in Eq. (10.12) vanish, and O•V =

w

8(z,E,w,'ljJ)

oz

.

(10.13)

This particularly simple form will be used throughout this chapter to illustrate various approximation techniques for treating the transport equation. For other geometries the explicit form of the streaming term in the transport equation is more complicated. In Table 10.1 explicit forms for the streaming term are given for the three widely used geometries shown in Fig. 10.3.

10.1.2 Integral Form of the Transport Equation The differential-integral form of the transport equation, Eq. (10.9), can be converted into a pure integral equation for the angular flux density or even for the scalar flux density. Since this alternate integral form of the transport equation is equivalent to the original differential-integral form, exact solutions to it are also not available, except for the simplest of cases. The utility of this alternate integral form is that different analytical and numerical techniques can be developed from it for obtaining approximate solutions for realistic problems. In particular, approximations to the integral form of the transport equation (see Section 10.6) are capable of treating more accurately situations involving highly anisotropic angular flux densities and scattering cross sections than are approximations based directly on the differential­ integral form of the transport equation. Integral Equation for the Angular Flux Density

To derive an integral equation for the angular flux density (r, E, 0), begin by writing Eq. (10.9) in the form O•V(r, E, 0) + µ(r, E)(r, E, 0) where q(r, E, 0)

= q(r, E, 0),

(10.14)

= 1 00 dE' 141'[ d!l' µ (r,E' -+ E,O' -+ O)(r, E',O') + S(r, E,0). 0

8

(10.15) From Eq. (10.10) it was seen that the streaming term O•V is simply the rate of change of along the direction ofn, namely, d/ds, wheres is measured along the direction ofn. Thus, if we define r = r' - RO, where R ( = -s) is measured along the direction opposite ton (see Fig. 10.4), then d/ds = -d/dR and Eq. (10.14) can be rewritten as - �(r'-RO,E,O)+µ(r'-RO,E) 1), interaction coeffi­ cients and volumetric source strengths (per unit unscaled volume) must be increased by a factor f However, the energy and directional dependence of the scaled and unscaled cross sections and sources must be the same. For example, substitution of iron in a scaled model of a concrete structure will approximately increase µ and µ 8 for photons by a constant factor, although the energy dependence will be changed slightly, particularly at low energies. 10.2.5 Volume-to-Surface Source Transformation A method was presented in Section 6.7.2 for obtaining the flux density or detector response in a restricted region of some larger space. In this method, the original problem, which allowed arbitrary source and material distributions throughout the space, was transformed into an equivalent problem in which the region of interest was surrounded by an effective surface source and the rest of the space not of interest was replaced by a perfect absorber. This equivalence seems physically reasonable since the distribution of particles in the smaller restricted region should be independent of the origin of the particles entering that restricted region. It is of no concern to the region of interest whether the particles were born at the surface from an effective surface source, or whether they were born far from the region and

376

Deterministic Transport Theory

Chap. 10

subsequently underwent many interactions before entering the region of interest. The connection between these two equivalent problems is that the incident fl.ow rate on the bounding surface of the restricted region, n•O(r s , E, !1), n•O < 0, in the original problem must equal the effective surface source strength S a (r s ,E, !1) in the equivalent problem, where n is the unit normal out of the restricted region at any given point r s of the surface. In this section a rigorous development [Case and Zweifel 1967] of this transformation is presented.

*

Sources

Incident flux density

c\>(r5,E, O), n.O < 0

' z

*

Sources

*

Sources

X

Figure 10.5. Geometry for the transformation to a smaller region of interest.

Consider a problem in which a region V, the "region of interest," has a surface source S a (r s ,E, !1) at a point on the boundary given by r s as in Fig. 10.5. If one chooses a local coordinate system with the origin on the surface, with x measured along the outward unit normal n, and y and z measured along ty and t z , unit normals tangential to the surface, the transport equation, in the neighborhood of a point on the bounding surface, can be written as 8¢> 8¢> 8¢> O•n ox + O•ty oy + O•tz oz + µ¢> = Sv(r,E, !1) + S a (r,E,!1)8(x) +

r lo

=

dE'1 dD.' µs (r,E'-+ E,O'-+ !1)(r,E',!1'), 4,r

(10.60)

Sec. 10.3. Approximations to the Transport Equation

377

where Sv describes any sources other than the surface source Sa . Upon integrating the equation along x from -f to E and taking the limit as E --+ 0, one obtains (10.61) where denotes the angular flux density on the positive (outward) and negative (inward) sides of the surface. Thus, it is seen that the effect of the surface source is to cause a discontinuity or "jump" in the rate at which particles flow across a unit area of its surface. In the original problem, there is no surface source on the boundary of the region of interest, but as a result of the sources inside and outside the region of interest, an angular flux density (r s , E, 0) is established at the boundary. If the material and sources outside the region of interest were now to be altered arbitrarily, but somehow the incident angular flux density at the boundary were still held equal to(r s , E, 0), n•O < 0, the flux density inside the region of interest would be unchanged since the solution of the transport equation in a region is determined uniquely by the material and sources in the region and by the incident boundary conditions [Case and Zweifel 1967]. In particular, if the medium outside the region of interest were replaced by a perfect absorber, the flux density would vanish outside the region of interest. To maintain the same incident neutron profile at the boundary of the region of interest, it is thus necessary to place a surface source on the boundary whose strength from Eq. (10.61) must be ±

(10.62) since";;,(r s , E, 0) = 0. It should be noted that the terms inside and outside in the discussion above are quite relative. The region of interest could equally well be the region outside the closed boundary of Fig. 10.5, while the region inside the closed boundary can be replace by a perfect absorber with a surface source, given by Eq. (10.62), placed on the boundary. This volume-to-surface source transformation is a very useful device for simpli­ fying practical transport calculations. For example, the calculation of the radiation field in a biological shield caused by radiation leaking from a reactor core would generally be very difficult if the core and shield had to be treated simultaneously. However, the foregoing transformation would allow one to decouple the core region from the shield region if the radiation leaking from the core (and incident on the shield) could first be determined. Often, only an approximate knowledge of the incident particle flow rate from the core is sufficient to obtain an adequate estimate of an effective surface source strength. The core then is replaced conceptually by a perfect absorber, so that calculations need be performed only in the biological shield. 10.3

APPROXIMATIONS TO THE TRANSPOR T EQUATION

The transport equation discussed in Section 10.2 presents a very detailed descrip­ tion of the radiation field in a medium. Often, such detail is not necessary since one

378

Deterministic Transport Theory

Chap. 10

seldom needs the complete spatial, directional, and energy dependence of the radi­ ation. Consequently, several approximations of the transport equation are widely used. These approximations, although providing less information about the ra­ diation field, require far less analytical or computational effort to estimate flux densities or dose rates compared to use of the transport equation. In this section three important approximations to the transport _equation are reviewed.

10.3.1

Exponential Attenuation

Many of the techniques presented in this book are based on the exponential at­ tenuation of radiation as it traverses a medium. This exponential attenuation is rigorously true only for the uncollided radiation-a fact readily inferred from the transport equation. For a homogeneous source-free medium, the transport equation for the uncollided flux density ¢> 0 can be written as 0 !l-"V¢>0 (r,E,!l) + µ(E)¢> (r,E,!l)

= 0,

(10.63)

=

since no scattered particles are included in the uncollided flux density. In Sec­ tion 10.1.1 it was shown that !l•V d /ds, wheres is measured along the direction !l. Thus, if the uncollided flux density is known at point r, the flux density at any other point r' (= r + s!l) is found from Eq. (10.63) to be (10.64) Although the total flux density (uncollided plus scattered) is not generally at­ tenuated in a purely exponential manner, there are several situations in which ex­ ponential attenuation may be a good approximation. For high-energy photon and neutron problems, the scattering cross sections tend to be very anisotropic and peaked in the forward direction and thus may be approximated by µ.(r,E'-+ E,!l'•!l) '.:::'. µ.(r,E'-+ E)o(!l'•!l-1).

(10.65)

This "straight-ahead approximation" implies that a particle does not appreciably change its direction of travel upon scattering. With this approximation, the trans­ port equation, Eq. (10.9), for a source-free medium becomes !l•Vef>(r,E, !l) + µ(r,E)ef>(r,E, !l) 00

= f dE' { dO.' µ.(r,E'-+ E)¢>(r,E',!l')8(!l'•!l-l) 14"

lo

=

1

00

dE' µ.(r,E'-+ E)¢>(r,E',!l).

(10.66)

If one is interested in a dose rate D, given the detector response function �(E), that is, D(r,!l)

=1

00

dE �(E)¢>(r,E,!l),

(10.67)

379

Sec. 10.3. Approximations to the Transport Equation

then integration of Eq. (10.66) with respect to energy, after multiplication by 1?.(E), gives f!•V .D(r,f!) +

1 = dE' 1?.(E')[µ(r,E')-µ.(r,E')](r,E', f!) = 0,

(10.68)

where a weighted scattering coefficient is defined as � (r,E') = _ µ.

ro = dE 1?.1?.(E) • (r,E, --+ E). (E t

J

'

(10.69)

Equation (10.68) may be rewritten as f!•V.D(r,f!) + fl a (r,f!).D(r,f!)

= 0,

(10.70)

with an effective attenuation coefficient defined as fl a (r,f!)

= D(. r,1 f!) 1o = dEJ?.(E)[µ(r,E)-µ.(r,E)](r,E,f!).

(10.71)

If the energy dependence of ¢(r,E, f!) is approximately separable from the angular dependence, then fl a is independent of f! and can be evaluated knowing only the energy dependence of ¢. In practice, one could approximate fl a from its definition by assuming some reasonable energy spectrum. This problem of calculating fl a without first knowing ¢(r,E, f!) is a special case of a macroscopic group-cross­ section (coefficient) evaluation, which is discussed in greater detail in Section 10.3.3. With s measured along the direction of f!, Eq. (10.70) can be integrated along f! to give .D(r + sf!,f!)

= .D(r,f!) exp

[-1•

fla(r + s'f!,f!) ds']

(10.72)

which for a homogeneous medium reduces to .D(r + sf!,f!)

= .D(r,f!) exp[-fl a s].

(10.73)

Thus in the straight-ahead approximation, the detector response .D is seen to at­ tenuate exponentially along the path of particle travel. This straight-ahead approximation leads one to expect that a beam of radiation would be attenuated in an approximately exponential fashion whenever the pene­ tration of radiation is dominated by small-angle scattering. For example, the pen­ etration of fast neutrons in an hydrogenous medium is controlled by the streaming of uncollided and small-angle-scattered neutrons both because the scattering cross sections tend to be peaked in the forward scattering direction in the laboratory coordinate system and because any large-angle scatter degrades the neutron energy to such an extent that it is subsequently thermalized rapidly and removed from the fast-energy region. Hence, one would expect the removal of fast neutrons in such

380

Deterministic Transport Theory

Chap. 10

a medium to exhibit an exponential behavior. Similarly, those high-energy gamma photons (whose the scattering cross sections are highly forward peaked) that are scattered through large angles are degraded in energy and subsequently attenuated rapidly. Thus, the deep penetration of these photons is governed by small-angle­ scattered photons, which in turn can be expected to attenuate exponentially with some effective attenuation coefficient. Finally, an exponential attenuation behavior can often be expected deep within a shield from more fundamental considerations. Many investigations have explored the characteristics of the transport equation; and for certain transport approxima­ tions, such as constant cross sections or the multigroup model (see Section 10.3.3), the flux density, sufficiently far from boundaries or sources, inherently attenuates in an exponential manner. However, the calculation of the exact value of the ap­ propriate attenuation coefficient in terms of the material cross sections is generally very complicated.

10.3.2 Diffusion Approximation For most shielding applications the angular distribution of the flux density is not needed. Rather, the directionally integrated flux density (i.e., the scalar flux den­ sity) (r, E) is all that is often desired. Although no rigorous equation exists for this quantity, it is possible to obtain an approximate equation for it. This approxima­ tion, known as the diffusion equation, is generally much simpler to solve numerically or analytically than is the rigorous transport equation; consequently, diffusion the­ ory is sometimes applied to shielding problems, particularly those involving neutron transport. Although the diffusion approximation may be derived directly from a particle balance requirement [Lamarsh 1966], it is more instructive to derive it from the transport equation and to see explicitly the inherent approximation. For simplicity, consider the transport equation in plane geometry in which the angular flux density is azimuthally symmetric, i.e., (x, E, w ,'l/;) = (x, E, w ). If the sources are then assumed to be isotropic so that S (x, E, w, 'l/;) = S (x, E) / 41r, the transport equation assumes the form w

=S(x, E ) 8(x, E, w ) 'I' x, E, w ) + µ(x, E),1,( ax 41r +

100 0

d E'

11-1 dw'

1271' 0

d'l/;' µ.(x, E '-+

E,f!'•f!)(x, E ',w').

(10.74)

To eliminate thew dependence, integrate over all directions to obtain the continuity equation

8. ( E) + µ(x, E)(x, E) = S(x, E) + Jx a:

100 0

d E ' µ.(x, E '-+

E )(x, E'),

(10.75)

where the following directionally integrated quantities have been defined (see Sec-

381

Sec. 10.3. Approximations to the Transport Equation

tions 2.3.1 and 2.3.3): ix (x,E)

3 ] l dww2 -+- WJx =-(x,E). 3 -1 4Jr 41T

(10.87)

The last term in the current equation, Eq. (10.83), can be written more compactly by defining the mean cosine of the scattering angle, W 8 as ,

µ8(x,E)w8(x,E)

= Jx. (x,l E) laf

00

dE' µs 1(x,E'--+ E)jx(x,E').

(10.88)

To evaluate this expression approximately without first knowing Jx(x,E), the energy transfer is commonly neglected in the anisotropic scattering component, namely µ81(x,E' --+ E) '.'.:='. µ81(x,E')8(E' - E), so that the right-hand side of Eq. (10.88) becomes simply µs 1(x,E). With Eqs. (10.87) and (10.88), the current equation finally assumes the form . _ [µ(x,E) - µs (x,E)w8(x,E)]Jx(x,E)

(x,E) = -18 3 o , x

(10.89)

383

Sec. 10.3. Approximations to the Transport Equation

or, equivalently,

. (X, )x

- -D(X, E) E) -

8(x, E) OX

,

(10.90)

where the diffusion coefficient D(x, E) is defined by (10.91) This relation between the particle current component and the integrated flux den­ sity, known as Fick's law of diffusion, states that the flow or diffusion of particles is away from regions with high particle densities. Finally, elimination of Jx (x, E) be­ tween Eqs. (10.90) and (10.75) gives the desired equation for the scalar flux density, namely,

E) - : [n(x, E)° �; ] + µ(x, E)(x, E) x

= S(x, E) +

100

dE' µs (x, E' -+ E)(x, E'). (10.92)

This result, called the energy-dependent diffusion equation, is considerable sim­ pler, and hence easier to solve, than the original transport equation. Although the derivation above is for the simplest geometry, the same derivation can be extended to more general geometries [Lamarsh 1966; Duderstadt and Martin 1979]. In general geometry, the diffusion equation assumes the form -V•D(r,E)V(r,E) + µ(r,E)(r,E)

= S(r, E) +

100

dE' µ8 (r, E'-+ E)(r, E'), (10.93)

with the approximate relation between the current density vector and the flux den­ sity given by Fick's law: j(r, E)

= -D(r,E)V(r, E).

(10.94)

The essential approximation of diffusion theory, Eq. (10.84), constrains the an­ gular flux density to vary at most linearly with w for Fick's law to apply. Deep in a medium, the flux density is often composed of multiply scattered particles and hence can be expected to be nearly isotropic in direction. For such cases the diffu­ sion approximation should be valid. However, near a free surface or a source, the flux density is quite anisotropic in direction and not at all well represented by a simple linear variation. For these situations the diffusion approximation may be poor. For example, near a point source, the flux density as calculated by diffusion theory varies as 1/r, while transport theory gives the correct 1/r2 variation [Case and Zweifel 1966].

384

Deterministic Transport Theory

Chap. 10

The energy-dependent diffusion approximation is often combined with an en­ ergy multigroup approximation (see Section 10.3.3 ) to eliminate the energy vari­ able, hereby making this model even more amenable to numerical solution. The multigroup diffusion model is widely used because of its simplicity and it often gives remarkably good results, particularly for neutron problems (see the removal­ diffusion models in Chapter 8). However, for problems near interfaces or sources, where the approximation of Eq. (10.84) is poor, or for particles which undergo few scatters and have highly anisotropic scattering cross sections, this diffusion model can result in serious errors in the flux densities.

10.3.3 Multigroup Approximation The steady-state transport equation of Eq. (10.9) is generally far too complex to solve even by numerical techniques. Even if the number of spatial and angular variables were reduced with a simple geometry, the energy variability of the cross sections generally precludes solving the transport equation except for unrealistically simple cross-section models such as constant cross sections. To make the transport equation more amenable to numerical and analytical treatment, it is usual to con­ sider an approximation of the transport equation in which the energy variable has been removed. By far the most common method to eliminate the energy variable in the trans­ port equation is to use the energy multigroup approximation. In this method the en­ tire particle energy range (0, E0 ), where Eo is the highest energy for the particles, is subdivided into G contiguous energy intervals or groups (Eg , Eg -l), g = l, 2, ..., G, in which Eg ( < Eg -l) is the lowest energy of the gth group. Equation (10.9) is then integrated over the gth energy interval to obtain

Ll1

{l-'Vq'.>g (r, !l) + µg q>g (r, !l)

=

G

g '=

df!'µg g (r, n' -+ 1

471"

!l )q'.>g , (r, !l')

+ Sg (r, !l), (10.95)

where the gth-group flux density and source are defined as

Sg (r, !l)

=1

(10.96) Eg-l

Eg

dE S(r, E, !l),

(10.97)

and the group total and scattering transfer coefficients are defined as µg

=

{E 9-1 1 dE µ(r, E )q'.>(r, E, !l ) q>g (r, !l ) }E.

(10.98)

385

Sec. 10.3. Approximations to the Transport Equation

and µ9 , 9

=

1

(

'Pg' r,

eg-1

O') 1eg

dE f

eg'-1

e1g

dE' µ 8 (r,E' ---t E,O.' ---t O)cp(r,E',O').

(10.99) Equation (10.95) is an exact result and this multigroup equation is equivalent to the original energy-dependent transport equation. However, to evaluate the "group constants" µ9 and µ9,9 from Eqs. (10.98) and (10.99), it is seen that the energy­ dependent flux density must be known first! Moreover, these group cross sections acquire a spatial and directional dependence from both the flux density and the energy-dependent interaction coefficients. The great value of the multigroup equations arises when the group cross sections can b� approximated without first knowing the flux density cp(r, E, 0.). For example, it is often assumed that the energy dependence of¢ can be separated as cp(r, E, 0.) � W(E)cp(r,0.), so that the group coefficients become eg eg µ9 (r) � f -i dE µ(r, E)W(E) / f l dE W(E) le g le g

(10.100)

and

For a very fine energy mesh, the weight function W(E) may be assumed constant over each group so that the group coefficients may be computed directly from these approximations, and the weight W in Eqs. (10.100) and (10.101) is seen to cancel out. For broader energy groups, various approximations for W(E) are used. In the MeV energy range, W(E) is often taken as the fission spectrum if fission neutrons are being considered, while in the epithermal range an E- 1 slowing-down spectrum is often chosen for W(E) [Bell and Glasstone 1970]. A more accurate way to evaluate the group coefficients for a particular problem is first to simplify the geometry of the problem and use a very fine multigroup mesh on this associated simplified problem, with constant weighting for evaluation of the fine-group coefficients. The simplified problem is then solved and the resulting flux density is then used as the weighting function in the computation of coarse-group coefficients for the actual problem. In effect, the fine-group coefficients are "collapsed" or reduced to a broad-group set. The same multigroup approximation can be used to eliminate the energy variable of the energy-dependent diffusion equation. If Eq. (10.93) is integrated over each energy group, one obtains - V•D 9 (r)V¢9 (r) + µ9 (r)cp9 (r)

=

Lµ G

g'=l

9,9 (r)cp9, (r)

+ S9 (r),

(10.102)

386

Deterministic Transport Theory

Chap. 10

where the following group constants have been defined: (10.103) µ g (r) µg ' g (r)

1

{Eo-1

= -;:- } 'f'g

ig E

Eg-1

= -:;:-' 1 1

'f'g

dE µ(r, E)(r, E),

E9

E•'-1 dE 1 dE' µ s (r, E' __, E)(r, E'), Eg '

(10.104)

(10.105)

(10.106) Again, if these group constants are to be evaluated a priori, some approximation for the flux density must be made. The generation of group coefficients or macroscopic cross sections for photon­ and neutron-transport problems is an important research area which blends both art and science. Many different codes have been developed to compute group con­ stants for various transport and diffusion problems. The elimination of the energy variable by the multigroup approximation enables numerical solutions for realis­ tic transport or diffusion problems to be obtained from Eqs. (10.95) and (10.102). Many powerful multigroup transport and diffusion codes for one-, two-, and even three-dimensional geometries are available. Nevertheless, the user of these codes should always remember that group coefficients are only approximations, and that the results of a multigroup transport calculation should be verified by experience, experiment, or an analysis of the sensitivity of the results to changes in the group coefficients.

10.4 METHOD OF MOMENTS In most calculations based on the transport equation, the energy dependence of the particles is treated in an approximate manner, usually with an energy multigroup approximation, as explained in the preceding section. Such an approximate energy treatment is necessary to reduce the complexity of the transport equation so that sufficient detail in the spatial dependence of the flux density can be obtained. However, there are transport problems with geometries sufficiently simplified that it is possible to obtain much more detail in the energy dependence of the flux density. A powerful method to determine the spatial and energy distribution of neutrons or photons originating from point, plane, or line sources in an infinite medium is the method of moments [Goldstein and Wilkins 1954; Goldstein 1959; Fano, Spencer, and Berger 1959]. In this computational method, the transport equation is first used to obtain the spatial and angular moments of the flux density, and then the flux density is reconstituted from these moments.

387

Sec. 10.4. Method of Moments

To illustrate the basics of this method, consider the transport of neutrons or gamma rays in an infinite homogeneous medium with an infinite, uniform, plane source at x = 0. If the source depends only on the polar angle, cos- 1 w, the angular flux density must be azimuthally invariant, namely, cp(x,E,w,'1/;) = cp(x,E,w ). For this situation the transport equation may be written as w

ocp(x

+

;:

,w ) + µ(E)cp(x,E,w )

100 0

= S(E,w)8(x)

dE' f dD' µ.(E'---+ E)/5(0'•0 - w.(E',E))cp(x,E',w'),

14-rr

(10.107)

where µ. and w8 are, respectively, the singly differential scattering coefficient and that function of the particle energies before and after scattering which equals the cosine of the scattering angle, given by Eq. (3.24) for neutron scattering and readily derivable from Eq. (3.16) for photon scattering. For neutron slowing-down problems or photon scattering, the scattering coefficient µ.(E'---+ E) vanishes if E > E' since particles always lose energy upon scattering. Thus, in Eq. (10.107) the lower limit of the E' integral may be replaced by E without any loss of generality. Further, for any realistic source there is always a maximum source energy Ea . Thus, cp(x,E,w) = 0 if E > Ea , and the upper limit of the E' integral may be replaced by Ea . Rather than calculate cp(x,E,w ) from the transport equation above, the method of moments seeks the angular Legendre moments cpz ( x,E) defined such that cp(x,E,w )

1 = -1)2l + l)cp 1(x,E)P,,(w ), 41r 00

(10.108)

l=O

where P,,(w ) is the Legendre polynomial of order l (see Appendix B.4). The or­ thogonality properties of these polynomials give the moments as, after multiplying Eq. (10.108) by Pm (w ) and integrating over 41r steradians, (10.109) To obtain an equation for these Legendre moments, multiply the transport equa­ tion by P,,(w ) and integrate over all w , that is,

1

1

_ 1

dw Pz(w ) x

E ocp { a dE' µ.(E'---+ E) + µcp - S 8(x)] = x }E

[w o

11_ d 11_ dw' 12-rr d'l/;'P1( 1

w

1

0

w )8(0•0'- w.(E',E))cp(x,E',w ').

(10.110)

Substitute for cp from the expansion of Eq. (10.108) and use the recursion relation for Legendre polynomials, Eq. (B.30), to evaluate the left-hand side of Eq. (10.110) in terms of the Legendre moments ¢>1• To evaluate the right-hand side, first expand

388

Deterministic Transport Theory

Chap. 10

the delta function as a Legendre polynomial series using Eqs. (B.36) and (B.37). The result is (10.111) Since the cosine of the scattering angle is given by (10.112) the addition theorem for Legendre polynomials, Eq. (B.39), can be used to expand _Pz(fM1') in Eq. (10.111). By thus expressing the delta function in terms of _Pz(w), Pz(w'), Pzm (w), Pzm (w'), and cos[m(v, - v,')], the right-hand side of Eq. (10.110) is readily evaluated by using the orthogonality properties of the Legendre polynomials. In this manner Eq. (10.110) reduces to � 8¢1+1(x,E) _l _ 0¢1-1(x,E) + µ(E),i.'1-'l (x ' E) + 2l 1 OX 21 + 1 ox + {Eo

= }E

dE' P1(ws(E',E))µs(E'-+ E)cpz(x,E') + Sz(E)8(x), (10.113)

where the source angular moments are defined by (10.114) The next step in the moments method is to reduce Eq. (10.113) further by introducing the following spatial moments of ¢1(x,E), that is, n,l � 0,

(10.115)

=

with the definition ¢-i,1 ¢n ,-l = 0. Thus, multiplication of Eq. (10.113) by xn /n! and integration of the result over all x yields for n = 0,1,2,. . . and l = 0,1,2,. . . , µ(E)¢n 1(E)

{Eo

= }E

dE' P1(ws(E',E))µs(E'-+ E)¢n 1(E') + S1(E)8n o

+ 21

1

+l

[(l + 1)¢n -1,l+l(E) + l¢n -1,l-l(E)].

(10.116)

To obtain this result, the spatial derivative terms have been integrated by parts; and the fact that the flux density exponentially vanishes at large distances from the source implies that limlxl->oo xn cpz(x) = 0. It is this requirement that the integrand in Eq. (10.115) vanish at the endpoints of the spatial integration that makes the

389

Sec. 10.4. Method of Moments

extension of the method of moments to finite systems impossible, since for such systems the integrand does not vanish at the surface. Equation (10.116) represents an infinite set of interlinked integral equations for the moments 0 a,b< 0.

(B.12)

Tables and graphs of the exponential integrals functions are widely available in the shielding literature, and from Figs. B.2 and B.3 it is seen that these func­ tions vary rapidly with their arguments. For large arguments, the principal func­ tional dependence of En (x) is x-1c "' [x-1e "' for Ei(x)]; and thus, by extracting these asymptotic dependencies from the functions, much more slowly varying func-

438

Mathematical Tidbits

Appendix B

tions, suitable for interpolation, result (Shure and Wallace 1975]. In Table B.2 the following slowly varying functions are tabulated: Ei(x) � xe-x Ei(x) and E n (x) � xeX En (x),n = 1,2,3.

Table B.2. Normalized exponential integral functions.

X

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.08 0.10 0.15 0.30 0.40 0.60 0.80 1.00 1.25 1.50 1.75 2.00 2.50 3.00 3.50 4.00 5.00 6.00 8.00 10.00 12.00 15.00 20.00 25.00 35.00 50.00 75.00 100.00 150.00 250.00 500.00 1000.00 00

Ei(x) = xe- "' E;(x) 0.00000 -0.03978 -0.06498 -0.08440 -0.09997 -0.11262 -0.12292 -0.13787 -0.14684 -0.15029 -0.06727 0.02809 0.25351 0.48434 0.69717 0.92435 1.10492 1.24186 1.34096 1.45162 1.48373 1.47178 1.43821 1.35383 1.27888 1.18184 1.13147 1.10297 1.07810 1.05595 1.04365 1.03036 1.02084 1.01369 1.01019 1.00673 1.00400 1.00196 1.00100 1.00000

Ei(x)= xe"' Ei(x)

E2(x) = xe"' E2(x)

E3(x) = xe"' E3(x)

0.00000 0.04079 0.06845 0.09148 0.11163 0.12972 0.14623 0.17566 0.20146 0.25522 0.36676 0.41913 0.49676 0.55300 0.59635 0.63879 0.67239 0.69979 0.72266 0.75881 0.78625 0.80787 0.82538 0.85211 0.87161 0.89824 0.91563 0.92791 0.94080 0.95437 0.96287 0.97294 0.98076 0.98701 0.99019 0.99342 0.99603 0.99801 0.99900 1.00000

0.00000 0.00959 0.01863 0.02726 0.03553 0.04351 0.05123 0.06595 0.07985 0.11172 0.18997 0.23235 0.30194 0.35760 0.40365 0.45151 0.49142 0.52537 0.55469 0.60296 0.64125 0.67246 0.69847 0.73945 0.77037 0.81410 0.84367 0.86504 0.88794 0.91258 0.92831 0.94726 0.96222 0.97435 0.98058 0.98692 0.99210 0.99602 0.99801 1.00000

0.00000 0.00495 0.00981 0.01459 0.01929 0.02391 0.02846 0.03736 0.04601 0.06662 0.12150 0.15353 0.20942 0.25696 0.29817 0.34281 0.38143 0.41530 0.44531 0.49630 0.53813 0.57319 0.60306 0.65139 0.68890 0.74359 0.78167 0.80978 0.84047 0.87418 0.89608 0.92288 0.94438 0.96205 0.97125 0.98076 0.98800 0.99400 0.99700 1.00000

439

Sec. B.3. Exponential Integral Function

Special values.

(B.1 3) n > 1. For n > 0, and especially useful for x

Series representations.

E( n X)

(B.14)

» l,

00 i i -1)! x +L(-l) (n+ = X-1 e-[l . I ]. .

x•(n - l).

i=l

(B.1 5)

For x «: 1 and n > 0, auseful series representation is oo (-x) n-l (-xr A + , lnx ) En(x) = ((B.16) n , _ ! (n l) (m _ n+l) m! m=O m# n-l where 1 = 0.577 216 ..., A1 = 0, and , for n > l, An = z:::;;::,�\ m-.1 Note that for X > 0, oo n = - 1- lnx - (-x) E(x) (B.17) 1 L., nn! n =l

L

'°' --,

whereas

�· L =l n n. oo

lnx+ Ei(x)=+ ,

n

(B.18)

n

Derivatives and integrals.

J

dEn(x) _ -En-( l X) dx du En(x)

100

'

n 2: l,

= -En+i(u)+constant, n 2 0,

du En(u)

= En+i(x), n 2 0,

(B.19) (B.20) (B.21)

where x and u are positive . Integration of Eq. (B.1 1) by parts leads to En(x) =(n -1)-1 [e-x - x En-(x) 1 ], n > 1. Successive integration by parts leads to

Recurrence relationships.

En(x)

=

'°'6

[n-m-1 (

where n > m 2 1.

-x) i e-x(n - 2 - i)" '. (-x) n-m(m - 1)1. Em(X) ' ]+ (n -1)! (n -1)!

(B.22)

(B.23)

440

Mathematical Tidbits

Direct application of Eq. (B.23) leads to the result that for ac

Special integrals.

i

and ad positive, c

d

dx x-m En (ax) -

=

[

(n - m -l)!(-ar-l E n-m+i(ax) (n -l)!

� (n - 2 - i)!(-a)i E 1(ax) ] (n -l)!xm-i-l m� i-0

Id

,

n>m2'.l.

C

Repeated integrations by parts leads to the result that for ad k/a < l, and a# k# 0:

1 d dx e

kx

Appendix B

En (ax + b)

=

[k-le kx �

- k- 1

(i)

n-l

(if

(B.24)

> -b, ac > -b,

En-i(ax + b)

e- kb/a E 1 ([1 - k/al[ax + bl)]

1:.

(B.25)

For k/a>l, the E1 function in the last term is replaced by -Ei([k/a -l][ax + bl). Finally, for kd > -b, kc > -b, and k = a# 0:

t

dx e'" E, (kx+b)

= [k-' e'" � E,-;(kx + b) + k-' e-b ln(kx + b)] [. (B.26)

Numerical evaluation. The first-order exponential integral for 0 ::; x ::; 1 can be approximated with an error jf(x)I < 2 x 10-7 by [Abramowitz and Stegun 1964)

(B.27) where a0 a3

= -0.57721 566 = 0.05519 968

a1

a4

= 0.99999 193 = -0.00976 004

a2

a5

= -0.24991 055 = 0.00107 857.

For 1::; x < oo with IE(x)I < 5 x 10-5, xex E1(x)

=

x2 + a1x + a2 + E(x), 2 X + b 1X + b 2

(B.28)

where a1 = 2.334733, a2 = 0.250621, b1 = 3.330657, and b2 = 1.681534. The higher-order exponential integral functions can readily be evaluated from the recurrence relation of Eq. (B.22).

8.4

LEGENDRE POLYNOMIALS

The Legendre polynomial u differential equation

= Pn (x),

n

= 0,l, 2, . . . , is one set of solutions to the

(1 - x 2 )u" - 2xu' + n(n + l)u = 0.

(B.29)

441

Sec. 8.4. Legendre Polynomials

The first few Legendre polynomials are (see Fig. B.4)

=

3x2 -1 2

Po (x)

= 1,

Pi ( X )

5x 3 - 3x . = x, p.3 ( X ) _ 2

P2 (x)

1. ........ -c Q.

·e

C:

., ., C:

0.5

0.0 -0.5 -1.

-1.

-0.5

0.0

0.5

1.

X

Figure B.4. First four Legendre polynomials, Pn (x).

The higher-order polynomials may be generated by the use of the recursion rela­ tionship n 2 -1 n-1 Pn (x) = --XPn -i(x) - --Pn - 2 (x) (B.30)

n

n

or the series

(B.31) where misn/2 or (n-1) /2 , whichever is integral. Note that asn is even or odd, Pn (x) contains only even or odd powers of x up to the nth power. The Legendre polynomials are orthogonal with unit weight factors over the interval (-1, 1) , such that

J_1

1

dxPn (x)Pm(x)

=

{ 0,

2 __ 2n+ 1'

m#n, m=n.

(B.32)

442

Mathematical Tidbits

Appendix B

Equation (B.31) may be written as Pn(X)

= On L ')';;:,xm,

(B.33)

m=O

and its inverse as Xn = /3n

n

L (;:.Pm(x),

(B.34)

m=O

where a n , f3n , ,,;;-,, and (;;-, are constants. As n is even or odd, only even or odd m-indexed values of,,;;-, and(;;-, are nonzero. Using Eq. (B.34) and the orthogonality property, it follows that

11

-1

mPn(x)

dx x

= 0,

n > m.

(B.35)

The Legendre polynomials are complete on the interval (-1,1); that is, an ar­ bitrary function f(x) can be expanded uniquely as f(x) =

f:

n=O

2 ) ( n: l fn Pn(x)

(B.36)

and it follows from the orthogonality property, Eq. (B.35), that the expansion co­ efficients fn are given by (B.37) In certain radiation shielding problems there arise the associated Legendre func­ tions of the first kind, P;:'(x), which may be generated using the relationship (B.38) These functions are needed, for example, in evaluating Pn(cos '19s ), where cos '195 fM)' = Pn(cos '19)(cos '19') +sin '19 sin '19' cos('ljJ - '1/J'). In this case Pn(cos'l9.)

=

= Pn(cos'l9)Pn(cos'l9')

� (n- m)1 ;P::'(cos'l9)P;:'(cos'l9') cos[m('I/J - '1/J')]. (B.39) (n +m) . m=l

+ 2� 8.5

CHANDRASEKHAR'S H FUNCTION

The function tabulated in Table B.3 is one of a number of specialized functions described by Chandrasekhar [1960] in his treatise on radiative transfer. The function H(,.,,,cos '19) is the solution of the nonlinear integral equation = � [ 1 uH(,.,,,u) 1 �+ du _ u+cos'l9 H(,.,,,cos'l9) 2}0

(B.40)

443

Sec. B.6. Dirac Delta Function

Table B.3. Chandrasekhar's H(r;,,cos?J) function. K,

cos0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

1.0000 1.0124 1.0186 1.0230 1.0263 1.0289 1.0311 1.0328 1.0344 1.0357 1.0368

1.0000 1.0256 1.0389 1.0483 1.0555 1.0612 1.0659 1.0698 1.0732 1.0761 1.0786

1.0000 1.0399 1.0611 1.0764 1.0881 1.0976 1.1054 1.1120 1.1176 1.1225 1.1268

1.0000 1.0554 1.0858 1.1079 1.1252 1.1392 1.1509 1.1608 1.1694 1.1768 1.1834

1.0000 1.0724 1.1135 1.1439 1.1680 1.1877 1.2043 1.2186 1.2309 1.2417 1.2513

1.0000 1.0913 1.1452 1.1859 1.2186 1.2458 1.2689 1.2889 1.3063 1.3217 1.3354

1.0000 1.1130 1.1825 1.2364 1.2806 1.3179 1.3501 1.3781 1.4029 1.4250 1.4447

cos0

0.8

0.85

0.9

0.925

0.95

0.975

1.0

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

1.0000 1.1388 1.2286 1.3006 1.3611 1.4133 1.4590 1.4995 1.5358 1.5685 1.5982

1.0000 1.1541 1.2570 1.3411 1.4129 1.4758 1.5315 1.5814 1.6265 1.6675 1.7050

1.0000 1.1721 1.2914 1.3914 1.4785 1.5560 1.6259 1.6893 1.7474 1.8008 1.8501

1.0000 1.1828 1.3123 1.4225 1.5198 1.6073 1.6870 1.7601 1.8274 1.8899 1.9480

1.0000 1.1952 1.3373 1.4605 1.5710 1.6718 1.7647 1.8509 1.9313 2.0065 2.0771

1.0000 1.2111 1.3703 1.5117 1.6414 1.7621 1.8753 1.9822 2.0834 2.1795 2.2710

1.0000 1.2474 1.4504 1.6425 1.8293 2.0128 2.1941 2.3740 2.5527 2.7306 2.9078

K,

It may be evaluated by using an iterative solution of Eq. (B.40) (using numeri­ cal quadrature for the integral) or by using numerical integration to evaluate the following exact expression [Mendelson and Summerfield 1964]: H("',cos{))

1 = ----------1 �2

(v0 + cos{J)(l + cos{J)(l - "')

[ 111

x exp - 7f

where

B.6

Vo

du _ --- tan o u + cos{)

1 1

is the positive root of 1 - /.;2 (s-1)

(Mevs-1)

(s-1)

1.867(-16)° 7.847(-11) 2.332(-05) 5.407(-05) 2.131(-04) 9.512(-05)

2.646(-04) 1.799(-03) 1.287(-02) 7.512(-02) 3.736(-01) 2.275(+00)

3.746(-13) 1.690(-11) 2.011(-08) 3.071(-06) 1.319(-04) 2.159(-04) 1.606(-04) 1.930(-05)

1.551(-05) 8.283(-05) 3.646(-04) 4.032(-03) 2.218(-02) 1.219(-01) 8.296(-01) 4.867(+oo)

1.086(-14) 5.064(-12) 1.246(-07) 1.476(-06) 3.387(-05) 3.933(-04) 3.374(-04) 6.828(-05)

3.767(-06) 1.431(-05) 1.717(-04) 5.364(-04) 5.157(-03) 4.566(-02) 4.092(-01) 3.064(+oo)

°'il

Group 4 (N4 = 9) 5-7.5 MeV

Group 5 (Ns = 9) 4-5 MeV

0:i3

,\;3

Group 6 (N5 = 11) 3-4 MeV

a;4 (Mevs-1)

,\;4 (s-1)

a; 5 (MeVs-1)

(s-1 )

(MeVs-1)

°'i6

,\;5 (s-1)

8.365(-09) 4.819(-10) 1.014(-06) 3.541(-06) 3.897(-05) 6.004(-04) 1.149(-03) 7.649(-04) 8.963(-05)

2.456(-06) 2.760(-06) 5.871(-05) 2.015(-04) 1.528(-03) 1.561(-02) 1.205(-01) 8.295(-01) 4.917(+00)

4.024(-08) 5.327(-09) 2.446(-06) 9.731(-06) 6.733(-05) 1.790(-03) 2.975(-03) 2.518(-03) 3.125(-04)

2.456(-06) 3.154(-06) 2.846(-05) 1.617(-04) 2.322(-03) 1.635(-02) 1.387(-01) 8.389(-01) 4.866(+00)

5.703(-15) 2.692(-18) 1.594(-08) 2.830(-07) 9.359(-07) 2.550(-05) 8.799(-05) 1.083(-03) 4.705(-03) 7.543(-03) 1.482(-03)

2.046(-09) 4.900(-09) 9.912(-07) 2.501(-06) 1.395(-05) 1.523(-04) 3.690(-03) 2.219(-02) 1.706(-01) 1.158(+00) 4.931(+00)

a Read as 1.867 X 10-16, etc. Source: LaBauve et al. [1980).

,\;5

Appendix G

513

Fission-Product Source Parameters

Table G.6. Total gamma- and beta-decay constants for fis­ sion products resulting from the thermal-neutron-induced fission of 239Pu.

All fission product betas

= 11)

(Nt

All fission product gammas

(Nt

= 11)

Ait

a ;t

Ait

(MeV s-1)

(s-1)

(MeV s-1)

(s-1)

2.835(-ll)a 2.776(-09) 1.811(-08) 2.485(-07) 7.942(-06) 1.512(-04) 1.869(-03) 2.543(-02) 1.434(-01) 2.938(-01) 6.142(-02)

8.016(-10) 2.425(-08) 2.399(-07) 1.959(-06) 2.208(-05) 2.548(-04) 1.833(-03) 1.786(-02) 1.530(-01) 1.206(+00) 5.098(+00)

3.062(-11) 4.761(-10) 3.390(-08) 4.775(-07) 7.251(-06) 2.278(-04) 1.336(-03) 2.076(-02) 9.920(-02) 2.223(-01) 4.919(-02)

7.441(-10) 2.856(-08) 2.052(-07) 1.772(-06) 2.071(-05) 2.323(-04) 1.727(-03) 1.699(-02) 1.566(-01) 1.213( +oo) 5.126(+oo)

O.it

All gaseous fission products betas

(Nt

= 11)

Ait

O.it

All gasseous fission product gammas O:it

(Nt

= 11)

Ait

(MeV s-1)

(s-1)

(MeV s-1)

(s-1)

6.411(-13) 4.791(-18) 9.763(-09) 9.019(-08) 1.183(-06) 1.375(-05) 4.336(-04) 3.841(-03) 1.059(-02) 8.181(-03) 7.977(-04)

2.047(-09) 5.532(-09) 1.027(-06) 2.384(-06) 1.585(-05) 1.703(-04) 2.761(-03) 1.776(-02) 1.439(-01) 7.672(-01) 5.105( +oo)

5.703(-15) 2.692(-18) 1.601(-08) 3.369( .,--07) 3.263(-06) 4.000(-05) 1.796(-04) 3.772(-03) 9.534(-03) 1.144(-02) 2.034(-03)

2.046(-09) 4.900(-09) 9.916(-07) 2.501(-06) 2.158(-05) 1.539(-04) . 2.280(-03) 1.764(-02) 1.464(-01) 1.021( +oo) 4.890(+oo)

a

Read as

2.835

X

10-11, etc. [1980].

Source: LaBauve et al.

Appendix H

Gamma and X-Ray Photons Emitted by Selected Radionuclides This appendix lists the principal x-ray, gamma, and annihilation photons emitted by important radionuclides. The half-life is given in units of years (y), days (d), hours (h), minutes (m), or seconds (s). For each radionuclide selected , the principal decay mechanisms [beta minus decay (/3-), positron decay (/3+), internal conversion (IC), electron capture (EC), or alpha decay (a)] are indicated. Also listed with the principal decay modes is the total kinetic energy of all emitted charged particles (in units of keV /decay). These charged particles include all beta electrons, positrons, Auger electrons, internal conversion electrons, and alpha particles. The bulk of Table H.l is a listing of the energy (in keV) and frequency (% per decay) of the principal photons emitted by each radionuclide. For a particular photon to be listed, its energy must be greater than 10 keV, and either its frequency be greater than 1% per decay or its contribution be greater than 1% of the sum of all photon energy emitted per decay, E-y ,tot• Finally, a residual gamma entry, such as R(2.15), is made if the sum of unlisted photon energies (each weighted by its frequency) is greater than 1% of E-y ,tot• The value given with each gamma residual is the percent of E-y ,tot contributed by unlisted photons. The following tabulation is extracted from comprehensive data files generated by Eckerman et al. 1 For the most part, data files prepared for the MIRD compendium2 of radioactive decay data were used. For nuclides not in the MIRD document, data files for the ICRP compendium3 were used. An exception is the nuclide 16 N for which data were taken from Kocher's tabulation.4 1 K.F. Eckerman, R.J. Westfall, J.C. Ryman, and M. Cristy, Nuclear Decay Data Files of the Dosimetry Research Group, ORNL/TM-12350, Oak Ridge National Laboratory, Oak Ridge, TN, 1993. Also available from the ORNL Radiation Shielding Information Center as DCL-172 NUCDECA Y: Nuclear Decay Data for Radiation Dosimetry Calculations for ICRP and MIRD. 2D.A. Weber, K.F. Eckerman, L.T. Dillman, and J.C. Ryman, MIRD: Radionuclide Data and Decay Schemes, Society of Nuclear Medicine, New York, 1989. 3 Radionuclide Transformations, Publication 38, Annals of the ICRP, Vols. 11-13, International Commission on Radiological Protection, Pergamon Press, Elmsford, NY, 1983. 4Kocher, D.C., Radioactive Decay Tables, Report DOE/TIC 11026, Technical Information Center, U.S. Department of Energy, Washington, DC, 1981.

514

Appendix H

515

Photons Emitted by Selected Radionuclides Table H.1. Principal photons emitted by selected radionuclides.

Nuclide 3H uc 14c 13N 16N 150 190 18p 22Na 24Na 28Al 31Si 32p 35g 36Cl 3sc1 41Ar 38K 40K 42K 43K 45Ca 49ca 47Ca 468c 47sc 51Cr 54Mn 56Mn 52pe 55pe 59pe 57Co 58Co

Halflife

Mode and average energy (keV /decay)°

Principal gamma, x-ray, and annihilation photons (keV [% frequency])

none 511.0[199.52] none 511.0[199.64] 1754.8[0.13] 2741.2[0.76] 6129.2[69.0] 7115.2[5.0] 511.0[199.77] 110.0[3.45] 197.0[90.34] 1356.0[50.34] 1444.0[3.35] 1550.0[2.20] R(l.07) 511.0[200.00] 109.8m 13+ EC 249.8 511.0[179.80] 1274.5[99.94] 2.602y 13+ EC 194.1 1368.5[100.00] 2754.1[99.94] 15.02h ;3- 553.6 1778.8[100.00] 2.240m ;3- 1242.3 1266.1[0.07] 157.3m ;3- 595.4 14.26d ;3- 694.7 none none 87.44d ;3- 48.8 3.01E5y ;3-13+ EC 273.6 511.0[0.03] R(l.96) 37.21m ;3- 1529.4 1642.4[32.50] 2167.5[44.00] 1293.6[99.16] 1.827h ;3- 463.8 7.636m 13+ EC 1208.6 511.0[198.93] 2167.0[99.80] 1.28E9y ;3- EC 522.6 1460.8[10.70] 12.36h ;3- 1429.5 1524.7[17.90] R(l.10} 22.6h ;3- 309.1 220.6[4.11] 372.8[87.27] 396.9[11.43] 593.4[11.03] 617.5[80.51] 1021.8[1.88] none 163.8d ;3- 77.2 8.715m ;3- 869.9 3084.4[92.10] 4071.9[7.00] R(l.32) 4.536d ;3- 350.6 489.2[6.51] 807.9[6.51] 1297.1[74.00] 83.83d ;3- 112.2 889.3[99.98] 1120.5[99.99] 3.345d ;3- 162.8 159.4[67.90] 27.70d EC 3.87 320.1[10.08] R(3.39) 312.5d EC 4.22 834.8[99.98] 2.5785h ;3- 829.9 846.8[98.87] 1810.7[27.19] 2113.1[14.34] 2522.9[0.99] 2657.4[0.65] 8.275h 13+ EC 193.5 168.7[99.23] 511.0[112.00] 2.73y EC 4.20 R(J00.0: 1.69 keV/decay) 44.50d ;3- 117.8 142.7[1.02] 192.3[3.08] 1099.3[56.50] 1291.6[43.20] 271.8d EC 18.6 14.4[9.23] 122.1[85.95] 136.5[10.33] R(3.96) 70.92d 13+ EC 34.2 511.0[29.98] 810.8[99.45] R(l.66} 12.33y 20.39m 5730y 9.965m 7.13s 122.2s 26.91s

;3- 5.67 13+ EC 384.6 ;3- 49.5 13+ EC 490.9 ;3- 2695.1 13+ EC 734.5 ;3- 1733.1

(cont.)

aonly principal decay modes are indicated. The average decay energy is the average kinetic energy per decay from all emitted charged particles (/3-, 13+, a, Auger, and internal conversion electrons). Subsequent photon energy is given separately by the next entries.

516

Photons Emitted by Selected Radionuclides Table H.1.

Nuclide 60-mco 60co 63Ni 62zn

Half­ life

(cont.)

Principal photons emitted by selected radionuclides.

Mode and average energy (keV / decay ) a

10.47m 135.270y 13100.ly 139.26h 13+

IT 57.8 96.4 17.1 EC 32.6

243.9d 13+ EC 6.8

Appendix H

Principal gamma, x-ray, and annihilation photons (keV [% frequency])

58.6[2.07] 1330.0[0.25] R(34.06) 1173.2[99.90] 1332.5[99.99] none

40.8[25.19] 243.4[2.49] 246.9[1.88] 260.4[1.34] 394.0[2.21] 507.6[14.65] 511.0[16.78] 548.3[15.16] 596.6[25.70] R(2.11) 511.0[2.83] 1115.6[50.70]

13.76h 13- IT 22.3 9.49h 13+ EC 983.3

67Ga

72As 74As 76As

438.6[94.87] 511.0(113.1] 833.5[6.03] 1039.2[37.90] 1333.1[1.23] 1918.6(2.14] 2189.9[5.71] 2422.8[1.96] 2752.0(23.19] 3229.2(1.48] 3381.3[1.40] 3422.8[0.83] 3791.6[1.02] 4086.3[1.14] 4295.9[3.49] 4462.l[0.72] 4806.6(1.48] R(4.19) 3.261d EC 34.4 91.3[2.96] 93.3(37.00] 184.6[20.40] 209.0[2.33] 300.2(16.60] 393.5[4.64] R(4.51) 68.06m 13+ EC 739.4 511.0(178.08] 1077.4[3.00] 14.lOh 13- 502.0 601.0[5.54] 630.0[24.77] 786.4[3.20] 810.2[2.0l] 834.0(95.63] 894.3[9.88] 970.5[1.10] 1050.7[6.91] 1230.9(1.45] 1230.9[1.45] 1260.l[l.13] 1276.8(1.56] 1464.0[3.55] 1596.7[4.24] 1861.1[5.25] 2109.5(1.04] 2201.7[25.9] 2491.0[7.68] 2507.8[12.8] R(4.87) 26.0h 13+ EC 1033.3 511.0[175.27] 629.9[7.92] 834.0(79.50] 1464.0(1.11] R(8.54) 17.76d 13-13+ EC 268.2 11.0[1.45] 511.0(58.20] 595.8(59.22] 634.8(15.39] R{l.38) 559.1[44.70] 563.2(1.17] 657.0[6.08] 1212.7[1.63] 1216.0(3.84] 26.32h 13- 1064.4 1228.5(1.39] 1439.1[0.33] 1787.7[0.33] 2096.3[0.66] 2110.8(0.39] 119.8d EC 14.7 16.2h

13+ EC 645.7

57.04h 13+ EC 9.5

35.30h 13- 139.0 81=Kr 13s

IT EC 59.0

R(4-02)

10.5(16.50] 10.5(32.11] 11.7[2.41] 11.7(4.72] 66.1(1.14] 96.7(3.48] 121.1(17.32] 136.0(58.98] 198.6(1.47] 264.7[59.10] 279.5(25.18] 303.9[1.34] 400.7[11.56] 11.2[6.98] 11.2[13.55] 12.5[1.04] 12.5[2.03] 472.9[1.86] 511.0(109.53] 559.1(74.00] 563.2[3.55] 657.0(15.91] 1129.8[4.59] 1213.1[1.70] 1216.1[8.81] 1228.7[2.09] 1380.5[2.52] 1471.1[2.31] 1853.7[14.65] 2096.7[1.36] 2111.2[2.49] 2391.3[4.74] 2510.8(1.95] 2792.7[5.62] 2950.5[7.40] 2997.3[0.96] 3604.0[1.55] R(15.48) 11.2(15.48] 11.2(30.06] 12.5[2.31] 12.5[4.51] 87.6(1.45] 161.8[1.14] 200.4[1.25] 239.0(23.89] 249.8(3.08] 281.6(2.37] 297.2[4.30] 303.8(1.22] 385.0(0.86] 439.5(1.62] 484.6(1.03] 511.0[1.48] 520.7[23.17] 567.9[0.89] 574.6(1.23] 578.9[3.06] 585.5[1.62] 755.3[1.72] 817.8[2.15] 1005.0[0.96] R(2. 63) 221.4[2.26] 554.3(70.78] 606.3[1.17] 619.1(43.45] 698.3(28.49] 776.5(83.56] 827.8(24.04] 1007.6[1.27] 1044.0(27.20] 1317.5(26.52] 1474.8(16.32] R(l.32) 12.6(5.011 12.7[9.81l 14.1(1.50] 19o.4[61.10J

(cont.)

aonly principal decay modes are indicated. The average decay energy is the average kinetic energy per decay from all emitted charged particles (/3-, 13+, a, Auger, and internal conversion electrons). Subsequent photon energy is given separately by the next entries.

Appendix H

517

Photons Emitted by Selected Radionuclides Table H.1.

(cont.)

Mode and Half­ Nuclide average energy life (keV /decay ) a

Principal photons emitted by selected radionuclides. Principal gamma, x-ray, and annihilation photons (keV [% frequency])

85mKr 4.480h /3.- IT 255.5 85Kr 10.72y 13- 250.5 87Kr 76.3m 13- 1323.8

12.6[1.13] 12.7[2.19] 13.4(1.18] 151.2[75.37] 304.9[13.76] 517.0[0.43] 402.6[49.50] 673.9[1.91] 845.4(7.28] 1175.4(1.12] 1338.0[0.65] 1740.5(2.05] 2011.9[2.90] 2554.8(9.31] 2558.1[3.91] 2811.4[0.32] 3308.5(0.45] R(S.30)

88Kr

2.84h

13.3[2.40] 13.4[4.63] 27.5[2.07] 166.0[3.10] 196.3[25.98] 362.2[2.25] 834.8(12.98] 985.8(1.31] 1141.3(1.28] 1250.7[1.12] 1369.5[1.48] 1518.4[2.15] 1529.8(10.93] 2029.8(4.53] 2035.4[3.74] 2195.8[13.18] 2231.8[3.39] 2392.1[34.60] R(B.07)

s1Rb

4.576h 13+ EC 184.3

84Rb

32.87d 13-13+ EC 161,6 18.66d 13- 667.5 64.84d EC 9.0 2.81h IT EC 67.1 50.5d 13- 583.2 29.12y 13- 195.7 64.0h 13- 934.9 58.51d 13- 602.2 64.02d 13- 117.9 16.90h 13- 700.4

86Rb sssr 87msr sgsr gosr 90y 91y 95zr 97zr

95mNb 86.6h

13- 364.3

13- IT 174.5

95Nb 99Mo

35.02d 13- 44.6 65.94h 13- 392.7

99mTc 103 Ru 106Ru io3mRh

6.0lh 39.26d 368.2d 56.llm

13- IT 16.1 13- 66.6 13- 10.0 . IT 37.1

106Rh

29.9s

13- 1413.1

12.6[16.73] 12.7[32.35] 14.1[2.54] 14.1[4.95] 190.3(64.03] 446.1[23.20] 456.7[3.02] 510.5[5.34] 511.0[57.05] 537.6[2.23] 803.7(0.83] 834.7[0.81] 1368.1[0.95] R(6.39) 12.6[11.60] 12.7(22.44] 14.1[1.76] 14.1(3.43] 511.0[51.82] 881.5[67.87] 1897.0[0.75] 1076.6[8.78] 13.3[17.10] 13.4[32.98] 15.0[2.64] 15.0[5.15] 514.0[98.30] 14.1[3.0l] 14.2(5.79] 388.4[82.26] 909.1(0.01] none R(J00.0: 0.00169 kev/decay)

1204.9[0.30] 724.2[44.15] 756.7[54.50] 254.2[1.25] 355.4[2.27] 507.6[5.05] 513.4[0.56] 602.4[1.39] 703.8[0.93] 804.5[0.65] 829.8[0.22] 854.9[0.33] 971.4(0.29] 1021.3(1.34] 1148.0(2.64] 1276.1(0.97] 1362.7[1.34] 1750.5(1.34] 1851.5[0.35] R(3.93) 16.5[11.71] 16.6(22.35] 18.6[1.99] 18.6(3.73] 204.1(2.24] 235.7(24.07] 765.8[99.79] 18.3(1.06] 18.4(2.01] 40.6[1.05] 140.5[4.52] 181.1[6.08] 366.4(1.15] 739.6[12.13] 778.0[4.34] R(2.11) 18.3[2.10] 18.4(3.99] 140.5[89.06] 497.1(90.90] 610.3(5.73] R(l.60) none 20.1[2.20] 20.2(4.17] 22.7(0.39] 22.7(0.73] 23.2(0.19] 39.8[0.07] R(S.85)

511.8[20.60] 616.2(0.70] 621.8(9.81] 873.1[0.42] 1050.1[1.46] 1128.0(0.39] 1562.0(0.15] R(4.16) (cont.)

aonly principal decay modes are indicated. The average decay energy is the average kinetic energy per decay from all emitted charged particles (/3-, 13+, a, Auger, and internal conversion electrons). Subsequent photon energy is given separately by the next entries.

518

Photons Emitted by Selected Radionuclides

Appendix H

Table H.1. (cont.) Principal photons emitted by selected radionuclides. Nuclide

Mode and Half­ average energy life (keV /decay)°

Principal gamma, x-ray, and annihilation photons (keV [% frequency])

llOmAg 249.9d ;3- IT 72.2

446.8[3.66] 620.3[2.78] 657.7(94.74] 677.6(10.72] 687.0(6.49] 706.7(16.74] 744.3(4.66] 763.9(22.36] 818.0(7.32] 884.7(72.86] 937.5(34.32] 1384.3(24.35] 1475.8(3.99] 1505.0(13.11] 1562.3(1.18] 1562.3[1.18] llOAg 24.6s ;3- EC 1181.5 657.7(4.50] 815.3(0.04] R(2.46) 109Cd 462.9d EC 82.6 22.0(28.47] 22.2(53.66] 24.9[4.95] 24.9(9.66] 25.5(2.71] 88.0(3.61] 111 In

R(l.12)

2.83d

EC 34.7

113mln 1.658h IT 134.0 115mrn 4.486h ;3- IT 172.4 125Sb 2.77y ;3- 100.2 1231

13.2h

EC 28.4

1241

4.18d

;3+ EC 195.6

60.14d EC 19.5 l.57E7y ;3- 63.2 8.04d 20.8h 6.61h

127Xe

36.4d

EC 32.4

l33mxe 2.188d IT 192.4 133Xe

5.245d ;3- 135.7

135mxe 15.29m ;3- IT 97.6 135Xe 9.09h ;3- 317.2

23.0(23.61] 23.2(44.38] 26.1[4.15] 26.1(8.08] 26.6(2.35] 171.3(90.24] 245.4(94.00] 24.0[6.99] 24.2[13.12] 27.2(1.24] 27.3(2.40] 391.7(64.23] 24.0(9.54] 24.2(17.90] 27.2(1.69] 27.3(3.28] 336.2(45.37] 27.2(13.43] 27.5(25.01] 30.9(2.43] 31.0(4.73] 31.7(1.45] 35.5(4.31] 176.3(6.70] 380.5(1.50] 427.9(29.50] 463.4(10.32] 600.6(17.64] 606.7(4.84] 636.0(11.33] 671.5(1.72] R(l.29) 27.2(24.65] 27.5(45.90] 30.9(4.46] 31.0(8.68] 31.7(2.66] 159.0(83.30] 440.0(0.43] 529.0[l.39] 538.5(0.38] R(2. 74) 27.2(16.47] 27.5[30.67] 30.9(2.98] 31.0(5.80] 31.7[1.78] 511.0(45.73] 602.7(60.50] 722.8[9.98] 1325.5(1.43] 1376.0(1.66] 1509.5[2.99] 1691.0(10.41] 2091.0(0.57] 2232.3[0.57] 2283.3(0.66] 2746.9(0.46] R(7.51} 27.2(39.71] 27.5(73.95] 30.9(7.19] 31.0(13.98] 31.7(4.29] 35.5(6.65] R(l.35) 29.5(19.96] 29.8(37.02] 33.6(3.65] 33.6[7.11] 34.4[2.37] 37.6(7.50] R(l.38}

29.5(1.40] 29.8(2.59] 80.2(2.62] 284.3[6.06] 364.5(81.24] 637.0(7.27] 722.9(1.80] R(J .47) 510.5(1.81] 529.9(86.31] 706.6(1.49] 856.3(1.23] 875.3(4.47] 1236.4(1.49] 1298.2(2.33] R(5.21} 220.5(1.75] 288.5(3.09] 417.6[3.52] 546.6(7.13] 836.8(6.67] 972.6(1.20] 1038.8[7.93] 1101.6(1.60] 1124.0(3.61] 1131.5(22.53] 1260.4(28.63] 1457.6[8.65] 1502.8(1.07] 1566.4(1.29] 1678.0(9.53] 1706.5(4.09] 1791.2(7.70] 2045.9[0.87] 2408.6(0.95] R(7.30} 28.3[24.98] 28.6(46.43] 32.2(4.55] 32.3(8.82] 33.1[2.83] 57.6(1.23] 145.3[4.29] 172.1(25.54] 202.9(68.30] 375.0[17.21] 29.5(16.05] 29.8(29.78] 33.6(2.94] 33.6(5.72] 34.4[1.91] 233.2(9.99] 30.6(13.11l 31.0(25.34] 34.9(2.52] 35.0[4.89] 35.8(1.10] 81.0(37.42] R(l.19} 29.5(3.94] 29.8(7.31] 33.6(1.40] 526.6(80.66] 30.6(1.46] 31.0[2.70] 249.8(90.13] 608.2(2.90] R(l.82) (cont.)

aonly principal decay modes are indicated. The average decay energy is the average kinetic energy per decay from all emitted charged particles (/3-, ;3+, a, Auger, and internal conversion electrons). Subsequent photon energy is given separately by the next entries.

Appendix H

Photons Emitted by Selected Radionuclides

519

Table H.1. (cont.) Principal photons emitted by selected radionuclides. Nuclide

Mode and Half­ average energy life (keV /decay)°

138Xe

14.17m 13- 672.6

129Cs

32.06h EC 17.8

13ocs

29.9m 13+ EC 401.1

131cs 9.69d 134mcs 2.91h

EC 6.6 IT 112.4

134Cs

2.062y 13- EC 164.3

137Cs 131Ba

30.0 y 13- 187.4 11.8d 13+ EC 45.8

133mBa 38.9h

IT 221.3

i35mBa 28.7h

IT 208.1

137mBa 2.552m IT 65.1 140Ba 12.74d 13- 312.8 1401a

40.27h 13- 535.1

141Ce 144Ce

32.50d 13- 170.7 284.3d 13- 92.2

144pr 17.28m 13- 1207.5 147pm 2.623y 13- 62.0 169yb 32.02d EC 123.2 198 Au

2.696d 13- 327.0

Principal gamma, x-ray, and annihilation photons (keV [% frequency]) 30.6(1.15] 31.0(2.12] 153.8(5.95] 242.6(3.50] 258.3(31.50] 396.4(6.30] 401.4(2.17] 434.5(20.32] 1114.3(1.47] 1768.3(16.73] 1850.9(1.42] 2004.8(5.36] 2015.8(12.25] 2079.2(1.44] 2252.3(2.29] 2321.9(0.62] R(10.15} 29.5(29.74] 29.8(55.18] 33.6(5.45] 33.6(10.60] 34.4(3.53] 39.6(2.99] 278.6(1.33] 318.2(2.46] 371.9(30.80] 411.5(22.45] 548.9(3.42] 588.5(0.61] R(2.64} 29.5(11.95] 29.8(22.16] 33.6(2.19] 33.6(4.26] 34.4(1.42] 511.0(89.29] 536.1(4.10] R(4.98} 29.5(21.02] 29.8(38.99] 33.6(3.85] 33.6(7.49] 34.4(2.50] R(1.60) 30.6(9.03] 31.0(16.68] 34.9(1.66] 35.0(3.22] 35.8[1.12] 127.5(12.70] R(3.04} 475.4(1.46] 563.2(8.38] 569.3(15.43] 604.7(97.56] 795.8(85.44] 801.9(8.73] 1167.9(1.80] 1365.2(3.04] none 30.6(27.72] 31.0(51.24] 34.9(5.10] 35.0(9.89] 35.8(3.43] 123.8(29.05] 133.6(2.19] 216.1(19.90] 239.6(2.41] 249.4(2.81] 373.3(13.33] 404.0(1.29] 486.5(1.89] 496.3(43.78] 585.0(1.23] 620.0(1.57] 923.9(0.70] 1047.6(1.19] R(3.50} 12.3(1.39] 31.8(15.18] 32.2(27.95] 36.3(2.80] 36.4(5.42] 37.3(1.96] 276.1(17.51] R(l.16} 31.8(15.41l 32.2(28.39] 36.3(2.84] 36.4(5.51l 37.3(1.99] 268.2(15.64] 31.8(2.13] 32.2(3.92] 661.6(89.78] 13.8(1.22] 30.0(13.78] 33.4(1.01] 162.6(6.21] 304.9(4.30] 423.7(3.15] 437.6(1.93] 537.3(24.39] R{l .12} 34.7(1.05] 328.8(20.74] 432.5(2.99] 487.0(45.94] 751.8[4.41] 815.8(23.64] 867.8(5.59] 919.6(2.68] 925.2(7.05] 1596.5(95.40] 2521.7(3.43] R(l.58) 35.6(4.86] 36.0(8.87] 40.7(1.76] 145.4(48.20] R(l.04) 35.6(2.96] 36.0(5.40] 40.7(0.55] 40.7(1.07] 80.1(1.64] 86.5(0.35] 91.0(0.35] 133.5(10.80] R(3.37) 696.5(1.48] 1489.2(0.30] 2185.7[0.77] R(100.0: 0.0044 keV/decay

49.8(53.00] 50.7(93.73] 57.3[9.94] 57.5(19.17] 59.1(8.20] 63.1(43.74] 93.6(2.66] 109.8(17.37] 118.2(1.88] 130.5(11.11] 177.2(21.45] 198.0(34.94] 261.1(1.90] 307.7(10.80] R(l.50) 10.8(1.38] 411.8(95.51l 675.9(1.06]

(cont.) aonly principal decay modes are indicated. The average decay energy is the average kinetic energy per decay from all emitted charged particles (/3-, 13+, a, Auger, and internal conversion electrons). Subsequent photon energy is given separately by the next entries.

520

Photons Emitted by Selected Radionuclides

Appendix H

Table H.1. (cont.) Principal photons emitted by selected radionuclides. Nuclide

Mode and Half­ average energy life (keV /decay) °

197Hg

64.lh

203Hg 201Tl

46.61d 13- 99.0 73.lh EC 43.4

204Tl

3.779y (3- EC 237.9

2osT1

3.053m 13- 611.0

210pb

22.3y

212Pb

10.64h (3- 175.4

214Pb

26.8m 13- 293.2

210Bi 212Bi

5.013d a 13- 388.9 60.55m a 13- 2678.3

214Bi

19.9m a 13- 660.5

210p0 212p0 214p0 21sp0 220Rn 222Rn 224Ra 226Ra

138.4d 0.298 s 164.3 s 3.05m 55.6s 3.824d 3.66d 1600y

EC 66.5

a (3- 37.9

Principal gamma, x-ray, and annihilation photons (keV [% frequency]) 11.2[1.81] ll.4[12.84] 11.6[4.39] 11.6[2.31] 13.4[2.57] 67.0[20.47] 68.8(34.96] 77.3[18.00] 77.6(4.14] 78.0(7.94] 80.2(3.27] 191.4(0.61] R(3.77) 10.3(2.17] 12.2(1.62] 70.8(3.86] 72.9(6.53] 82.6(1.49] 279.2(81.46] 11.6(1.21] 11.8(10.55] 11.9(4.19] 12.0(1.51] 13.8(2.15] 68.9[27.15] 70.8(46.18] 79.8(5.48] 80.3(10.53] 82.6(4.43] 135.3(2.65] 167.4(10.00] R(3.11) 11.8(0.17] 68.9(0.42] 70.8(0.71] 79.8(0.08] 80.3(0.16] 82.6(0.07] R(5.53)

10.6(1.23] 12.8(2.23] 75.0(3. 76] 277.4(6.31 l 510.8(22.61l 583.2(84.48] 763.1(1.81] 860.6(12.42] 2614.5(99.16] 10.7(0.93] 10.8(8.37] 12.7(2.13] 13.0(2.09] 13.0(4.11] 13.2(2.46] 15.2(0.87] 15.6(0.74] 15.7(0.90] 46.5(4.05] R(2.66} 10.8(5.84] 13.0(1.46] 13.0(3.75] 74.8(10.42] 77.1(17.51] 86.8(2.08] 87.3(3.99] 89.9(1.84] 238.6[43.65] 300.1(3.34] 10.8(5.13] 13.0(1.28] 13.0(3.08] 53.2(1.10] 74.8(6.39] 77.1(10.73] 86.8(1.28] 87.3(2.45] 89.9(1.13] 241.9(7.46] 295.2(19.17] 351.9(37.06] 785.9(1.09] 839.0(0.59] R(4.88} none 10.3(2.82] 12.2(1.50] 39.9(1.09] 452.8(0.31] 727.2(6.65] 785.4[1.11] 893.4(0.37] 952.1(0.18] 1078.6(0.54] 1512.8(0.31] 1620.6(1.51] 1679.5(0.07] 1806.0(0.11] R(3.60} 609.3(46.09] 665.5(1.56] 768.4(4.88] 806.2(1.23] 934.1(3.16] 1120.3(15.04] 1155.2(1.69] 1238.1(5.92] 1281.0(1.47] 1377.7(4.02] 1401.5(1.39] 1408.0(2.48] 1509.2(2.19] 1661.3(1.15] 1729.6(3.05] 1764.5(15.92] 1847.4[2.12] 2118.5(1.21] 2204.1(4.99] 2447.7(1.55] R(J0.04)

a a a a a a a a

5304.6 8784.3 7687.1 13- 6001.4 6288.0 5489.3 5676.1 4778.0

803.0(0.0012] none 799.7(0.010] none 549.7(0.070] 510.0(0.078] 81.1(0.12] 83.8(0.20] 241.0(3.90] R(2.32) 81.1[0.18] 83.8(0.30] 186.0(3.28] R(3.83} (cont.)

0nly principal decay modes are indicated. The average decay energy is the average kinetic energy per decay from all emitted charged particles (/3-, 13+, a, Auger, and internal conversion electrons). Subsequent photon energy is given separately by the next entries. 0

Appendix H

Photons Emitted by Selected Radionuclides

521

Table H.1. (cont.) Principal photons emitted by selected radionuclides. Nuclide

Mode and Halfaverage energy life (keV/decay)0

233pa

27.0d

239 u

23.54m 13- 412.0

239Np

2.355d 13- 260.2

13- 196.3

241 Am 432.2y a 5537.7

Principal gamma, x-ray, and annihilation photons (keV [% frequency]) 13.4[1.74] 13.6[15.59] 16.4[4.11] 16.6(2.70] 17.2[10.02] 11.5[2.45] 20.2(2.41l 20.5[1.10] 20.1[1.04] 75.3[1.17] 86.6(1.76] 94.7(10.06] 98.4[16.25] 110.4(2.00] 111.3[3.80] 114.7(2.00] 300.1[6.19] 312.0[36.00] 340.5(4.21] 375.4(0.58] 398.6(1.19] 415.8[1.51] R(1.18) 13.9[5.21] 16.8[1.38] 17.1[1.28] 17.8[3.07] 18.0[l.13] 43.5[4.45] 74.7[50.00] 662.2[0.19] 748.1[0.10] 812.9[0.08] 819.2[0.15] 844.1[0.17] 964.3[0.09] R(10. 07) 12.1(1.21] 14.1[2.38] 14.3[21.31] 17.3(5.68] 17.6[1.59] 17.9[1.26] 18.3(16.27] 18.5[1.37] 21.4[3.97] 99.6[14.62] 103.8(23.39] 106.1[22.70] 116.3(2.89] 117.3[5.47] 120.8[2.92] 209.8(3.24] 228.2[10.72] 277.6[14.10] 285.4[0.78] 315.9[1.59] 334.3[2.03] R(1.99} 11.9(1.38] 13.8[2.75] 13.9[24.59] 16.8(6.53] 17.1[3.19] 17.5(1.41] 17.8[20.02] 18.0[2.82] 20.8(4.86] 21.1[1.33] 21.3[1.22] 21.5[1.03] 26.3[2.41] 59.5[35.90] R(1.16)

0 0nly principal decay modes are indicated. The average decay energy is the average kinetic energy per decay from all emitted charged particles (/3-, 13+, a, Auger, and internal conversion electrons). Subsequent photon energy is given separately by the next entries.

Index

A Absorbed dose, 124, 129 relation to kerma, 133 Absorbed fraction, 180 Absorption coefficient, ( see Interaction coefficient) Activation gamma photons, 103-104 Activation neutrons, 91 Activity, 24 (see also Radiation sources) Addition theorem, 388, 442 Agreement state, 10 Air kerma, (see Kerma) Air: air kerma buildup factor, 479 Berger parameters, 484 geometric progression parameters, 485-487 Taylor parameters, 483 electron range in, 70 energy for ion pair, 141 gamma-ray interaction coefficients, 452 medium for exposure, 141 proton range in, 70 radiation yield in, for electrons, 49 skyshine gamma photons, 256-259 neutrons, 322-323 ALBEDO-ALBEZ code, 244 Albedo: concept, 235-240 neutron, 312-318 examples, 313 fast neutrons, 312-315 secondary photons, 317-318 thermal neutrons, 317 photon, 235-242 Chilton-Huddleston formula, 240 data, 241-243

differential dose, 237 differential number, 236 dose reflection factor, 237 number reflection factor, 237 single scatter, 239-240 Alpha decay, 333 Alpha particles: decay-data tables, 514-521 use in neutron sources, 86-91 Aluminum: electron range in, 70 fast-neutron initial buildup factors, 289 fast-neutron relaxation distance, 289 fast-neutron albedo, 314 neutron cross section, 52 neutron nonelastic cross section, 57 proton range in, 70 radiation yield in, for electrons, 49 stopping power in, 67 Ambient dose equivalent, 146, 151 Ambient radiation, 144 American Nuclear Society, 7 Analog Monte Carlo, 409 Angular flow: area source, 24 definition, 21 Angular fluence, (see Fluence) Angular flux density, (see also Flux density) Legendre expansion, 382 Angular moments, 388 Anisotropic transport, 361 Annihilation radiation, 45, 104, 368 ASFIT code, 217 Atomic Energy Commission, 9 Attenuation coefficients, ( see Interaction co­ efficients) Attenuation factor: broad-beam, 229-235, 420 broad beams of x rays, 231-233

523

524

Index data tables, 484 boundary effects, 225-227, 271-272 broad beam attenuation, 488-492 definition, 215 empirical formulas, 220-223 finite and infinite media compared, 220, 225-226 Foderaro-Hall approximation, 221 geometric-progression approximation: data tables, 485-487 definition, 222-223 initial, 289 interface effects, 228 neutron, 271-272, 285, 295 point and plane sources compared, 218220 point-kernel method, 222-225 stratified media, Kalos formula, 227 Taylor approximation: definition, 221 data table, 483

definition, 229 oblique photon beams, 230-231 photons in lead and concrete, 230 Attenuation, geometric and material, 158 Auger electrons, 44, 105

B Backscattering, ( see Albedo) Barn, definition, 30 Beam transmission, (see broad-beam attenuation) Becquerel, definition, 24 Berger buildup approximation, 222, 304 Beryllium: converter material, 85 photoneutron cross section, 86 use in (a,n) sources, 89-91 Beta decay, 334-336 energy spectra, 334-336 Fermi function, 335 shape function, 335 transition probabilities, 334-336 Beta particles: decay-data tables, 514-521 line kernel, 342-334 plane kernel, 336-338, 344-347 point kernel, 336-338 Bhaba cross section, 63 Boltzmann equation, ( see Transport equa­ tion) Boundary effects, on buildup, 225-227, 271272 BREESE code, 244, 312 Bremsstrahlung, 67-69, 108-111, 366, 368 (see also Radiation yield) angular distribution, 110 from beta particles, 111 from monoenergetic electrons, 109 inner, 111 production by electrons, 48, 67-69, 109111 thick target yield, 109 yield, 47, 109 Broad-beam attenuation: gamma and x rays, 229-235 neutron transmission, 307-308 oblique beams of neutrons, 310 oblique beams of photons, 231-234 photon attenuation and buildup, 231234 x-ray transmission, 231-235 Broder formula, 228 Buildup factor, 176, 215-229, 478-492 Berger approximation, 222, 304

C Capture gamma photons, 101-103, 317, 323 ( see also Gamma photon sources) attenuation of, 300-304 cross sections, 450 data sources, 12 number, 450 Carbon: radiation yield in, for electrons, 49 removal cross section, 282, 287 Cartesian coordinates, ( see Rectangular co­ ordinates) CDF, (see Cumulative probability distribution) Center-of-mass coordinates, 37 Central-limit theorem, 421 Chandrasekhar's H-function, 317, 407, 442443 Channeling, 253 Characteristic x rays, 105-108 energies, 107-108 intensity, 108 nomenclature, 105-106 production, 105-108 Charged particles, 61-72 bremsstrahlung production, 67-69, 108-111 (see also LET, radiative) CSDA point kernel, 338 electrons, 45 range formulas, 71 range, 69-72, 177 residual range concept, 72

Index shielding, 61-72 stopping power, 66-70 track structure, 339 transmitted energy spectrum, 72 Charged-particle equilibrium, 131-134 conditions for, 132-133 effect of bremsstrahlung, 134 kerma vs. absorbed dose, 133 Chilton-Huddleston formula, 240-241 Classical electron radius, 40, 63 Coherent photon scattering, 42-44 data tables, 451-456 Collided dose, 173-177 Collisional stopping power, (see Stopping power) Composition method, ( see Random variable selection) Compton scattering, 33-34, 366, 417-418 Compton unit, 33, 366 Compton wavelength, (see Wavelength) Concrete: air kerma buildup factor, 481 Berger parameters, 484 geometric progression parameters, 486 Taylor parameters, 483 albedo, Chilton-Huddleston parameters, 242-243 calcareous, 307 compositions, 305 exposure buildup factor, broad beam attenuation, 488, 491 fast-neutron albedo, 314, 316 gamma-ray interaction coefficients, 454 half-value thickness, 235 intermediate-energy neutron albedo, 315-317 neutron shielding, 304-312 effect of aggregate, 311 effect of response function, 311 effect of water content, 307, 309 normal incidence, 307-308 oblique incidence, 307, 310 transmitted dose, 307-312 photon albedo data, 241-243 point kernel for Cf fission neutrons, 306 siliceous, 311 transmitted neutron dose oblique incidence, 307, 310 perpendicular incidence, 307-308 types, 305 x-ray half-value thickness, 233 Constants and conversion factors, 431-432 Continuous slowing-down approximation, (see CSDA)

525 Conversion coefficient, 121, 129 (see also Response function) Coordinate systems Cartesian, 362-363 center of mass, 37 cylindrical, 362-363 laboratory, 37 spherical polar, 16 spherical, 362-363 three basic geometries, 362-363 Corresponding positions, 374 Cosine distribution, 23 Coulomb barrier, 87 Cross section effectiveness ratio, 254-255 Cross sections, ( see also Removal cross sections) electrons, 12 fusion reactions, 93 Legendre expansion, 58, 369, 370, 381 macroscopic, 29 microscopic, 30-31 minima, 288 multigroup, 297-298, 384-386 neutron, 50-61 ( see also Neutron scattering) activation, 103-104, 449 Bragg cutoff, 51 Bragg scattering, 50 capture, 448, 450 characteristics, 52-55 differential scattering, 57-60, 368371 elastic scattering, 369-371 fission, 448 high-energy, 56 inelastic scattering, 56-61, 103, 369370 Legendre expansion, 58, 369 low-energy, 51 nonelastic, 57 ( n, p) reactions, 448 radiative capture, 60-61, 101-103 resonances, 51 scattering, 448 secondary-photon production, 6061, 101-103 shielding importance, 50 thermal averaged, 102 thermal capture, 60-61, 102-103 total, 52-55 types of interactions, 50 variation with nuclear mass, 52-55 photon, 39-49, 366-367 Compton, 40-42, 46-47 data sources, 11-12

526 energy scattering, 47 incoherent scattering, 40-42 Klein-Nishina, 40-42, 366 photoelectric absorption, 47 photoelectric effect, 43 photoneutron reactions, 86 Rayleigh, 42-43 scattering, 366 secondary-photon production, 366 Thomson, 40 photoneutron production, 86 scattering: Legendre expansion, 58, 58, 369 notation, 28-31 CSDA: charged particles, 338, 353 electron fluence, 349-351, 354 Cumulative distribution method, ( see Ran­ dom variable selection) Cumulative probability distribution: definition, 411 examples, 428-429 random variable selection, 412 Curie, definition, 24 Current density, (see Vector current density) Current equation, 381 Cy lindrical G-functions, 172 Cy lindrical coordinates, 362-363 Cylindrical volume source, 171-173

D Data libraries, 11-12 DCTDOS code, 244 Deep dose equivalent index, 145 Delayed neutrons, 81 Delta function, 25, 443-444 Density variations, treatment of, 195-202, 372-374 Density, effect on radiation field, 406 Dept. of Health and Human Services, 10 Detector response, (see Response function) Deterministic quantities, 123 absorbed dose, 124 exposure, 124 kerma, 124 Deuterium: converter material, 85 fusion reactions, 92 photoneutron cross section, 86 Diffusion coefficient, 383, 386 Diffusion theory, 291-292 energy dependent, 380-384 multigroup approximation, 297-299 multigroup, 297, 385-386 Dirac delta function, 25, 443-444

Index Directional dose equivalent, 146 Directly ionizing radiation, 1 Discrete-ordinates method, 392-397 analytical solution, 394 derivation, 393-397 finite-difference approximation, 395 limitations, 397 numerical solution, 395-396 one-speed model, 407 shielding importance, 397 stability, 397 Disk source, 164-167, 189, 218-220 anisotropic emission, 167 Displacement distance, 295 Distributed sources, 161-173 Dose calculation: averaged over target, 174, 178 generalized methods, 155, 177-202 advantages, 181-182 multiregions, 191-195 point-kernel method, 160-177 source superposition method, 164 Dose equivalent quantities, 126 dose equivalent index, 146 dose equivalent, 127-128, 131 maximum dose equivalent index, 146 quality factor, 126-129 variation with LET, 127-129 relative biological effectiveness, 126 Dose point kernel, 174 Dose, (see also Uncollided dose) measured vs. calculated, 156 stochastic vs. mechanistic, 155 Dosimetric quantities, 15 absorbed dose, 124 energy imparted, 123 exposure, 124 kerma, 124 lineal energy, 123 specific energy, 123 Duct penetration, ( see Duct streaming) Duct streaming: characterization of incident radiation, 244-245 gamma rays, 242-252 LeDoux-Chilton method, 250-252 two-legged rectangular duct, 250252 neutrons, 318-322, 318-322 empirical results, 322 multiple bends, 322 Simmon-Clifford model, 321 two-legged ducts, 320-321 straight ducts line-of-sight dose, 245-248, 318

527

Index multiple wall scatter, 319-320 single wall scatter, 318-319 wall-penetration dose, 248, 318 wall-reflection dose, 248-249 two-legged ducts, 250-252, 320-321

E Edge energies, 107 Effective dose equivalent, 148, 151 Effective dose, 148 EGS4 code, 216, 227, 339, 409 Electron capture, 107 Electron shells, 105, 105-107 Electronic equilibrium, ( see Chargedparticle equilibrium) Electrons: annihilation, 104 binding effects, 41 binding energy, 44, 105-107 bound, 41 dose distribution, 338-340 edge energies, 106-107 free, 40-41 decay-data tables, 514-521 line kernel, 342-344 line sources, 342-344 plane kernel, 336-338, 344-347 plane sources, 336-338, 344-347 point kernel, 336-338 production of x rays, 105-115 radiation yield, 48-49, 72, 109 radius, 40 range data, 70 range formulas, 71 recoil, 337 scattering angle, 33-34 skin dose, 346-347 slowing down theory, 348-352, 354 stopping power, 65-66 vacancy production, 106-107 volume source, 347-348, 348-352 Elliptic integrals, 433 ENDF, 50 Energy absorption coefficient, ( see Interac­ tion coefficients) Energy conservation, 32 Energy deposition coefficient, ( see Interaction coefficients) Energy distribution, 20 Energy fluence, 20 Energy flux density, doubly differential, 365 Energy imparted, 123 Energy multigroup approximation, 297-299, 384-386, 392-394

Energy transfer coefficient, ( see Interaction coefficients) Environmental Protection Agency, 6, 9 Euler's constant, 303, 439 Evaluated Nuclear Data Files (ENDF), 50 Excitation energy, 32, 35-36, 65, 103, 369 Exponential attenuation, 2, 156, 270, 378380 fast neutrons, 289, 290 Exponential integral, 166, 303, 435-440 data, 437-438 properties, 439-440 Exponential transformation, 424-425 Exposure, 124

F Fano's theorem, 196, 372 Fast neutron attenuation, 271-291 (see also Neutron shielding and Re­ moval theory) Albert-Welton method, 273-278 correction for nonhydrogen compo­ nent, 277 hydrogenous medium, 273-288 non-hydrogenous medium, 288-290 Federal Radiation Council, 9 Fermi age, 293-295, 297 Fermi function, (see Beta decay) Fick's law, 383 Finite-difference approximation, 395-396 Fission neutrons, 80-84 (see also Neutron sources) attenuation, 273, 273-278 energy spectrum, 82-84, 273 number per fission, 81 spontaneous fission, 82 yield, 81 Fission product gamma photons, ( see Gamma photon sources) Flow rate, definition, 19 Fluence rate, definition, 18 Fluence-to-dose conversion factor, 121 (see also Response function) Fluence: angular dependence, 20, 21-23 definition, 17-18 differential energy and directional, 20 energy and wave number, 21 energy spectrum, 20 Legendre expansion, 22 transport equation, 404 Fluorescence, 44, 107-108, 366, 368 Flux density, (see also Fluence) angular moments, 388 definition, 18

528 differential energy and directional, 356 integral equation, 365 scalar, 365 Foderaro-Hall buildup approximation, 221 Food and Agriculture Organization, 9 Free field, 144 Frustrum of a cone, 169 Fusion neutrons, 92 Fusion reactors, 4

G Gamma interactions, ( see Photon interac­ tions) Gamma photon sources, 92-103 activation nuclides, 103-104 annihilation radiation, 45, 104 fission product decay, 94-101 approximate formulas, 99 data tables, 506-513 decay rate, 96-101 delayed emission, 96-101 examples, 100-101 gamma yield, 97 total energy, 96-97 inelastic neutron scattering, 56, 103, 269, 451 neutron capture, 61, 101-103, 269, 291, 300-304, 450 energy spectrum, 61, 102, 450 source strength, 102 prompt fission photons, 93-94 average number, 94 energies, 94 decay-data tables, 514-521 radionuclide decay, 92-93 Gamma rays, ( see Photon production and Gamma-photon sources) Gamma-ray shielding, 214-264 Gauss quadrature, 393, 398 Gaussian approximation, 27-276 Generalized dose method, 177-202 Geometric progression buildup approximation, 222-223 Geometric transformations, 203-206 (see also Monte Carlo method) volume to surface, 375-377 Geometry factor, (see also Point-pair distribution) basic geometries, 192-193 calculation of, 185-190 Cartesian coordinates, 192 circular cylinder, 211 cylindrical coordinates, 193 cylindrical source, 180 definition, 180

Index effect of density variations, 195-202 graphical examples, 191, 201 infinite slab source, 187 infinite slab, 212 modified, 199 Monte Carlo calculation, 189-191 multiple-target property, 191 multiregion example, 193-195 multiregion geometries, 191 plane source, 212 reciprocity relation, 191 slab source and target, 212 slab source, 212 spherical coordinates, 192 spherical surface source, 213 square cylinder, 211 symmetry, 180 with density variations, 199 Glass, x-ray half-value thickness, 233 Gold, stopping power, 67 Graphite, energy-dependent removal cross sections, 287 Gray, 140 Green's function, 173 (see also Point kernel) Group constants, 297-299, 385-386 Gypsum, x-ray half-value thickness, 233

H H-function, 317, 407, 442-443 Half-lives for selected radioisotopes, 449, 514-521 Half-space source, 171 Half-value layer, (see Half-value thickness) Half-value thickness, 3, 157, 234-235 x-ray, ( see entry for material) Health Physics Society, 8 Heterogeneous shields, ( see Shield hetero­ geneities) HVL, (see Half-value thickness) Hydrogen effect on removal cross section, 283 fast-neutron kernel, 284, 285 neutron cross section, 57, 58, 274 neutron removal, 273-278 neutron scattering, 57

I ICRP, ( see International Commission on Ra­ diological Protection) ICRU, ( see International Commission on Ra­ diation Units and Measurements) Importance sampling, 421-423 Incoherent photon scattering, 40-42

529

Index Indirectly ionizing radiation,2 Inelastic gamma photons, 34-35, 57, 103, 269,369,451 (see also Gamma photon sources) Inelastic scattering,35-36,369-370 (see also Neutron scattering) photon production,56,103 Infinite slab source,179,189 Inner bremsstrahlung,111 Integral transport equation: applications,402 approximate solutions,398-402 derivation,361-365 Interaction coefficients,28-31,157 (see also Cross sections) absorption,3 compounds and mixture,31 diffusion theory, 291,293,297-300, 385-386 effective attenuation,379 group transfer,385 linear,28-30 mass,31 multigroup diffusion theory, 297-298, 385-386 multigroup transport theory,384-385 neutron: (see also Neutron interactions and Removal cross sections) absorption,102,291 group transfer,298,385-386 thermal averaged,102-103 photon,45,366-367 attenuation, 45 coherent scattering,452-456 energy absorption,452-467 energy transfer,452-467 incoherent scattering,452-456 linear absorption,137 linear energy absorption,139 linear energy deposition,137 linear energy scattering coefficient, 41,240 linear energy transfer,137 pair production,45,452-456 photoelectric effect,452-456 photoelectric,45 total-coherent, 452-467 total,452-456 photoneutron reactions,85 relation to mean free path,157 scattering, 29-30 weighted scattering, 379 Interaction forcing, 424 Interaction probability,156-157

Interface effects: neutron buildup,228 photon buildup,228 Internal conversion,107 International Atomic Energy Agency,7,9 International Commission on Radiological Protection, 2,6,8 International Commission on Radiological Units and Measurements,6,7,8 Ionization energy, ( see Excitation energy) Ionizing radiation,1 Iron: air kerma buildup factor,Berger pa­ rameters,483 geometric progression parameters, 487 albedo,Chilton-Huddleston parame­ ters,242-243 energy-dependent removal cross sec­ tions,287 exposure buildup factor,broad beam attenuation,492 fast-neutron initial buildup factors,289 fast-neutron relaxation distance,289 fast-neutron albedo,314 gamma-ray interaction coefficients,455 neutron cross section,53 radiation yield in,for electrons,49 x-ray half-value thickness,233 Isotopic abundances,448-449 ITS code group,409

K

K-edge,106-108,218 Kalos formula,227-228 Kerma,124,131 buildup factors,479-482 neutron,tissue,134-136 relation to absorbed dose,133 Kernel,(see Point kernel) Klein-Nishina cross section,40-42,41 sampling,417-418

L

Laboratory coordinates,37 Laminate shields,227-228,280 Lead: air kerma buildup factor,482 Berger parameters,484 geometric progression parameters, 487 Taylor parameters,483 albedo,Chilton-Huddleston parame­ ters,242-243

530 exposure buildup factor, broad beam attenuation, 492 fast-neutron initial buildup factors, 289 fast-neutron relaxation distance, 289 gamma-ray interaction coefficients, 456 half-value thickness, 235 neutron cross section, 54 neutron nonelastic cross section, 57 radiation yield in, for electrons, 49 x-ray energy levels, 106 x-ray half-value thickness, 233 Leakage estimators, 425 Legendre expansion, 22, 58, 58, 370, 381, 387 avoiding, 398 Legendre moments, (see Moments method) Legendre polynomials, 440-442 associated, 442 orthogonality property, 441 LET, 63-69 (see also Stopping power) collisional, 62-67, 125-126 delta rays, 126 CSDA point kernel radiative, 67-69 restricted, 66-67, 126 Lethargy, 371, 392 Limiting source and target volume, 182 Line source, 161-164, 188, 342-344 Line-beam response function, 256-257, 323 Lineal energy, 123 Linear absorption coefficient, ( see Interac­ tion coefficients) Linear attenuation coefficient, ( see Interac­ tion coefficients) Linear energy absorption coefficient, (see In­ teraction coefficients) Linear energy deposition coefficient, ( see In­ teraction coefficients) Linear energy transfer coefficient, ( see Inter­ action coefficients) Linear energy transfer, (see LET) Linear interaction coefficients, ( see Interac­ tion coefficients) Linear momentum conservation, 31

M MAC code, 299 Macroscopic cross section, 29 Magic nuclei, 35-36 Mass coefficients, ( see Interaction coeffi­ cients) Mass energy deposition coefficient, ( see In­ teraction coefficients) Mass thickness, 176, 198, 373 Material kerma, (see Kerma)

Index Maximum dose equivalent, 146 Maximum permissible dose, 234 Maxwellian distribution, 82, 297 Mean chord length, 26 Mean free path, 157 Mechanistic doses, 155 Medical facilities, shielding techniques for, 231-234 Medium kerma, (see Kerma) Microscopic cross sections, ( see Cross sec­ tions) MICROSKYSHINE code, 259 Moments method, 216-217, 279, 386-392 derivation of moments equations, 387388 reconstruction of flux density, 390-391 shielding importance, 391-392 solution of moment equations, 389 Monte Carlo application: geometry factor, 189-191 point-pair distribution, 189-191 Monte Carlo method, 355, 408-430 central-limit theorem, 421 computations, 408-426 exponential transformation, 424-425 geometric transformations, 415-416 importance sampling, 421-423 interaction forcing, 424 leakage estimators, 425 next-event estimators, 426 particle tracking, 416-419 Russian roulette, 423-424 sample calculation, 415-420, 430 scoring, 419-420 splitting, 423-424 tallying, 419-420 truncation methods, 423 Multigroup diffusion method, 297-299, 384386 Multigroup transport approximation, 384385, 392-394 Multiregion geometry, 191-195 example, 193-195 M!1lller cross section, 63

N NAS-BEIR, 2, 8 Nat. Acad. of Sciences, Biological Effects of Ionizing Radiation, 2, 9 National Council on Radiation Protection and Measurements, 2, 6, 7, 8 National Nuclear Data Center, 7 National Radiological Protection Board, 8 NCRP, 2, 6, 7, 8 Neutron capture, 448, 450

531

Index high energy, 60 (n,a) reactions, 56 thermal, 60-61, 102, 300, 448-449 Neutron cross sections, (see Cross sec­ tions) Neutron generators, 92 Neutron interactions, ( see also Cross sections) activation nuclides, 103-104 capture gamma photons, 322 capture, 34 fission, 80-81, 273 inelastic scattering, 322 radiative capture, 60-61, 101-103, 269, 301 scattering, 34-39, 271, 369-371 types, 50 Neutron scattering, 34-39 average energy loss, 58-60 average logarithmic energy loss, 293, 297 center-of-mass system, 37 cross sections, 49-61, 369-371 cutoff energy, 36 double-valued region, 36 doubly differential cross section, 369370 elastic, 34, 369-370 energy transfer, 135 final energy, 38, 58 inelastic, 34-36, 103, 369-370, 451 energy transfer, 135 kinematics, 31-32, 35-38 laboratory system, 37 potential, 34 recoil nucleus, 37 scattering angle, 36-37, 57-58, 57-58 secondary photons, 56, 103 single-valued region, 36 threshold energy, 35-36 types, 34 Neutron shielding: associated problems, 269-271 buildup factors, 271 by concrete, 304-312 dose units, 272 fast neutrons, ( see Fast neutron atten­ uation and Removal theory) intermediate energy neutrons, 269, 291, 293-295 need for simplified methods, 270-271 thermal neutron capture, 291 thermal neutrons, 291-295 Neutron sky shine, (see Skyshine) Neutron sources , 80-92

(a,n)

reactions, 86-91 example spectra, 90 fabrication, 88 important nuclides, 87 in nuclear waste, 88 neutron energy, 90-91 neutron yield, 89 source strengths, 8-89 threshold energy, 88 activation reactions, 91 decay of 17N, 91 fission neutrons, 80-84 energy spectrum, 82-84, 273 number per fission, 81 fusion reactions, 92 cross sections, 93 neutron energy, 92 principal reactions, 92 photoneutrons, 84-87 cross sections, 86 energy spectrum, 85 hazards, 86 important nuclides, 87 source characteristics, 87 source strength, 87 threshold, 85 spontaneous fission, 82 Neutron streaming, 318-322 Neutron, discovery, 4 Next-event estimators, 426 Nitrogen: 16N production, 104 neutron emission from 17N, 91 Nonanalog Monte Carlo, 409 Nonstochastic quantities, (see Deterministic quantities) Nonuniform sources, 183-184 Normalized shielded output factor, 232 NRN removal-diffusion method, 298-300 Nuclear decay data sources, 11 Nuclear excitation levels, 35-36 Nuclear Regulatory Commission, 9 Nuclear weapons, shielding methodology, 4 Numerical quadrature, 393, 398

0 OECD Nuclear Energy Agency, 7 Optical depth, 159 Optical distance, 399 Optical thickness, 159, 176 ORIGEN code, 97-98 Oxygen, activation, 91, 104 energy-dependent removal cross sec­ tions, 287

532

p Pair production, 45, 366 PALLAS code, 217, 219 Particle balance, 356 Particle density, 355 number distribution, 356 Particle flow, (see also Radiation field) angular dependence, 21-22 definition, 18-19 PDF, (see Probability density function) Phantoms: anthropomorphic, 147 ICRU sphere, 147 slab and cylinder, 147 Photoelectric effect, 43, 44, 106, 366 Photoelectrons, 44 Photon buildup factors, 12, 479-492 Photon buildup, general properties, 214-220 Photon cross sections, (see Cross sections) Photon interactions, (see also Interaction coefficients and Cross sections) coherent scattering, , 42-44, 43 bound electrons, 43 Compton scattering, 33-34, 40-42, 366 incoherent scattering bound electrons, 41 free electrons, 40-41 pair production, 45, 45, 366 photoelectric effect, 44, 106, 366 photoneutron production, 85-86 types, 40 x-ray production, 105-115 Photon production, ( see also Gamma­ photon sources and X-ray sources) activation nuclides, 103-104 coefficient for secondary photons, 366368 inelastic neutron scattering, 56, 103, 269, 300, 451 neutron capture, 60-61, 300-301, 450 positron annihilation, 104, 366 x-ray machines, 111-115 x-ray sources, 105-115 Photon shielding, 214-264 ( see also Capture gamma photons, attenuation of) Photoneutrons, ( see Neutron sources) Physical constants, 431 Plane source, 336-338, 344-347 Plane-density variations, 196 Point kernel: Cf fission neutrons in concrete, 306 in water, 270

Index collided dose, 173-177 electrons, 177 examples, 176-177 fast neutrons in water, 285 fast neutrons, 285 in hydrogen, 284 in water, 285, 286 fission neutrons, , 177 Albert-Welton, 273-278 Casper, 278 effect of fissioning isotope, 275 in hydrogen, 273-277 gamma photons, 176 heavy charged particles, 177 scaled, 177 uncollided dose, 160-161, 176 with density variations, 198 Point source, 157-161 Point-kernel method, 222-225 buildup factors, 222-225 for a disk source, 203 general application, 203 line and plane sources with buildup, 222-225 Point-pair distribution, ( see also Geometry factor) calculation of, 185-190 definition, 179 disk source, 186 examples, 200, 201 line source, 185 modified, 199 Monte Carlo calculation, 189-191 spherical source, 186 symmetry, 179 with density variations, 199 Polyenergetic sources, 159-160 Positrons, 45, 104 annihilation, 45, 104 pair production, 45 Prescribed dose equivalent, (see Dose equiv­ alent) Probability density function: definition, 411 examples, 428-429 Prompt fission gamma photons, 93-94 ( see also Gamma photon sources) Prompt fission neutrons, 80 Protons: range data, 70 range formulas, 71 Pseudorandom number, (see Random num­ ber) Public Health Service, 9

Index

Q

Q-value: (a, n) reactions, 88 beta decay, 334 definition, 32 inelastic scattering, 32, 35-36, 369 photoneutron production, 85 Quality factor, 126-129, 131, 148, 150 versus LET, 127-129 1966 ICRP formulation, 127-129 1991 ICRP formulation, 127-129

R

Radiation field: angular differential, 20 angular flow, 21 angular representation, 22 concepts, 15-23 definition, 15 directional properties, 19-20 effect of density, 406 energy distribution, 20 energy fluence, 20 existence and uniqueness, 371 flow, angular properties, 21-23 fluence energy spectrum, 20 flux density, 18 fundamental variables, 17-19 in medium with plane density variations, 372-374 Legendre expansion, 22 net particle flow, 19, 21 scaling, 374-375, 406 spatially uniform flux density, 371-372 spectral distribution, 20 total fluence, 17-18, 20 variables, 15-23 vector current density, 22 vector flow, 22 volume-to-surface source transforma­ tion, 375-377 Radiation intensity, 365-366 Radiation protection: books, 10 history, 2-6 institutions, 8-10 Radiation Shielding Information Center, 7 Radiation sources, 80-115 ( see also specific type) activity, 24 area, from angular flow, 24 discrete vs. distributed, 24-25 energy-angular dependence, 24 general discussion, 23-24 point approximation, 23

533 point, line, area, 24 shielding importance, 80 strength, 24 Radiation streaming, ( see Duct streaming) Radiation transport, ( see Transport equation and Monte Carlo method) Radiation weighting factor, 150 Radiation yield, electron, 48-49 Radiation, ( see specific type) directly ionizing, 1 hazards, 3 indirectly ionizing, 2 obsolete units, 3 regulation, 6 summary of biological effects, 2 Radiative capture, 60-61, 101-103, 300-304 Radiological units, 431 Radiometic quantities, 15 Radiometric quantities, aligned field, 145 angular flow, 21 angular fluence, 21 expanded field, 145 fluence, 17 net flow, 18 Random number, (see also Random vari­ able) selection, 409-410, 428 Random variable selection: composition method, 414 composition-rejection method, 414 cumulative distribution method, 412 discrete variables, 411 rejection method, 413-414 Range, 177 (see also Charged particles) RASH-E code, 299 Ray analysis, 2 shield discontinuities, 255 vs. broad beam attenuation, 231 Rayleigh cross section, 43 Rayleigh scattering, 42-44 RBE, (see Relative biological effectiveness) Reciprocity theorem, 182 Recoil electron, 33 Recoil nucleus, 37 Rectangular coordinates, 362-363 Rectangular source, 167-169 Reduction factor, 180, 181 charged particle sources, 348 Reflection factor, (see Albedo) Regulatory organizations, 6, 8-10 Rejection method, ( see Random variable se­ lection) Relative biological effectiveness, 126 Relaxation constant, 292

534 Relaxation length, fast neutrons, 289, 290 Removal bands, 296 Removal coefficients, 277, 279, 290 (see also Removal cross sections) Removal cross sections, 277, 279-288 approximations, 282 empirical formulas, 282 energy-dependent, 287, 287-288 fission neutrons, 279-282 laminate shields, 280 microscopic, 277, 282 mixtures, 280, 283 Removal theory, 279-288 energy-dependent, 284-288 homogeneous shields, 283 hydrogen importance, 280, 282, 284 laminate shields, 280 limitations, 279, 282-284 non-hydrogenous media, 290 Removal-diffusion theory, 295-300 improved methods, 298-300 Spinney method, 296-297 Response function, 121-151, 468-477 ambient dose equivalent, 146 deep dose equivalent index, 145 definition, 129 directional dose equivalent, 146 effective dose equivalent, 148 effective dose, 148 for limiting dose, 144 for operational dose, 144 local, 130-131, 134-144 neutrons, 134-136 photons, 137-144 neutron: ambient dose equivalent, 473 deep dose equivalent, 473-474 effective dose equivalent, 475-476 local tissue kerma, 136 shallow dose equivalent, 473 tissue kerma, 477 phantom related, 144-151 photon: ambient dose equivalent, 470 deep dose equivalent, 470 effective dose equivalent, 471-472 local absorbed dose, 140 local dose equivalent, 140 local exposure, 141 local kerma, 140 selection of local, 141-144 shallow dose equivalent, 470 shallow dose equivalent index, 146 Roentgen, 141 RSIC, 7

Index Russian roulette, 423-424 Rutherford cross section, 63

s

SAIL code, 312 Scalar flux density, 365 Scaled point kernel, 177 Scaling theorem, 196 Scaling, 374-375, 406 Scattering: elastic, 34, 38-39 inelastic, 34-36, 35-36 mean cosine of angle, 59, 135 neutron, (see Neutron scattering) photon, (see Photon interactions) Secant integral, 163, 433-436 Secondary radiation, ( see also Fluorescence and Gamma photon sources) photons, 317, 323, 366-368 production coefficient, 366-367 neutrons, 269 Shallow dose equivalent index, 146 Shape function, (see Beta decay) Shield analysis, ( see also Shielding) definition, 1 history, 2-6 impact of computers, 4-5 Shield heterogeneities, 252-256 channeling, 253 cross section effectiveness ratio, 254 ray theory approximation, 255 transmission factor, 253 Shielding: books, 10 data sources, 11-12 definition, 1 development of methods, 3-6 for medical facilities, 231-234 history, 2-6 information sources, 10-12 institutions, 7-8 journals, 8 societies, 8 SI prefixes, 432 Sievert integral, 163, 433-436 Sievert, 140 Similar media, 197 Simplified methods, 155 Skin dose, 346-347 Skyshine: gamma-ray, 256-259 ground correction factors, 257 line beam response function, 256257, 500-501 shielded source, 259

535

Index upward-collimated source, 257-258 neutron, line beam response function data, 502-505 neutrons, 322-323 line-beam response function, 323 secondary gamma-rays, 322-323 secondary gamma-ray, line beam re­ sponse function data, 502-505 Slowing down density, 293 Solid angle, 16 differential, 16 subtended by disk, 26 subtended by rectangular area, 26 Source superposition technique, 164 Source term: capture gamma calculations, 300-301 diffusion group, 297 exponential representation, 292, 302 in age-diffusion theory, 294 in diffusion theory, 291 in removal-diffusion theory, 296, 297, 300 in transport equation, 356 Source transformation: disk to point source, 203 volume to area source, 204 Sources, (see also Radiation sources) cylindrical, 184-185 equivalent area source, 376-377 nonuniform, 183-184 volumetric, 178 Space vehicles, shielding methodology, 4 Spatial moments, (see Moments method) Specific absorbed fraction, 181 Specific energy, 123 Specific gamma-ray constant, 229-230 Spectrum weighting, 297 Speed of light, 32 Spherical coordinates, 362-363 Spherical harmonics, ( see Legendre expansion) Spherical polar coordinates, 362-363 Spherical surface source, 169 Splitting, 423-424 Spontaneous fission, 81-82 Standards, 11 Statistical average, 155 Steel, ( see Iron) Steradian, 16 Stochastic quantities: doses, 155 energy imparted, 123 lineal energy, 123 Monte Carlo method, 408-426 specific energy, 123

Stopping power, (see also LET) 63-67, 89, 177 collisional, 62-67, 108-109 data sources, 12 radiative, 67-69, 108-109 restricted, 66-67 STORM code, 244 Straight-ahead approximation, 378-380 Stratified shields, 280 buildup factor, 227-228 Streaming, ( see duct streaming) Superposition technique, 164 Surface source, from volume source, 204

T Taylor buildup factor approximation, 221 Tenth-value thickness, 157 Thermonuclear reactions, 92 Thomson cross section, 40 TIGER code, 346, 409 Tissue kerma, (see Kerma) Tissue weight factor, 149-150 Tissue, absorbed dose, (see Absorbed dose) ICRU 4-element approximation, 136 kerma, (see Kerma) Transfer coefficients, 298, 385-386 Transmission factor, (see also Attenuation factor) broad beams of neutrons, 493-497 heterogeneous shields, 253 Transport equation, 355-371 approximations, 377-386 ( see also Moments method and Discrete-ordinates method) diffusion, 380-384 discrete ordinates, 392-397 exponential attenuation, 378-380 moments method, 386-392 multigroup diffusion, 385-386 multigroup transport, 384-385, 392394 with integral form, 398-402 assumptions, 358 derivation, 356-357 explicit form, 359-361 for neutrons, 368-371 for photons, 365-368 infinite medium, 405 integral form, 361-365, 405 integrodifferential form, 355-361 iterative solution, 399-400 one-speed approximation, 407 plane geometry, 361, 392-393 properties, 371-377 existence and uniqueness, 371

536 plane-density variations, 372-374 scaling of radiation fields, 374-375 spatially uniform flux density, 371372 volume-to-surface source transformation, 375-377 spherical geometry, 404 streaming term, 359-362 uncollided flux density, 405 Transport equations, iterative solution, 396 Transport theory, ( see Transport equation) deterministic, 355 Monte Carlo, 355 Transuranic isotopes: fission neutron energy spectrum, 81-82 fission neutron yields, 81-82 properties, 82 Trial function, 390 Tritium: fusion reactions, 92 production, 4 Truncation methods, 423 Tungsten, x-ray target material, 112

u

UN Scientific Committee on the Effects of Atomic Radiation, 2, 9 Uncollided dose: conical frustrum source, 169-171 constant half-space source, 210 cylindrical volume source, 171-173 disk source, 164-167, 189 distributed source, 161-173 exponential half-space source, 210 from capture gamma photons, 303 from geometry factor, 188-189 geometric attenuation, 158, 158 half-space source, 171 infinite slab source, 170, 189 line source, 161-164, 188 point kernel, 160-161 point source, 157-161 rectangular source, 167-169 spherical source, 210, 211 spherical surface source, 167-169 Uncollided radiation, 155 Unit conversions, 431-432 United Nations organizations, 9 UNSCEAR, 2, 9 Uranium: fission neutron energy spectrum, 82-84 fission neutron yield, 81-82 neutron cross section, 55 neutron nonelastic cross section, 57 radiation yield for electrons, 49

Index

V Vector current density, 22 Vector flow, 22 Volume source, 375-377 Volume to surface source transformation, 204

w

W-value for air, 141 Wall scattering, 248-250, 318-320 Water: air kerma buildup factor, 480 Berger parameters, 484 geometric progression parameters, 485 Taylor parameters, 483 albedo, Chilton-Huddleston parame­ ters, 242-243 electron range in, 70 exposure buildup factor, broad beam attenuation, 491 fast-neutron dose kernels, 278, 285, 286 gamma-ray interaction coefficients, 453 of hydration, 305 point kernel for Cf fissions, 270 proton range in, 70 radiation' yield in, for electrons, 49 stopping power, 67 thermal neutron diffusion constants, 293 water kerma buildup factor, geometric progression parameters, 485 Watt distribution, 82-83 Wavelength: Compton, 33, 366, 392 photon, 366 Weight factor, (see Tissue weight factor or Radiation weighting factor) Wood, x-ray half-value thickness, 233 World Health Organization, 9

X X ray sources, 105-115 ( see also Fluorescence) bremsstrahlung, 108-111 characteristic x rays, 105-108 X rays: broad beam attenuation, 231-235 discovery, 2 filtration, 233 fluorescence yield, 108 from radioactive decay, data tables, 514-521 half-value thickness, 234-235

Index

machines, 111-115 filters, 112 photon spectrum, 113-115 radiation output, 115 target materials, 112 maximum permissible dose, 234 occupancy factor, 234 production, 44 radiation output factor, 232 use factor, 234 workload, 234 X-ray shielding, 214-264

537