Radiation Mechanics: Principles & Practice

Radiation Mechanics PRINCIPLES AND PRACTICE

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Radiation Mechanics PRINCIPLES AND PRACTICE

ESAM M. A. HUSSEIN Department of Mechanical Engineering University of New Brunswick Fredericton, N.B. Canada

Amsterdam • Boston • Heidelberg • London • New York • Oxford Paris • San Diego • San Francisco • Singapore • Sydney • Tokyo

Elsevier The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK Radarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands First edition 2007 Copyright © 2007 Elsevier Ltd. All rights reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333; email: [email protected]. Alternatively you can submit your request online by visiting the Elsevier web site at http://elsevier.com/locate/permissions, and selecting Obtaining permission to use Elsevier material Notice No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress For information on all Elsevier publications visit our web site at books.elsevier.com Typeset by Charon Tec Ltd (A Macmillan Company), Chennai, India www.charontec.com Printed and bound in Great Britain 07 08 09 10 10 9 8 7 6 5 4 3 2 1 ISBN: 978-0-0804-5053-7

To my two Amina’s: Mother and Daughter Dedicated in memory of: Uncle Ahmed Sabry Abdel-Ghaffar Uncle Mohammad Ali Abu-Hussein

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CONTENTS

Preface List of Algorithms

xi xv

1. Mechanisms

1

1.1 1.2

1.3

1.4

1.5

1.6

1.7

1.8

Introduction Radiation 1.2.1 Neutral particles 1.2.2 Charged particles 1.2.3 Photons Wave–Particle Duality 1.3.1 Corpuscular nature of waves 1.3.2 Wave nature of particles 1.3.3 Uncertainty principle Nuclear/Atomic Fields 1.4.1 Potential field 1.4.2 Nuclear strong-force field 1.4.3 Nuclear weak-force field 1.4.4 Electromagnetic field 1.4.5 Quantum states Atom and Nucleus 1.5.1 Atomic structure 1.5.2 Nuclear structure Nuclear Decay 1.6.1 Kinetics 1.6.2 Statistics 1.6.3 Alpha decay 1.6.4 Beta decay 1.6.5 Gamma decay 1.6.6 Internal conversion 1.6.7 Spontaneous fission 1.6.8 Decay by neutron or proton emission Reactions and Interactions 1.7.1 Interaction with atomic electrons 1.7.2 Interaction with electric field of atom 1.7.3 Nuclear interactions Macroscopic Field 1.8.1 Transport space 1.8.2 Particle density and flux

1 3 3 5 5 7 8 9 10 11 12 13 14 14 16 19 19 20 26 27 28 29 32 36 37 38 39 40 40 45 46 53 54 55 vii

viii

Contents

1.9

1.8.3 Atomic/nuclear density 1.8.4 Interaction rate Problems

2. Collision Kinematics 2.1 2.2 2.3

2.4

2.5

2.6 2.7

2.8

2.9

Overview Center-of-Mass System Relativity 2.3.1 Special theory of relativity 2.3.2 Center of relativistic mass 2.3.3 Lorentz transformation of momentum and energy Conservation Laws 2.4.1 Stoichiometric conservation 2.4.2 Intrinsic conservation 2.4.3 Kinematical conservation Einsteinian Kinematics 2.5.1 Two-body kinematics 2.5.2 Analysis using invariants 2.5.3 Non-elastic interactions 2.5.4 Non-relativistic approximation Newtonian Kinematics Specific Interactions 2.7.1 Elastic scattering 2.7.2 Inelastic scattering 2.7.3 Non-elastic collisions Electromagnetic Interactions 2.8.1 Coulomb scattering 2.8.2 Radiative collisions 2.8.3 Diffraction Problems

3. Cross Sections 3.1 3.2

3.3

Introduction Nuclear Cross-Section Models 3.2.1 Optical model 3.2.2 Compound nucleus 3.2.3 Continuum theory 3.2.4 Evaporation 3.2.5 Stripping 3.2.6 Photonuclear reactions 3.2.7 Nucleonic collisions Neutron Cross Sections 3.3.1 Elastic scattering 3.3.2 Inelastic scattering

57 58 60

67 67 68 74 74 78 79 80 81 81 82 83 83 93 98 103 104 108 109 114 116 123 123 133 144 146

153 153 156 156 162 165 165 166 166 168 168 169 172

ix

Contents

3.4

3.5

3.6

3.7

3.8

3.3.3 Radiative capture 3.3.4 Fission 3.3.5 Charged-particle production 3.3.6 Energy and angular distribution 3.3.7 Thermal neutrons Electrodynamics 3.4.1 Quantum electrodynamics 3.4.2 Feynman diagrams Photon Cross Sections 3.5.1 Thomson scattering 3.5.2 Compton scattering 3.5.3 Rayleigh scattering 3.5.4 Diffraction 3.5.5 Photoelectric effect 3.5.6 Pair production 3.5.7 Triplet production 3.5.8 Delbruck scattering Charged-Particle Cross Sections 3.6.1 Coulomb scattering 3.6.2 Rutherford scattering 3.6.3 Mott scattering 3.6.4 Bremsstrahlung 3.6.5 Moller scattering 3.6.6 Bhabha scattering 3.6.7 Pair annihilation Data Libraries and Processing 3.7.1 Libraries 3.7.2 Processing and manipulation 3.7.3 Compound and mixture cross sections Problems

4. Transport 4.1

4.2

Boltzmann Transport Equation 4.1.1 Basics 4.1.2 Transport in void 4.1.3 Divergence law 4.1.4 Attenuation law 4.1.5 Point kernel 4.1.6 Diffusion theory 4.1.7 Adjoint transport equation Modal Solution Methods 4.2.1 P 1 approximation 4.2.2 Diffusion equation 4.2.3 Numerical solution and computer codes

173 174 174 175 179 183 183 189 194 195 197 203 204 205 208 213 213 215 215 217 219 220 224 226 227 229 229 232 237 238

247 247 247 251 252 254 255 256 257 259 262 264 266

x

Contents

4.3

4.4

4.5

4.6

Nodal Solution Methods 4.3.1 Discretization of directions: discrete ordinates 4.3.2 Discretization of time, energy, and space 4.3.3 Multigroup approximation 4.3.4 Discretization of transport equation 4.3.5 Curved geometries 4.3.6 Source term 4.3.7 Solution of S n equations 4.3.8 Computer codes Stochastic Methods 4.4.1 Introduction 4.4.2 Random variables and statistical basis 4.4.3 Abstract analysis 4.4.4 Random numbers 4.4.5 Random number generation 4.4.6 Sampling 4.4.7 Particle transport 4.4.8 Example 4.4.9 Computer codes Transport of Charged Particles 4.5.1 Special features 4.5.2 Stopping power and range 4.5.3 Transport Problems

266 267 270 273 273 275 278 279 283 284 284 285 286 292 292 293 296 302 305 306 306 307 307 308

Bibliography Constants and Units

311 315

Useful Web Sites Glossary Index

319 321 323

PREFACE

The word “radiation’’ refers to electromagnetic waves (at various frequencies), atomic emissions (X-rays), or nuclear decay and reaction products (alpha and beta particles, gamma rays, neutrons, positrons, etc.). Conventional optical principles are used to describe the behavior of electromagnetic radiation in the form of visible light, while the concepts of radiative heat transfer are utilized when dealing with thermal (infrared) radiation. These relatively simplistic principles along with the more elegant analysis of electromagnetic radiation using the Maxwell equations are appropriate when a large number of photons are involved, and wave characteristics are the norm. At very high frequencies (in the range of Xand γ-rays), electromagnetic radiation exhibits corpuscular properties, and conventional particle mechanics (conservation of energy and momentum) become directly applicable. The transport, as a collective, of these particle-like photons is governed by the Boltzmann transport equation. Similarly neutral radiation particles (neutrons) abide by conservation laws and the transport equation. However, when characterizing the dynamics of neutron interactions with the nucleus, wave (quantum) mechanics is utilized. At low energy, neutrons exhibit wave properties, and the wave nature of radiation still prevails. Particles carrying an electric charge (such as alpha and beta particles, or protons) are affected by the Coulomb forces of the atom and its nucleus as they traverse matter, and as such do not penetrate deep into matter. They can, however, trigger the generation of a chain of electrons in the form of a “shower’’ that can propagate further into matter. Electrons, being light in mass, can acquire a speed that approaches the speed of light, then relativistic effects become pronounced and must be taken into account. It is obvious from the above preamble that the mechanics of atomic/nuclear radiation involves many physical effects. At the time of writing this book and to the author’s knowledge, there was no single textbook that covers all these aspects. While a classical book such as that of Evans [1] covers the basic interactions and mechanisms, it does not tackle the transport theory and in essence considers a single interaction of a radiation entity with matter at a time. On the other hand, a book on transport theory, such as that of Davison [2], concerns itself with the mathematical aspects of the transport theory. Applied textbooks, such as those concerned with reactor theory [3, 4], radiation detection and measurement [5, 6], radiation shielding [7, 8], or radiation-based devices [9], tend to focus on the aspects that specifically relate to the subject of interest. Students of the field formulate an overall understanding of radiation behavior from individual subjects, ranging from basic nuclear physics and quantum mechanics to radiation transport theory and computations. This work integrates these aspects of radiation behavior in a single treatise under the framework of “radiation mechanics’’, in the same manner all aspects of fluid flow are covered under the subject of “fluid xi

xii

Preface

mechanics’’, and stress analysis is addressed under “solid mechanics’’ or “mechanics of materials’’. Mechanics is the science of studying energy and forces, and their effects on matter. It involves (1) mechanisms, (2) kinematics, (3) cross sections, and (4) transport. Mechanisms describe how various types of radiation interact with different targets (atoms and nuclei). Kinematics studies particle motion via conservation of energy and momentum, albeit taking into account energy stored within the target, along with any relativistic effects which become pronounced at high particle speeds. Therefore, kinematics determines the energy and direction of radiation following a certain interaction. A reaction cross section is a measure of the probability of occurrence of a certain interaction, at given kinematic (energy and direction) attributes. Interaction cross sections are determined by the interaction “amplitudes’’, as dictated by the “potential field’’ of the target and its effect on the incoming radiation. Quantum mechanics provides a mathematical framework for obtaining these amplitudes, the square of which (properly normalized) defines a cross section. The transport (spread and distribution of radiation from one location to another and its evolution with time) is determined by bookkeeping principles via the particle transport theory. The book addresses the above four aspects of radiation mechanics in four separate chapters. The first two chapters can be covered in a one-semester course, and the latter two chapters in a subsequent semester. However, students with some background in modern and nuclear physics can skip Chapter 1, and each of the other three chapters can be presented in any order, since they are reasonably independent of each other. At the end of each chapter a set of problems is presented to further assist students in understanding the underlying concepts. Use of computations and Internet resources1 are included in the problems as much as possible. Instructors can approach the author at [email protected] for a copy of the solutions manual for the problems in this book. In order to enable the reader to navigate through this book, one interaction at a time,Table 1 below provides a summary of all the relevant interactions and refers to the pages in this book where they are defined, their kinematics are addressed, and their cross sections are presented. This book was written using LATEX, based on MiKTEXplatform, with WinTEX XP as the interfacing editor. Special thanks to Mr. John T. Bowles for proof reading the text of the four chapters of the book. Esam Hussein Fredericton, N.B. Canada December, 2006

1

Uniform resource locators (URL’S) in this book refer to web sites that were active at the time of finalizing this work. Such web sites may change location and content, or disappear altogether. Readers are advised to do an Internet search under the relevant topic, if a URL ceases to be accessible.

xiii

Preface

Table 1 List of radiation interactions: numbers indicate the page number in which a particular aspect of an interaction is addressed

Interaction

Mechanism

Kinematics

Anomalous scattering

Cross section 196

Bremsstrahlung

46

136

220

Cerenkov radiation

45

141

Compton scattering

42

112

197

Delbruck scattering

53

143

213

Diffraction

43

144

183, 204

Elastic scattering: • Anomalous • Bhabha • Coulomb • Moller • Mott • Neutron/nuclear • Potential • Rayleigh • Rutherford • Thomson

43 43, 48 43 43, 49 47 47 42 49 43, 49

113 123, 129 113 113 109 140 131 139

196 226 215 224 219,225 156,169 159 203 217 195

Fission

51

122

174

Inelastic Scattering: • Compton • Coulomb • Neutron/nuclear • Thermal neutron

42 44 50 179

112 132 114 179

197 215 172, 180 179

Pair production

45, 53

118

208

Photoabsorption (Photoelectric effect)

41

117

205

Positron annihilation

44

116

227

Potential scattering

47

Production (nuclear): • Charged particle • Neutron

52 52

159 119 120

174, 179 177 (Continued )

xiv

Preface

Table 1

(Cont.)

Interaction

Mechanism

Kinematics

Cross section

Radiative capture

51

119

173, 178

Rayleigh scattering

42

140

203

Resonance scattering

49

Spallation

51

Thomson scattering

43, 49

139

Transition radiation

46

143

Triplet production

42

118

169

195

213

LIST OF ALGORITHMS

• •

Relativistic kinematics of a two-body interaction: 2(1,3)4 Invariant-based kinematics of a two-body interaction: 2(1,3)4

92 96

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C H A P T E R

O N E

Mechanisms

Contents 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

Introduction Radiation Wave–Particle Duality Nuclear/Atomic Fields Atom and Nucleus Nuclear Decay Reactions and Interactions Macroscopic Field Problems

1 3 7 11 19 26 40 53 60

1.1 Introduction Mechanics is the study of forces and energy and their effect on matter, it is also the study of mechanisms. Radiation mechanics then involves: 1. Understanding the mechanisms via which radiation interacts with a target atom/nucleus and how the target reacts. 2. Studying the kinematics of an interaction via the conservation of momentum and energy. 3. Determining the probability of interaction of a certain radiation entity, having a certain energy with a particular atom/nucleus (cross sections). 4. Modeling the transport of a flux of these entities into a medium. Figure 1.1 illustrates these concepts schematically for a scattering mechanism. In the Figure, a particle (or an equivalent wave packet) approaches the nuclear field, and its associated electric (Coulomb) field. These fields change the particle’s energy and direction, i.e. the particle is scattered. The energy of the emerging particle is determined by the kinematics of scattering. The probability that the particle will emerge in any specific direction is dictated by the potential field of the target nucleus and its effect on the incoming particle. This particle will continue to move in the medium causing different types of interactions with many target nuclei, until it is absorbed or escapes from the domain of interest. The Radiation Mechanics: Principles & Practice ISBN-13: 978-0-08-045053-7

© 2007 Elsevier Ltd. All rights reserved.

1

2

Chapter 1 Mechanisms

E(ϑ) Mechanism (scattering) E0

Particle (wave packet)

ϑ

Nuclear field

Kinematics: E(ϑ)

Coulomb field

Excited state Potential field Ground state

Transport

Figure 1.1 A schematic view of the mechanics of a radiation interaction: a scattering mechanism, the kinematics of which determines the outgoing energy, E(ϑ), based on the incoming energy, E0 , and the angle of scattering, ϑ; while the potential field governs the probability of scattering by ϑ to E(ϑ). A sequence of scattering events results in the transport of many particles in a medium of many atoms/nuclei.

collective movement of these particles in a medium is called particle transport. In Fig. 1.1, the particle could have been replaced with a photon and the target with an atom, or even a single atomic electron, and the same three aspects of particle mechanics will still emerge. These aspects are discussed in this book. Chapter 2 deals with interaction kinematics. The cross sections of interactions are discussed in Chapter 3. The particle transport process is analyzed in Chapter 4. In this chapter, the basic mechanisms affecting the behavior of each type of radiation addressed in this book are examined. The chapter begins by introducing the various forms of radiation, and identifying the ones that evince corpuscular (that of a minute particle) properties, which are the main subject of this book. Since nuclear/atomic radiation can assume both wave and particle characteristics, the concept of wave–particle duality is introduced. The information given in this chapter is quite basic and can be found in more detail in nuclear and atomic physics textbooks such as those of [1, 10–16]. The natural atomic/nuclear fields involved in the formation of the atom and its nucleus, to which radiation particles

3

1.2 Radiation

are exposed, are then reviewed, followed by an examination of the structure of the atom and the nucleus, and the nuclear decay processes. The types of interactions radiation can encounter upon approaching these fields are subsequently classified. After studying individual radiation particles and the individual target atoms/nuclei they interact with, we take a macroscopic point-of-view by involving many radiation particles with a medium containing many atoms/nuclei. This defines the space within which radiation transport takes place. Note that this book is not concerned with high-energy particle physics, a field that deals with studying the fundamental constituents of matter, though some of the concepts introduced are suited for use with high-energy particles.

1.2 Radiation This book deals with three types of radiation as outlined in Table 1.1.

1.2.1 Neutral particles A particle by definition carries a mass. Einstein (1905), through the special theory of relativity, introduced an energy term corresponding to the mass, m0 , of a particle when it is at rest; (see Section 2.3), the so-called rest-mass energy: Rest-mass energy = m0 c 2

(fundamental equation)

(1.1)

where c is the speed of light. This is a fundamental relationship, not derivable from other relations, just like Newton’s laws of motion. A number of these fundamental relationships will be identified throughout this chapter. The introduction of the rest-mass energy enables the accommodation of changes within the target atom and nucleus that involve changes in mass and energy.

Neutrinos and antineutrinos The neutral particle of interest in this book is the neutron. There is, however, another elementary particle called the neutrino, which is discussed here because Table 1.1 Types of nuclear/atomic radiation

Radiation

Rest mass

Electric charge

Neutral particles

>0

=0

Neutrinos (ν), neutrons (n)

Charged particles

>0

 =0

Beta electrons (β− ), beta positrons (β+ ), alpha (α), protons (p), ions

Photons

=0

=0

X-rays, gamma-rays (γ)

4

Chapter 1 Mechanisms

of the role it plays in the β-decay process. A neutrino (referred to as ν) has very little mass (less then a few eV rest-mass energy)1 , if any, and is not affected much by the strong nuclear forces (see Section 1.4). As such, neutrinos are highly penetrating as they interact weakly with matter. It is, therefore, difficult to detect neutrinos, or harness their use in practical applications. A neutrino serves to conserve momentum and energy in decay processes involving the emission of positive (positron, β+ ) or negative (electron, β− ) beta rays. These nuclear decay processes, unlike those of gamma decay, exhibit a continuous, rather than a discrete, energy distribution, although both decay processes are associated with discrete transitions in energy from one nuclear state to another. The neutrino shares a portion of the released energy with the emitted beta particle, causing the energy distribution to be continuous, as that portion can vary from nothing to the entire decay energy. The conservation of momentum in the β-decay process requires also the emission of a neutrino at 180◦ from the emitted β-particle. The neutrino associated with β− decay is known as an antineutrino, while that associated with β+ decay is called a neutrino. In fact the antineutrino is the antimatter counterpart of the neutrino, in the sense that the particles annihilate each other if they ever coexist at the same location and the same time. The spin of a neutrino is opposite its linear momentum, i.e. it follows the orientation of the fingers of the left hand when wrapped around the vector of the linear momentum with the thumb pointing along the direction of the linear momentum. On the other hand, the spin of the antineutrino is “right-handed’’.

Neutrons The neutron is an elementary particle with a mass, mn , of about 1.675 × 10−27 kg or 1.0087 u, where u is the atomic mass unit2 , and an equivalent rest-mass energy, mn c 2 = 939.6 MeV (≈1 GeV). A free neutron has a half-life of about 615 s, as it decays by emitting a β− particle and an antineutrino, and in the process transmutes itself into a proton. Therefore, neutrons do not naturally exist, but are produced as a result of other nuclear interactions, nuclear decay, or fission. Because of its neutral charge, a neutron easily reaches the nucleus, without being obstructed by electric forces. Neutron reactions are, therefore, interactions with the nucleus, rather than with atomic electrons. Though the neutron has no electrical charge, it has an intrinsic angular momentum and a magnetic moment. These minute magnetic properties are due to the electric charge of the three fundamental moving particles (quarks), from which the neutron is assumed to be constituted. The neutron is composed of two down quarks and one up quark. An up (U) quark has a charge of 23 e and a down (D) quark has a − 13 e charge, where e is the electronic charge (see Glossary for the definition of quarks). At very low energy (in the meV range), neutrons exhibit strong wave characteristics that are employed in 1 eV

is unit of energy (see Glossary at the end of the book) conventionally used in radiation mechanics to reflect the small amount of energy carried by radiation entities (1 eV = 1.602 × 10−19 J). 2 1 u = 1 th of the mass of 12 C ≡ 931.494 MeV in rest-mass energy. 12

5

1.2 Radiation

studying the crystal structure of matter. These neutrons are known as “cold neutrons’’, to distinguish them from the “thermal’’ neutrons, which have an energy corresponding to the temperature of the medium in which they propagate (see Section 3.3.7).

1.2.2 Charged particles The term charged particles refers to electrons (e− ), positrons (e+ ), protons (p), and the positive ions of any atom, including that of the 4 He atom which is known as the alpha (α) particle. The electric charge of each particle is defined in terms of the charge of one electron (e); a fundamental constant called the elementary charge (1 e = 1.602 × 10−19 C). Charged particles are classified as light particles (electrons, positrons) and heavy particles (protons, α particles, and other ions). The Glossary at the end of this book gives the charge and mass of common charged particles. When these particles move at a speed much larger than the velocities of thermal motion, they are referred to as “swift’’ particles. These swift particles are those of interest in this book.

1.2.3 Photons A photon is a quantum of electromagnetic energy. It can be thought of as a packet of waves consolidating together within a confined space as shown in Fig. 1.2. This packet of energy has no mass, no electric charge, and an indefinitely long lifetime, but it exhibits the characteristics of a discrete particle. The energy of a photon, E, is related to the frequency, ν, of its electromagnetic wave by the Planck constant, h (= 4.135667 × 10−15 eV/s), so that: E = hν

(fundamental equation)

(1.2)

The photons we are interested in here are those that have sufficient energy to change the physical status of an atom, say by exciting its electrons or ionizing it by stripping one of its electrons. The electron-binding energy of the tightest bound electron (those in the shell closest to the nucleus, K shell) is in the range

Figure 1.2 A wave packet formed by waves consolidating together within a confined space.

6

Chapter 1 Mechanisms

Table 1.2 Wavelength, frequency, and photon energy of various forms of electromagnetic radiation

Radiation

Wavelength (m)

Frequency (Hz)

Energy (eV)

ULFa

108

3 × 100

10−14

ELFb

107

3 × 101

10−13

VFc

106

3 × 102

10−12

Communicationsd

105

3 × 103

10−11

Microwave

10−2

3 × 1010

10−4

Infrared

10−3

3 × 1011

10−3

Visible light

7.6 × 10−7

4 × 1014

1.63

Ultraviolet

3.8 × 10−7

8 × 1014

3.26

Soft X-ray

10−8

3 × 1016

102

Hard X-raye

10−10

3 × 1018

104

Gamma-raye

10−12

3 × 1020

106

a

Ultra low frequency. Extremely low frequency. Voice (telephone) frequency. d Includes VLF (very low frequency) waves, used for instance in submarine communications, radio waves, and the UHF (ultra high frequency) waves of TV channels. e X- and γ-rays are distinguished by their origin; the former from electronic effects while the latter is due to nuclear excitation. The given range is only indicative, as the two types of radiation overlap in range. b c

from 13.6 eV (for H) to 121.76 eV (for Pu)3 . Therefore, our interest here is in electromagnetic radiation with an energy in the tens of eVs, so that they can reach the bound electron in the inner electronic shells. Obviously, for photons to affect the nucleus, they must have much higher energy to overcome the barrier of atomic electrons. As Table 1.2 shows, only radiation in the upper range of the ultraviolet (UV) waves and above has sufficient energy to affect bound atomic electrons. However, given the relatively long wavelength of ultraviolet radiation and soft (low energy) X-rays (10−8 to 10−10 m), in comparison to the size of the atom (about 10−11 m), UV and soft X-ray photons tend to behave more like waves than particles when encountering the atom. With this wave behavior, no energy is deposited within the atom and the waves are reflected, refracted, diffracted, or Doppler shifted in 3 See

http://www.webelements.com/webelements/elements/text/periodic-table/bind.html for the electron-binding energies of atomic shells in various elements, which reports a maximum value of 115.606 keV for U, and http://www.csrri.iit.edu/periodic-table.html which includes information on the energy edges for X-ray emission up to Pu.

7

1.3 Wave–Particle Duality

a manner similar to an optical wave. However, in the smaller wavelength of soft X-rays, the radiation wavelength becomes comparable in value to the spacing distance between atoms. Therefore, soft X-rays can provide useful information on lattice structures from the diffraction patterns of incident waves. This raises, however, the question of whether photons are particles or waves (see Section 1.3).

X- and γ-rays X-rays are the photons produced electronically, by fast electrons bombarding an electron-rich target. X-ray generators can be common X-ray tubes or powerful electron accelerators (linear accelerators (linacs) or synchrotrons). Gamma (γ) rays are the photons emitted during the decay of a radioactive material. Since an electron bombarding a target slows down gradually, it emits a continuous spectrum extending in energy from an energy equal to the energy of the incident electron to zero energy. X-rays have, therefore, a continuous energy distribution. γ-ray photons have, on the other hand, discrete energies, as they correspond to certain transitions between the excitation levels of a nucleus.

1.3 Wave–Particle Duality Before discussing the concept of wave–particle duality, let us examine the general characteristics of waves and particles, and the concept of duality.

Waves A wave is characterized by a frequency, ν, and a wavelength, λ, related such that: c = νλ

(fundamental equation)

(1.3)

where c is the speed of the wave, which is equal to the speed of light in the case of electromagnetic radiation. A wave propagates and exists in all locations and at all times. For instance a sine wave is expressed by the function A sin(ωt + kx), where A is the amplitude of the wave, k = 2π λ is its wave number, and ω = 2πν, is its angular velocity. This wave repeats itself every 2π within a period of T = 2π ν . A more complex waveform can be constructed by a linear summation of sine waves, with different frequencies. When two waves interact, they can interfere with each other constructively (increasing the resultant amplitude) or destructively. A wave can also change its amplitude when encountering a change in the medium in which it propagates, and it can be phase-shifted (its period displaced) in the process. Subsequently, the phenomena of reflection, refraction, diffraction, and Doppler shifting are observed with waves.

8

Chapter 1 Mechanisms

Particles Unlike a wave, a particle is a consolidated discrete uniform entity with an energy concentrated within a well-defined finite and confined space with definite boundaries.Therefore, a particle exists at a specific location at a certain time, and can only move to a new position in space under the external influence of a force or a potential field. Hence, a particle, unlike a wave, can change its speed, and consequently can be accelerated or decelerated. When a particle collides with another particle, the interaction between the two is governed by the principles of conservation of momentum and energy.

Duality Duality of wave and particle means that an entity can possess the quality or state of having corpuscular or wave properties. This has been observed, for instance, in the case of light photons which at the frequency range of ultraviolet radiation or higher can behave as particles that collide with atomic electrons and liberate them. Then an electric current can be driven by applying an external voltage, as in the case of photocells. On the other hand, particles such as electrons are known to produce diffraction patterns in a grating, similar to those observed with light. In fact, electron microscopes work on the premise that electrons function as waves providing a resolution on the order of their wave length, 10−12 m or less; a resolution much better than that the 10−6 m, or so, of an optical microscope. In order to be able to express the corpuscular behavior of electromagnetic waves and the wave behavior of particles, the concept of wave–particle duality was devised.

1.3.1 Corpuscular nature of waves The concept of a wave exhibiting particle behavior is better explained by the expanded definition of energy, E, to include the rest-mass energy of Eq. (1.1), so that: E = T + m0 c 2

(1.4)

where T refers to the kinetic energy. This expression accommodates a particle with zero mass. A zero-mass particle can be used to express the corpuscular properties of a wave. This “particle’’ is called a photon, and it has only a kinetic energy equal, according to Eq. (1.2), to: T = E = hν =

hc λ

(1.5)

While this takes care of the energy of the photon, we must also give that photon a momentum, p, so that it can possess all the attributes of a particle. Since this photon “particle’’ has no mass, we cannot use the classical definition of moment as mass × velocity. We can rely instead on the relativistic definition of momentum,

9

1.3 Wave–Particle Duality

discussed in Section 2.3, pc =

 E 2 − (m0 c 2 )2

(1.6)

For a photon: E hν h = = (1.7) c νλ λ Equations (1.5) and (1.7) give an electromagnetic wave kinetic energy and momentum values, which in turn enable us to use the laws of conservation of energy and momentum of particles. Note that the momentum is a vector with a direction corresponding to the direction of wave propagation. p=

1.3.2 Wave nature of particles In order for a particle to behave like a wave, it needs to have a frequency, ν, and a wavelength, λ, so that νλ = v, where v is the particle’s velocity. One must also accommodate the fact that while a wave tends to extend across the entire space, as in the case of a sine wave, a particle is concentrated within a small region in space. This confinement of space can be accommodated using the fact that waves interfere constructively and destructively, and that interference can be formed such that the waves combine into a packet (or a beat, as it is called in the case of sound waves), as shown in Fig. 1.2; see also Problem 1.2 and the Wave Packet Explorer: http://phys.educ.ksu.edu/vqm/html/wpe.html on the Internet. We will demonstrate this by simply taking two sinusoidal waves, sin(k − k)x and sin(k + k)x, propagating in the x direction with wave numbers k − k and k + k, respectively. The combination of these two waves gives: sin(k − k)x + sin(k + k)x = 2 sin(kx) cos( k x)

(1.8)

It is obvious that the combined wave has a wave number equal to the average of the two waves, and consequently oscillates at the average of the two frequencies, 2πν since k = 2π λ = v . The cosine term in the combined wave can be seen as a gradual modulator of the amplitude of the sine wave, which initially strengthens the combined wave, but tends to destroy it as the two combined waves become completely out of phase. The result is that the energy of the two waves is consolidated into a “beat’’ within a distance of about x. Note that the two combined waves will again become in phase, and the beat will periodically re-emerge. If a wide range of wave numbers is taken, the quality (concentration) of the beat is improved, and its length, x, is reduced. In the limit, when a continuous distribution of waves of various values of k, spread over a range, k, are combined, one obtains a wave packet similar to that shown in Fig. 1.2, and the combined waves will never become in phase again since they have many different values of k. That is, the wave packet will not be repeated and will be confined within a distance of x, so that x k ∼ 1. Now, we have a localized wave packet that resembles a particle in its confinement to a finite space. What remains is to relate the kinetic energy and momentum of the particle to its wave packet. Before we do this, it

10

Chapter 1 Mechanisms

should be kept in mind that the above argument can be repeated for a wave propagating in time (using waves of the form: sin ωt), resulting in the confinement of the wave packet within a time interval, t ω ∼ 1. We can link the mass and velocity of a particle to the wavelength, λ, and the corresponding wave number, k, of the associated wave, using de Broglie’s relationship (L. de Broglie, 1924) of duality between particle and wave: λ=

2π h = k p

(fundamental equation)

(1.9)

where p is the momentum of the particle. Note that this relationship is identical to that of Eq. (1.2) for photons, though the latter was derived for photons (zero mass), and that Eq. (1.9) cannot be derived from Eq. (1.6) due to the non-zero particle mass, m0 . As such Eq. (1.9) is another fundamental relationship. Similarly, the energy of the particle, E, can be related to the frequency of its associated wave using Eq. (1.2): hv E = hν = (1.10) λ

1.3.3 Uncertainty principle There is a problem with using a single wave number, and frequency, as Eqs (1.9) and (1.10) indicate, while using a wave packet to resemble a particle. A wave packet, as indicated earlier, is the result of the combination of many waves of various wave numbers within k around k; while the above-mentioned equations designate one value of k; a particle can have only one momentum, p, at a given point in space. This dilemma is resolved by the Heisenberg’s uncertainty principle, which states that a particle’s momentum is uncertain until it is measured. This is a logical statement, as it says one cannot know the value of the momentum or any other physical property until one measures it. The uncertainty principle has profound implications. It enables a wave packet to possess the many values of k needed for its formation, while only a single value of k, hence a single value for p, can be measured. This is because the measurement process is an intrusive process that requires some perturbation of the physical property of the particle in order to measure its attributes. For instance, in order to measure the position of a particle such as an electron, one can send a high-energy photon (say an X-ray) and observe its reflection. This X-ray wave will, however, give energy to the electron and in effect change its momentum. On the other hand, if we send a low-energy photon (e.g. a light wave) to precisely measure the momentum, one would get a fuzzy picture of the particle and its position becomes more uncertain. Therefore, in order to know exactly the value of x, one cannot determine the value of k (the range of the wave numbers forming the wave packet), or equivalently the particle’s momentum. The opposite is also true, to measure k, the value of x will be uncertain. This effect is already demonstrated by the fact that x k ∼ 1. 2π From Eq. (1.9), k = 2πp h , hence k = h p. The uncertainty (or indeterminacy)

11

1.4 Nuclear/Atomic Fields

principle (W. Heisenberg, 1927) states that:

x p ≥

h 2π

(1.11)

h is often referred to as . Note that the above inequality The quantity 2π confines the precision with which a quantity can be measured along with its complementary variable4 . Another form of this principle is:

t E ≥ 

(1.12)

The above form of Heisenberg’s principle indicates that certainty in time comes at the expense of uncertainty in energy, and vice versa. The small value of  makes this principle practically irrelevant when dealing with large objects. Given the above discussion one may ask the questions: Is a diffracted neutron a wave? Is a scattered photon imparting kinetic energy to an electron a particle? The concept of wave–particle duality answers these questions by the fact that all radiation entities (particles or photons) evince at times wave-like characteristics and at other times corpuscular behavior. In other words, these entities appear to us, the observers, as particles in some interactions and as waves in others. The wave nature of particles is used to explain the interaction probabilities as discussed in Chapter 3. The term particle is, therefore, often used metaphorically to refer to all subatomic entities.

1.4 Nuclear/Atomic Fields There are four basic natural forces: (1) the force of gravity, (2) electromagnetic forces (3) the strong nuclear force, and (4) the weak nuclear force. Since the particles we deal with here are minute, the effect of gravity is negligible, hence the use of the term corpuscular properties. The other forces create potential fields that can affect approaching nuclear particles. A natural potential field is seen as an interaction between two entities mediated by some exchange particle. For the field of gravity, the exchange particle is thought to be yet undetectable virtual particle with zero mass called the “graviton’’. Photons are the particles mediating the electromagnetic field. The strong nuclear forces are mediated by unstable nuclear particles called pions (a type of mesons; see Glossary), with an energy of 135 MeV, while intermediate vector bosons of energy of about 80 GeV mediate the weak nuclear forces. The intermediate vector bosons are particles that exist for an extremely short period of 4 If the uncertainties x and p are taken as the standard deviation (±σ) of the position and momentum measurements, h respectively, then x p = 4π . In Eq. (1.11) the uncertainty is taken to be the smallest range in a normal distribution that contains 50% of the observed values (i.e. between ±0.67σ). The ratio between the certainty levels should be 0.672 = 0.45, but a value of 0.5 is tolerated by the inequality.

12

Chapter 1 Mechanisms

time to facilitate an interaction. They erupt from the vacuum momentarily, borrowing energy which is paid back upon their annihilation. The mean lifetime of these exchange particles is limited by the uncertainty principle, the inequality of (1.12), with E equal to the exchanged energy. Taking the speed of light as the ultimate speed of these exchange particles, one can determine the range of each of these natural forces. It can be easily shown that the range, R, is given by: R=

c 0.2 × 10−12 (MeVm) ≈ mc 2 mc 2

(1.13)

where mc 2 is the rest-mass energy of the exchange particle. Since the exchange forces in the gravitational and electromagnetic forces have a zero mass, the range of these forces is infinite. For the strong nuclear force, the range is about 10−15 m (1 fm) (equal to the diameter of a medium-size nucleus), while that of the weak force is 10−18 m (1 am).

1.4.1 Potential field The potential energy, V (r), between two entities separated by a distance, r, from any of these four natural forces can, in general, be expressed as: V (r) = ∓A2

exp (−r) r

(1.14)

where the negative value signifies a potential well (producing a force of attraction) while the positive sign indicates a repulsion force, A2 is the strength of the interaction, called the coupling constant, r refers to the distance from the center of the field, and m is the mass of the mediating particle. Table 1.3 gives A2 values for the various forces between two protons. A dimensionless fine structure constant5 can be used to define the inherent strength of a field. For the strong nuclear force, the fine structure constant is equal to about unity, while that for the weak fore is in the range from 10−6 to 10−7 . For the electromagnetic forces, the A2 1 , with A2em being fine structure constant is denoted by α, given by α = emc = 137 the coupling constant. For gravity, a fine structure constant can be defined in a A2

manner analogous to α as cg ≈ 5.9 × 10−39 , where A2g is the coupling constant for gravity between two protons. It is obvious that for radiation particles such as protons, the effect of gravity is so weak in comparison to the effect of the other three forces that it is ignored. Therefore, only the other fields are discussed in associated with the potential energy some details below. Note that the force, F, is determined by V = ∇ · F. 5 The

value of the fine structure constant is not directly related to any obvious physical effect, but is observed when examining finer detail of the physical process involved.

13

1.4 Nuclear/Atomic Fields

Table 1.3 Strength and range of natural fields between two protons∗

∗

Field

Exchange particle (mc2 )

Coupling constant, A2 (eV m)

Range (m)

Fine structure constant

Gravity

Graviton (0)

1.17 × 10−45

∞

Electromagnetic

Photon (0)

1.44 × 10−09

∞

5.9 × 10−39 1 137

Strong nuclear

Pion (135 MeV)

1.56 × 10−08

1.5 × 10−15

1

Weak nuclear

Intermediate vector Boson (91 GeV)

2.01 × 10−12

2 × 10−18

3 × 10−7

Guide to Nuclear Wallchart, Chapter 4, Four Fundamental Interactions, Lawrence Berkeley National Laboratory, 2000: http://www.lbl.gov/abc/wallchart/chapters/04/0.html, accessed October, 2004.

Pion

Nucleon

Nucleon

Figure 1.3 A schematic of the nuclear strong force.

1.4.2 Nuclear strong-force field This field exists between the nucleons (protons and neutrons) of the nucleus, and produces a force stronger than the Coulomb repulsion force between protons. The field has, however, a very short range (about 10−15 mm), and as such affects only particles approaching a nucleus. The strong nuclear force is what keeps the nucleons together. The strong nuclear force is created by the exchange of a meson (pion) between two nucleons, as schematically shown in Fig. 1.3. This field can affect an incoming radiation particle as long as the particle can reach within the range of the field around the nucleus. The neutron is the particle with greatest ability to reach the nucleus within the range of this field, since being a neutral particle it is not affected by the electromagnetic force created by the protons of the atom. The very short range of this potential has the effect that a nucleon inside the nucleus is primarily affected by its immediate neighbor. In essence, every nucleon inside a nucleus is subjected to the same field strength, and subsequently the field of the strong nuclear force can be represented by a flat potential well. The short range of the strong nuclear forces tends also to produce a uniform distribution of nuclear matter inside the nucleus, and as such the volume of a nucleus (which defines the width of its potential field) is proportional to the number of nucleons in the nucleus, i.e. its mass number (A). Subsequently, the radius, R, of a nucleus

14

Chapter 1 Mechanisms 1

is proportional to A 3 , or: 1

R = R0 A 3

(1.15)

where R0 = 1.2−15 m = 1.2 fm. This assumption of uniform distribution of nuclear matter leads to a nuclear density of 2.3 × 1017 kg/m3 .

1.4.3 Nuclear weak-force field The weak nuclear force is mediated by the exchange of massive fundamental particles known as the intermediate vector bosons (fundamental particles of a rest-mass energy greater than 80 GeV). Their range, according to Eq. (1.13), is on the order of 10−18 m (about 0.1% of the diameter of a proton). The weak nuclear force, as the name implies, is more feeble than that of the strong force. These weak forces are responsible for the decay of free neutrons and for beta decay; both involve the emergence of neutrinos and electrons or positrons.

1.4.4 Electromagnetic field The electromagnetic (EM) force is a force between two charged bodies (Coulomb force), or a force induced on a moving electrically charged body by a magnetic field. Magnetic and electric forces are the result of the exchange of a quanta of EM energy (photons), i.e. a photon mediates the exchanged force. The quantum approach to the electromagnetic force is dealt with in quantum electrodynamics (as discussed in Section 3.4). Since photons can travel indefinitely, the EM force has an infinite range, but its magnitude decreases with the inverse of the square of the distance between the affected bodies. At the atomic/nuclear level, the EM 2 1 force is weaker than the strong nuclear force by a factor of 4πεe 0 c = 137 , which is the fine structure constant, with ε0 being the electric permittivity of free space (= 8.85 × 10−12 F/m (or C2 /J m)).

Electric field The Coulomb force between atomic electrons is the dominant force in atomic and molecular structures, since these electrons are too far from the nucleus to be affected by the strong and weak nuclear forces. Magnetic effects provide refinements that are detectable only at high-resolution measurements, or under the influence of very strong magnetic fields (as is the case in magnetic resonance imaging, MRI). The Coulomb (electric) potential energy, V (r), between charged particles, having charges of ze and Ze, where e is the electronic charge and z and Z refer to the atomic number of the particles, is expressed as: V (r) = ∓

1 Zze 2 4πε0 r

(1.16)

15

1.4 Nuclear/Atomic Fields

where r is the distance of approach between the two particles, and the ∓ sign signifies an attraction (−) or repulsion (+), depending, respectively, on whether the electric charges have a different or same charge sign.

Magnetic field defined by the force, F, on a moving The magnetic field is a directional vector, B, charge in accordance to Lorentz’s law as:

F = eZ( v × B)

(1.17)

where eZ is a charge equal to that of Z electrons, v is the velocity of the charge Recall and the operator × is the cross product between the two vectors, v and B. that the direction of the force is given by the right-hand rule. The units of the are N s/(C m), or N/(A m), or simply Tesla (T). The effect of magnetic field, B, the magnetic field is combined with that of other fields by vectorially combining the forces resulting from each field. The potential energy of the magnetic field is ; a vector quantity of the current, i, defined by the magnetic dipole moment, M in a loop (coil) times the loop’s area, A, in the direction perpendicular to the = iAˆn, nˆ is the unit vector current’s loop in the right-hand rule direction, i.e. M normal to the loop’s plane (see Fig. 1.4). The magnetic potential energy, Vm (r, ϕ), is expressed by the dot product: (r) · B Vm (r, ϑ) = −M

(1.18)

where r is the loop’s radius, ϑ is the angle between the plane of the loop and the i.e. cos ϑ = nˆ · nˆ B where nˆ B is the unit vector in the direction of B. direction of B, Equation (1.18) shows that when M is aligned with B, the potential energy is low and B are opposite to each other. The est (most negative), but is highest when M difference between the highest and lowest potential energies is 2|M ||B|, where |·| designates magnitude. These relationships have relevance when dealing with the magnetic dipoles of electrons orbiting an atom, and with the intrinsic magnetic moment associated with the spin of the electron and nucleus. The magnetic e moment of an electron is given by μB = 2m , where e and me are, respectively, e the charge and mass of the electron. This is a fundamental constant known as nˆ

M = iAnˆ

A = Area of loop i

Figure 1.4

, produced by a current, i, in a loop of area, A. Magnetic moment, M

16

Chapter 1 Mechanisms

e the Bohr magneton and is equal to 5.788 × 10−5 eV/T6 . The quantity μB = 2m , p where mp is the mass of the proton, is another physical constant known as the nuclear magneton, = 3.152 × 10−8 eV/T. The magnetic moment of the proton is equal to 2.793μN , while that of the neutron is −1.913μN (with the negative sign indicating that the spin of the neutron is in opposite direction to that of the proton). The fact that the magnitudes of the magnetic moments of the neutron and the proton are greater than μN , and that the neutron (a neutral particle) has a magnetic moment, suggest a more complex internal structure than a mere single entity. In fact, the proton consists of two up quarks and one down quark, while the neutron is composed of two down quarks and one up quark (see Glossary for the definition of quarks).

1.4.5 Quantum states The effect of the atomic and nuclear fields can be studied by both classical continuum mechanics and quantum mechanics. The latter deals probabilistically with the wave nature of particles, while classical mechanics addresses the corpuscular nature of matter. Quantum mechanics is fundamental to the understanding of the atom and its nucleus, and the interaction of radiation with matter. A brief conceptual introduction is given here. As discussed in Section 1.3, a particle of mass m moving with a momentum p has associated with it a wave packet of wavelength λ = hp . Let us indicate this wave by the function ( r , t); a function of the position of the particle in space, r , and the time, t. The intensity of a wave is proportional to the square of its amplitudes. As such, ||2 is indicative  of the density of a particle at a given point in space and time. In other words, ||2 dV is the probability of finding  ∞ a particle within a volume dV at a certain point in time. Consequently, −∞ ||2 dV = 1. In essence, ||2 is a probability density function. In order to determine  one must have a basic wave equation to solve. This equation is the Schrödinger equation: i

∂ 2 = − ∇ 2  + V (r) ∂t 2m

(fundamental equation)

(1.19)

for a particle of mass m approaching a field V (r), where r is the radial distance from the field’s center.This is a fundamental equation (like Newton’s laws, it needs no proof, but its consequences are supported by physical observations). Note that in the above equation, the potential is assumed to be a function of position only, as normally is the case with nuclear fields. For the function  to be physically meaningful, the function itself and its derivatives must be finite, continuous and single valued at all points in space, and it must vanish at infinity. It should be measured value of the magnetic moment of an electron is equal to −1.00116μB (see http://physics. nist.gov/cuu/Constants/index.html), under “electron magnetic moment to Bohr magneton ratio’’ for the precise value.

6 The

17

1.4 Nuclear/Atomic Fields

kept in mind that the function  represents a single particle, not the statistical distribution of particles. However, it is possible for this wave to interfere with itself, since the wave exists at all times and at all points in space. Note that for a free particle, the potential is zero, i.e. V (r) = 0, but when a particle approaches an atom or a nucleus V (r) assumes one or more of the potential field values described in Section 1.4.

Quantum numbers The solution of the wave equation is more readily obtained in the spherical spatial coordinates (r, ϑ, ϕ) using the method of separation of variables. This method of solution necessitates equating the separated functions to constants (to a total of three constants), since they are independent of each other. Each of these independent equations can then be solved separately. The physical acceptability of the solutions, according to the above-mentioned restrictions on , necessitates that the three constants assume certain values, known as the quantum numbers. The three quantum numbers are designated as n, l, and m and are such that: n = 1, 2, 3, . . .

(1.20)

l = 0, 1, 2, . . . , (n − 2), (n − 1)

(1.21)

m = −l, (−l + 1), (−l + 2), . . . , −1, 0, 1, 2, . . . , (l − 1), l

(1.22)

The quantum number n is known as the principal quantum number and determines √ the particle’s energy, E, while l specifies the angular momentum = l(l + 1) and m is the magnetic quantum number, as it controls the energy split when a particle is placed in a magnetic field (the Zeeman effect). The angular momentum number, l, is specified by a certain letter, according to spectroscopy terminology as shown in Table 1.4. In addition to the above quantum numbers, a spin quantum number, s, is added to take into account that a particle rotates around an axis√ passing through itself.The angular momentum associated with this spin is equal to s(s + 1), with s assuming two possible values, ±s, corresponding to the two possible directions of rotation. Note that the proton, neutron, neutrino, and electron all have a spin of s = 12 , while an α particle has no spin and a photon has a spin of s = 1. Particles of half-integer spin are called fermions, while those with an integer spin are known as bosons. A quantum state is defined by a unique set of numbers. The concept of quantum states is not unique to atomic and nuclear states. For example, the harmonic analysis of a vibrating string or the conduction of heat under certain Table 1.4 Terminology for the angular momentum quantum number l

l

0

1

2

3

4

5

6

…

s

p

d

f

g

h

i

…

18

Chapter 1 Mechanisms

boundary conditions, dictate a set of finite convergent series of sine and cosine functions (Fourier series), comprising a series of “quantum’’ frequencies, whose magnitudes are integer multiples of the fundamental frequency.

Pauli exclusion principle The Pauli exclusion principle stipulates that no two particles of half-integer spin (fermions) can occupy the same quantum state. Let us consider two identical particles 1 and 2 with wave functions 1 and 2 occupying two quantum states, i and j (with i and j being sets of two quantum numbers identifying the two states). The wave function of the system could then be ij = 1 (i)2 ( j). Since the two particles are identical, the same system can be arrived at if particle 1 occupies state j and particle 2 in state i, resulting in another system wave function ji = 1 ( j)2 (i). That is, the same system is represented by two different wave functions,  and  , producing two different physical properties, which is physically unreasonable. However, a linear combination of ij and ji leads to a system wave function:  = 12 [ij − ji ] = 12 [1 (i)2 (j) − 1 ( j)2 (i)]. A weight of 12 is used for normalization, and equal weights are employed, since the two particles are identical. The negative sign is used here so that if the two states i and j are identical, then the system’s wave function will be zero and the system cannot exist. That is, for the combined system of the two particles to coexist, the two identical particles must be in different quantum states, i.e. with different quantum numbers. If the two identical particles interchange their quantum states, ij must be equal to −ji , which leads to  = ij = −ji . A system of particles whose wave function flips sign if any of two identical particles in the system interchange positions is said to follow the Fermi–Dirac (anti-symmetric) statistics, or simply Fermi statistics, and the particles are called fermions. A system in which the interchange of the quantum states of two identical particles leaves the system’s wave function unaffected follows the symmetric Bose–Einstein statistics. Particles that follow this statistics are called bosons. Fermons (such as electrons, protons, neutrons, and neutrinos), which have half-integer spin, abide by the Pauli exclusion principle, while bosons (alpha particles and photons, with integer spin) do not. That is, more than one boson can occupy the same quantum state. The ability of bosons to occupy the same quantum state, hence energy state, can lead to concentration, or condensation, of bosons. However, for this to be possible, the bosons must be identical and their waves must overlap and exist at high concentration. This is achievable in electromagnetic radiation as in the case of laser cohesiveness and blackbody radiation, otherwise such conditions are very difficult to attain and occur only at extremely low temperatures7 . Note that Fermi–Dirac statistics is 7 The

phenomenon of superfluidity (zero viscosity) of helium cooled to 2.17 K can be explained by Bose–Einstein statistics. Superfluidity is caused by the condensation of many atoms to the lowest possible energy. The same condensation effect is behind superconductivity, where it is theorized that pairs of electrons coupled by lattice interactions behave like bosons and condensate into a state of zero electrical resistance. Ultra-cold atoms also follow Bose–Einstein statistics and can condensate.

1.5 Atom and Nucleus

19

used to describe the physical properties of metal, such as electrical conductivity, paramagnetism, thermal conductivity, etc.

Parity Another useful and important concept that arises from quantum mechanisms is that of parity, denoted by  (upper case“pi’’).The wave function, ( r ), describing a particular system, e.g. an atom or a nucleus, may or may not change its parity, when r reverses direction to − r . If ( r ) = −(− r ), the system is said to have an “odd’’ or a negative parity,  = −1. On the other hand, if ( r ) = (− r ), the wave function has an “even’’ or a positive parity,  = +1. The concept of parity has no analogous formulation in classical mechanics, but can be viewed as a property that describes the reflection of a particle along a plane passing through its origin, and whether it maintains its  as is (even parity), or alters its sign (odd parity). Parity is conserved, like energy and momentum, in processes involving strong nuclear forces and electromagnetic interactions, as long as the system remains isolated from external effects. The parity of a particle is determined by its orbital angular momentum and is equal to (−1)l , i.e.  = +1 if l is even and  = −1 for odd values of l. As such, the parity is a quantum number. The parity of an electron is considered to be even, and the same applies to the proton, neutron, and neutrino. The wave function of a system consisting of a number of particles, e.g. an atom or a nucleus, is the product of the wave functions of the individual  particles, i.e.  = 1 2 . . . N = N i=1 i , for a system of N particles. Therefore, the parity of  depends on the parity of individual wave functions, and   = 1 2 . . . N = N i=1 i . In defining a quantum state, the parity is given as a  superscript of the total angular momentum of the system, i.e. J  , with J= (l + s)i where i refers to particle i in the system, and  = −1 or +1. Nuclides with even number of protons and neutrons have an even parity, since nucleons pair with each other (a neutron with a neutron and a proton with a proton). When two nucleons pair to produce a net angular momentum of zero, their energy is lowered and as such nucleons occupying the ground state in an even–even nucleus have a zero spin. For a nucleus of an odd mass number, one nucleon would remain unpaired, and the spin and parity of that nucleon (called the valence nucleon) determines the spin and parity of the nucleus.The parity and spin of a nucleus with an odd number of neutrons and an odd number of protons is determined by the two (one proton and one neutron) unpaired valence nucleons.

1.5 Atom and Nucleus 1.5.1 Atomic structure An atom consists of a nucleus surrounded with Z electrons, where Z is the atomic number. The chemical name of the element defines Z, since each element has a

20

Chapter 1 Mechanisms

unique value of Z; Z = 1 for hydrogen (H) and Z = 2 for helium (He), etc. These Z electrons rotate around the nucleus in orbits or shells, called Bohr orbits, of definite energies. Each shell is defined by the principal quantum number n which determines its energy; with n = 1 corresponding to the inner most orbit (K shell), n = 2 to the next orbit (M shell), etc. For the hydrogen atom, the energy of level n, E n , can be explicitly expressed by the equation: En =

Ry n2

(1.23)

where Ry is known as the Rydberg8 energy and is equal to 13.61 eV.The angular √ momentum of the orbiting electron is l(l + 1), where the angular quantum number, l, is such that 0 ≤ l ≤ n − 1. The magnetic quantum number, m, takes an integer value from −l to l, including zero, and defines the components of the angular momentum in a certain direction (observed under the application of an external magnetic field). The electron also has two spin states that produce a nonorbiting angular momentum, designated by the spin quantum number: s = + 12 and − 12 . The electrons are distributed in each orbit according to these quantum numbers, such that they obey the Pauli exclusion principle (see Section 1.4.5). Table 1.5 shows the quantum numbers for the first two energy states. Electrons are filled in the order shown in the table. Atoms whose shells are completely filled, such as helium in the K shell and neon in the L shell, are chemically inert as they have a stable (fully occupied) orbit arrangement that does not need additional electrons. On the other hand, elements with incomplete shells are chemically active. Elements which have one electron in a shell (such as lithium in the L shell) or need an electron to fill the shell (as in the case of fluorine with a missing electron in the L shell) are the most chemically reactive; they can easily either lose the lone electron in the shell or acquire an electron from any other element to fill up the shell. The two angular moment numbers, l and s, are combined into a single quantum number j that defines the total angular momentum such that j = l + s = l ± 12 . The designation nsj defines an electron with a principal quantum number n, l = 0 (for s) and j = 12 , i.e. s = j − l. A 2p 1 is one for which 2

n = 2, l = 1, and s = 12 − 1 = − 12 ; there are three electrons with this designation depending on whether m = −1, m = 0, or m = +1. The designation nf 7 indicates that shell n for l = 3 contains seven electrons.

1.5.2 Nuclear structure The nucleus contains Z protons (same number as the atomic electrons), and A − Z neutrons, with A being the number of nucleons (protons and neutrons), known 8 Rydberg

(1889) obtained empirically a constant, R∞ , while arriving at a formula to determine the wavelength of the optical spectra of elements. This constant was shown later by Bohr’s (1913) model of the hydrogen atom to be equal 4 ee to R∞ = cε2m(4π = 1.097 × 107 m−1 . The constant Ry is the constant that defines the energy level of the spectral )3 line. Therefore, Ry = R∞ hc.

21

1.5 Atom and Nucleus

Table 1.5

Order

Quantum numbers for the first two energy states (K and L shells)

n

l

m

s

K shell 1st

1

0

0

+ 12

2nd

1

0

0

− 12

L shell 3rd

2

0

0

+ 12

4th

2

0

0

− 12

5th

2

1

−1

+ 12

6th

2

1

0

+ 12

7th

2

1

+1

+ 12

8th

2

1

−1

− 12

9th

2

1

0

− 12

10th

2

1

+1

− 12

as the mass number. The nucleus is designated by A Z X , where X is the name of the element; or simply A X , since for each element X , there is a unique value of Z. For example 1 H, 2 H, and 3 H indicate three forms (called isotopes9 ) of hydrogen, with A = 1, 2, and 3, respectively (2 H is also known as deuterium (D) and 3 H as tritium (T)). In all cases, it is necessary that (A − Z) ≥ Z for a nucleus to be stable. For A ≤ 40, Z and A − Z are nearly equal for stable nuclides10 , but heavier nuclides have (A − Z) > Z, since more neutrons are needed to provide nuclear forces that can overcome the increasing repulsive Coulomb force caused by the elevated nuclear charge.

Mass defect 1 The mass of a nuclide is typically given in atomic mass units (u), 1 u = 12 th of the 12 mass of C ≡ 931.493 MeV in rest-mass energy. In all cases, the mass of a nuclide, M , is less than the sum of the mass of its individual nucleons and electrons (the mass of the electrons is so small, compared to that of a nucleon, and is normally nuclides of the same A are called isobars, while those with the same number of neutrons (A − Z) are known as isotones. 10 An atom identified by its nuclear constituents is called a nuclide. 9 Two

22

Chapter 1 Mechanisms

neglected). That is, M = Zmp + (A − Z)mn − M

(1.24)

Mc = Zmp c + (A − Z)mn c − B

(1.25)

2

2

2

where mp is the mass of a proton and mn is that of a neutron, the difference,

M , is called the mass defect, as it reflects the deficiency in mass between the mass of the individual constitutes of the nucleus and its collective mass. The energy corresponding to M , according to Eq. (1.1), is B = Mc 2 , and is called the binding energy. This is the energy associated with the work done by the nuclear forces between nucleons, and it is the energy that would be released if an atom were formed from its constituents11 . Any two particles that are no longer free create a negative potential field, i.e. energy that can be released if the two particles become free. In the nucleus, therefore, the depth of the potential field shown in Fig. 1.1 is equal to −B. As in all natural systems, stability favors the lowest possible potential energy. Therefore, nuclear decay and energy-producing (exoergic or exothermic) nuclear interactions generally move toward a state of lower energy, i.e. the final state of the process will have more binding energy than the initial state, with some exceptions. If the binding energy of the products in a nuclear reaction is lower than that of the reactants, the reaction is only possible if an external additional energy is added to the initial state to make up for the difference; the reaction is then called an endoergic (or endothermic) reaction.

Binding energy The binding energy is indicative of how strong the nucleus is bound. This binding effect is due to the strong nuclear forces between neighboring nucleons (both neutrons and protons). However, this force is subdued by a number of factors. First, the nuclear force of nucleons near the surface of the nucleus is less than that for those in its interior, due to the lower number of neighboring nucleons at the surface. Second, the longer range Coulomb forces between protons act against the short range nuclear force of attraction. Third, an odd number of protons or neutrons tends to weaken the nuclear bond than an even number of either, since the last odd number nucleon does not share its energy state with any other nucleon and as such occupies a higher-energy state. Fifth, nature favors an equal number of protons and neutrons to minimize the total energy of the nucleus (according to Pauli exclusion principle which applies to protons and neutrons, separately). The only factor that can enhance the bonding caused by the nuclear factors is the third factor in reverse, i.e., when an even number of neutrons and protons are present. These trends combined tend to favor an increasing neutron/proton ratio for high mass number nuclides, and equal number of protons and neutrons for light nuclides. Too many neutrons or protons can cause the binding energy to be negative, and the nucleus to become unstable. 11 A

stable nucleus has a higher binding energy than a less stable one, in the same manner the ashes of a burned matter are more strongly bound than the original matter to the extent that no more energy can be extracted from the ashes.

23

1.5 Atom and Nucleus

Nuclear shell model B The binding energy per nucleons, A (see Problem 1.13), generally increases with the mass number, A, until it reaches a maximum value of about 9 MeV in the region of A = 60. It then decreases slowly with A for heavier elements. This decrease is indicative of the fact that heavy nucleons are less tightly bound, due to the increasing effect of the Coulomb force between protons. This general trend is not, however, smooth, but is interrupted by sharp spikes for elements having Z or A − Z equal to 2, 8, 20, 28, 50, 82, and 126. These numbers of protons or neutrons are known as the magic numbers, as they provide the most stable nuclear configuration; analogous to an inert atom which has its atomic shells filled. This observation leads to the establishment of the nuclear shell model, in which the nucleons in the nucleus, like the electrons in the atom, are arranged in discrete energy levels in accordance to their quantum numbers and the Pauli exclusion principle. Unlike the atom, there is no principal quantum number that determines the energy level in the nucleus, rather those levels are determined by the angular momentum quantum number, l. The other significant quantum numbers are the spin number, s = ± 12 , the total angular momentum, j = l + s = l ± 12 , the magnetic angular momentum, m (= −l, −l + 1, . . . , l − 1, l) and the magnetic total angular momentum, mj (= −j, −j + 1, . . . , j − 1, j). The energy levels are determined by the l and j values. The label (2g 7 )8 designates the second energy level for l = 4 2

(for g), with a total angular momentum j of 72 , i.e. with a spin quantum number s = j − l = − 12 , and a room for (2j + 1 = 8) nucleons (8 multiplicity states) each with a different value of mj .The nuclear shell structure is shown inTable 1.5. Note that because of the Coulomb force, the proton energy levels differ from those of the neutron, but the sequence of the energy levels in the nucleus is hardly affected (Fig. 1.5). The highest level in the nucleus occupied by a nucleon is called the Fermi level, and the energy, EF corresponding to this level is the Fermi energy. The energy required to remove a nucleon from the Fermi level is equal to the binding energy of a nucleon to the nucleus, about EAB , where EB is the binding energy and A is the mass number. The value of EAB is about 7–9 MeV. Therefore, the Fermi level lies at about 7–9 MeV below the zero potential energy level (level after which the Coulomb field arises, shown in Fig. 1.1 by the horizontal solid line.) The value of EF for a nucleus of a mass number A and an atomic number Z can be estimated as [17]:  2 Z 3 EF (protons) = 53 MeV A 2  A−Z 3 EF (neutrons) = 53 MeV A

(1.26) (1.27)

24

Chapter 1 Mechanisms

Energy level

Quantum state

Cumulative number of nuclides

(2g7)8 2

4 1g (2p9)10 2

3 2p

50 (Closed shell)

(2p1

40

2

38

)6

(1f5

2

3 1f

)2 )4

(2p3

32

2

(1f7)8

28 (Closed shell)

2

2 2s 2 1d

(1d3)4 2

(2s1)2 2

)6

(1d5

2

(1p1)2 1 1p

20 (Closed shell) 16 14 8 (Closed shell)

2

)4

(1p3

6

2

0 1s

)2

(1s1

2 (Closed shell)

2

Figure 1.5 A schematic of lower-energy levels and quantum states (magic numbers correspond to closed shells). For a complete set, see [12].

The average energy of a nucleon in any of the available energy states is about 0.6 EF . For nuclides of ZA = 0.5, EF ≈ 33 MeV for both protons and neutrons, and the average energy is about 20 MeV. Then the ground state is about 8 MeV below EF , i.e. at about 40 MeV below the zero potential level. Equations (1.26) and (1.27) show that for hydrogen, as one would expect, EF (neutron) = 0. For a heavy nucleus rich in neutrons, EF (neutron) > EF (proton), due to the fact that the highest-filled proton level is at a lower energy since there are more neutrons than protons, hence the additional energy levels for neutrons.

Nuclear excitation state In a given nuclide, the lowest possible energy levels are filled first. Note that the energy level for the l − 12 states is lower than that for the l + 12 states. A nucleon moving to an unfilled state brings the nucleus to an excited state. There are specific excitation levels for each nuclide, in the same manner there are specific electron orbits for an atom. These excitation levels are available on the Internet (NuDat 2.0: http://www.nndc.bnl.gov/nudat2/). Figures 1.6 and 1.7 show for the sake of demonstration the excitation levels for a light isotope, 12 C, and a heavy one, 235 U, respectively (note the difference in scale). Nucleons in excited states will

25

1.5 Atom and Nucleus

12C

Level energy (MeV)

↑ etc.

16.11 15.11 14.08 12.71 11.16 9.64 7.65 4.44

0.00

Figure 1.6

Ground level

Excitation levels in

12 C.

Level energy (keV)

↑ etc.

235U

1028.00 927.21 826.64 680.11 518.10 307.18 148.38 44.92 0.00

Figure 1.7

Excitation levels in

Ground level 238 U.

decay eventually to more stable states. It is obviously easier (takes less energy) to excite a nucleus to its first excitation level above the ground level.

Nuclear collective models Nuclear transitions, particularly in heavy nuclides, can involve nucleons moving collectively to higher excited states. This “collective’’ transition resembles the movement of a drop of liquid (in which molecules move together). Therefore, this model of the nucleus is also known as the liquid drop model, and is used to describe the process of nuclear fission, in which a heavy nucleus divides into two smaller nuclides; a drastic change from the more common nuclear excitation to

26

Chapter 1 Mechanisms

higher energy levels. Using this model, if a nucleus were spherical in shape, its 1 1 radius R would be proportional to A 3 , where A is the mass number, or R = R0A 3 with R0 = 1.2 fm; as indicated by Eq. (1.15). This liquid drop model is also known as the vibrational model, as it assumes that the liquid drop nucleus oscillates between oblate and prolate shapes, due to the competition between the collective Coulomb force amongst protons and the attraction force of the nucleons on the surface of the nucleus (the “surface tension’’). These vibrations affect the excitation energy state of the nucleus. A characteristic quantum number, K , is introduced to describe the projection of the total angular momentum of the nucleus, J , on its symmetry axis. The balance between these two collective forces can break spontaneously in some nuclides, as liquid droplets tend to do, causing spontaneous fission. A gentle prodding of the liquid drop can also cause it to break up. In the same manner, a slow (thermal) neutron can cause the fission of a heavy nucleus. However, when the surface tension force is strong, more energy would be required to destroy the droplet, as such fast-neutron fission can also take place. Nevertheless, this collective action of the nucleons occurs only in certain nuclides, those with many nucleons occupying shells far away from closed shells (i.e. with number of neutrons and protons far removed from the magic number). Such nucleons tend to act as a group. Isotopes of uranium and plutonium fit this pattern, as well as 252 Cf (a common source of neutrons). The Fermi gas model of the nucleus assumes that nucleons are not as tightly bound to each other as in the liquid drop model, but they are always in motion in a manner similar to the molecules of a gas. The gas model describes the ground state, which has the minimum energy, much like a gas in a state of equilibrium. The model considers two gases, a proton gas and a neutron gas, with both having an equal tendency to occupy the lowest possible energy level. However, due to the repulsive force between protons, the lowest energy of the proton gas is slightly higher than that of the neutron gas, which explains why for heavy nuclides, where the electric repulsion between the protons is strong, there are more neutrons to achieve a stable (minimum energy) state. In lighter nuclides, the ground energy for both the ground states of the protons and neutrons are not very different from each other, and the number of neutrons and that of protons tend to be equal.

1.6 Nuclear Decay The stability of a nucleus is not absolute, and can be disrupted by the addition of external energy as in the case of nuclear interactions, or by spontaneous radioactive disintegration as a nucleus decays in attempt to reach a more stable state. This can involve the release of alpha or beta particles, γ radiation, and some other particles, as explained in the following sections. We begin, however, with an overall (macroscopic) analysis of the decay process.

27

1.6 Nuclear Decay

1.6.1 Kinetics Consider a radioactive material containing N0 nuclei at time t = 0. Let λ be the probability per unit time that this material will decay, one way or another.The probability that the material will not decay within a short time interval t is then q = 1 − λ t. That is, qN0 nuclei will remain without decay after t, q2 N0 after 2 t, and so on. Therefore, within n sequential time intervals, adding up to a total time period, t, the number of nuclei remaining without decay in the nth time interval, N (t), is then N (n t) = N (t) = qn N0 . As t → 0, n → ∞, one has12 :   t n N n n = lim q = lim (1 − λ t) = lim 1 − λ n→∞ n→∞ n→∞ N0 n = e−λt = exp(−λt)

(1.28)

This is the exponential law of decay, and λ is the decay constant of the considered nuclide. The rate of decay (or disintegration) is called the activity, A:    dN   = Nλ  (1.29) A= dt  The activity is expressed in becquerels (Bq), disintegration per second13 . The half-time, t 1 , is the time it takes a material to lose by decay half of its 2

original quantity, i.e. the time at which t1 = 2

N N0

= 12 . It can be easily shown that:

ln 2 0.693 = λ λ

(1.30)

If a nuclide decays by more than one mode, a partial half-life, τ 1 , for the ith type of 2 decay (or a particular transition with the same type of decay), which has a decay constant λi , is defined as: t1 ln 2 τ1 = = 2 (1.31) 2 λi fi where fi is the ratio of the partial to the total decay constant, called the branching ratio. The partial half-life is the half-life if the material were to decay exclusively by the prescribed mode of decay. The daughter in a decay process may also be radioactive. Then, the rate of accumulation of a daughter nuclei is equal to its rate of production minus its rate of decay, i.e.: dNd = λp Np − λd Nd = λp Np (0) exp(−λp t) − λd Nd dt

b 1 + ab a = e. 13 The old unit is called Curie (Ci), 1 Ci = 3.7 × 1010 Bq = 37 GBq. 12 lim



b a →∞

(1.32)

28

Chapter 1 Mechanisms

where p refers to the parent nuclide and d to the daughter nuclide. The solution of Eq. (1.32) is:  λp Np (0) Nd (t) = Nd (0) − exp(−λd t) λd − λp +

λp Np (0) exp(−λp t) λd − λp

(1.33)

d When λp < λd , i.e. the parent had a longer half-life, N Np will reach a constant value with time; the parent and daughter are then in a state of transient equilibrium. If λp << λd , so that the decay of the parent during the accumulation of the daughter nuclei is negligible, Nd will eventually approach a saturation value, establishing a state of secular equilibrium. There are a number of long decay sequences involving many progenies; those of 232Th, 235 U, 238 U, and 241 Pu, known, respectively, as the thorium, actinium, uranium, and neptunium decay series. An interactive demonstration of these decay series can be found on the Internet (e.g. Natural Radioactive Series: http://www.eserc.stonybrook.edu/ProjectJava/Radiation/).

1.6.2 Statistics The decay process is a discrete process in the sense that it involves the decay of one nucleus at a time. Such discrete events are statistically governed by the binomial distribution. Let N be the population from which n decay events occur within a time interval t. Within this period, t, the probability that no decay takes place, q, is exp(−λt), as evident from Eq. (1.28). Then the probability that decay occurs is p = 1 − exp(−λ t). This is while according to the binomial distribution, the probability P(n) of n decay events is: N! pn qN −n n!(N − n)! N! [1 − exp(−λ t)]n [exp(−λ t)]N −n = n!(N − n)!

P(n, t) =

(1.34)

The mean of this distribution is n¯ = Np = N [1 − exp(−λ t)], which is the average number of particles emitted within t. When λ t << 1 (or equivalently, when the observation time is much less than the half-life), then n << N , and one can take the limit N → ∞ of the distribution (1.34) as a reasonable representation of the decay statistics. Then: P(n, t) =

(λ t)n exp(−λ t) n!

(1.35)

This is the Poisson distribution, which has a mean equal to n¯ = λt (corresponding to Np in the binomial distribution) and a variance, σ 2 , equal to its mean, that is

29

1.6 Nuclear Decay

σ 2 = n¯ . Although, N is not infinity, it is quite large compared to the number of decays observed within a certain time period, and the Poisson distribution is used to statistically analyze radioactive decay. Another interesting feature of Poisson distribution is that it is also a special case of the normal (Gaussian) distribution, when n is sufficiently large (> 20 or so). Then:  (¯n − n)2 P(n) = √ exp − 2¯n 2πn¯ 1

(1.36)

The use of the normal distribution facilitates the definition of confidence intervals, one can state that 50% of the measurements lie between m ± 0.67 σ, 68% within m ± σ, 95% within m + 2 σ, etc. It should be noted though while the Poisson distribution is a distribution of discrete events, the normal distribution is a continuous distribution.

1.6.3 Alpha decay The decay of a nucleus by the emission of an α particle requires firstly that two protons and two neutrons combine within the nucleus, and secondly the formed positively charged particle overcomes the Coulomb barrier in the field of the nucleus (about 20 MeV for a typical heavy nucleus). For an α particle (a 4 He nucleus) to form within a nucleus, the nucleus must be rich in both neutrons and protons. Therefore, α decay occurs in nuclides heavier than lead (possible for A ≥ 150 and is the dominant decay mode for A ≥ 210, where A is the mass number). In such a heavy nuclide, many nucleons can exist far away from the closed nuclear shell, with some freedom to collectively combine with each other. The formation of an α particle is favored over the formation of the lighter 2 H nucleus, since the binding energy of 4 He is much higher than that for 2 H (28 MeV compared to only 2 MeV); since the most stable forms have the highest binding energy per nucleon. Also, the probability of α particle formation in a nuclide with even number of nucleons is higher than in those containing odd numbers of nucleons, resulting in a lower decay rate for the latter. An α particle formed in the negative portion of the nuclear potential field, i.e. below the energy level at which the Coulomb field begins to appear (see Fig. 1.1), cannot leave the nucleus. However, an α particle in a positive energy state can “tunnel’’ through the Coulomb barrier. This is possible as an α particle can possess wave properties; a portion of the wave incident on the inside of the Coulomb barrier can be transmitted through to the outside of the barrier, while the remaining portion is reflected back. That is, there is a finite probability that an α particle can exist outside the Coulomb barrier; hence a finite probability exists for α emission out of the nucleus. The α decay of a parent nuclide X with a mass number A and an atomic A−4 number Z, designated here as A Z X, to a daughter nuclide Z−2Y, can leave the latter

30

Chapter 1 Mechanisms

in an excited state. The decay process can then be expressed in two stages as: A ZX A−4 ∗ Z−2 Y

→ →

A−4 ∗ 4 Z−2Y + 2 α + Eα A−4 Z−2Y + Eγ

whereY∗ indicates the excited state of the product nucleus (if any, see below), Eα is the kinetic energy carried by the α particle and the recoil nucleus, and Eγ is the energy of the γ radiation resulting from de-excitation. The binding energy for this combined decay process is such that: [Zmp + (A − Z)mn ]c 2 − B1 = [(Z − 2)mp + (A − Z − 2)mn ]c 2 − B2 + [(2mp + 2mn )c 2 ] − Bα + Eα + Eγ Qα = Eα + Eγ = Bα + (B2 − B1 )

(1.37)

where mp and mn are, respectively, the mass of a proton and a neutron, and B1 and B2 refer to the binding energy of the original and product nuclei, respectively. The energy corresponding to the difference in mass between the parent nucleus and the decay products (daughter nucleus and the α particle), which is equal to the value of Eα + Eγ , is called the Q-value of the decay. The high binding energy of α particles, about 28 MeV, makes α decay an exoergic process in many nuclides, since the energy consumed in removing the particle from the nucleus is so small. Therefore, this decay can take place spontaneously, without leaving the nucleus in an excited state, i.e. with Eγ = 0. Alpha decay also lowers the Coulomb energy of the daughter nucleus (by removing two protons), while not having much effect on the binding energy per nucleon. Therefore, the daughter nucleus tends to be more stable than the parent nucleus. Note that all α particles emitted from a certain nuclide have the same kinetic energy, since the change in binding energy is always the same.

Alpha transitions The conservation of the total angular momentum and parity restricts the energy levels from which α decay can occur. A parent nucleus with a total angular momentum given by Jp decays to a daughter of a total angular momentum determined by Jd such that the total angular momentum of the α particle, jα , is the vector difference between the angular momentum of the initial and final states; i.e. Jα = | Jp − Jd |. The latter vector difference can have any absolute value from | Jp − Jd | to | Jp + Jd |, depending on the relative orientation of Jp and Jd . Since the α particle has zero spin, its total angular momentum jα is equal to its angular momentum lα (recall that j = l + s). Therefore, conservation of total angular momentum for α decay requires that: | Jp − Jd | ≤ lα ≤ | Jp + Jd |. The parity of an α particle is equal to (−1)l , since its spin is zero. Conservation  of parity dictates that Jp p = (−1)l Jdd . If for example the parent is 0+ , i.e. has an

31

1.6 Nuclear Decay

even parity with Jp = 0, then the daughter should be such that Jd is equal to 0+ for l = 0; 1− for l = 1; 2+ for l = 2, and so on; all other transitions are forbidden. Alpha decay tends to leave the daughter in the ground state, so that the α particle carries a maximum amount of energy. For nuclides with even number of neutrons and protons, the spin of the ground state is zero, consequently α decay tends also to result in a daughter in a zero-spin ground state. However, in nuclides with an even mass number, the daughter nucleus can be left in an excited state (though close to the ground state) following α decay. This leads to the emission of α particles with different distinct energies depending on the level of excitation of the product nucleus. Such energy variation is known as the fine structure of the α particle energy spectrum.

Hindrance factor The half-life for α decay, t 1 α , is empirically related to the Q-value of the α decay, 2 Qα , by the Keller–Munzel relationship14 :   2 Z 3 +b (1.38) −Z log 10 t 1 α = a √ 2 Qα with Qα in MeV, and a and b are best-fit constants for t 1 (in seconds): 2

Z

A−Z

a

b

Even

Even

1.61

−20.261

Even

Odd

1.65

20.238

Odd

Even

1.66

20.726

Odd

Odd

1.77

20.657

It is obvious that the lowest half-life, hence highest rate of decay, is achieved in even–even nuclides. The α decay of such a nuclide is a transition to another even–even nuclide. Since the total angular momentum for even–even nuclide is zero, and their parity is even, the ground state to ground state (gs–gs) α transitions are 0+ → 0+ transitions. The half-life of this type of decay can be theoretically calculated using the so-called “one-body’’ model of α particles, which assumes that the α particle is pre-formed (as one body) within the nucleus before penetrating the Coulomb potential barrier by the tunneling effect. In this model, the probability of decay (i.e. release of the pre-formed α particle) in the nucleus depends on the probability of the particle hitting the Coulomb barrier times 14 K.

H. Keller and H. Munzel. Shell effect and α-decay probability, Zeitschrift fur Physik, Vol. 255, 1972, pp. 419–424.

32

Chapter 1 Mechanisms

the probability of penetrating the barrier. The ratio between the measured partial half-life of a particular transition and that calculated using the one-particle model for an even–even nuclei at the energy of α decay is known as the hindrance factor (HF).

1.6.4 Beta decay Beta decay involves the conversion of a neutron to a proton, or a proton to a neutron, i.e. the transformation of a nucleus to an isobar (as the mass number does not change). This conversion process is not surprising given the fact that protons and neutrons are both nucleons that abide by the same set of quantum numbers, and as such are subject to transformation from one state to another within the nucleus. The result of this conversion process is that the neutron-to-proton ratio changes; increasing in a neutron-rich nucleus and decreasing in a proton-rich nucleus. This mode of interaction occurs in the majority of unstable nuclides, and can take one of the three forms: β− decay, β+ decay, and the capture of an atomic electron.

β− decay The conversion of a neutron to a proton requires the release of an electron (a β− particle) for charge balance, such that: 1 0n

→11 p +−10 e + 00 ν¯ + Eβ + Eν¯

where ν¯ refers to an antineutrino and Eβ and Eν¯ are the kinetic energies of the emitted electron (β− particle) and antineutrino (¯ν). The presence of the almost massless neutrino is required to satisfy momentum conservation, as well as to meet the experimentally observed variation in the energy spectrum of the emitted β− particle, which varies from zero to a maximum energy, Eβmax . The neutron, the proton, and the electron are all fermions with a spin number of 12 . Therefore, without the neutrino, the spin (hence angular momentum) will not be conserved. It is also necessary to balance the lepton number15 . Therefore, an antineutrino (not a neutrino) is assigned to this decay process. The β− decay of a parent nuclide A Z X may leave the nucleus in an excited ∗ . Therefore, this combined decay process is expressed as: Y state, A Z−1 A ZX

A ∗ →Z+1 Y +−10 e + 00 ν¯ + Eβmax

A ∗ Z+1Y 15 A

A →Z+1 Y + Eγ

lepton is particle that is not involved in the strong nuclear forces, but participates in the weak ones (see Glossary). A lepton number of 1 is assigned to both the electron and the neutrino and −1 is associated with both the antineutrino and the positron.

33

1.6 Nuclear Decay

where Eγ is the energy of γ radiation. In terms of binding energy for the final state, after de-excitation, [Zmp + (A − Z)mn ]c 2 − B1 = [(Z + 1)mp + (A − Z − 1)mn ]c 2 −B2 + me c 2 + Eβmax + Eγ Qβ− = Eβmax + Eγ = (B2 − B1 ) + (mn − mp − me )c 2

(1.39)

where mp , mn , and me are, respectively, the mass of a proton, a neutron, and an electron, c is the speed of light, and B1 and B2 refer to the binding energy of the original and product nuclei, respectively. Note that (mn − mp − me )c 2 ≈ 782 keV. Equation (1.39) indicates that a β− decay processes in which Eβmax + Eγ < 782 keV ≈ (mn − mp − me )c 2 has to involve a decrease in binding energy (contrary to the general trend of increasing binding energy encountered in most decay processes). Note that if the product nucleus after β− decay is in the ground state, Eγ will be equal to zero.

β+ decay The β− decay process is anticipated, and encountered in the disintegration of natural nuclides, as they all have a neutron-to-proton ratio greater than one in order for them to exist in stable states. Since the neutron is heavier than the proton, it is also natural to expect that a neutron would shed some mass to reach a lower and more stable energy state. In fact, a free neutron would eventually decay to a proton. On the other hand, the proton is lighter than the neutron, and is a stable free particle. Therefore, for the positron emission process: [E] + 11 p →10 n + 01 e + 00 ν + Eβ + Eν to take place, where ν designates a neutrino, the proton has to acquire some energy, [E], from another nucleon in the nucleus. This decay occurs in artificially produced proton-rich isotopes, such as 13 N and 15 O, which has proton-to-neutron ratios of, respectively, 76 and 87 . Like in β− decay, the neutrino is emitted to conserve linear and angular momentum and the lepton number, which also explains the observed energy distribution of the emitted positron. The β+ decay of a parent nuclide A Z X is such that: A ZX

A ∗ →Z−1 Y + 01 e + 00 ν + Eβmax

A ∗ Z−1Y

A →Z−1 Y + Eγ

34

Chapter 1 Mechanisms

Naturally, if after β+ decay the nucleus is not excited, Eγ will be equal to zero. Then, [Zmp + (A − Z)mn ]c 2 − B1 = [(Z − 1)mp + (A − Z + 1)mn ]c 2 − B2 + me c 2 + Eβmax + Eγ Qβ+ = Eβmax + Eγ = (B2 − B1 ) − (mn − mp + me )c 2

(1.40)

With (mn − mp + me )c 2 ≈ 1.804 MeV, β+ decay requires the product nucleus to have a binding energy higher than that of the product nucleus by more than 1.8 MeV. The positron generated in this reaction will travel through the sea of electrons surrounding the nucleus and will combine with one of these electrons causing their mutual annihilation and the production of two photons, each of energy of me c 2 = 511 keV, emitted in opposite directions to conserve linear momentum. This positron annihilation process is the essence of the positron emission tomography (PET) technique used in medical imaging.

Electron capture A proton can also be converted to a neutron by capturing an electron from the atomic electrons nearby, e.g. from the K shell. In this manner, there is no need for a proton to acquire energy from within the nucleus, hence no positron emission. The electron capture (EC) process (also referred to as ε decay) is such that: 0 1 1 p +−1 e

→10 n +00 ν + Eν

where Eν is the kinetic energy carried by the neutrino and the nucleus as it recoils (the latter is a negligible quantity). The emission of the neutrino is again essential, to conserve momentum, as the nucleus recoils and compensates for the addition of an electron of a lepton number of 1 to the parent nucleus. Electron capture creates a vacancy in the affected electron shell, which is subsequently filled by an electron from a higher shell. This causes the emission of X-rays and electrons characteristic of the product nucleus, called Auger electrons. Mass– energy balance alone is not sufficient to produce the above interactions, since mp + me < mn . Additional energy has, therefore, to be acquired from the nucleus, by the change in the binding energy. The nuclear binding energy for a parent A ZX A Y, is as follows: nucleus decaying by electron capture to a daughter nucleus, Z−1 [Zmp + (A − Z)mn ]c 2 − B1 + me c 2 

= (Z − 1)mp + (A − Z + 1)mn c 2 − B2 + Eν Eν = (B2 − B1 ) + (mn − mp − me )c 2

(1.41)

The energy of binding the atomic electron is not included here, since it is not part of the nucleus, and is at any rate quite small compared to the binding energy

1.6 Nuclear Decay

35

of a nucleon in the nucleus. The available decay energy (i.e. the energy aside from the change in binding energy) is similar in value to that of Eq. (1.39) for β− decay. However, by comparison to Eq. (1.40), the decay energy available for EC is 1.022 MeV (= 2me c 2 ) lower than that for positron decay.Therefore, if the available decay energy is less than 2me c 2 , positron emission is impossible and a nucleus can decrease its number of protons by EC. Of course, if the available energy is higher than 2me c 2 , both β+ decay and EC can take place. For example, 22 Na decays 9% of the time by EC (Auger electrons) and 91% by positron emission, while in heavy elements EC is highly favored over positron emission; e.g. 100% of the decay of 125 I to 125Te and 207 Bi to 207 Pb is by EC. Since electron capture produces the same change in the nucleus as a positron emission, decay information for these two types of decay are often included together. Decay information is available on the Internet (NuDat 2.0: http://www.nndc.bnl.gov/nudat2).

Beta transitions The emitted beta particle and neutrino each have a spin of ± 21 , which can combine vectorially parallel to each other resulting in a net spin S = ±1, or anti-parallel with a net spin S = 0. Therefore, nuclides most readily decay when | Jp − Jd | = 1 or 0, and their is no change in parity between the parent and daughter nuclei, where Jp and Jd refer to the total angular momentum of the parent and daughter nuclei, respectively. Nuclear transitions associated with beta decays are called allowed transitions. Decay with | Jp − Jd | = 1 is known as Gamow–Teller transitions, while those with | Jp − Jd | are called Fermi transitions. The probability of emission depends on the value of | Jp − Jd |, being highest when | Jp − Jd | = 0. The probability of beta emission is expressed through a function known as the comparative half-life or ft value, where t is the half-life of the decaying nucleus, and f is a function of the atomic number of the decaying nucleus and the decay energy, Eβ + Eν . The log ft values are reported in the Isotope Explorer: http://ie.lbl.gov/ensdf/ under tables. The smallest ft values are those of mirror nuclides, i.e. those whose number of neutrons and protons are interchangeable, 23 23 25 25 e.g. (n,p), (31 H, 32 He), (178 O, 17 9 F), (11 F, 12 Mg), and (12 Mg, 13Al). Nuclear transitions leading to beta decay are classified according to | Jp − Jd |, and the value of log ft as: superallowed, allowed first forbidden, second forbidden, and so on. Superallowed transitions occur when both the parent and daughter have the same total angular momentum parity, i.e. Jp − Jd = 0 and p = d . Conservation of parity is not necessary in beta decay, since it involves the weak nuclear forces. These superallowed transitions are typically those of light elements, and include the decay of a free neutron. A transition with Jp − Jd = ±1, with the parent and the daughter having the same parity, is called an allowed transition, as it still has a high probability of emission, but is more likely than the superallowed transition as the electron carries a higher angular momentum. Transitions with larger values of | Jp − Jd | have much lower emission rates, since this implies that the emitted electron carries a large momentum which is quite unlikely given the small

36

Chapter 1 Mechanisms

mass of the electron in comparison to the nucleus. Such forbidden transitions are transitions in which the parent and the daughter nuclei may not have the same parity. An nth transition is such that | Jp − Jd | = n, n + 1 and p d = (−1)n . The only exception to this rule is that of the first forbidden state, in which parent and daughter have different parties, | Jp − Jd | = 0, 1, or 2. Note that the above rules for | Jp − Jd | apply to the three forms of beta decay (β− , β+ , and ε), since they pertain to the state of the isomers before (parent) and after (daughter) beta decay. Since the values of | Jp − Jd | overlap different degrees of forbiddenness, the value log ft determines whether a certain decay process belongs to a particular class, the lower log ft the lower is the degree of forbiddenness. Transitions in which |Lp − Ld | > | Jp − Jd |, where L is the angular momentum (excluding spin of the nucleus) are called l forbidden. In a deformed (non-spherical nucleus), if |Kp − Kd | > | Jp − Jd |, the transition is called K -forbidden, where K is a quantum number that describes the deformation (from spherical symmetry) of a nucleus (see Section 1.5.2). There is a very small probability that a nucleus can decay to a stable state by emitting two beta particles simultaneously. This double-beta decay process is quite rare.

1.6.5 Gamma decay Gamma (γ) decay is caused by the de-excitation of the nucleus, either to the ground state or to another exited state. The γ-ray energy can be as small as a few keV (e.g. 38.9 keV for 172W γ decay) or as high as a few MeV (e.g. 4.443 MeV for 11 Be γ decay). If the excited state has a measurable lifetime, the nuclide is in a metastable state, called an isomeric state and is identified by adding the letter m after its mass number, e.g. 99m Tc emits 142.7 keV γ (used as a radiopharmaceutical in medical nuclear imaging). The decay process is then referred to as an isomeric transition (IT). As indicated above, γ decay can accompany other types of radiation decay which leaves the nucleus in an excited state. To conserve momentum, the emitting nucleus recoils, gaining kinetic energy, but this energy is so low that it is negligible. Therefore, the energy of the emitted γ-rays is slightly lower than the difference in energy between the initial and final states.

Gamma transitions The energy of an emitted γ-ray, Eγ , is equal to the difference between the initial and final states of the isomers, Ei − Ef , i.e.: Eγ = hν = Ei − Ef

(1.42)

where ν is the frequency of the γ-ray and h is Planck’s constant. This transition is accompanied by a change in the total angular momentum and parity, i.e. from J i to J f and from i to f . The γ photon acquires an angular momentum defined

37

1.6 Nuclear Decay

by the quantum number ł, which is such that | Ji − Jf | ≤ l ≤ | Ji + Jf |, due to the vector nature of J i and J f . However, transitions take the lowest change in angular momentum, as such l = | Ji − Jf |. Gamma nuclear transitions are classified according to the value of | Ji − Jf |, and whether i is equal to f or not. If i = f , an odd value for | Ji − Jf | refers to a magnetic multipole transition, M| Ji − Jf | , and an even value results in electric multipole transition, E| Ji −Jf | ; the opposite is true for f = |i . The degree of multiplicity of a pole is given by 2l , with l = 1 resulting in a dipole, l = 2, a quadruple, l = 3 an octupole, and so on. That is: |Ji − Jf |

1

2

3

4

...

Parity

i  = f

i = f

i  = f

i = f

...

Electric

E1

E2

E3

E4

...

Parity

i = f

i  = f

i = f

i  = f

...

Magnetic

M1

M2

M3

M4

...

The terminology “electric’’ and “magnetic’’ describes the type of electromagnetic field associated with radiation16 . Favored transitions have a small | Ji − Jf | and high transition energy, while all other transitions are either unfavorable or do not occur. Energy levels close to the ground state tend to have the same parity as the ground state, and such transitions from those states to the ground state or in between themselves are the M1 or E2 transitions. Often though mixed M1 /E2 transitions are observed, due to the enhancement of E2 caused by the deformed (non-spherical) shape of the nucleus. Note that | Ji − Jf | = 0 is not permitted, since a released photon requires an angular momentum of 1. However, nuclear vibrations (see the collective model of the nucleus in Section 1.5.2), can transfer energy to a surrounding atomic electron, even at | Ji − Jf | = 0, ejecting it from the atom. This is the process of internal conversion described below.

1.6.6 Internal conversion The excitation energy of a nucleus can be transferred into an orbital electron. This can be thought of as a “shivering’’ nucleus affecting the electrons surrounding it. As indicated above this process can take place without change in the total angular momentum of the nucleus. An affected atomic electron will in turn be liberated, carrying an energy equal to the nuclear transition energy minus the electron’s binding energy. If the same transition energy can be released as a γ-ray, then both processes can take place at the same time. The ratio between the two processes is 16 Electric

fields are produced by oscillating charges, while magnetic ones are due to oscillating currents.

38

Chapter 1 Mechanisms

called the conversion coefficient, α, defined as: α=

Number of internal conversion decays Number of γ-ray decays

(1.43)

Since the ejected electrons can arise from any electron orbit, an orbit-specific conversion coefficient can be defined, then α = αK + αL + αM , . . . , where the subscript refers to the shell from which the electron is emitted. The electron with the highest energy will then be emitted from the outer atomic shell, which has the lowest binding energy, while the K shell gives rise to the lowest electron energy. Most internal conversion takes place at the K shell, and the value of α tends to increase with increasing atomic number and decreasing nuclear excitation energy. A vacancy left in an outer shell by internal conversion (IC) is subsequently filled with an electron from an outer shell. This is either followed by the release of Auger electrons or characteristic X-rays, in a manner similar to the electron capture process of beta decay.

1.6.7 Spontaneous fission According to the liquid drop model of the nucleus, indicated in Section 1.5.2, a gentle perturbation of a large nucleus should split it into two small droplets, causing division or fission. One could argue that superheavy nuclides do not exist because of the instability against spontaneous fission. This instability is caused by the large repulsive force between the protons in heavy nuclei, which drives the droplet to a non-spherical shape that is vulnerable to division, but is tampered by the binding energy stabilizing effect associated with the shell structure of the nucleus. The latter effect, in many cases, stabilizes most heavy nuclides against spontaneous fission. Fission is resisted by a potential barrier of about 5–8 MeV for nuclides of mass number from 232 to 242. If a nucleus is excited to an energy equal or above this barrier energy it will undergo fission, most likely before it releases its excitation energy by γ emission. Nuclides at the ground state can still undergo spontaneous fission by tunneling through the fission barrier, in a manner similar to α decay. Therefore, nuclides that undergo spontaneous fission are also α emitters. Some of the nuclides that exhibit spontaneous fission with significant intensity are: 248 Cm, 252 Cf, and 254 Cf. Californium-252 is a common isotropic source of neutrons. Other less significant spontaneous fission occurs in 208 Po, 231Th, 235 U, 238 U, 239 Pu, 240 Pu, 250 Cm, 252 Fm, 254 Fm, 256 Fm, and 258 Fm. The fission process of a heavy nucleus results in the release of ν neutrons (0 ≤ ν ≤ ∼6 or so, with an average value, ν¯ , of about 2.5) and two medium-size nuclides, with mass numbers in the range of 70–160. This process for a nucleus A Z X can be represented as:  A ZX

1 →A Z1 X1

+

A2 Z2 X2

+ ν01 n + γ

with

A1 + A2 + ν = A Z1 + Z2 = Z

39

1.6 Nuclear Decay

The γ-rays released simultaneously as the nucleus divides, is called prompt γ-rays A2 1 and can have an energy up to 8 MeV. The fission products, A Z1 X1 and Z2 X2 , are typically left in an excited state and emit γ-rays within less that 1 ms of their formation. Since fission takes place in heavy nuclei that are neutron rich, the fission products are also neutron rich, and as such decay by a series of β− emissions to reduce their proton-to-neutron ratio. Such β− decays typically leave the nucleus in an excited state and is usually accompanied by γ decay. Some of the nuclides created by fission also decay by neutron emission, producing the so-called delayed neutrons to distinguish them from the prompt neutrons produced during the fission process17 . The emitted neutrons acquire a kinetic energy over a wide energy range, up to about 14 MeV. The energy distribution of the neutrons is typically expressed by the Watt distribution18 : N (E)dE = a exp(−E) sinh

√ bE dE neutrons/MeV

(1.44)

where N (E)dE is the number of neutrons per fission emitted with an energy between E and E + dE, and a and b are nuclide-dependent constants. The average kinetic energy of a fission neutron is about 2 MeV, though the most probable energy is about one-third the average value.

1.6.8 Decay by neutron or proton emission Since the nucleus is composed of protons and neutrons, one might expect that decay by neutron or proton emission would be quite common. However, this type of decay is forbidden by energy conservation, that is the binding energy of the product nucleus of this type of decay would be less than that of the parent nucleus. Therefore, such decay requires the addition of energy to the parent nucleus, to enable one or more of its nucleons to overcome the potential barrier and leave the nucleus. This requirement prevents the decay of natural nuclides by this type of emission. However, decay by proton and neutron emission is observed in the isomeric decay of excited nuclides, typically following a β decay. For example, after the β+ decay of 111Te, the daughter 111 Sb decays by proton emission to 110 Sn, while the 87 Kr isomer, resulting from the β− decay of 87 Br (about 2% of the time), decays to 86 Kr by neutron emission. These processes are known as betadelayed proton or neutron emissions. Note also that these beta-delayed decays can release α particles, and other types of particles. The metastable isomer 53m Co emits (1.5% of the time) a proton directly (without prior beta decay), decaying to 52 Fe. Some other heavy nuclei also exhibit proton decay from the ground state, such as 147Tm and 151 Lu. When proton emission takes place, it must tunnel 17 Delayed

neutrons are relied upon to control the fission process in nuclear reactors, by ensuring that a reactor does not become critical without their contribution. 18 B. E.Watt. Energy spectrum of neutrons from thermal fission of U235 , Physical Review, Vol. 87, 1952, pp. 1037–1041.

40

Chapter 1 Mechanisms

through the Coulomb potential barrier, unless it has an energy greater than that of the barrier.

1.7 Reactions and Interactions When a radiation particle approaches an atom/nucleus it can be affected by the associated fields in a number of ways. It is then said that the particle has “interacted’’. The target atom/nucleus, on the other hand, may suffer some change, i.e. “react’’, or be left unaffected, or so barley affected that the reaction is inconsequential. Due to the short range of nuclear forces, interactions with the nucleus are quite different in nature from those with the atom as a whole, or with its individual atomic electrons. Therefore, these two types of interactions are discussed separately in the ensuing sections. One can classify these interactions using both the corpuscular nature and the wave nature of radiation, or based on the nature of the affecting atomic/nuclear field. However, it is more practical to classify these interactions by the effect the fields have on the incident radiation. Most textbooks address all aspects of one interaction at a time; presenting the mechanism of an interaction then directly examining its kinematics and the probability of its encounter, all at once. This book takes a broader perspective, given the fact that most of these interactions can be described by generalized kinematics and interaction mechanics. Therefore, in this section a compilation of all possible reactions are presented, while the kinematics of all interaction are discussed in Chapter 2 and the probability of interaction are introduced in Chapter 3. In each of these Chapters, a generalized analysis is first presented, and is subsequently applied to specific interactions. This gives the reader a broader view and avoids the need to consider one interaction in isolation of others. Readers interested in one type of interaction can consult either the Index at the end of the book or Table 1 in the Preface. Some simplified schematic diagrams, using obvious notations, are presented below to facilitate understanding of some interaction mechanisms. Note that the symbol γ is used to represent an X-ray or a γ-ray photon.

1.7.1 Interaction with atomic electrons The first target a radiation particle encounters is the atomic electrons, which affect the incoming radiation by virtue of their charge and the associated electromagnetic field. Neutrons are obviously not affected by the electrons as they have no charge. On the other hand, both photons and charged particles are strongly influenced by the atomic electrons. The mechanisms affecting both types of radiation are quite different, since photons carry no charge. Figure 1.8 provides a general classification of the interactions of photons and charged particles with one or more of the atomic electrons. The interactions of each of these two types of radiation are discussed below.

41

1.7 Reactions and Interactions

Photon

Charged particle ±

Absorption

Photoelectric effect, triplet production

Atomic electron(s)

Compton Incoherent scattering Rayleigh, Thomson Coherent

Figure 1.8

Coulomb, Moller, Bhabha

Elastic scattering

Soft, hard Neutralization, Annihilation

Inelastic scattering

Absorption

Interactions with atomic electron(s) for photons and charged particles.

Photon–electron interactions Photoabsorption Photon absorption by an atomic electron occurs in the photoelectric effect process, in which the photon loses its entire energy to an atomic electron which is in turn e− liberated from the atom. This process requires the incident Atomic electron photon to have an energy greater than the binding energy of an orbital electron. For X- and γ-ray photons of sufficient energy, photoelectric absorption is most likely to be caused γ by the most tightly bound electrons, i.e. those of the K shell, because the concentration of electrons is highest in this shell. Note that the photoelectric effect cannot occur if the electron is unbound, as it will not be possible to conserve both energy and momentum. The reaction is also favored at low photon energy and for atoms with a large atomic number, where there are many electron shells for the incoming photon to interact with, and to match in energy. The incident photon is completely absorbed in this process and the electron is released carrying a kinetic energy equal to that of the incident photon minus its binding energy. Although, the atom recoils in this process to conserve momentum, the kinetic energy it carries is negligible. The released electron leaves behind a vacancy in the inner orbit it occupied. This vacancy is subsequently filled by an electron from a shell with a lower binding energy (an outer orbit, e.g. from an L to a K shell). The difference between the two binding energies of the electron filling the vacancy is released in the form of an X-ray photon, known as fluorescent radiation. Alternatively, the energy may lead to the ejection from the atom of another orbital electron, called the Auger electron. This electron in turn leaves a vacancy and the process of emission of X-ray photons or Auger electrons is repeated, and so on. In atoms with high atomic number, X-ray emission is usually favored over the releases of Auger electrons, and the opposite is true for light elements. In essence, while in the photoelectric effect the original photon is absorbed, other photons (X-rays) and electrons (the first liberated electron and any Auger electrons) are released.

42

Triplet production In the electric field of the electron, a photon of an energy greater than four times the rest mass of the electron (4 × 0.511 = 2.044 MeV) can disappear and be replaced by a positron–electron pair.The target electron also recoils, and because of its small mass it can acquire a significant amount of energy. One then observes three particles (two electrons and a positron) replacing the absorbed photon.

Chapter 1 Mechanisms

e− e−

e+

Atomic electron γ E > 2.044 MeV

Compton (incoherent) scattering A more likely scenario is that the incident photon gives some of its γ energy to an atomic electron, while remaining as a e− photon with a reduced energy. In essence, the inciAtomic electron dent photon acts like a “particle’’ colliding with a γ “free’’ (and at rest) electron. The electron is considered here to be free, since this process, which is called E > Be Compton scattering, occurs at a photon energy much higher than the binding energy, Be , of atomic electrons, to the extent that the electron is considered to be practically “unbound’’. This is an interaction in which the photon loses energy and changes direction, but the total energy and momentum of the two colliding particles is conserved. The interaction is considered to be an incoherent scattering, since the photon possesses in this process corpuscular properties, rather than coherent wave properties. However, Compton scattering is an elastic process, as far as the interacting photon and electron are concerned. From the atom’s point of view, this is an inelastic scattering, because the atom loses an electron and can become ionized, or at least excited. Following the reaction, recoil electrons are transported through matter and interact as charged particles, and ultimately dissipate their energy as heat before coming to rest. Note that inverse Compton scattering can also take place, that is a high-energy free electron can scatter with a photon of lower energy. This interaction is important in astrophysics. Rayleigh (coherent) scattering This is an interaction with bound atomic electrons (as a collective) in a coherent fashion, in the sense that photons behave as waves and interact elastically, because the nature of the atom is not altered19 . The entire atom recoils to conserve momentum. As a result, the deflected photon emerges with an energy almost equal to the incident energy, and the photon scatters by a very small angle. The practical impact of this interaction is a slight change in angle in the forward direction, with almost no reduction in energy. 19 The

γ

Atom γ

classical definition of Rayleigh scattering is the scattering of electromagnetic waves by particles smaller in size than the wavelength. The scattering of such waves with particles larger in size than the wavelength is called Mie scattering.

43

1.7 Reactions and Interactions

Diffraction When the radiation wavelength is about equal in value to inter-atomic spacing (lattice pitch in crystallized structures), wave interference effects produce a diffraction pattern (fringes), due to the coherence between incident and reflected waves at some discrete directions. The structure and identity of the crystal can be deduced from these patterns. The same effect can also be used to study chemical bonds and biological structures, that exhibit regular patterns of atomic arrangements. Both low energy X-rays and cold neutrons produce this effect. At this range of wavelength and higher, X-rays and cold neutrons also exhibit other optical properties such as reflection, refraction, and polarization. Thomson electron scattering This is a coherent, hence elastic, interaction of a photon with a single unbound electron. As such, the photon energy does not change, but the photon can scatter in any direction (forward and backward). This reaction is, however, not as probable as Rayleigh scattering, and the latter itself has generally a low probability in comparison to other interactions.

γ γ

Atomic electron

Charged-particle interactions with atomic electron Elastic scattering A charged particle can be deflected by the Coulomb forces of the atomic electrons in such a way that energy and momentum are conserved. Heavy charged particles are hardly affected by electrons because of the greatness of their mass relative to that of the electron. Elastic scattering of charged particles with the atom as a whole is in effect a scattering by the positive electric charge of the nucleus screened by the negative charge of the atomic electrons. The reaction is an elastic one when the atomic electrons do not receive any excitation energy in the process. When the incident particles are electrons or positrons, they can interact elastically with individual atomic electrons by, respectively, repulsive or attractive Coulomb forces. If the Atomic incident particle is an electron, the projectile electron and the target become identical and interference between the waves describing their interaction affects their behavior. The interaction is then called Mott scattering between Moller Bhabha identical particles20 . When the energy loss per collision for the incident particle is >0.255 MeV, electron–electron scattering is called Moller scattering, while the positron–electron interaction is known as Bhabha 20 The

Coulomb elastic scattering of an electron by the atomic nucleus, when dealt with using quantum mechanics, is also called Mott scattering.

44

Chapter 1 Mechanisms

scattering. Obviously, for an electron (or a positron) to lose energy > 0.255 MeV (half its rest-mass energy), it must possess a high velocity; which necessitates the relativistic treatment of Moller (and Bhabha) scattering. Inelastic scattering The predominant interaction of charged particles with atomic electrons is an inelastic one, in which Atomic the atomic electrons receive energy that electrons exceeds their binding energy. The target atom then becomes excited, or even ionized if an atomic electron leaves the nucleus and Repulsion Attraction becomes unbound. Given the small mass of the electron, the kinetic energy received by an ejected electron in the ionization process can be quite large. These swift electrons are called delta (δ) rays, and their energy can be sufficiently high to cause further (secondary) ionization before they lose their entire energy and are subsequently re-absorbed in the atom. When the amount of energy transferred to the atomic electron is small, on the order of its excitation or ionization energy, the interaction is called a “soft collision’’. Obviously, the interaction is a “hard collision’’ if the energy transferred to the electron is such that the effect of its binding energy is negligible, so that the atomic electron involved in the collision can be considered to be initially free. Absorption As charged particles slow-down they may have sufficient energy to break chemical bonds, but they also dissipate their kinetic energy in the form of heat until they reach a state of thermal equilibrium within the medium. A positively heavy charged particle eventually captures an electron and becomes neutral. Then, its effect on the atom is considerably reduced. In solids, the absorption of a charged particle can lead to lattice defects, when atoms are displaced from their lattice. An electron can be captured by an atom ionized in a previous interaction, changing it from an ion to a neutral atom, or it can attach itself to a neutral atom creating a negative ion. These neutralized atoms will also attain an equilibrium condition. Atomic shells act, therefore, as donors of electrons to positively charged particles, or as receptors of electrons, in effect absorbing these particles. Positron annihilation As a positron comes to rest, it is absorbed by a free or loosely bound atomic electron, in a process known as annihilation, in which the two particles mutually self-destruct. Their rest mass is then converted into two photons each possessing an energy of 511 keV (equivalent to the rest mass of an electron or a positron). To conserve momentum these two photons are emitted in two opposite directions.

γ

Nucleus

45

1.7 Reactions and Interactions

Photon

Charged particle ±

Absorption

Figure 1.9 particles.

Pair production

Electric field of atom/nucleus

Cerenkov, bremsstrahlung

Radiative collisions

Interactions with the electric atomic/nuclear field for photons and charged

1.7.2 Interaction with electric field of atom The electric field created by the charge and motion of atomic electrons and the nucleus and its nucleons provides an environment that enables some interactions to take place. These interactions are outlined in Fig. 1.9 and introduced above.

Photon interactions Pair production This is an absorption process in which a photon disintegrates into an electron and a positron. e− e+ For this reaction to take place the energy of the photon Atom has to be greater than the rest-mass energy of the pair nucleus γ produced, i.e. 2 × 0.511 = 1.022 MeV. Photon energy in excess of this rest-mass energy is shared as kinetic E > 1.022 MeV energy between the electron and the positron. Obviously this pair has to have some kinetic energy, otherwise the two particles will recombine. Simultaneous conservation of momentum and energy requires the presence of a third body/field, which in this case is the atomic electron or the nucleus, that recoils to conserve momentum. Pair production in the field of the atomic electrons is much less significant than that within the field of the nucleus. In the electron field, the target electron recoils with significant momentum, due to its small mass, and the process is referred to as triplet production (see Section 1.7.1).

Charged-particle interactions Cerenkov radiation When a charged particle moves in a medium at a speed greater than the phase velocity of light in this medium21 , the electric field of the particle is subjected to a strong perturbation. If v < nc , the perturbation is cancelled by destructive wave interferences in all directions. However, if v > nc , the waves constructively interfere,

γ ± n<1 c > n

phase velocity of light in a medium is equal to nc , where c is the speed of light in vacuum and n is the medium’s optical index of refraction.

21 The

46

Chapter 1 Mechanisms

producing an optical “shock wave’’ (in the same manner a supersonic jet generates very intense waves: the Mach waves). The result is the emission of electromagnetic radiation in the visible range called the Cerenkov radiation. This phenomenon is typically observed with electrons as they can easily acquire speeds larger than the speed of light in the medium. Transition radiation When a high-energy charged particle crosses a boundary between two media of different dielectric permittivity, hence different atomic structure, a sudden change in the rate of energy loss occurs. This mismatch results in an acceleration or deceleration of the charged particle, and is compensated by the emission of X-rays, called the X-ray transition radiation.

γ

Bremsstrahlung When a free electron approaches the positively charged nucleus, it experiences an attraction force that deflects its electron and causes the electron to accelerate. This sudden acceleration causes the emission of a pulse of radiation in the form of a photon. This in turn causes the electron to lose kinetic energy, and it slows down. The emitted radiation is, therefore, called the braking or impulse radiation, and more commonly the bremsstrahlung22 .The bremsstrahlung produced by electrons in the field of the nucleus of heavy elements is the main source of photons in X-ray tubes. The acceleration of a positron in the field of a free electron also produces bremsstrahlung. However, the bremsstrahlung emitted as a result of acceleration of an electron in the field of another electron (by the repulsion force), tends to be weak, since the bremsstrahlung electric fields of the incident electron and the target electron are out of phase with each other, resulting in a destructive interference. This interaction, in the field of the nucleus, is quite pronounced in electrons, much more than heavy charged particles, because of the small mass of the electron, which makes it quite susceptible to accelerating and deceleration. This effect is, therefore, negligible for all heavy charged particles. However, when a charged particle collides with a free electron, that electron can itself produce its own bremsstrahlung.

1.7.3 Nuclear interactions Notation Nuclear interactions/reactions are defined by the notation a X(z, y)b Y, where z designates the projectile radiation particle, a X refers to a target nucleus of element X which has a mass number a, and b Y is the residual nucleus of mass number b, left after the interaction. Since the number of nucleons in this interaction must be conserved, the mass number of all involved radiation particles must also be conserved. In addition, the charge must be conserved. For example, 1 H(n,n)1 H 22 Bremsstrahlung

is German, from “bremse’’ (brake) and “strahlung’’ (radiation).

1.7 Reactions and Interactions

47

is an interaction where the neutron simply scatters without affecting the target nucleus; hydrogen with mass number 1 in this case. Therefore, the latter reaction is simply referred to as 1 H(n,n). Also in the reaction 56 Fe(n,n )56 Fe∗ , the target element stays the same (iron with mass number 56) but is in an excited state (indicated by ∗) after emitting a neutron. Since the emitted neutron may not necessarily be the same as the incident neutron, it is designated as n . The same reaction can be expressed as 56 Fe(n,nγ), since the excited nucleus subsequently releases immediately the excitation energy in the form of a γ-ray. It is then understood that the outgoing neutron is not necessarily the incident neutron (from a practical point of view, the two are indistinguishable), and that the nucleus did not elementally change, and it has been in an excited state. A third example is 10 B(n,α)7 Li; note the conservation of the mass number with an α particle having a mass number of 4. The last example also implicitly conserves the number of protons via the explicit designation of the elements and particles (5 + 0 = 2 + 3). Nuclear interactions involve not only neutrons, but also photons and charged particles that have sufficient energy to overcome the atomic electric field in order to reach the nucleus. The following are other observed interactions, aside from elastic and inelastic scattering: (n,γ), (n,p), (n,2n), (γ,n), (γ,p), (p,α), (p,d), (p,n), (p,γ), (d,α), (d,p), (d,n), (d,2n), (n,α), (α,p), and (α,n). The discussion below considers a general incident particle, z.

Reaction types When a radiation particle approaches a nucleus it is affected by its strong forces and its electric field. However, in many cases a simple collision between two solid particles occurs, resulting in the so-called “hard-ball’’ collisions. If the incident particle succeeds in penetrating the nucleus, the target nucleus can be viewed as a “soft ball’’. A more formal way of describing these reactions is by referring to them as elastic and non-elastic, since in the hard-ball collision the target nucleus returns to its original state after the interaction, while some “deformation’’ takes place in the case of soft-ball interactions. In other words, the nucleus acquires some internal energy in the case of non-elastic scattering. With the incident radiation being inside the potential well of the target nucleus, a “compound nucleus’’ is formed, typically in an excited state. If this new excited state is stable, the nucleus can remain in this state indefinitely. In many cases though, the excited state is metastable and the compound nucleus attempts to reach a more stable state by emitting gamma rays, neutrons, charged particles, or a combination of thereof. If the de-excitation is not immediate, i.e. prompt, subsequent de-excitation becomes in effect a nuclear decay process (see Section 1.6). Figure 1.10 provides a flowchart of various possible nuclear interactions, a brief description of each interaction is given below.

Elastic and ground-state scattering Potential scattering When a radiation particle encounters a potential field and does not affect it, or if the effect is so negligible to be observable, the interaction

48

Chapter 1 Mechanisms

Total (z,total)

Ground state reaction (2) (z,z0)

Inelastic scattering (4,102–107) (z,z)

Excited state reaction (3) (z,non-elastic)

Absorption (27) (z,abs)

Fission (18) (z,fission)

Electric field interactions (517)

Production of neutrons (16,17,37) (z,kn)

Disappearance (101) (z,disap)

Production of neutrons and charged particles (11, 22–37,41–45) (z,jnjq)

Radiative capture (102) (n,γ) Production of charged particles (103–117) (n,jq)

Figure 1.10 Types of nuclear interactions for a radiation particle z, where j, j  are integers ≥1, and k is also an integer ≥1 expect for z = n where k ≥ 2, q designates a charged particle different from z (if z was charged), and the numbers in parenthesis are the ENDF MT numbers described in Table 3.2.

would simply be between the solid matter of the proPlane wave jectile and target. This occurs, for instance, when a (scattered particle) neutron collides with a nucleus without penetrating Nucleus it. The process then resembles a “hard-ball’’ collision, and its kinematics can be described by conventional Plane wave conservations laws of momentum and energy. When (incident particle) the radiation particle is viewed as a plane wave, the wave will be reflected on the solid surface of the field (i.e. that of the target it represents). The scattering is then called potential scattering. In effect, the incident particle is deflected by the short-range strong nuclear forces as it approaches the nucleus, without touching the nucleus. No compound nucleus is formed in this process. Coulomb elastic scattering Charged particles can scatter elastically by the force between the electric fields of an incident particle and a target nucleus. Direct collision, or contact, between the incident particle and the nucleus is, therefore,

1.7 Reactions and Interactions

49

not necessary. In this type of collision, the charged particle Nucleus is deflected without exciting the nucleus and without being accompanied with the release of electromagnetic radiation. The incident particle loses only the kinetic energy needed for conservation of momentum. The scattering of slow charged Repulsive particles by heavy nuclei is called Rutherford scattering 23 . In quantum mechanics, the elastic scattering of an electron with Nucleus the Coulomb field of the nucleus is called Mott scattering. When the incident particles and the target are identical, e.g. a proAttractive ton on a hydrogen nucleus, the incident and target particles, become indistinguishable, and quantum treatment of the scattering between the two particles requires accounting for the interference between their waves. The interaction is then known as Mott scattering between identical particles. Thomson nucleus scattering Thomson scattering refers to the scattering of electromagnetic radiation by a charged particle. The nucleus being a charged particle subjects incident photons to this type of elastic (coherent) scattering. This scattering combines coherently with Rayleigh scattering, and its effect is quite small due to the large mass of the nucleus. Resonance scattering Elastic scattering can also take place by the formation of a compound nucleus, Nucleus with the subsequent re-emergence of a particle of the same type as the incident particle (no other Target Compound Residual types of radiation is emitted). That is, the total kinetic energy of the incident particle and the nucleus are conserved, and the nucleus stays at its ground state. This is known as compound elastic scattering or resonance elastic scattering, since this process favors particles with energies that resonate with (match) one of the energy levels in the nucleus. At high particle energy, where the energy levels are so close that they appear to be continuous, all particle energies become susceptible to this type of scattering, which can then be called smooth or unresolved resonance scattering; with the word “unresolved’’ reflecting the fact that the resonance levels are no longer distinguishable. Since in this scattering process the target nucleus remains in its original state, without experiencing any nuclear excitation, the reaction is designated as (z, z0 ) or simply (z, z), where z refers to the incident particle and z0 signifies that the target nucleus remains in the ground state after the particle z re-emerges. Resonance (resolved or unresolved) neutron elastic scattering occurs with all nuclei, since there is no Coulomb barrier it is easier for the compound nucleus to emit a neutron than to 23 Rutherford

scattering, in general, is the scattering of heavy charged particles under the Coulomb field of the nucleus. Therefore, the term is also used to describe high-energy distant collisions in which direct contact between the interacting particles does not take place.

50

Chapter 1 Mechanisms

emit a charged particle. Moreover, γ emission is a slow process, in comparison to the emission of particles, and as such the latter usually takes precedence over the emission of γ radiation. In practice, all above elastic scattering processes amount to the same effect: conservation of both the total kinetic energy and momentum of the colliding bodies, with the target nucleus staying at its ground state. In most cases, the energy of the radiation particle is much higher than the energy of the target nucleus, and the target can be considered in practice to be stationary (i.e. at rest). In effect, a target nucleus possesses the thermal energy of its atom, which typically results in very small vibration (kinetic) energy, in the meV range (see Section 3.3.7). The target nucleus then receives some kinetic energy, which causes it to “recoil’’. Inelastic scattering In this process, a particle of the γ Nucleus same type as the incident particle re-emerges. Since in practice the same type of particle reappears, the process is considered a scattering process in which Target Compound Residual the particle changes both its direction and energy. This process involves the formation of a compound nucleus, which immediately releases a particle of the same type as the incident particle leaving a nucleus in the excited state. The resulting nucleus is referred to as the residual nucleus to distinguish it from the original target nucleus. This nucleus in turn releases its excitation energy in the form of a γ-ray with an energy that depends on the excitation level from which the nucleus is de-excited. The reaction is designated as (z,zi ) to indicate that the residual nucleus is left in the ith state, or (z,z ) or (z,zγ) when all the excited states are considered. The set of nuclear levels, including the ground state, is referred to as the exit channels. Note that the γ radiation emitted in this reaction is due to the de-excitation of the residual nucleus, unlike the radiation capture gammas which are a direct product of the interaction. Therefore, the incident particle must have sufficient energy to bring the nucleus to its first excitation state, unlike radiative capture which can take place even at very low particle energy. The minimum energy required for inelastic scattering is called the threshold energy, and it is on the order of a few MeV for light nuclei, but is only around 100 keV or less for heavy nuclei. This reaction is commonly observed with fast neutrons, as in the case of the 12 C(n,n )12 C∗ and the 238 U(n,n )238 U∗ reactions, requiring, respectively, threshold energies of 4.8 MeV and 45 keV24 .This reaction also occurs when a charged particle passes sufficiently close to the nucleus to be influenced by the nuclear forces, and transfers a sufficient amount of its kinetic energy to the nucleus to bring it to an excited state. The charged particle also changes its direction of motion. Whether the incident particle is neutral or 24 The

threshold energy is greater than the energy of the first excitation level for reasons explained in Section 2.7.2, Eq. (2.148).

51

1.7 Reactions and Interactions

charged, the total kinetic energy of the interaction is not conserved, since the energy associated with nuclear excitation comes at the expense of the kinetic energy of the incident particle. Non-elastic reactions If the target atom/nucleus is viewed as a “soft ball’’, then an incoming radiation particle can penetrate it, and in essence provide it with some additional “internal’’ energy. This in effect causes the particle to “fall ’’ into the potential well, increasing its energy. The target nucleus becomes excited, a state that is not sustainable, since stability is durable only at the ground state (i.e. at minimum potential energy). Then, a number of processes can take place as discussed below. γ γ Nucleus

Absorption Fission or spallation

Radiative Charged- Neutron capture particle production production

Absorption An incident particle can be absorbed by the nucleus in a non-elastic interaction. While such absorption leads to the disappearance of the incident radiation, it also adds energy into the potential field that can destabilize the nucleus. In some heavy nuclides, such as those of uranium and plutonium, an absorbed neutron can deposit sufficient energy to break up the nucleus into smaller nuclides, causing a fission process that is also accompanied by the release of a few neutrons. Highly energetic25 charged particles can also shatter nuclides with intermediate mass leading to the emission of several particles, in a process known as spallation. In fission and in spallation, a few more particles of the same type as the original incident radiation re-emerge, leading in effect to “apparent’’ multiplication of radiation. Radiative capture Radiation absorption is often accompanied by the release of γ radiation, hence the process is called radiative capture and denoted by (z,γ). The emission of γ-rays is not accompanied by particle emission, and is a direct result of the interaction, unlike that associated with inelastic-scattering γ-ray which is produced as the compound nucleus decays. The captured incident radiation, in effect, becomes bound to the target nucleus, forming a new nuclide. 25 In

the domain of high-energy physics: 10 MeV to the GeV range.

52

Chapter 1 Mechanisms

The γ-ray is released as a result of the excess binding energy. However, the newly formed nuclide may itself be radioactive, and decays and releases radiation. Examples of radiative-capture reactions include 1 H(n,γ)2 H, 238 U(n,γ)239 U∗ , and 12 C(p,γ)13 N∗ . The asterisk (∗) in these reactions indicate that the product residual is unstable and will subsequently decay; by β− in the case of 235 U to 239 Np (which also decays to 239 Pu after another β− emission) and by β+ emission in the case of 13 N to 13 C. Notice that neutron capture results in the production of an isotope of the same element as that of the target nucleus, while the absorption of a charged particle gives rise to a new element. Charged-particle production Here reactions that produce a charged particle different from the incident particle are considered. If upon the absorption of an incident particle the release of γ-ray energy does not result in a stable nucleus, charged particles (one or more) can be ejected. When an incident particle enters a nucleus, it can be captured in one of the excitation levels of the nucleus.Whenever the excitation energy is larger than the binding energy of a proton, plus the energy required to overcome the Coulomb barrier, a proton is emitted from the nucleus. If more than one excited state is created by the incident particle, a group of particles can be emitted, usually accompanied by γ radiation that brings the residual nucleus to the ground state. If the energy of the incident particle is very high, it can directly knock out individual nucleons from the nucleus. A third charged-particle mechanism is called stripping, in which multi-nucleon incident particles, particularly deuterons, are broken up by the nuclear forces as they approach the nucleus. In the case of the deuteron, it splits into a neutron and a proton. The neutron, being neutral, is easily absorbed by the nucleus, while the proton is repelled by the Coulomb field of the nucleus and appears as an ejected particle. That is, in (d,p) reactions, the emerging proton does not arise from the target nucleus, but only appears to. The reverse of this stripping process occurs in the proton–deuteron, (p,d), and proton–triton, (p,t) reactions, where the incident particle acquires a nucleon or more, rather than losing one. When several particles are emitted, the reaction is called “spallation’’. Examples of charged particle producing reactions include: 3 He(n,p)3 H, 26 Mg(γ,p)25 Na∗ , 10 B(n,α)7 Li, 16 O(n,p)16 N, 63 Cu(p,n)63 Zn, 209 Bi(d,p)210 Bi∗ , and 27Al(α,p)30 Si. Neutron production This reaction does not include fission, which is considered as part of the neutron absorption process, since it can take place at low particle (neutron) energy. It does not include the (n,n) and (n,n ) reactions, which are not obviously production reactions. A charged particle or a photon can release a neutron, from a compound nucleus or by direct interaction, when they have an energy larger than the neutron’s binding energy in the nucleus. Examples of these (z,n) reactions, with z = n, include: 2 H(γ,n)1 H , 9 Be(γ,n)8 Be, 2 H(d,n)3 He, and 3 H(d,n)4 He. At high incident-particle energy, the probability of emitting a nucleon from the compound nucleus, or by direct interaction, decreases; otherwise this single nucleon will carry an excessively large energy.Therefore, the probability

1.8 Macroscopic Field

53

of emitting two nucleons begins to increase as the probability of ejecting a single nucleon decreases. At even higher energy, the release of two nucleons decreases, as the emission probability of three nucleon increases, and so on. The emission of a neutron is favored over the release of a charged particle, since there is no Coulomb field to overcome; and the reactions (z,2n), and (z,3n) are usually more probable (but not always) than the emission of charged particles (alone or in combination of one neutron or more). In either case, the incident particle must have an energy greater than the threshold energy required to liberate each of the emitted particles. As such, these reactions appear at high incident energy. For example, the reaction 209 Bi(α,2n)211At requires α particles with more than 21.1 MeV, while the 209 Bi(α,3n)210At reaction becomes possible at an energy of almost 30 MeV. Simultaneous production of neutrons and charged particles In some cases, combinations of neutrons and charged particles are ejected from the compound nucleus, or as a result of a direct interaction, though neutron production is favored over the generation of charged particles. This typically happens when a nucleus with more dense energy levels is produced. For example, the reaction 63 Cu(p,pn)62 Cu is more likely than the 63 Cu(p,2n)62 Zn reaction.

Electric field interactions Pair production The production of a positron and an electron resulting from the disintegration of a photon of energy greater than the combined rest mass of this pair of charged particles (2 × 0.511 MeV) is in effect a photon absorption process. This interaction, known as pair production, requires the presence of an external charged particle, which recoils to enable the simultaneous conservation of momentum and energy. Therefore, it also occurs in the presence of the field of a nucleus, as well as in the field of atomic electrons. However, pair production in the field of the nucleus is much more likely than that with the field of the atomic electrons. Delbruck scattering This is an elastic scattering caused by the potential electric field of the nucleus. Although the photon has no charge, it is affected by the electric field of the nucleus, if it is viewed as consisting of a “virtual’’ electron– positron pair. This charge pair is then scattered in the Coulomb field of the nucleus, and the two virtual particles recombine to form a photon of the same energy as that of the original photon. The only observed change is then a change in angle. As such, this interaction is often thought of as a coherent scattering. Its effect is, however, very small, and is hardly detectable.

1.8 Macroscopic Field So far we have considered a single projectile (a neutron, a photon, or a charged particle), a single target (an atomic electron, an atom, or a nucleus), and

54

Chapter 1 Mechanisms

the interaction between one projectile and one target. In reality these radiation projectiles travel in a flux of many entities, and they can encounter an enormous number of targets as they travel through a medium. One, therefore, must take a macroscopic view of radiation and the medium it interacts within, while accommodating the microscopic “one-projectile on one-target’’ mechanisms. These aspects are considered here, after examining the nature of the space within which radiation travels.

1.8.1 Transport space As explained in Section 1.4, targets with which radiation interacts are presented directly by the atom or the nucleus or the electric field associated with them. These effects are only felt by a projectile, if it passes in the proximity of a target, since the range of nuclear forces is limited while the Coulomb force decreases rapidly with distance, as explained in Section 1.4. However, the targets occupy only a small fraction of the available space, as demonstrated below. While charged particles are continuously affected by surrounding electromagnetic fields, a neutral particle can travel some distance before encountering a target to interact with. A neutral projectile may also miss a target altogether. This affects the interaction rate as discussed in Section 1.8.4. We begin, however, by examining the space occupied by atoms in a medium.

Sparseness As shown in Section 1.4, following Eq. (1.15), the mass density of the matter from which the nucleus is made is equal to 2.3 × 1017 kg/m3 . This is obviously an extremely high density compared to the material density normally observed26 . This analysis is indicative of the fact that the nuclei are widely distributed over the volume and the space in between them is vacant. Even for the most dense material, the nuclei occupy about 10−14 of the total volume. The space that the atomic electrons occupy is quite small, because of their very small mass. However, the electrons are spread around the nucleus over a larger distance. For example, the radius of the smallest atom, hydrogen, as given by Bohr constant, is 5.29 × 10−11 m, compared to a nucleus radius of 1.25 × 10−15 m, according to Eq. (1.15). For a heavy element like iron, the ionic and atomic radii27 are 6.45 × 10−11 and 1.72 × 10−10 m, respectively, compared with the nucleus’s 3.7 × 10−15 m.Taking the atomic radius and assuming that each atoms is a cube of width twice the atomic radius and a weight of Au, the density of iron (A = 56) can be shown to be about 2284 kg/m3 . This is still lower than the nominal density of iron, which shows that there is some overlapping between atoms. However, using the covalent radius of 1.17 × 10−11 m, one arrives at a density of 7255 kg/m3 , which is closer to the nominal density. 26 The

most dense elements are osium and iridium, with a densities of 22,610 and 22,650 kg/m3 , respectively. ionic radius is the radius in a crystal where the ions are packed together so that their outermost electronic orbits are in contact with each other, while the atomic radius is half the distance between two adjacent atoms in a crystal.

27 The

1.8 Macroscopic Field

55

Penetrability A fundamental difference between the transport medium of radiation and that of classical continuum mechanics (e.g. fluid mechanics) is that radiation transport is not hindered by boundaries or interfaces. That is, fluids are governed by the principle of impenetrability, which causes them to form interfaces between each other and prevents them from penetrating solid barriers. Radiation particles do not respect such barriers or interfaces. Consequently, radiation does not form clouds, droplets, clusters, or the likes. The implication of this penetrability of radiation is that while a fluid can be confined to a certain direction of flow, radiation can move in all directions. The result is that radiation has a full velocity directional distribution. Therefore, radiation intensity can change with the magnitude of its velocity (or energy) and with direction. Therefore, the transport space for radiation has the velocity vector, v , as an independent variable, much like the position vector, r and time, t. That is, the transport phase space is a seven-dimensional space: three for position, (x, y, z in cartesian coordinates), one for the magnitude of velocity (|v|) or equivalently energy (E), two for the direction of motion (), and one for time (t).

1.8.2 Particle density and flux In order to examine the transport of many particles at a time, one must consider the particle density, that is the number of particles per unit phase-space element in the transport space. Given that the transport space is a seven-dimensional space, then an element in this space is given by dV dE d dt, where dV is an infinitesimal spatial volume, dE an infinitesimal energy interval, d an infinitesimal solid angle, and dt an infinitesimal time interval. We speak then of t)dV dE d as being the number of particles per unit time within dV n( r , E, , with an energy in the interval dE around E, and with a direction in d around Then, n( r , E, , t) is the particle density function, or simply the particle density. . However, the particle flux is more commonly used and is introduced below. The concept of flux is a simple one, but it can be misunderstood. In general, flux is the rate of fluid, particle, or energy, flow through a surface, and the flux density is simply the flux per unit area. This conventional definition of flux is suitable in directionally well-defined flows, such as a fluid in a pipe, particles in a chute, or the lines of a magnetic field, where a surface area can be designated at a direction normal to the flow direction. Radiation particles tend to move in all directions, making it difficult to define a particular orientation for the particles or the surface. However, the dimensions of flux density is the number of flowing entities per unit area per unit time. The same dimensions can be arrived at if we consider the number length per unit volume per unit time, where the number length is the summation of the track-lengths of all particles passing through the unit volume, as schematically shown in Fig. 1.11. This definition better suits radiation particles as it removes any directional dependence, while maintaining the concept of flux as pertaining to flow. The higher the flux, the more the

56

Chapter 1 Mechanisms

li f= i

∑ li V

per unit time

li = length of radiation track i within volume V

Figure 1.11 A schematic showing the definition of flux.

density of the tracks crossing the volume, the more particles are passing through the volume. The flux density, φ (called simply flux), is then related to the particle density, n(v), for particles moving at a certain velocity, v, by the relationship:    li li per unit time v φ(v) = per unit time = = V V V Total number of particles of speed v × v = n(v)v (1.45) φ(v) = V where the notations are as shown in Fig. 1.11, and v in the parentheses indicates that the associated quantity is defined for particles moving at a velocity v. For photons, v = c, where c is the speed of light; then φ = nc. However, this definition of flux does not take into consideration the directionality of the flowing particle. the flux of particles of energy E A better definition is then given by φ(E; ), moving in the direction .This definition accommodates both photons and other charged particles, by considering energy, rather than velocity, while accounting only for one direction (yet not focusing on a particular surface area).The following relationships can then be established between the flux, φ, and the particle density, n, starting from the most basic definition of the angular flux density (also called t), i.e. the flux in a given direction  for particles with a pointance), φ( r , E, , certain energy E at given instant in time, t: t) = vn( r , E, , t) φ( r , E, ,  t)d φ( r , E, t) = φ( r , E, ,

(1.46) (1.47)

4π

t) = φ( r , ,



t)dE φ( r , E, ,

(1.48)

φ( r , E, t)dE

(1.49)

φ( r , E, t)dt

(1.50)

 φ( r , t) =  φ( r , E) =

57

1.8 Macroscopic Field

where r designates some point in the space at a distance and location defined by the vector r . The flux integrated over time, φ( r , E), is known as the particle fluence, or simply fluence; also called irradiation, irradiance, or exposure. The above definitions of flux enable one to specify the flux at any level of detail over the basic seven coordinates of the transport space. The particle current density, or simply current, J , is given by the relationship: t) = v n( r , E, , t) J ( r , E, , vn( r , E, , t) =  φ( r , E, , t) = 

(1.51)

Therefore, the current is the directed flow of particles in a given direction , and like flux it can be integrated over energy and time. The current density, J , can be used as a measure of the number of particles crossing a surface in a given direction, by simply projecting the vector J on the desired direction across the surface. Current, therefore, includes directional characteristics, while the flux, φ, is simply a scalar quantity.

1.8.3 Atomic/nuclear density In order to account for the presence of many atoms/nuclei in a medium, one must determine the atomic density, that is the number of atoms per unit volume, N , which is simply expressed as: N =

ρ Mass of material per unit volume (density) = Mass of a single atom Au

(1.52)

where ρ is the material density (mass per unit volume) and Au is the mass of an atom of mass number A, with u being the atomic mass unit (= 1.6605 × 10−27 kg). Equation (1.52) is often expressed as N = Aρ A0 , with ρ in g/cm3 and A0 being Avogadro’s number (= 0.6022 × 1024 g/mol). For a mixture of density ρm , the density of each element in the mixture, wρm , should be used, with w being the weight fraction of the element in the matrix. For a compound of a molecular weight, M , one calculates the number of molecules per unit volume, Nm , as: Nm =

Mass of material per unit volume (density) ρ = Mass of a single molecule Mu

(1.53)

Then one can determine the atomic density for a particular element in the molecule from chemical composition. For example, H2 O has two hydrogen atoms and one oxygen atom, as such the atomic density of H in H2 O is equal to 2Nm and that of O is Nm , where Nm is calculated using Eq. (1.53) at the proper water density and with M = 2 × 1 + 1 × 16 = 18.

58

Chapter 1 Mechanisms

1.8.4 Interaction rate The introduction of the particle flux density along with the atomic density enables direct evaluation of the interaction rate, R, per unit volume. The number of interactions a particle i encounters has to be proportional to the distance, li , it travels within a volume V , the longer the distance the higher the number of interactions. More interactions will also occur if there are more targets, i.e. more atoms or more nuclei per unit volume, N . In addition, the higher the probability of interaction, σ, of each particle with each target, the larger is the interaction probability. Therefore, in an infinitesimal volume, V , the interaction rate per unit time per unit volume for all particles can be expressed as: d R=N dt



i li σ





V

=N

i vi σ

V

(1.54)

where vi is the velocity of particle i. For particles of the same velocity, v, or energy, E, Eq. (1.54) can be expressed as: R(E) = N σ(E)nv(E) = N σ(E)φ(v)

(1.55)

where n(E) is the number of particles per unit volume that have a velocity v, or energy E, and use was made of Eq. (1.45) to introduce the flux, φ(v). Matching the dimensions in the two sides of Eq. (1.55) dictates that σ(E) has dimensions of area in order for R to be the number of interactions per unit volume per unit time. Recalling that σ designates the interaction probability of one radiation particle with one target, it is referred to as the microscopic cross section. In essence, σ represents the area projected by the target to the incoming projectile. Therefore, one would expect its value to be on the same order of magnitude as the crosssection area of the target, which for a nucleus, using Eq. (1.15), is on the order of 10−28 m2 , or so. Therefore, the microscopic cross section is reported in the unit of barn (b), with 1 b = 10−28 m2 = 10−24 cm2 . The value of σ depends on the nature of the target, as well as on the type of incident radiation and its energy and the nature of the interaction. In case of scattering, there is dependence on the angle of scattering. Cross-section libraries report these values, which are either measured or calculated using quantum mechanics (as discussed in Chapter 3).

Macroscopic cross section The quantity:  = Nσ

(1.56)

represents the overall target area projected by all nuclei in a unit volume, and as such is called the macroscopic cross section. From Eq. (1.56), it is evident that the macroscopic cross section is the summation of the microscopic cross sections

59

1.8 Macroscopic Field

of all targets per unit volume. Therefore, for a mixture, one can write:  (mixture) = αi i

(1.57)

i

where αi is the volume fraction occupied by species i and i is its total cross section. Note that wi = ρρi αi , where wi is the weight fraction and ρi is the density of the material of component i, and ρ is the mixture’s density. The macroscopic cross section of a compound can be calculated from the microscopic cross sections, σi s, of its elements as:  (compound) = Ni σi (1.58) i

where Ni is the target density of element i, which for atoms and nuclei is: Ni = wi

ρ Ai u

(1.59)

where Ai is the mass number of element i. Combining the above two equations gives:   ρ (compound) = σi = wi wi i (1.60) Ai u i i where i is the macroscopic cross section of element i, if it had the density of the compound. As evident from Eq. (1.56), the macroscopic cross section has dimensions of inverse distance. It can, therefore, be thought of as the probability of interaction per unit distance, in the same manner the decay constant λ (which has dimensions of inverse time) was shown in Section 1.6 to be the probability of decay per unit time. Therefore, the same probabilistic arguments used to derive the exponential law of decay can be used to derive the exponential law of attenuation (reduction in radiation intensity), so that: I = I0 exp(−x)

(1.61)

where I0 is the intensity of a narrow beam of radiation incident on material of thickness x, and I is the intensity of the radiation that succeeds in penetrating the material. The use of a narrow beam is necessary here due to the one-dimensionality of the relationship (only in x), which excludes any radiation transport in other directions; a process conceivable only in narrow beams. Nevertheless, this attenuation law elucidates the physical meaning of  as an attenuation coefficient. The value ρ is customarily reported for photons, as it provides a density-independent parameter, and is referred to as the mass attenuation coefficient. The one-dimensionality of the attenuation law makes it possible to describe the movement of particles in one specific direction, along some distance x, by exponential attenuation. Then one can state that the probability a

60

Chapter 1 Mechanisms

radiation particle traveling a distance x without interacting then interacting at a distance between x and x + dx is: p(x)dx = exp(−t x)t dx

(1.62)

where the exponential term is the survival probability to x and dx is the interaction probability within dx. Using the above equation, the mean distance of travel, or the mean-free-path (mfp), that is the average distance a radiation particle will travel between interactions, is given by:  ∞  ∞ 1 mfp = xp(x)dx = xexp(−x)dx = (1.63)  0 0 This gives the macroscopic cross section another useful meaning as the reciprocal of the mean-free-path. In other words, within a distance x, a radiation particle x = x interactions. That is,  can also be viewed as on average will encounter mfp the average number of interactions per unit length. The distance 1 is also called the relaxation distance, since according to Eq. (1.61), 1 is the distance required to attenuate a beam of radiation to 1e (= 0.3679) of its initial value.

1.9 Problems Section 1.3 1.1 Using Eq. (1.6), show that for a particle of v << c, the kinetic energy is given by the classical expression: 12 m0 v 2 . 1.2 Consider a light wave of wavelength λ = 5 × 10−7 m. 1. Plot this wave over a distance in the interval: [−5λ, +5λ], at a certain time. 2. Replot the above wave combined with a wave of a wave number of 1.1k, where k is the wave number of the original wave. 3. Combine the two waves with a third wave of wavenumber of 1.2k, and replot the combined wave. 4. Add an additional wave with a wave number of 1.3k to the above three waves and plot the combined wave. 5. Comment on the development of a wave packet as these waves are combined. 1.3 A 6.6 kg bowling ball moving at a speed of 10 m/s. 1. Calculate the width of the slit through which this ball can diffract. 2. If the slit width is 250 mm, what will be the particle speed necessary for the bowling ball to exhibit diffraction? Comment on your answers.

61

1.9 Problems

1.4 The kinetic energy of a particle is T = 12 mv 2 , while that of its associated wave is T = hν. The momentum of the particle is p = mv and that of the associated 1

mv 2

h 2 1 wave is p = λh . The speed of the wave is c  = λν = mv h = 2 v. This wave speed seems to be only half that of the particle it is associated with. If this is the case, the particle and its associated wave would not stay together. What is wrong with this analysis? Propose an alternative analysis that ensures that the particle and its associated wave have the same speed. 1.5 If the two sinusoidal waves sin[(k − k)x − (ω − ω)t] and sin[(k + k)x − (ω + ω)t] are combined, show that the shape they form has a velocity of w

k . Using the argument that when a group of continuous waves are combined, the velocity of the wave packet would be w

k , show that the velocity of the wave packet will be identical to the particle it is associated with. 1.6 A group of waves of wavelengths around λ + δλ are grouped into a wave packet to resemble a particle of velocity v. Show that the velocity of the wave packet is equal to that of the particle. du , where λ is the Hint:The group velocity, w, of waves is given by w = u −λ dλ wavelength of an individual wave and u is the velocity of that wave (called the phase velocity) = νu , with ν being the frequency of the individual wave. 1.7 A ball with a mass of 0.10 kg is approaching a player at a speed of 45 ± 1 m/s. What is the uncertainty in determining its position at a given instant. Comment on the results.

Section 1.4 1.8 Using the uncertainty principle determine the range of each of the four natural forces. 1.9 Prove that the quantity: r0 =

μ0 e 2 4πme

(1.64)

has dimensions of meters and determine its value, where μ0 is the vacuum permeability, me is the rest mass of the electron and e is its charge. The above quantity is called the classical radius of the electron. Rewrite the above expression in terms of the permittivity in vacuum, ε0 , and elaborate on the 2 physical meaning of 4πεe 0 r0 . 1.10 For an electron evolving around a nucleus, assuming a circular orbit, show that the electron’s magnetic moment Mm = − 2me e ωr , where ωr is the angular moment of the electron, e its charge and me its mass.  1.11 Show that if the sum of the quantum angular momentum, l, for a system of particles is a positive number, the system has an even parity, and if the sum is an odd number, the system has an odd parity.

62

Chapter 1 Mechanisms

1.12 Two particles, a and b, occupy any of two states. Distribute these two particles according to classical statistics (Maxwell–Boltzmann), Bose–Einstein statistics, and Fermi–Dirac statistics. Explain your logic. Hint: in the latter two distributions, the two particles are considered to be indistinguishable from each other.

Section 1.5 1.13 The binding energy, B(Z, A), can be represented as a function of the atomic number, Z, and the mass number, A, by the expression: B(Z, A) = a1 A − a2 A 3 − a3 Z(Z − 1)A− 3 − a4 (2Z − A)2 A−1 2

1

+ a5 A− 4 3

(1.65)

where the a’s are constants that can be obtained by best curve fitting of the above expression to actual binding energy data. 1. Explain the physical basis behind each term. 2. Calculate the binding energy for the following nuclides: 4 He (4.0026032), 6 Li (6.0151223), 8 Be (8.0053051), 12 C (12), 16 O (15.9949146), 20 Ne (19.9924402), 28 Si (27.9769265), 40 Ca (39.9625912), 58 Ni (57.9353479), 56 Fe (55.9349421), 100 Zr (99.9177617), 116 Sn (115.9017441), 208 Pb (207.9766359), and 209 Bi (208.9803832), using both Eqs (1.65) and (1.25), and compare the results. The numbers in parenthesis are the atomic masses, with mp = 1.00727647u, mn = 1.008665012u and u = 931.4943 MeV. Use the following values for the constants (all in MeV)28 : a1 = 15.5 a2 = 16.8 a3 = 0.72 a4 = 23,   34 even N and even Z 0 for odd N or odd Z a5 = −34 odd N and odd Z 3. Identify the nuclide(s) with the maximum binding energy. 4. Plot the binding energy per nucleon using the results obtained above, and compare the two curves.

Section 1.6 1.14 1. Use Poisson distribution to determine the expected (average) number of decays within a time interval t.  nxn exp (−x) = x. Hint: ∞ n=0 n! 28 The semi-empirical mass formula (SEMF) Isobar mass chains (http://www.pp.rhul.ac.uk/ptd/TEACHING/PH2510/

nuclear-2-d.pdf).

63

1.9 Problems

2. What is the probability of a nucleus surviving a time t then decaying within t + t? 3. What is the mean-life time of a radionuclide that has a decay constant λ? 4. Can equilibrium conditions be reached when a parent nuclide decays to a daughter of about the same half-life? 1.15 Show that for α decay to take place the condition below must be satisfied:   B1 B2 B2 1 Bα A1 − A2 − < (1.66) A1 − A2 A1 4 A2

1.16 1.17 1.18 1.19

1.20

where B refers to the binding energy, A to the mass number, and the subscript 1, 2 and α to the parent and daughter nuclei, and the α particle, respectively. Comment on the value of the change in the binding energy per nucleon, and whether α decay leads to a more stable nucleus. Using Eq. (1.65) show that nuclides of mass number greater than about 150 are all energetically unstable against α decay. α Are any of the following parity transitions of decay possible: (1) 1+ → 0+ , α + α − (2) 2− → 0+ , (3) 52 → 52 ? Express the binding energy conditions of β− , β+ and electron capture decay, i.e. Eqs (1.39)–(1.41) in terms of the atomic masses of the original and product nuclei. Comment on the results. The nuclide 64 Cu has all the three beta decay modes. Using the Isotope Explorer: http://ie.lbl.gov/ensdf/, identify these modes and for each mode report (1) the change in total angular momentum, (2) the change in parity, (3) the relative emission intensity, (4) the half-life, (5) the decay energy, and (6) the ft value. Comments on the relationship between these four properties of emission. Using the Isotope Explorer: http://ie.lbl.gov/ensdf/, under Level Table and the appropriate decay table, classify the following beta decay processes. Use the following log ft values as a guide: Superallowed 3.2–3.8, Allowed 3.5– 7.5, First-Forbidden 6–9, Second-Forbidden 10–13, Third-Forbidden 14– 20, Fourth-Forbidden ≈ 23. Recall that there are parity change conditions associated with each category. 1

n → H 3 H → He 6 He → Li 7 Be → Li

13

N→C

36

Cl → Ar

40

37

Rb → Sr

89

139

14

Ba → La

C→N

14

O→N

K → Ca

60

C→Y

115

141

17

F→O

11

C→B

35

S → Cl

Co → Ni

75

Ge → As

In → Sn

137

Cs → Ba

87

Kr → Rb

Ce → Pr

1.21 Examine the beta transitions for 65 Ni → Ga for the change in the angular momentum quantum number l, and identify which one can be classified as l forbidden. (Use the Isotope Explorer: http://ie.lbl.gov/ensdf/).

64

Chapter 1 Mechanisms

1.22 Using the Isotope Explorer: http://ie.lbl.gov/ensdf/, examine and comment on the decay of 129Te to the ground state, in terms of the change in total angular momentum, parity, transition type (electric or magnetic multipole), internal conversion ratio (α), and isomeric transitions. 1.23 NuDat (http://www.nndc.bnl.gov/nudat2/) reports two modes for the decay of 252 Cf: 3.1% by spontaneous fission and 96.9% by α decay, both at a half-life of 2.645 y. If 252 Cf were to decay by spontaneous fission only, what would be its half-life? In other words, what is the partial half-life for 252 Cf decay by spontaneous fission? 1.24 Identify five delayed-beta neutron emissions (βn) and five delayed-beta proton (βp) emissions, with at least one example for ε-delayed decay. Comment on the trend. Hint: use NuDat: http://www.nndc.bnl.gov/nudat2/

Section 1.8 1.25 Consider water at the normal density of 1000 kg/m3 : 1. Calculate the number of molecules of H2 O in 1 m3 . 2. Assuming a spherical nucleus, calculate the total condensed volume and mass occupied by the nuclei of one H2 O molecule. Assume a condensed nuclear density of 2 × 1017 kg/m3 . From this, determine the packing factor of H2 O nuclei. 3. Assuming a mean radius of a water molecule, or in other words, the mean separation distance between the O and H atoms in the H O H atomic bond (van der Waal’s radius) to be 1.41 × 10−10 m, calculate the total condensed volume occupied by the H2 O molecules in 1 m3 . From this, determine the packing factor of H2 O molecules. 4. Calculate the maximum number of molecules that can be packed in a 1 m3 (i.e. at 100% packing factor). 5. Comment on the above volume-based results. 1.26 Particle flux can be defined as the number of particles per unit area per unit time, or as the number of particles that enter a sphere of unit cross-sectional area. Relate these definitions to the definition of flux density as the number of track lengths per unit volume. 1.27 The following cross sections are symbolically given: Radiation

Target

Interaction

Cross section

γ

Electron

Compton

σe

γ

Atom

Photoelectric

σpe

Neutron

Nucleus

Absorption

σa

Neutron

Molecule Thermal scattering

σs

65

1.9 Problems

Using these symbols, write expressions for the macroscopic cross sections for H2 O. 1.28 Consider that the microscopic cross section can be viewed as the area projected by the atom to an incoming beam of radiation, the probability that radiation will interact with the atoms in a layer of thickness x normal to the incident beam can be expressed as29 : p =

Area projected by atoms in layer Area of layer normal to incident beam

=

Area projected by one atom × number of atoms Area of layer normal to incident beam

=

σt × NA x = N σt x = t x A

(1.67)

where σt and t are the microscopic and macroscopic cross sections, respectively, A is the area of the layer, and N is the number of atoms per unit volume in the layer. The probability of radiation leaving the layer without interaction is then: q = 1 − p = 1 − t x

(1.68)

1. If n layers are staggered against each other, each of thickness x containing the same material, show that the flux leaving the nth layer, φn , is given by: φn = qn φ0

(1.69)

where φn is the flux of the incident beam. 2. As x → 0, n → ∞, shows that: φ = φ0 exp[−t x]

(1.70)

where φ is the flux incident on a layer of thickness x.  Hint :

lim

n→∞

1+

a  ab =e b

(1.71)

1.29 Prove that the mean-free-path is equal to the reciprocal of the linear attenuation coefficient. Explain why this relationship is valid only for radiation that can travel some distance without suffering any interactions with matter.

29 Problem

attributed to P. J. Arsenault.

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C H A P T E R

TW O

Collision Kinematics

Contents 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9

Overview Center-of-Mass System Relativity Conservation Laws Einsteinian Kinematics Newtonian Kinematics Specific Interactions Electromagnetic Interactions Problems

67 68 74 80 83 104 108 123 146

2.1 Overview Kinematics is the study of motion of a system of bodies without directly considering the forces or potential fields affecting the motion. In other words, kinematics examines how the momentum and energy are shared among interacting bodies. Many of the interactions discussed in Chapter 1 are two-body interactions, in the sense that they involve two interacting entities: a projectile on a target. One important implication of two-body interactions is that there is a single plane that contains the two interacting particles. As such, there will be no momentum component off the plane of interaction. Therefore, the two bodies emerging from the interaction must be also on the same plane. Two-body interactions are, therefore, two-dimensional. In addition, with only two bodies, one can easily define a center-of-mass around which the incident and emerging particles will revolve. The center-of-mass is such that the total moment of the mass (mass × distance from the center-of-mass) is zero. Consequently, the center-ofmass is also such that the total momentum (mass × velocity) is also equal to zero. This facilitates considerably the performance of momentum balance. As such, kinematic analysis is performed in the center-of-mass. In Section 2.2, we discuss the relationship between the center-of-mass frame of reference and the laboratory frame of reference (in which observations are recorded). In addition to conserving Radiation Mechanics: Principles & Practice ISBN-13: 978-0-08-045053-7

© 2007 Elsevier Ltd. All rights reserved.

67

68

Chapter 2 Collision Kinematics

energy and momentum, interactions must also abide by some stoichiometric and intrinsic conservation principles which are summarized in Section 2.4. When dealing with radiation particles, the small size of the particle can lead to magnitudes of velocities close to the speed of light. Photons always move at the speed light. These high speeds introduce special relativistic effects that must be taken into consideration. Therefore, the special theory of relativity is reviewed in Section 2.3, with particular attention to its impact on the conservation of energy and momentum. Of course, there are many instances in which relativistic effects are not important. Nevertheless, the use of relativistic kinematic analysis will lead to results that are valid both at high and low speeds. In essence, classical kinematics is a special case (a first order approximation) of relativistic mechanics, as demonstrated by some examples in this chapter. A detailed mathematical analysis of relativistic particle kinematics is given in [18]. Relativistic kinematics, often referred to as Einsteinian kinematics, is discussed in Section 2.5, while the non-relativistic classical Newtonian kinematics is addressed in Section 2.6. In Section 2.7, the kinematic analysis is applied to those interactions discussed in Chapter 1 that are not affected by electromagnetic fields. In most cases, the kinematics of radiation interactions concerns itself with the interaction of one projectile with a single target, unlike in the interaction of the molecules of a gas where the collective movement of the gas is considered. Gas dynamics necessitates an overall analysis of the motion of gas molecules, since these molecules interact with each other. In the case of radiation, the field density is generally quite low and interaction amongst radiation particles is quite improbable, and is simply ignored. Therefore, radiation interactions are considered as a one-on-one (projectile on a target) process, in which the incoming radiation imparts energy to the target, while not receiving energy from it. However, when radiation particles are in thermal equilibrium with the atoms of the medium, particles can gain energy from the target atoms; a process that is quite analogous with that of gases. However, the collective moment of radiation particles is dealt with in a probabilistic manner, as discussed in Section 3.3.7 for the thermal equilibrium of neutrons. As Chapter 1 showed, there are some radiation interactions that are governed by electromagnetic wave behavior, rather than by the corpuscular nature of radiation. The kinematics of these wave interactions are discussed in Section 2.8.

2.2 Center-of-Mass System Definition In the ordinary laboratory frame of reference, events of an interaction are described from the point of view of an external observer. When the observer is located at the center-of-mass of the two interacting bodies, the interaction can be described from the view point of an “internal’’ observer. The relationship between the two

69

2.2 Center-of-Mass System

is governed by a transformation between the two frames of reference. In a static system of multiple bodies, the center-of-mass measured from some reference point is such that:  mi ri r0 =  (2.1) mi where r0 is the distance of the center-of-mass from the point of origin, ri is the distance of a body of mass mi from the origin. In a dynamic system, we differentiate with respect to time to arrive at:  dri  mi dt dr0 mi vi  v0 = = =  mi mi dt

(2.2)

where v0 is the velocity of the center-of-mass (also called the center of momentum) system with respect to the lab system where the point of origin is located. The velocity vector vi of body i is along ri , and is considered to be positive if the particle is moving away from the point of origin of the lab system. Transformation between the center-of-mass frame of reference (C ) and the laboratory frame of reference (L) is then done using the relationship: v  = v − v0

(2.3)

where v  and v are velocity vectors in C and L, respectively. In the analysis below, the magnitude of a vector is designated by simply dropping off the vector notation. Note that the vectorial relationship of Eq. (2.3), applied to a certain particle, enables relating the angles of scattering in L and in C to each other using the law of sines on the triangle formed by this vector equation; i.e.: sin(ϑ − ϑ) sin(π − ϑ ) sin ϑ = = v v0 v

(2.4)

where the angles are as designated in Fig. 2.1.

υ

ϑ

υ ϑ

υ0

Figure 2.1 Velocity diagram relating C to L, where v is the velocity of a particle in L, v  is its velocity in C and v 0 is the velocity of C itself.

70

Chapter 2 Collision Kinematics

ϑ4

m2, υ2

m2, υ2 ϑ4

ϑ3

m3, υ3

m4, υ4

m4, υ4 m3, υ3 ϑ3

C

m1, υ1

m1, υ1 L (Point of origin) Laboratory frame of reference (L)

Center-of-mass frame of reference (C)

Figure 2.2 Two particles approaching each other in the laboratory and the center-of-mass frames of reference.

Before an interaction Figure 2.2 schematically shows the relationship between C and L for the twobody interaction of a projectile 1 of mass m1 colliding with a target with mass m2 , resulting in the emergence of a particle 3 with mass m3 and particle 4 with mass m4 . Of course, if the reaction is an elastic one, then m1 = m3 and m2 = m4 . If a photon is emitted, m3 has to be replaced by an energy term, as will be shown in Section 2.3. If the initial particle velocities are, respectively, v1 and v2 , at some point in time, the definition of C requires that: m1 v1 − m2 v2 = (m1 + m2 )v0

(2.5)

which is equivalent to equating the rate of moment of the two masses dx0 dx2 1 (m1 dx dt + m2 dt ) to that of the center-of-mass ((m1 + m2 ) dt ) around a point of origin located somewhere along the line connecting the two particles, where dx2 1 x refers to distance from the origin in L. Note that if v1 = − dx dt , then v2 = − dt , since if particle 1 is moving away from the origin, particle 2 will be moving toward it, and vice versa. m1 (v1 − v0 ) − m2 (v2 + v0 ) = 0 (2.6) Since v0 is a vector along the direction of the incident particle (i.e. opposite to that of the target), the velocities of the projectile and the target in C are: v 1 = v1 − v0 v 2

= v2 − v0

v1 = v1 − v0 v2

= −(v2 + v0 )

(2.7) (2.8)

Therefore, Eq. (2.6) becomes: m1 v1 + m2 v2 = 0

(2.9)

71

2.2 Center-of-Mass System

The same relationship can    be arrived at by equating the rate of moment of the dx1 dx2 two masses dt + dt around the center-of-mass to zero, where x refers to distance of particle from C , keeping in mind that both particles are moving in opposite directions toward C and x1 and x2 are opposite in sign. The important implication of Eq. (2.9) is that the total momentum in C is equal to zero. Comparing Eqs (2.5) to (2.9), one notices that in the latter equation the right-hand side is zero, and in determining the center-of-mass the momenta are added. These are due to the fact that the observer is at C (hence the zero value), and the projectile and target are moving toward C (hence the addition of momentum).

Reduced mass From Eq. (2.5):   v1 m1 v1 − m2 v2 v2 v0 = = μ1,2 − m1 + m 2 m2 m1 where: μ1,2 =

m1 m2 m1 + m2

(2.10)

(2.11)

is known as the reduced mass. Note that if m2 >> m1 , μ1,2 ≈ m1 , and v0 ≈ v2 , i.e. C coincides with m2 as one would expect. Equation (2.10) can be expressed as: v0 v1 v2 = − μ1,2 m2 m1

(2.12)

Let us now consider the momentum of the reduced mass obtained by multiplying μ1,2 by the relative velocity, vr , of particle 1 with respect to the target 2, where vr = v1 − v2 . Then, vr = v1 − (−v2 ) = v1 + v2 ; with a negative sign associated with v2 because it is opposite to v1 . Using Eq. (2.12), it can then be shown that: ⎧   m1 v1 m1 v0 ⎪  ⎪ ⎪ ⎨ μ1,2 v1 + m2 − μ1,2 = (v1 − v0 )m1 = v1 m1 μ1,2 vr = μ1,2 (v1 +v2 ) =   ⎪ m2 v0 m2 v2 ⎪ ⎪ + + v2 = (v2 + v0 )m2 = −v2 m2 ⎩ μ1,2 μ1,2 m1 (2.13) The above equation shows that the momentum of the reduced mass in terms of the relative velocity is equal to the momentum of the projectile or the target in C .That is, with the reduced mass and the relative velocity, one can easily determine the momentum of the projectile and the target, which arises of the fact that the total momentum in C is zero. In other words, one can reduce the two-body system to a single-body equivalent system of a mass equal to the reduced mass and a velocity

72

Chapter 2 Collision Kinematics

equal to the relative velocity. The latter velocity, vr , has the same value in both systems, since the relative velocity is not affected by the transformation from one frame of reference to another. That is: vr = v1 − v2 = v1 − v2

vr = v1 + v2 = v1 − v2

(2.14)

In turn, the momentum μ1,2 vr has the same value in either system, which is a useful tool for relating the two frames of references to each other.

After the interaction Let us now turn our attention to the emerging particles 3 and 4 in the C system. The center-of-mass of the emerging particle must be on the line joining the two particles, which necessitates that: m3 v3 + m4 v4 = 0

(2.15)

since the velocity of the center-of-mass is zero in C . This also shows that the total momentum remains zero in C after the interaction. Analogous to Eq. (2.13), one can state that: μ3,4 (v3 + v4 ) = v3 m3 = −v4 m2 (2.16) m4 . If the interaction is an elastic scattering, μ1,2 = μ3,4 , and the where μ3,4 = mm33+m 4 equivalent one-body system maintains the same momentum. Since the relative velocity does not change with system transformation, v3 + v4 = v1 + v2 . Then using Eq. (2.16), it becomes possible to determine both v3 and v4 in terms of v1 − v2 , for elastic scattering. The next step is to transform these values back to L. Let, as shown in Fig. 2.2, ϑ3 be the angle between the directions of particles 1 and 3 in L, and ϑ3 be the corresponding angle in C . Then one can project back v3 to v3 by adding to the former vectorially the velocity of C , as Eq. (2.3) indicates. That is, v3 = v3 + v0 (2.17) The velocity vectors can be expressed as:

v3 = v3 (cos ϑ3ˆi + sin ϑ3ˆj )

(2.18)

v 3 = v3 (cos ϑ3 ˆi + sin ϑ3 ˆj )

(2.19)

v0 = v0ˆi + 0ˆj

(2.20)

where ˆi and ˆj are, respectively, unit vectors in the direction cosines of the initial direction of incidence of particle 1 and the axis normal to it. Applying the above to Eq. (2.17) and equating the ˆi and ˆj components leads to: v3 cos ϑ3 = v3 cos ϑ3 + v0

(2.21)

v3 sin ϑ3 =

(2.22)

v3 sin ϑ3

73

2.2 Center-of-Mass System

Combining the above equations gives: tan ϑ3 =

sin ϑ3 cos ϑ3 +

(2.23)

v0 v3

With v3 known from Eq. (2.16) and with v0 determined by Eq. (2.10), the relationship between ϑ3 and ϑ3 is established by Eq. (2.23). That is, if one of those angles is known, the other is determined. Then, one can evaluate v3 using Eq. (2.22). The velocity, v4 , can then be also determined using Eq. (2.16).

Angular momentum and moment of inertia Another interesting kinematic quantity that maintains the same value in L and C is the angular momentum, J , around the center-of-mass. The angular momentum of a particle of mass m around a point is a vector product J = r × p, where r is the vector connecting the particle to the point and p is the linear momentum. The magnitude of J is | J | = mvr sin ψ, where ψ is the angle between r and v .Therefore, for the configuration of Fig. 2.3, the total angular momentum of particles 1 and 2 around C is:   x | J | = [m1 (v1 − v0 )r1 + m2 (v2 + v0 )r2 ] r1 + r2   x = [m1 v1 r1 − m2 v2 r2 ] = μ1,2 vr x = μ1,2 (v1 + v2 )x (2.24) r1 + r 2 where use was made of Eq. (2.13).This shows that | J| stays the same in both frames of reference, which is expected, since they all differ by the translational velocity, v0 , that plays no part in the angular momentum.The distance x is known as the impact parameter, and is zero if the two particles are approaching each other head on. m1, z 1e

υ1 r1

r2

x

C

υ2

m2, z 2e

Figure 2.3 A schematic showing parameters for calculating the angular momentum around C.

74

Chapter 2 Collision Kinematics

The moment of inertia, I , around C for the configuration of Fig. 2.3 is given by: I = m1 r12 + m2 r22 = μ1,2 (r1 + r1 )2 =

m12 2 m2 r1 = 2 r22 μ1,2 μ1,2

(2.25)

where use is made of the definition of C which is such that m1 r1 = m2 r2 = μ1,2 (r1 + r2 ).

2.3 Relativity 2.3.1 Special theory of relativity The theory of relativity deals with the viewing of events by observers moving at different speeds. Its effect is quite significant for particles moving at speeds approaching the speed of light. This can happen for example to electrons, that easily acquire high speeds at low kinetic energy because of their very small mass. The special theory of relativity deals with systems at rest or those moving at a constant speed (without acceleration, i.e. not subjected to field effects). Therefore, this theory, first formulated by Einstein (1905), is quite relevant when dealing with radiation kinematics. The theory considers coordinate systems that are in uniform rectilinear translational motion relative to each other, and assumes in these systems that: 1. Physical laws maintain the same mathematical form. 2. The speed of light, c, is constant, and is independent of the motion of the source of light (or electromagnetic radiation). We will apply first the classical Galilean transformation, named x3 x 3 after Galileo (1564–1642), betυ0 ween two coordinate systems K (x1 , x2 , x3 ) and K  (x1 , x2 , x3 ), with the latter moving at a conx2 x2 stant speed v0 in the direction of x3 of the K system. Such frames of reference are called inertial frames, x1 x 1 since they move at a constant speed in straight lines with respect to each other and are non-accelerating. We assume that at time t = t  = 0, the systems coincided. We are using here subscripts to denote the spatial coordinates, rather than the traditional x, y, z notation, since the theory of relativity requires the expansion of this three-dimensional space to a four-dimensional space. For the same reason, time is given different notations in the two coordinate systems.

75

2.3 Relativity

Galilean transformation Consider the emission of a pulse of light at the origin of the two systems at time t = t  = 0, i.e. when the systems coincide with each other. Since light travels at the speed of light, c, and behaves as a wave, at time, t, the observer in the K system will see a spherical wavefront of radius ct, which is mathematically describable by the equations x21 + x22 + x23 = (ct)2 (2.26) The observer in K  would express the spherical wavefront of radiation by: 2 2  2 x2 1 + x2 + x3 = (ct )

(2.27)

Using the classical Galilean transformation between the coordinates of the two systems: x1 = x1 x2 = x2 x3 = x3 − v0 t t  = t (2.28) in Eq. (2.27), one gets x21 + x22 + x23 = (ct  )2 − v02 t 2 + 2v0 t. This implies that the wavefront moves at a speed different from the speed of light, which is an obvious violation of physical laws. Therefore, to accept that Galilean transformation is to admit that either the speed of light depends on the frame of reference (a violation of the first postulate of the theory of relativity), or that the mathematical law that describes the waveform is system-dependent (a contravention of the second postulate); accepting either would violate physical evidence.

Lorentz transformation The above difficulty with the Galilean transformation is resolved by considering √ a fourth imaginary independent coordinate: x4 = ict, x4 = ict  , where i = −1. The wavefront in the two systems is then described by: x21 + x22 + x23 + x24 = 0 x2 1

+ x2 2

+ x2 3

+ x2 4

(2.29)

=0

(2.30)

The Galilean transformation needs then to be expanded, and modified, to accommodate the fourth variable. This is achieved by Lorentz (1895) via the transformation: x1 = x1

x2 = x2

x1 = x1

= x2

x2

x3 = γ(x3 − βct) x3

=

γ(x3

+ βct)

x4 = γ(x4 − iβx3 )

(2.31)

x4 =

(2.32)

γ(x4

+ iβx3 )

where β = vc0 and γ = √

1 . With this transformation, the waveform maintains 1 − β2 the same spherical shape and the same speed of propagation in both the K and K 

frames of reference. This transformation makes time, t, an independent variable, orthogonal to the other three spatial independent variables (x1 , x2 , x3 ). This introduces difficulties in defining velocity (and other time-dependent variables

76

Chapter 2 Collision Kinematics

such as momentum and force), since one cannot differentiate two independent variables (e.g. space with respect to time) to obtain a dependent variable (like velocity). Therefore, a new time-like variable is introduced to define momentum, which replaces velocity as the main fundamental variable.

Proper time Let us consider the rotation (rather than the translation as considered above) of coordinate systems with a common origin. In an ordinary transformation, such rotation leaves the Euclidean distance r 2 = x21 + x22 + x23 unchanged, i.e. invariant. Similarly, the rotation of the four-dimensional space (often called the space–time continuum, or Minkowski space, named after Minkowski (1908) who first introduced the concept of space–time) should have s2 = x21 + x22 + x23 + x24 invariant. Since x4 is imaginary, one would expect s to be imaginary, so let us set s = icτ, in analogy with x4 = ict, where τ is an invariant (since c is constant) with the units of time. Since t is now an independent variable, τ is a dependent time-like variable, called the “proper time’’. Then: r2 s2 = r 2 − c 2 t 2 = −c 2 τ 2 τ 2 = t 2 − 2 (2.33) c The metric s2 in Eq. (2.33) does not have to be a positive quantity, since r 2 can be less that c 2 t 2 . In fact, in the above analysis for the light pulse, s2 = 0, hence τ = 0. Events with τ = 0 (s2 = 0) are said to have a “light-like’’ separation from the origin of the space–time continuum. When τ is imaginary (s2 > 0) events have a “space-like’’ separation, as the spatial distance, r, dominates. A real value of τ refers to events in which time is dominant, s2 < 0, which are said to have a time-like separation. In the latter separation, τ can be positive (forward) or negative (backward).

Four-vector momentum The proper time enables the definition of four components of velocity (U1 , U2 , U3 , U4 ) and momentum (P1 , P2 , P3 , P4 ) in the four-dimensional space (x1 , x2 , x3 , x4 ) as follows1 : dx1 dx1 dt = = γ x˙3 dτ dt dτ dx2 dt dx2 = U2 = = γ x˙2 dτ dt dτ dx3 dt dx3 = = γ x˙3 U3 = dτ dt dτ dt = iγc U4 = ic dτ

U1 =

1

t dτ = =  dt τ

1 r2 1− 2 2 t c

= 

1 v2 1− 2 c

= 

1 1 − β2

= γ.

P1 = γm0 x˙1 P2 = γm0 x˙2 (2.34) P3 = γm0 x˙3 P4 = iγm0 c

77

2.3 Relativity

v β = , v is the ordinary three-vector velocity, and m0 is c the “rest’’ (at zero speed) mass of the body for which the momentum is evaluated. The introduced fourth momentum, P4 , has a physical significance that can be demonstrated by considering the low-speed case, v << c, where relativistic effects can be neglected. Then, the Taylor series expansion2 of γ is:  1 2 3 4 5 6 1 = 1 + β + β + β + ··· (2.35) γ= 2 8 16 1 − β2 where γ = √

1 , with 1 − β2

Then, at v << c, Eq. (2.34) becomes:   1 1  v 2 (2.36) P4 = im0 c 1 + β2 = im0 c 1 + 2 2 c i The second term in the left-hand side is times the well-known non-relativistic c c kinetic energy, 12 m0 v 2 . Analogously, P4 is an energy term, called the total energy, i E, defined such that: E=

cP4 = γm0 c 2 i

(2.37)

Total energy The term m0 c 2 in Eq. (2.37), is called self-energy, rest-mass energy, or simply restenergy. Equation (2.37) indicates that the total energy is greater than the rest energy. Since so far we considered free bodies, the difference between E and m0 c 2 has to be attributed to the kinetic energy, T . That is, E = m0 c 2 + T

(2.38)

Therefore, the total energy consists of an intrinsic constant component, m0 c 2 , and a variable component acquired by motion, T , as was already shown by Eq. (1.4). Using the Taylor series expansion of for γ, given by Eq. (2.35), then:  3 4 5 6 2 1 2 β + β + β + ··· (2.39) T = m0 c 2 8 16 which shows that the conventional expression of T = 12 m0 v 2 is only valid when v << c. The total energy, E, can also be expressed in terms of a “relativistic mass’’, m, so that: m0 (2.40) E = mc 2 m =  1 − β2 2 Taylor

series expansion: f (x) = f (0) +

∞ xn ∂n f (x)|x=0 . n=1 n! ∂xn

78

Chapter 2 Collision Kinematics

With this definition of E, the four-vector momentum of Eq. (2.34) becomes: P1 = mx˙1

P2 = mx˙2

P3 = mx˙3

P4 = imc

(2.41)

Momentum The ordinary momentum, p, is equal to p = mv = mβc, with v being the ordinary three-vector velocity. Then from the definition of m in Eq. (2.40) one gets: m2 =

m02 1−

p2 m2 c 2

(2.42)

which leads to: (pc)2 = (mc 2 )2 − (m0 c 2 )2 = E 2 − (m0 c 2 )2

(2.43)

With E = T + m0 c 2 , then: p2 c 2 = T 2 + 2m0 c 2 T

(2.44)

The above equation demonstrates that when the rest-mass energy is zero, as in the case of photons, p = Tc and from Eq. (2.38), T = E. Then, p = Tc , which is the same as given in Eq. (1.7). One final useful expression for the total energy is obtained by replacing T in Eq. (2.44) by E − m0 c 2 , leading to: E 2 = p2 c 2 + (m0 c 2 )2

(2.45)

2.3.2 Center of relativistic mass With the relativistic mass defined by Eq. (2.40) as mc 2 , one must revisit the concept of the center-of-mass described in Section 2.2 for the rest mass. The center for relativistic mass is referred to here by C r to distinguish it from the center of rest mass (C ).The relationship between the C r frame of reference and that of laboratory (L) system can be established using Lorentz transformation with the system K  given a relative velocity β0 in the x1 axis of the K system. Then, the K  system corresponds with the C r system, while that of K coincides with the L system. By using the ordinary velocity, we can restrict the transformation to the three spatial coordinates. Then differentiating the spatial coordinates in Eqs (2.31) and (2.32) with respect to t and multiplying each side by the relativistic mass m, one obtains:   β0 p1 = p1 p2 = p2 p3 = γ0 p3 − E (2.46) c   β0     (2.47) p1 = p1 p2 = p2 p3 = γ0 p3 + E c where γ0 = √

1 1 − β02

and use is made of the fact that E = mc 2 , with the linear

momentum evaluated using the relativistic mass, m. The value of β0 is obtained

79

2.3 Relativity

using of the fact that the total momentum in C r is always zero. Applying this fact to the interacting bodies and transforming the momentum in C r back to the L system using Eq. (2.46), for the problem described in Fig. 2.2, one has: p1 + p2 = 0  β0 β0 p1 − E1 + γ0 p2 − E2 = 0 c c p1 + p2 p 1 + p2 β0 = = c E1 + E2 m1 c 2 + m2 c 2

 γ0





(2.48)

where 1 and 2 refer here to the projectile and the target, respectively. Notice here that attention is not given to assigning a negative value for p2 or p2 although both oppose p1 and p2 , respectively, to avoid assigning a wrong sign for energy in the transformation. The direction of the momentum should be accounted for at the very end of the analysis. Comparing Eqs (2.48) to (2.10), one can see that the relativistic mass simply replaces the rest mass in determining the speed of the center-of-mass system. While it is not possible to create a simple velocity diagram for C r , analogous to that of Fig. 2.1 for C , interaction kinematics can be formulated vectorially as shown in Section 2.5.

2.3.3 Lorentz transformation of momentum and energy Lorentz transformation, thought applied above to the space–time fourcomponent vector: ( r , ict), can also be applied to the four-component momentum-energy vector: ( p, iE), with x1 ≡ cp1 , x2 ≡ cp2 , x3 ≡ cp3 , and x4 ≡ iE; with p1 , p2 , and p3 being the three orthogonal components of p, and c is introduced for consistency of units. Guided by Eq. (2.31), momentum and energy can be transformed between the two frames of references K and K  moving with a velocity v0 (=β0 c) in the direction of p3 , as follows: cp sin ϑ cos ϕ = cp1 sin ϑ cos ϕ cp cos ϑ = γ0 (cp cos ϑ − β0 E)

cp sin ϑ sin ϕ = cp sin ϑ sin ϕ E  = γ0 (E − β0 cp cos ϑ)

where ϑ and φ are the polar and azimuthal angles of the vector p, shown in Fig. 2.4, and the primed values are the corresponding ones, in the K  system. Equivalently, one can write: ϕ = ϕ cp cos ϑ = γ0 (cp cos ϑ − β0 E)

p sin ϑ = p sin ϑ E  = γ0 (E − β0 cp cos ϑ)

(2.49)

The reverse transformation, from K  to K , is simply performed by replacing β0 with −β0 . This facilities greatly the transformation of momentum and energy between the two frames of reference, particularly when K is taken as the L system, and K  is considered to be the center-of-mass system for the relativistic mass (C r ), then v0 = βc. For the two-body problem of Fig. 2.2, β0 can be calculated using

80

Chapter 2 Collision Kinematics

x3

cp cos ϑ

cp

υ ϑ β0 = c0

x2

cp sin ϑ sin ϕ

ϕ x1

Figure 2.4

cp sin ϑ cos ϕ

Polar (ϑ) and azimuthal (ϕ) angles of a momentum vector cp .

Eq. (2.48), which can also be arrived at by setting the total momentum for particles 1 and 2 in C r (K  ) equal to zero and transforming it back to L(K ) using Eq. (2.49); keeping in mind that ϑ1 = ϑ1 = 0 and ϑ2 = ϑ2 = −π, while the momentum of one particle is opposite in direction to that of the other. That is, cp1 + cp2 = 0 = (cp1 − β0 E1 ) + (cp2 − β0 E2 )

2.4 Conservation Laws Interaction kinematics is concerned with the laws of conservation of momentum and energy. However, interactions also adhere to some other conservation principles, in addition to the kinematical ones. We will classify these other principles as stoichiometric and intrinsic. Stoichiometric conservation is an elemental balance that assures equality of charge and nucleons between the reactants and the products. Intrinsic balance determines the internal state of the products, i.e. whether they are in stable or excited states and what quantum numbers define these states. Both stoichiometric and intrinsic conservation affect the kinematics of interaction as they determine consecutively the mass and charge of the reaction products and the amount of internal energy stored, which affects in turn the momentum and energy of the reaction products. These conservation laws are presented here. Let us consider the generic reaction between a projectile X1 and a target nucleus X2 that results in the emission of a particle X3 and a product nuclide X4 : A1 Z1X1

2 +A Z2X2 →

A3 Z3X3

4 +A Z4X4

(2.50)

where the subscripts and the superscripts designate, respectively, the atomic and mass numbers.

81

2.4 Conservation Laws

2.4.1 Stoichiometric conservation Charge Z1 + Z2 = Z3 + Z4 , total electric charge is conserved, i.e.



Z = constant.

Lepton number A lepton is a particle not affected by the strong nuclear forces, but is only subjected to the weak forces. As such, electrons and neutrinos are leptons. A lepton number of 1 is assigned to both the electron and the neutrino and −1 to the antineutrino and the positron. Conservation of the lepton number is useful in studying interactions, and decay processes, in which β particles (positive or negative) are produced. This conservation principle is also useful in high-energy physics.

Mass number A1 + A2 = A1 + A4 , number of nucleons is conserved, i.e.



A = constant.

Baryon number A general form for the conservation of the mass number is the conservation of the baryon number. A baryon is any particle that consists of three quarks, such as the proton and neutron. A baryon number B = 1 is assigned to each baryon, subsequently a quark has a baryon number of 13 . Conservation of the baryon number, B = constant, is particularly useful when dealing with high-energy physics reactions.

Statistics

  As a consequence of A = constant, if A is odd, it stays odd and the interaction is governed by Fermi–Dirac statistics, while if A is even, Bose–Einstein statistics prevails throughout the interaction. See Section 1.4 for the definition of each type of statistics.

2.4.2 Intrinsic conservation Angular momentum The total angular momentum number is defined by the quantum number J and is conserved vectorially, i.e. J1 + J2 = J3 + J4 . The reactants can have a combined magnitude of the total angular momentum that vary in magnitude from | J1 − J2 | to | J1 + J2 |, depending on the relative orientation of J1 and J2 . Similarly the products have angular momentum values from | J3 − J4 | to | J3 + J4 |.

82

Chapter 2 Collision Kinematics

Conservation of the total angular momentum restricts the values of J for the interaction products within the same range of the reactants.

Parity

 Conservation of parity, , requires that 1 2 = 3 4 , i.e.  = constant.That is, if the combined parity of the reactants is even, that of the products must stay even, and if the former is odd the latter is also odd. This adds further restrictions on the angular quantum numbers (the l quantum number) of the products, in addition to that imposed by the conservation of the total angular momentum J .

Isospin Although conservation of nucleons (Baryon number) is essential, its satisfaction does not assure that a certain excited state resulting from an interaction, or even the interaction itself, is allowed.This is because, while the proton and the neutron have nearly the same mass, and as such are subjected to almost the same strong nuclear field (which does not depend on the charge), they are still two separate entities, distinguished by a hypothetical spin number, analogous to the intrinsic spin, called the isobaric spin, or simply the isospin. It is set equal to + 12 for the proton and − 12 for the neutron. Interactions that result in excited states which do not conserve isospin are forbidden, even if such states conserve angular momentum and parity. This conservation principle can explain why some excited states are not observed, and in high-energy physics why some interactions are not possible.

2.4.3 Kinematical conservation Linear momentum The total linear momentum of the products must be equal to the total linear momentum of the reactants. Note that the linear moment (called simply momentum) is a vector quantity and is conserved at any direction. In the center-of-mass system (see Section 2.2), the total momentum is always zero, before and after the interaction, in any direction. This fact facilitates greatly the application of momentum conservation.

Total energy Neither kinetic energy nor mass are conserved on their own in radiation interactions, the conservation is that of the total energy (kinetic energy + rest-mass energy) (see Eq. (2.38)).The following section provides a detailed analysis of kinematic conservation of momentum and energy for interactions that are assumed to abide by the stoichiometric and intrinsic conservation laws given above. If the latter laws are not adhered to, the interaction would not take place and the application of kinematic analysis becomes meaningless.

83

2.5 Einsteinian Kinematics

2.5 Einsteinian Kinematics 2.5.1 Two-body kinematics Relativistic mechanics is applied here to a general two-body interaction. The analysis below is an extension of that given in [19, Section I.B.]. Consider the interaction 2(1,3)4, shown in Fig. 2.5, which involves a projectile of mass M1 and kinetic energy T1 bombarding a stationary target of mass M2 at rest, i.e. with a zero kinetic energy, T2 = 0. Let the emerging particle have a mass M3 and kinetic energy T3 and the residual target having a mass M4 and a kinetic energy T4 . In all cases, it is referred to the rest mass. In relativistic mechanics, the state of a body is defined by (E, P ), where E is its total energy and P is its momentum. These state variables are shown in Fig. 2.5 for a two-body interaction, identified for each body by the appropriate subscript.

Notation In order to facilitate analysis, we use here units of energy for both mass and momentum, by multiplying the former by c 2 and the latter by c and we use upper-case letters for both. Table 2.1 relates the notation of this section to the conventional one. Let us define the angles ϑj and ϕj as, respectively, the polar and azimuthal angles a body j ( j = 1–4) makes with the incident direction of particle 1 in L (i.e. ϑ1 = ϑ1 = 0), with ϑj and ϕj being the corresponding angles in C r . Since body 2 is stationary, then also ϑ2 = ϑ2 = 0.

Kinematical conservation in C r To take advantage of the fact that total momentum in the relativistic center-of-mass frame of reference (C r ) is zero, momentum and energy balance is performed in C r , E4, P4

E2, P2

E3, P3 ϑ4 ϑ3

ϑ3

E2, P2

E 3, P 3

ϑ4

b0 E 4, P 4 L

E1, P1

Cr

E1, P1

Figure 2.5 A schematic of a two-body interaction in the lab (L) and relativistic center-of-mass (C r ) frames of reference, with the state of each body defined by the total energy E and the linear momentum, P.

84

Chapter 2 Collision Kinematics

Table 2.1

Notation used in analysis

Notation Variable Conventional

Relativistic

Velocity

v

β = vc

Rest mass

m0

M = m0 c 2

Momentum

p = mv = m0 γv

P = pc = M βγ

Momentum vector

p = m v = m0 γ v

P = M γ β

Total energy

E = T + m0 c 2

E =T +M

Relationships

p2 c 2 = T 2 + 2m0 c 2 T

P 2 = T 2 + 2MT

E 2 = (pc)2 + (m0 c 2 )2

E2 = P 2 + M 2

v 1 v . T = kinetic energy; both M and P are in units of energy; c = speed of light; β = ; β = ; γ =  c c 1 − β2

and the results are then transformed back to the laboratory frame of reference (L). The speed of C r with respect to L is obtained using Eq. (2.48), which ensures that the total momentum in C r is always zero. After employing the notation of Table 2.1 and assigning P2 = 0 and E2 = M2 , one obtains: β0 =

P1 E1 + M2

(2.51)

Conservation of momentum and energy requires that:  i

P i =



P f 

and

f

 i

Ei =



Ef

(2.52)

f

where i and f refer to the initial and final states, respectively. Applying these conservations laws in C r results in the following. Momentum in direction of projectile in C r

P1 cos ϑ1 + P2 cos ϑ2 = P3 cos ϑ3 + P4 cos ϑ4 = 0

(2.53)

Momentum in direction ⊥ to projectile in C r Since in C r the total momentum is zero, the initial and final particles can assume any azimuthal direction. Therefore,

85

2.5 Einsteinian Kinematics

two orthogonal directions in the azimuthal planes are considered in this analysis, so that: P3 sin ϑ3 cos ϕ3 + P4 sin ϑ4 cos ϕ4 = 0 (2.54) P3 sin ϑ3 sin ϕ3 + P4 sin ϑ4 sin ϕ4 = 0 Total energy

E = E1 + E2 = E3 + E4

(2.55)

where E is the total energy of C r . Angles of Scattering in C r The two equations of (2.54) lead to:

tan ϕ3 = tan ϕ4 The above equation is satisfied when: |ϕ3 − ϕ4 | = π

(2.56)

which again describes the physical fact that the momentum vectors of particles 3 and 4 lie in a plane in C r , as is the case in L. Therefore, P3 sin ϑ3 = P4 sin ϑ4 . Equation (2.53) gives: P3 cos ϑ3 = −P4 cos ϑ4 , and as such tan ϑ3 = −tan ϑ4 . Therefore, ϑ4 = π − ϑ3 (2.57) This shows that particles 3 and 4 emerge in opposite directions in C r , as expected. Substituting Eq. (2.57) in Eq. (2.53) gives: P3 = P4 = P 

(2.58)

Energy and momentum in C r Using E 2 = P 2 + M 2 along with Eq. (2.58) results in:

E32 = E42 + M32 − M42

(2.59)

2

Adding E3 + 2E3 E4 to both sides of the above equation, then, 2E3 (E3 + E4 ) = (E3 + E4 )2 + M32 − M42 1 [(E  + E4 )2 + M32 − M42 ] E3 =  2(E3 + E4 ) 3 Consequently, E3 =

E 2 + M32 − M42 κ = 2E 2E

(2.60)

where, κ = (E3 + E4 )2 + M32 − M42 = E 2 + M32 − M42

(2.61)

86

Chapter 2 Collision Kinematics

The momentum of particle 3 is equal  to that of particle 4, as Eq. (2.58) shows,

and their value is obtained using P3 = E32 − M32 :  

P =

P3

=

P4

=

κ2 − M32 4E 2

(2.62)

The total energy of particle 4 is obtained from Eq. (2.55) as: E4 = E − E3 = E −

κ 2E 2 − κ κ = = 2E 2E 2E

(2.63)

where, κ = E 2 − (M32 − M42 )

(2.64)

Once the total energy of the two bodies (the value of E according to Eq. (2.55)) is determined, the energy and momentum of particles 3 and 4 become known. Their angles of scattering, ϑ3 and ϑ4 , can be determined from Eqs (2.53) and (2.57). Note that any two azimuthal angles ϕ3 and ϕ4 will satisfy momentum balance provided that |ϕ3 − ϕ4 | = π, as Eq. (2.56) indicates. In relation to incident particle in L In order to able to transform the results obtained above in C r back to L, we recall that C r is by definition moving in the direction of the incident particle, particle 1, and that ϑ1 = 0, P2 = T2 = 0. The velocity of C r is given by Eq. (2.48), which with the notation of Table 2.1 becomes (after setting E2 = T2 + M2 = M2 and P2 = 0):

β0 =

P1 E1 + M2

(2.65)

Therefore, β0 is the ratio between the system’s momentum and energy in L. Then, γ02 =

1 1 (E1 + M2 )2 = = 2 2 2 P1 1 − β0 M1 + 2M2 E1 + M22 1 − (E1 +M 2 ) 2

where use is made of E12 = P12 + M12 . Therefore, γ0 = where:

E1 + M2 E

(2.66)

 E = M12 + 2M2 E1 + M22

which is shown in Eq. (2.69) below as being the system’s energy in C r . As such, γ0 is the ratio between the energy of the incident particles in L and the C r energy.

87

2.5 Einsteinian Kinematics

Now we can transform the energy balance equation of (2.55) to L using the Lorentz transformation of momentum and energy, Eq. (2.49) adjusted for the notation of Table 2.1, to obtain: E1 + E2 = γ0 [(E1 − β0 P1 ) + (M2 )] E3

+ E4

= γ0 [(E3 − β0 P3 cos ϑ3 ) + (E4 − β0 P4 cos ϑ3 )]

(2.67) (2.68)

Then using Eqs (2.55), (2.65), and (2.66) along with P12 = E12 − M12 , one gets: E3 + E4 = E1 + E2 = γ0 [E1 − β0 P1 + M2 ]  P12 E1 + M2 E1 + M2 − = E E1 + M2  1 = [M12 + 2E1 M2 + M22 ] = M12 + 2M2 E1 + M22 E Therefore,  E3 + E4 = M12 + 2M2 E1 + M22 = E1 + E2 = E

(2.69)

since E = E3 + E4 = E1 + E2 , as defined by Eq. (2.55). The energy E is the total energy of the reactant particles 1 and 2, made available for the creation of the product √ particles 3 and 4. At very high energy, i.e. when E >> M1 , M2 , one has E ≈ 2M2 E1 . Then, a high incident energy E1 in L does not produce a high C r energy that can be used to supply particles 3 and 4 with high energy if the target particle 3 is at rest, due to the damping by the relatively smaller M2 value. In order to generate sufficient energy to produce new particles in high-energy studies, the use of fixed targets is avoided, instead intersecting, or colliding, particle beams are utilized. This has led into the concept of “colliders’’, which serve to maximize the energy available in C r , E , hence facilitating the creation of new particles with rest-mass energies much higher than the reactant particles. Now with E known in terms of the properties of the incident particles, all parameters in C r are fully determined. Transformation to L The transformation of Eq. (2.49), adjusted for the notation of Table 2.1, gives: E3 = γ0 (E3 − β0 P3 cos ϑ3 ) (2.70)  Given that E3 is now known, from Eq. (2.60), and keeping in mind that E32 = P32 + M32 , Eq. (2.70) becomes a quadratic equation that can be solved for the value of P3 , then for E3 . The possibility of having two values for E3 and P3 indicates that there are two possibilities for the angle ϑ , as discussed below. Reverse transformation L → C r Applying the transformation of Eq. (2.49) in reverse, with β0 replaced with −β0 , to particle 3, one obtains:

ϕ3 = ϕ3

(2.71)

88

Chapter 2 Collision Kinematics

P3 sin ϑ3 = P  sin ϑ3

(2.72)

P3 cos ϑ3 = γ0 (P3 cos ϑ3 + β0 E3 )

(2.73)

E3 = γ0 (E3 + β0 P  cos ϑ3 )

(2.74)

Equations (2.72) and (2.73) give: γ0 tan ϑ3 =

β3 sin ϑ3 β0 + β3 cos ϑ3

(2.75)

where: β3 =

P E3

(2.76)

Possible angles of scattering Equation (2.75) indicates that there can be restrictions on the range of values the angle ϑ3 can assume, depending on whether it is less than, greater than, or equal to β0 . These restrictions, or lack of, can be illustrated by vectorially combining β 0 and β 3 to yield a vector that defines the angle ϑ3 , or precisely the angle ϑ = tan−1 (γ0 tan ϑ3 ), with this angle being the angle the combined vector β 0 + β 3 makes with β 0 . The resulting vector diagrams are shown in Fig. (2.6), which indicates that Eq. (2.75) can be satisfied by two possible values of ϑ3 . These two possibilities are designated by the points 1 and 2 on the circle representing all possible values β3 for a determined (by the energy and momentum of the incident particle) value of β0 . The various possibilities are discussed below. When β3 < β0 , as Fig. 2.6 shows, the vector β 0 + β 3 produces two possible points 1 and 2 at the same angle ϑ. Therefore, for the same angle, ϑ3 , in L, there are two values of β3 , hence two possible energies. This is the so-called doubleenergy domain. Note however, in this domain, there is a maximum value of for ϑ3 , the value at which β 0 + β 3 becomes tangential to the circle. The corresponding maximum value of ϑ3 is such that: tan ϑ3 |max =

β π <  3 2 γ0 β02 − β32

(2.77)

When β3 > β0 , Eq. (2.77) indicates that there is no real-value for ϑ3 |max , i.e. there is no maximum, and therefore ϑ3 can assume any value within its range. Then the combination vector β 0 + β 3 produces two values for β3 , hence two values of ϑ3 which differ from each other by π. Accordingly two values of ϑ3 are possible.

89

2.5 Einsteinian Kinematics

1

1

β3

1

β3

β3

β0

β0

β0 2 β3

β3

β0

2 β0

2 β3  β0

Figure 2.6 A vector diagram showing the possible combinations of β0 and β3 , marked angles show limits of ϑ3 .

When β3 = β0 , the first solution is such that: sin ϑ3 γ0 (1 + cos ϑ3 )   1 1 − cos2 ϑ3 1 − cos ϑ3 = = γ0 (1 + cos ϑ3 )2 γ02 (1 + cos ϑ3 )

tan ϑ3 =

(2.78)

The second solution for ϑ (point 2 in the corresponding graph in Fig. 2.6) corresponds to ϑ3 = π, which according to Eq. (2.75), produces a value in L for ϑ3 = π2 . The momentum in direction of particle 1 in L is such that: P1 = P3 cos ϑ3 + P4 cos ϑ4

(2.79)

Then, at ϑ3 = π2 , P4 cos ϑ4 = P1 . Since the momentum is zero in L in the direction normal to that of the incident particle, a particle emerging in that direction must have a zero momentum. That is, P3 = 0 when ϑ3 = π2 . The momentum in the direction normal that of particle 1 is such that: P3 sin ϑ3 − P4 sin ϑ4 = 0

(2.80)

Therefore, with ϑ3 = π2 , and P3 = 0, P4 sin ϑ4 = 0, ϑ4 = 0, and P4 = P1 .

Q-value and excitation energy The Q-value, Q, of an interaction is defined as: Q = (M1 + M2 ) − (M3 + M4 ) = (T3 + T4 ) − (T1 + T2 ) = (T3 + T4) − (T1 + T2) (2.81) The proof that Q is the difference between the total kinetic energy of the products and the reactants, in either C r or L, is straightforward, since M is an invariant and

90

Chapter 2 Collision Kinematics

M = E − T = E  − T  . In other words, Q has the same value in C r and in L. In elastic scattering, Q = 0. A positive Q-value indicates that the reaction releases energy, since the total mass of the products is less than that of the reactants, with the difference transformed to energy. When Q is negative, the reaction is endoergeic and some minimum energy must be supplied in order for the reaction to take place; then [−Q] is referred to as the excitation energy of the reaction, or separation energy of the interaction. We will use the square brackets around Q to indicate the excitation or separation energy, and simply Q to refer to the Q-value. This minimum energy is referred to as the threshold energy, and is introduced below.

Forward threshold energy If the Q-value given by Eq. (2.81) is negative, the reaction in endoergeic, and T1 must exceed a threshold value, Tf , called the forward threshold energy. For the reaction to occur, the total kinetic energy of the product particles in C r must be nonnegative, or equivalently: E = E3 + E4 ≥ M3 + M4 . Then, E 2 = κ − M32 + M42 > (M3 + M4 )2 , or equivalently κ > 2M3 (M3 + M4 ). For particle 3 to emerge, M3 has to be greater than zero, i.e. κ > 0 is required for the two-body reaction to take place. An equivalent expression in terms of the Q-value, or the excitation energy, [−Q], is obtained using Eqs (2.69) and (2.81): E 2 = (M12 + M22 + 2M2 E1 ) ≥ (M3 + M4 )2

M12 + M22 + 2M2 T1 + 2M1 M2 ≥ (M3 + M4 )2   M1 1 Q (M3 + M4 )2 − (M1 + M2 )2 = [−Q] 1 + T1 ≥ − 2M2 M2 2M2 Therefore, Tf is given by:  1 M1 Q 2 2 [(M3 + M4 ) − (M1 + M2 ) ] = [−Q] 1 + − Tf = 2M2 M2 2M2

(2.82)

At T1 = Tf , E3 + E4 = M3 + M4 . Then, with M3  = 0 and M4 = 0, particles 3 and 4 have no kinetic energy in C r , hence in L they will both move together with the same velocity.

Back threshold energy At the scattering angle for particle 3 of ϑ3 = π2 , the corresponding kinetic energy of particle 1, T1 , is known as the back threshold energy, since then particle 3 cannot emerge at an angle greater than π2 , when β3 ≤ β0 . With ϑ3 = π2 , E3 = M3 , and E42 = P42 + M42 = P12 + M42 = T12 + 2T1 M1 + M42 , since P3 = 0, P4 sin ϑ4 = 0, ϑ4 = 0 and P4 = P1 . Therefore, from the conservation of energy in L: E1 + M2 = M3 + E4

(2.83)

2.5 Einsteinian Kinematics

91

one obtains: (E1 + M2 − M3 )2 = E42 = T12 + 2T1 M1 + M42 2(M1 + M2 − M3 )T1 + (M1 + M2 − M3 )2 = 2T1 M1 + M42 2T1 (M2 − M3 ) + (M1 + M2 − M3 )2 − M42 = 0 T1 = −

(M1 + M2 − M3 )2 − M42 2(M2 − M3 )

T1 = −

(M1 + M2 − M3 − M4 )(M1 + M2 − M3 + M4 ) 2(M2 − M3 )

T1 = −

Q(M1 + M2 − M3 + M4 + Q − Q) 2(M2 − M3 )

T1 = −

Q[2M1 + 2(M2 − M3 ) − Q] 2(M2 − M3 )

Since for β0 ≥ β3 , the maximum allowed angle for ϑ3 is π2 , the above value for T1 is the back threshold energy, because particle 3 cannot scatter by an angle greater than π2 . This threshold value is expressed as:  (M1 + M2 − M3 )2 − M42 Q M1 Tb = − − = [−Q] 1 + 2(M2 − M3 ) M2 − M3 2(M2 − M3 ) (2.84)

Reaction possibilities The forward and back threshold values, Eqs (2.82) and (2.84), respectively, define the threshold energies of the 2(1,3)4 reaction, so that if: T1 < Tf : reaction is energetically forbidden. Tf < T1 < Tb : ϑ3 < π2 and two distinct energies are observed at the same angle ϑ3 , corresponding to the two signs of the quadratic solution of Eq. (2.70). T1 > Tb : the energy of particle 3 is determined by Eq. (2.70), with the positive sign of its quadratic solution corresponding to ϑ3 < π2 , and the negative sign to ϑ3 > π2 . For elastic scattering, Q = 0, and the threshold energies are zero. The two energies of the quadratic solution of Eq. (2.70) correspond then to forward scattering ϑ3 < π2 , and backscattering ϑ3 > π2 .

Algorithm 1 Given all the rest masses and T1 , one can find T3 at all angles in C r , with 1 ≤ cos ϑ3 ≤ 1, using Algorithm 1. Then the corresponding values at the permissible range in L can be determined.

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Chapter 2 Collision Kinematics

Algorithm 1

Relativistic kinematics of a two-body interaction: 2(1,3)4

Require: Mi ≥ 0, where i = 1–4 Require: T1 > 0 Require: P2 = 0 Ensure: Ei ≥ Mi , where i = 3, 4 Ensure: Ei ≥ Mi , where i = 3, 4 Ensure: P3 = P4 Ensure: −1 ≤ cos ϑi ≤ 1, where i = 3, 4 Ensure: −1 ≤ cos ϑi ≤ 1, where i = 3, 4 1: E1 = T1 + M1 E2 = M2  P1 = T12 + 2T1 M1 P2 = 0 2: E 2 = M12 + 2M2 E1 + M22

3: E3 =

E 2 + M32 − M42 2E

4: P3 =

E 2 − M32 + M42 2E

E4 =

 E32 − M32

P4 =

 E42 − M42

5: Transfer back to L: β0 =

P1 E1 + M2

E1 + M2 E

γ0 =

6: for all ϑ3 such that −1 ≤ cos ϑ3 ≤ 1 do 7: E3 = γ0 [E3 + β0 P3 cos ϑ3 ] 8: P3 =

E4 = γ0 [E4 + β0 P4 cos(ϑ3 + π)]

 E32 − M32

P4 =

 E42 − M42

9: cos ϑ3 = 10: end for

γ0  [P3 cos ϑ3 + β0 E3 ] P3

cos ϑ4 =

γ0  [P4 cos (ϑ3 + π) + β0 E4 ] P4

93

2.5 Einsteinian Kinematics

2.5.2 Analysis using invariants Kinematical analysis can be considerably simplified using the concept of invariants3 . An invariant quantity maintains the same value in all frames of reference, but is not conserved. For instance, mass is an invariant, but is not conserved. In geometry, the square of the Euclidean distance ( r 2 ) is an invariant, and we have shown in Section 2.3, that the proper time (τ 2 ) in the space–time domain and the rest-mass energy in the four-dimensional space of mass and energy are invariants. The invariants are most readily defined if the following four-component energy–momentum vectors are introduced: ⎧ ⎪ ⎨

⎫ E ⎪ ⎬ P sin ϑ cos ϕ μ P = ⎪ ⎩ P sin ϑ sin ϕ ⎪ ⎭ P cos ϑ

⎧ ⎪ ⎨

⎫ E ⎪ ⎬ −P sin ϑ cos ϕ Pμ = ⎪ ⎩ −P sin ϑ sin ϕ ⎪ ⎭ −P cos ϑ

(2.85)

where the superscript μ is used to indicate the so-called contravariant components of the vector, while the subscript μ designates the covariant vector. Now the scalar (inner product) of the two vectors gives the value of: P 2 = (P μ ){Pμ } = E 2 − P 2 = M 2

(2.86)

where the parentheses indicate the transpose of the vector. This is an invariant, known as the Lorentz invariant. Note that this invariant is equal to zero when dealing with photons, since a photon has a zero rest mass. For the 2(1, 3)4 interaction, 16 invariants can be defined in principle, relating a particle to itself and to the three other particles. The four self-to-self invariants replicate the Lorentz invariant of Eq. (2.86) for each particle, and hence relate a particle’s energy to its momentum. The invariants are commutive, i.e. the invariant relating particle i to particle j is the same as that relating the latter particle to the former. Therefore, there are only six particle-to-particle invariants. Since the reactant particles 1 and 2 are independent from each other, one can only speak of their total energy–momentum vector, which provides the invariant s1,2 . Conservation of energy and momentum necessitates that s1,2 = s3,4 , i.e. the values of P μ and Pμ before and after the interaction must be equal. Therefore, μ

μ

μ

μ

P1 + P2 = P3 + P4

Pμ1 + Pμ2 = Pμ3 + Pμ4 μ

μ

μ

μ

s1,2 = (P1 + P2 ){Pμ1 + Pμ2 } = s3,4 = (P3 + P4 ){Pμ3 + Pμ4 } = s

3 In

this analysis, use was made of the lecture notes of W. von Schlippe, Relativistic kinematics of particle interactions, posted on http://www.phys.spbu.ru/Library/Lectures/Stud/Schlippe/Data/Kin_Rel/kin_rel.pdf, March 2002.

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Chapter 2 Collision Kinematics

This s invariant in L and C r (recall that two-body interactions are coplanar, i.e. all have the same azimuthal angle ϕ) can be expressed as: s = s1,2 = (E1 + E2 )2 − P12 − P22 − 2P1 P2 cos(ϑ2 − ϑ1 ) = s3,4 = (E3 + E4 )2 − P32 − P42 − 2P3 P4 cos(ϑ4 − ϑ3 )  = (E1 + E2 )2 − P12 − P22 − 2P1 P2 cos(ϑ2 − ϑ1 ) = s1,2  = (E3 + E4 )2 − P32 − P42 − 2P3 P4 cos(ϑ4 − ϑ3 ) = s3,4

(2.87)

Since in C r the total momentum is equal to naught, P1 = −P2 , ϑ2 − ϑ1 = π, P3 = −P4 , ϑ4 − ϑ3 = −π, then s = (E1 + E2 )2 = (E3 + E4 )2 . That is, s = E 2 , where E is the energy of C r ; same as Eq. (2.55). The threshold energy for an interaction can be directly obtained by equating s1,2 to s3,4 in L, with particles 3 and 4 having no momentum.This leads directly to:  Q2 (M3 + M4 )2 − (M12 + M22 ) M1 Ef = + = M1 − Q 1 + (2.88) 2M2 M2 2M2 with Ef being the threshold energy when particle 2 is at rest. Obviously when M1 = M3 and M2 = M4 , i.e. in the case of elastic scattering, Ef = M1 , i.e. Tf = Ef − M1 = 0. Another set of invariants can be established by relating a reactant particle to a product particle. Since the change in momentum and energy between a pair of particles in the interaction must be taken up by the other pair in this two-body interaction, one obtains: μ

μ

μ

μ

P1 − P3 = P2 − P4

t1,3 =

μ (P1

Pμ1 − Pμ3 μ − P3 ){Pμ1 − Pμ3 }

= Pμ2 − Pμ4 μ

μ

= t2,4 = (P2 − P4 ){Pμ2 − Pμ4 } = t

(2.89)

and μ

μ

μ

μ

P1 − P4 = P2 − P3

u1,4 =

μ (P1

Pμ1 − Pμ4 μ − P4 ){Pμ1 − Pμ3 }

= Pμ2 − Pμ3 μ

μ

= u2,3 = (P2 − P3 ){Pμ2 − Pμ4 } = u

(2.90)

The invariants of Eqs (2.89) and (2.90) can be expressed in L and C r as: t = t1,3 = (E1 − E3 )2 − P12 − P32 + 2P1 P3 cos(ϑ3 − ϑ1 ) = t2,4 = (E2 − E4 )2 − P22 − P42 + 2P2 P4 cos(ϑ4 − ϑ2 )  = (E1 − E3 )2 − P12 − P32 + 2P1 P3 cos(ϑ3 − ϑ1 ) = t1,3  = (E2 − E4 )2 − P22 − P42 + 2P2 P4 cos(ϑ4 − ϑ2 ) = t2,4

(2.91)

95

2.5 Einsteinian Kinematics

and u = u1,4 = (E1 − E4 )2 − P12 − P42 + 2P1 P4 cos(ϑ4 − ϑ1 ) = u2,3 = (E2 − E3 )2 − P22 − P32 + 2P2 P3 cos(ϑ3 − ϑ2 )  = u1,4 = (E1 − E4 )2 − P12 − P42 + 2P1 P4 cos(ϑ4 − ϑ1 )  = u2,3 = (E2 − E3 )2 − P22 − P32 + 2P2 P3 cos(ϑ3 − ϑ2 )

(2.92)

The invariants of Eqs (2.87), (2.91) and (2.92) for this two-body interaction are named after Mandelstam (1958) who originally introduced them. They always add to a constant, such that: s + t + u = M12 + M22 + M32 + M42

(2.93)

Therefore, only two of the invariants are independent. These variables facilitate relating the momentum and energy of interacting particles. For instances, in L where θ1 = 0, P2 = 0, and E2 = M2 , the invariant t gives: (E1 − E3 )2 − P12 − P32 + 2P1 P3 cos ϑ3 = (M2 − E4 )2 − P42 Total energy conservation requires E1 + M2 = E3 + E4 , then eliminating E4 from the above equation and making use of the fact that M 2 = E 2 − P 2 results in: M12 + M32 − 2(E1 E3 − P1 P3 cos ϑ3 ) = M42 − M22 − 2M2 (E1 − E3 )

(2.94)

From this equation one can directly determine the E and P of the outgoing particle 3 in terms of the incoming particle 1, without having to perform vectorial momentum balance or transformation from one frame reference to another.

Algorithm 2 Algorithm 2 gives a summary of equations that can be used to determine the energies and momenta of all particles in a two-body interaction, given E1 and P1 . Calculations should be carried over a range of cos ϑ3 values within [−1, 1] in C r , from which the range at various scattering angles, cos ϑ3 , in L can be found.

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Chapter 2 Collision Kinematics

Algorithm 2

Invariant-based kinematics of a two-body interaction: 2(1,3)4

Require: Mi ≥ 0, where i = 1–4 Require: T1 > 0 Require: P2 = 0 Ensure: Ei ≥ Mi , where i = 3, 4 Ensure: Ei ≥ Mi , where i = 3, 4 Ensure: −1 ≤ cos ϑi ≤ 1, where i = 3, 4 Ensure: −1 ≤ cos ϑi ≤ 1, where i = 3, 4 1: s = M12 + M22 + 2M2 (T1 + M1 ) 2: In C r : E1 =

s + M12 − M22 √ 2 s

E2 =

s + M22 − M12 √ 2 s

s + M32 − M42 s + M42 − M32 E4 = √ √ 2 s 2 s 1  P1 = P2 = √ [s − (M1 − M2 )2 ][s − (M1 + M2 )2 ] 2 s 1  P3 = P4 = √ [s − (M3 − M4 )2 ][s − (M3 + M4 )2 ] 2 s E3 =

3: for all ϑ3 such that −1 ≤ cos ϑ3 ≤ 1 do 4: t = M12 + M32 − 2(E1 E3 − P1 P3 cos θ3 ) u = M12 + M42 − 2(E1 E4 − P1 P4 cos θ4 ) 5: In L: E1 =

s − M12 − M22 2M2

E2 = M2

−u + M22 + M32 −t + M22 + M42 E4 = 2M2 2M2 1  P3 = [−u + (M2 − M3 )2 ][−u + (M2 + M3 )2 ] 2M2 1  [−u + (M2 − M4 )2 ][−u + (M2 − M4 )2 ] P4 = 2M2

E3 =

cos ϑ3 = 6: end for

t − M12 − M32 + 2E1 E3 2P1 P3

cos ϑ4 =

u − M12 − M42 + 2E1 E4 2P1 P4

97

2.5 Einsteinian Kinematics

A useful relationship between the energy and momentum of particle 3, as well as its scattering angle, can be obtained from the t invariant as: 2P1 P3 cos ϑ3 = [M42 − M12 − M22 − M32 − 2M2 E1 ] + 2(E1 + M2 )E3 = [M42 − M32 − s] + 2(E1 + M2 )E3

(2.95)

Isolating E3 on one side of the equation, squaring both sides, and substituting E32 = P32 + M32 , one obtains a quadratic equation of the form: aP32 + bP3 + c = 0 a = 4[(E1 + M2 )2 − P12 cos2 ϑ3 ] = 4(s + P12 sin2 ϑ3 ) b = −4(s + M32 − M42 )P1 cos ϑ3 c = 4M32 (E1 + M2 )2 − (s + M32 − M42 )2 b2 − 4ac = 16(E1 + M2 )2 {[s − (M3 − M4 )2 ][s − (M3 + M4 )2 ] − 4M32 P12 sin2 ϑ3 } = 64(E1 + M2 )2 {sP32 − M32 P12 sin2 ϑ3 } The solution of the above quadratic equation leads to:

P3 =

 (s + M32 − M42 )P1 cos ϑ3 ± 2(E1 + M2 ) sP32 − M32 P12 sin2 ϑ3 2(s + P12 sin2 ϑ3 )

(2.96)

A physically acceptable solution requires that sP32 ≥ M32 P12 sin ϑ32 . If angles of ϑ3 are permitted, but only the solution with the positive sign is acceptable, since P3 being the magnitude of the momentum should 2 2 2 be positive. Onthe other  hand, if sP3 < M3 P1 , then there is a maximum angle √ sP32 > M32 P12 , all

sP 

ϑ3 |max = sin−1 M3 P31 . Then for each value of ϑ3 > ϑ3 |max , there are two values for P3 , and consequently two values for P4 , corresponding to the two signs of Eq. (2.96). Note that a similar expression can be obtained for P4 by exchanging the subscripts 3 and 4. For non-elastic scattering, one can incorporate the Q-value of the reaction with the help of Eq. (2.88), to obtain: s + M32 − M42 = 2[M2 (E1 − Ef ) + M3 (M1 + M2 − Q)]

= 2[M2 T1 + (Q + M3 )(M1 + M2 ) − M3 Q] − Q 2 (2.97)

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Chapter 2 Collision Kinematics

2.5.3 Non-elastic interactions Interactions in which the product particles are not identical to the reactant particles can be analyzed either through an intermediary composite particle (or a compound nucleus) which subsequently decays, or as a direct interaction. In the latter case, if it is a two-body interaction, the analysis presented in the preceding sections can be directly applied. If more than two particles emerge, one can focus on observing the behavior of one particle and lump the attributes of all other product particles into one entity, to create an equivalent two-body interaction. This so-called inclusive collision is discussed below, followed by the analysis of the kinematics of the creation of the composite particle, and its decay to two or more particles. We will be using the notation of Table 2.1, unless otherwise mentioned.

Inclusive collisions Let us consider the interaction 2(1,3)4, . . . n, in which n − 2 particles are created (3–n), but our interest is in observing only a certain particle, which we will consider to be particle 3. For this interaction, the invariant s of Eq. (2.87) is such that: s = (M1 + M2 )2 + 2M2 T1 = (E3 + E4 + · · · + En )2 ≥ (M3 + M4 + · · · + Mn )2 (2.98) with the inequality made possible, since E ≥ M . From the above one can stipulate that the threshold total and kinetic energy for this interaction are: Ef =

(M3 + M4 + · · · + Mn )2 − (M12 + M22 ) 2M2

Tf = Ef − M1 =

(M3 + M4 + · · · + Mn )2 − (M1 + M2 )2 2M2

(2.99) (2.100)

Obviously if particle 1 carries only this threshold energy, all the created particles will have no kinetic energy in C r (except of course if a particle is massless, e.g. a photon). Therefore, more kinetic energy than Tf is needed for the created massive particles to emerge and be observed in L as individual distinguishable particles; otherwise they will appear as a single entity with a zero velocity. In order to determine the momentum and energy of one of the emerging particles, one can create the equivalent two-body inclusive collision: 2(1,3)b, where all particles 4–n are presented by one state, b, with a mass Mb = M4 + M5 + · · · + Mn . The two-body kinematics of Section 2.5 can then be applied. The process can be repeated for other particles, by lumping all particles other than the one of interest into one equivalent entity.

99

2.5 Einsteinian Kinematics

Formation of composite particle The energy of projectile particle, 1, and the target, 2, can be combined to form a composite particle a. Considering the target to be at rest, one can state that:  E = (M1 + M2 )2 + 2M2 T1 = Ta + Ma (2.101) where use was made of Eq. (2.69). Even if particle a does not acquire any kinetic energy, some kinetic energy must be supplied, in this case by particles 1, in order for the reaction to take place. From Eq. (2.101), with Ta = 0, the threshold energy, Tf , is then:  Ma2 − (M1 + M2 )2 M1 Q = [−Q] 1 + − (2.102) Tf = 2M2 M2 2M2 where use was made of Eq. (2.82) to arrive at the right-hand side of the equation. It is, therefore, obvious that non-elastic collisions require Ma > M1 + M2 , since an equal value will occur only if Q = 0, at which point no excitation of particle a takes place. This interaction can also be analyzed in L: E1 + M2 = Ea P1 = Pa Momentum balance forces particle a to move in the same direction as particle 1, since the momentum in any other direction is zero, and there is only one particle remaining after the interaction.

Two-body decay A composite particle, a, formed by the interaction of particles 1 and 2, can carry some kinetic energy, if T1 > Tf , with particle 2 being at rest and Tf given by Eq. (2.102). If this composite particle further decays, to say particles 3 and 4, it will then decay “on the fly’’. Let the total energy of the composite particle a be Ea (=E3 + E4 ) and its momentum, P a , along a direction defined by an angle, ϑa , with respect to particle 1, in the plane of particles, 1, 2, and a. Particle a must also be coplanar with the decay particles 3 and 4. Conservation of momentum in the azimuthal direction between the initial particles 1 and 2 and the final particles 3 and 4, requires that the four particles, along with the intermediate particle a, be coplanar. The invariant s of Eq. (2.87) becomes s = Ma2 , since there is no second reactant body along with a. With C r now coinciding with particle a, E3 + E4 = Ma , and s = E32 + E42 + 2E3 E4 , which after eliminating E4 and using E32 = P32 + M32 gives: 1  2 P = [Ma − (M3 − M4 )2 ][Ma2 − (M3 + M4 )2 ] (2.103) 2Ma

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Chapter 2 Collision Kinematics

From Eq. (2.96): P3 =

 (Ma2 + M32 − M42 )Pa cos ϑ3a ± 2Ea Ma2 P 2 − M32 Pa2 sin2 ϑ3a 2(Ma2 + Pa2 sin2 ϑ3a )

(2.104)

2 . If A physically acceptable solution requires that Ma2 P 2 ≥ M32 Pa2 sin ϑ3a 2 2 2 2 Ma P > M3 Pa , all angles of ϑ3a are satisfied, but only the solution with the positive sign is acceptable, since P3 being the magnitude of the momentum should be 2 2 2 2 positive. On the other  hand,  if Ma P < M3 Pa , then there is a maximum 

Ma P angle ϑ3a |max = sin−1 M . Then for each value of ϑ3a > ϑ3a |max , there are 3 Pa two values for P3 , and consequently two values for P4 , each corresponding to the two signs of Eq. (2.104). Note that:

 P4 = P32 + Pa2 − 2Pa P3 cos ϑ3a as can be easily shown by combining the two momentum balance equations in the direction of flight and perpendicular to it: Pa = P3 cos ϑ3a + P4 sin ϑ4a 0 = P3 sin ϑ3a + P4 sin ϑ4a If the composite particle a does not acquire in L any kinetic energy from particles 1 and 2, i.e. when only the threshold energy for the formation of the composite particle is supplied, then particle a coincides with C r of particles 3 and 4, and with Pa = 0 and Ea = Ma , Eq. (2.104) becomes P3 = P , and using Eq. (2.103): P3 = P4 = P  =

1  2 [Ma − (M3 − M4 )2 ][Ma2 − (M3 + M4 )2 ] 2Ma

(2.105)

where use was made of Eq. (2.103), and only the positive square roots are accepted in order to produce realistic momentum values. It is obvious from Eq. (2.105) that the momenta of particles 3 and 4 have a fixed value determined by the masses of particles a, 3, and 4, and the excitation energy of the reaction. If Ma = M3 + M4 , particles 3 and 4 will have no momentum and would not be observed in L as moving particles. When particle a is at rest, Eq. (2.105), unlike Eq. (2.104), does not show any angular dependence. Therefore, there is no preferred direction for the emergence of particles 3 and 4, i.e. their angular distribution is isotropic. However, since the momentum of the parent particle a is assumed to be zero in L, particles 3 and 4 must emerge in L at two opposite directions, i.e. back-to-back.

2.5 Einsteinian Kinematics

101

Radiative capture If one particle, say, 3, is massless, e.g. a photon, then the other particle 4, is such that M4 = M1 + M2 , as happens in radiative capture, then (keeping in mind that Q = M1 + M2 − Ma , and the value of Q is the same in both L and C r ), Eq. (2.103) gives: P3 |M3 =0

=

E3 |M3 =0

 1 Q 2 2 = [M − M4 ] = [−Q] 1 + 2Ma a 2(M1 + M2 − Q) (2.106)

If particle a is at rest, then Eq. (2.105) gives also the same above values for P3 |M3 =0 = E3 |M3 =0 in L. When Q << M1 + M2 , one gets P3 |M3 =0 = E3 |M3 =0 = [−Q], i.e. the excitation energy of the reaction. Note that this excitation energy may not be released immediately, but may keep the composite particle in an excited quantum state for delayed decay by γ emission. The excitation energy may also be transformed into rotational energy of the composite particle. Further decay If following radiative capture, the residual particle, particle 4 in the above discussion, further disintegrates to say particles 5 and 6, the two-body particle decay process can proceed as above with Ma replaced with Ma − E3 |M3 =0 , i.e. the C r energy is reduced by E3 |M3 =0 . Alternatively, one can consider the combined radiative capture decay and the two-body decay of particle a as a three-body decay process, as discussed below.

Three-body decay While the approach of inclusive collisions discussed above enables one to observe one product particle at a time, by creating a two-body equivalent collision process, it is often useful to examine more than one product particle simultaneously to determine the kinematical limits imposed by the presence of more than two product particles. Let us again examine the decay of a composite particle a, formed by particles 1 and 2, that subsequently decays to particles 3, 4, and 5. We will start the analysis in the C r frame of particle a (called here C r:a ), designated by the primed variables. Transformation back to the L frame can be done using the invariants P1 Ea or by Lorentz transformation with β0 = M and γ0 = M , where the variables are 1 a for particle a in L. We also perform the analysis in C r ’s for the particle pairs (3,4), (4,5) and (3,5), called here C r:3,4 , C r:4,5 , and C r:3,5 , with their variables designated with superscripts, ◦, •, and , respectively. In these so-called Jackson frames, the concerned particles have an equal momentum values and move in opposite directions. The use of these frames enables one to determine the kinematic limits of the invariants.

102

Chapter 2 Collision Kinematics

We will consider here the following set of invariants, expressed in C r:a : s = sa = Ma2 s3 = sa3 = (Ma − E3 )2 − P32 = Ma2 + M32 − 2Ma E3 = s45 = (E4 + E5 )2 − P42 − P52 + 2P4 P5 cos(ϑ4 + ϑ5 ) s4 = sa4 = (Ma − E4 )2 − P42 = Ma2 + M42 − 2Ma E4 = s35 = (E3 + E5 )2 − P32 − P52 + 2P3 P5 cos(ϑ3 + ϑ5 ) s5 = sa5 = (Ma − E5 )2 − P52 = Ma2 + M52 − 2Ma E5 = s34 = (E3 + E4 )2 − P32 − P42 + 2P3 P4 cos(ϑ3 + ϑ4 ) These invariants guarantee that energy and momentum are conserved, though the sa invariant here is trivial as it has no kinematical content. It is easy to show that s3 + s3 + s5 = Ma2 + M32 + M42 + M52 . Therefore, only two of these invariants are sufficient to fully determine the behavior of the interaction. The upper bound of s3 occurs when E3 = 0, as obvious from the definition of sa3 . An analogous situation exists for s4 and s5 . Therefore, s3 ≤ (Ma − M3 )2

s4 ≤ (Ma − M4 )2

s5 ≤ (Ma − M5 )2

(2.107)

The invariants evaluated in C r:4,5 , C r:3,5 , and C r:3,4 are: s3 = (E4• + E5• )2

s4 = (E3 + E5 )2

s5 = (E3◦ + E4◦ )2

Obviously the lower limit for the value of these invariants occurs when the total energy of the relevant particles is equal to their rest-mass energy: s3 ≥ (M4 + M5 )2

s4 ≥ (M3 + M5 )2

s5 ≤ (M3 + M4 )2

(2.108)

Equations (2.107) and (2.108) define the bounds the values of the invariants are confined within. However, not all possible values are attainable, as shown below. Consider C r:4,5 , at which P4• = P5• , and Pa• = P1• . Then: s3 = (Ea• − E3• )2 =

2   Ma2 + Pa•2 − M32 + P3•2

1 [s3 − (Ma − M3 )2 ][s3 − (Ma + M3 )2 ] 4s3 2   M42 + P4•2 + M52 + P5•2 s3 = (E4• + E5• )2 =

P3•2 =

P4•2 = P5•2 =

1 [s3 − (M4 − M5 )2 ][s3 − (M4 + M5 )2 ] 4s3

103

2.5 Einsteinian Kinematics

where the obtained momenta are direct solutions of their preceding equations. Then: 1 E3• = √ (s − s3 − M32 ) 2 s3 1 E4• = √ (s3 + M42 − M52 ) 2 s3 1 E5• = √ (s3 + M52 − M42 ) 2 s3 The invariant s3 evaluated in C r:3,4 does not show any angular dependence. We need to consider another invariant to complete the kinematic description of particles 4 and 5. Consider s4 : s4 = s35 = M32 + M52 + 2[E3• E5• − P3• P5• cos(ϑ3• + ϑ5• )] A fixed value for s3 sets fixed values for P3• , P4• , and P5• as shown above. Then s2 depends only on ϑ3• + ϑ5• , i.e. the angle between particles 4 and 5 in C r:3,4 . This in turn sets an upper bound (s4↑ ) and a lower bound (s4↓ ) on the value of s2 , since the cosine of an angle is limited within the interval [−1, 1]. Therefore, s4↓ = M32 + M52 + 2(E3• E5• − P3• P5• ) ≤ s4 ≤ M32 + M52 + 2(E3• E5• + P3• P5• ) = s4↑ (2.109) At another value of s3 , one would obtain another pair of limits on s4 , and so on. Covering all possible values of s3 within its limits, one obtains the domain of the range of the allowed values of s3 and s4 enclosed with the so-called Dalitz plot, as schematically shown in Fig. 2.7. Let us consider the special case where one of the particles is massless, say particle 3 is a photon, i.e. M3 = 0. Then (M4 + M5 )2 ≤ s3 ≤ Ma2 , and M 2 − (M + M )2

s3 = Ma2 − 2Ma P3 . Therefore, P3 ≤ a 2M4 a 5 , whose magnitude is identical to that obtained using two-body kinematics, Eq. (2.106), if particles 4 and 5 are lumped into a single equivalent particle. This shows that the concept of inclusive scattering is sufficient to determine the kinematic attributes of a certain particle. Therefore, in the case of multibody decay, one can either reduce the system to the equivalent of a two-particle decay or a three-particle decay, with the latter used if the interest is to observe simultaneously two particles.

2.5.4 Non-relativistic approximation When the energy of the projectile particle is much less than its rest-mass energy, i.e. T << M , it is reasonable then to set P 2 ≈ 2MT . By eliminating ϑ4 from the two momentum equations in L, Eqs (2.79) and (2.80), substituting all the P 2

104

Chapter 2 Collision Kinematics

S4 (Ma  M4)2 S4

↓

S4↓ (M3  M5)2

(M4  M5)2

S3 (Ma  M3)2

Figure 2.7 A representative Dalitz plot for three-body decay of permitted values of invariants.

terms by the corresponding 2MT terms, and then using the energy equation in the L, Eq. (2.81), one obtains the following non-relativistic value of T3 :  2 M1 M3 T1  2ϑ +η ± cos (2.110) cos ϑ T3 ≈ 3 3 (M3 + M4 )2 where:

  M1 + M2 M4 Q M4 Q M3 + M4 M4 − M1 + ≈ M4 − M1 + η= M1 M3 T1 M1 M3 T1 (2.111)

The approximation in the above value is valid when the Q-value is less than the Q << 1. Note that when M4 is comparable rest masses of all particles involved, i.e. M to M1 in value, the above approximation is not a very good one as the value of Q becomes dominant, which violates the condition under which the equation was arrived at. The relationship of Eq. (2.110) could have been also obtained from classical conservation of momentum and energy, without considering the rest-mass energy, as discussed in Section 2.6 below. Similar approximations can be obtained for the other kinematic parameters.

2.6 Newtonian Kinematics In interactions involving mass-carrying projectiles, where the speed of the interacting bodies is well below the speed of light, the use of classical Newtonian kinematics is usually adequate. Although one can arrive at the results of this

105

2.6 Newtonian Kinematics

m2, υ2 m4, υ4

m3, υ3

m3, υ3

ϑ3

ϑ4 ϑ3

ϑ4

m2, υ2 m4, υ4

υ0

L

m1, υ1

C

m1, υ1

Figure 2.8 A schematic of a two-body interaction in the lab (L) and the center-ofmass (C) frames of reference, with the state of each body defined by the mass, m, and velocity v.

Newtonian analysis as an approximation of relativistic kinematics, as was done at the end of Section 2.5, analysis using classical dynamics is presented here as it is commonly applied. The presented analysis is based on that given in [19, Section I.B]. We will resort back to the conventional notation for energy T = 12 mv 2 and the momentum being equal to m v , where m here refers to the rest mass and v to velocity. Therefore, the kinematic state of a particle is fully determined by m and v . In the laboratory frame-of-reference, the following conservation equations apply for the 2(1,3)4 interaction of Fig. 2.8: m1 v1 = m3 v3 cos ϑ3 + m4 v4 cos ϑ4

(2.112)

0 = m3 v3 sin ϑ3 − m4 v4 sin ϑ4

(2.113)

1 1 1 m1 v12 = m3 v32 + m4 v42 − Q 2 2 2 Eliminating ϑ4 and v4 from the above equations leads to:  1 m1 m3 T1  ± η + cos2 ϑ3 cos ϑ T3 = m3 v32 = 3 2 (m3 + m4 )2 where:

 m3 + m4 m4 Q η= m4 − m1 + (m1 m3 ) T1

(2.114)

2

(2.115)

(2.116)

Equation (2.115) is identical to the non-relativistic approximation, Eq. (2.110). The threshold limits Tf and Tb are obtained similar to the relativistic analysis of Section 2.5 by considering the problem in the center-of-mass system (C ), in which the total system momentum is zero (see Section 2.2). But due to the discrepancy in the total mass between in the initial state with a rest mass m1 + m2

106

Chapter 2 Collision Kinematics

and the final state with m3 + m4 , finding a C that accommodates both states is not possible, since there are two centers of mass, one before the interaction takes f place, v0i , and the other after the interaction, v0 . Note that in elastic scattering, m1 + m2 = m3 + m4 and this discrepancy does not exist. With the two velocities f v0i and v0 taken in the direction of particle 1, their values are: v0i = f

v0 =

m1 v1 m1 + m 2

(2.117)

m3 v3 cos ϑ3 + m4 v4 cos ϑ4 m1 v1 = m3 + m 4 m3 + m 4

(2.118)

f

where use was made of Eq. (2.112). However, v0i ≈ v0 when:    i f v0 − v0  ≈ 0     1 1   = m1 v1  − m1 + m2 m3 + m4     m1 + m2  Qv0i i  = = v0 1 − m3 + m4  (m3 + m4 )c 2   f  Qv0 f  m3 + m4 = v0  − 1 = m1 + m2 (m1 + m2 )c 2 f

Therefore, for the difference between v0i and v0 to be small, |Q| << (m1 + m2 )c 2 ≈ f (m3 + m4 )c 2 . Then, a velocity v0 ≈ v0i ≈ v0 can be taken for C so that: v0 =

m1 v1 m1 v1 m1 v1 ≈ ≈√ m1 + m 2 m3 + m4 (m1 + m2 )(m3 + m4 )

(2.119)

Conservation of momentum and energy in the zero-total-momentum, C , gives: m1 v 1 + m2 v 2 = m3 v 3 + m4 v 4 = 0 (2.120) T1 + T2 = T3 + T4 − Q Equation (2.120) with T = 12 mv 2 along with Eq. (2.119) leads to: v 1 = v 2 and v 3 = v 4 m12 v12 = m22 v22 and m32 v32 = m42 v42 m1 T1 = m2 T2 and m3 T3 = m4 T4 m1  m3  T1 and T4 = T T2 = m2 m4 3

(2.121)

107

2.6 Newtonian Kinematics

Note also that with: 1 1 m22 T1 T1 = m1 v1 2 = m1 (v1 − v0 )2 = 2 2 (m1 + m2 )2 which in Eq. (2.121) leads to: T3

 m2 T1 m4 = +Q m3 + m 4 m1 + m 2

(2.122)

The value of T1 corresponding to T3 = 0 leads to the forward threshold that makes the reaction possible: Tf = [−Q]

m1 + m2 m2

(2.123)

The condition for |v3 | < v0 leads to the second threshold, Tb , since according to Eq. (2.3) |v  | ≥ v0 : v3 2 = v32 + v02 − 2v3 v0 cos ϑ3

(2.124)

Therefore, the condition for v3 < v0 is cos ϑ3 > 2vv30 ≥ 0. If the condition is satisfied, particle 3 cannot emerge at an angle greater than π2 . The T1 energy corresponding to the v3 < v0 condition, Tb , is such that: m3 v32 = m3 v02 T3 =

(m1 v1 )2 1 m3 2 (m1 + m2 )(m3 + m4 )  m2 m3 m1 Tb m4 Tb +Q = m3 + m 4 m1 + m2 (m1 + m2 )(m3 + m4 )

where use was made of Eqs (2.119) and (2.122). The above leads to: Tb = −

m2 m4 Tf m4 (m1 + m2 )[−Q] = m4 m2 − m1 m3 m2 m4 − m1 m3

(2.125)

It can be easily shown that the total kinetic energy in C is: T1 + T2 =

1 m1 m2 2 1 v = μ1,2 v12 2 m1 + m 2 1 2

(2.126)

where μ1,2 is the reduced mass of particles 1 and 2. By analogy one can state that: T3 + T4 =

1 m3 m4 2 1 m1 m2 2 v¯ ≈ v¯ 2 m3 + m 3 2 m1 + m 2

(2.127)

108

Chapter 2 Collision Kinematics

with the value of v¯ obtained from the non-relativistic conservation of energy, T1 + T2 = T3 + T4 − Q, so that: 1 m1 m2 2 1 m1 m2 2 v = v¯ − Q 2 m1 + m 2 1 2 m1 + m2

(2.128)

One can then define a mass m¯ 2 such that: m¯ 2 = m2

v¯ v1

(2.129)

This can be seen as the modified mass of particle 2 that will make the interaction equivalent to elastic scattering. Eq. (2.129) in Eq. (2.128) gives:  Q (m1 + m2 ) 2 2 (2.130) m¯ 2 = m2 1 + T1 m2 Then Eq. (2.122) can be rewritten as: T3 =

m¯ 22 m2 m4 T1 (m1 + m2 )(m3 + m4 ) m22

(2.131)

Using the vector velocity relationship of Eq. (2.3), one has: v32 = v32 + v02 − 2v3 v0 cos (π − ϑ3 ) 1 T3 = T3 + m3 v02 + v3 m3 v0 cos ϑ3 2 1 m1 v1   m12 v12 T3 = T3 + m3 + 2T3 m3 cos ϑ3 2 (m1 + m2 )2 (m1 + m2 )

(2.132)

where use is made of Eq. (2.119). Now with aid of Eq. (2.122) one obtains: T3 =

T1 (m1 + m2 )(m3 + m4 )  ! " m¯ 22 m3 + m4 m¯ 2 m1 m2 m3 m4 (m3 + m4 ) × m2 m4 2 + m1 m3 +2 cos ϑ3 m1 + m2 m2 m1 + m2 m2 (2.133)

2.7 Specific Interactions We will apply the relationships developed in Sections 2.5 and 2.6 to the various two-body interactions described in Section 1.7. We will use Algorithms 1

109

2.7 Specific Interactions

and 2 for relativistic analysis, and Eq. (2.133) for the Newtonian analysis. For neutrons, Newtonian kinematics is usually sufficient, since in most situations the speed of a neutron does not approach that of the speed of light. Nevertheless, it is a good practice to employ relativistic kinematics, as it will lead to the same results as those of classical kinematics while conserving both mass and kinetic energy via the use of the total energy.

2.7.1 Elastic scattering In elastic scattering, one has the two body interaction 2(1,1)2, i.e. the projectile and the observed particles, 1 and 3 have the same rest mass, and the target is not altered. Therefore, M1 = M3 , M2 = M4 . Then, the Mandelstam variables, Eqs (2.87)–(2.92), add to the constant value s + t + u = 2(M12 + M22 ) (see Eq. (2.93)). Momentum conservation in C r requires that P1 = P2 and P3 = P4 = P  , ϑ2 = ϑ1 + π and ϑ4 = π − ϑ3 . Therefore, the s invariant of Eq. (2.87) applied in C r with E  = M + P  , results in P12 = P32 and P22 = P42 . Consequently, E3 = E1 and E4 = E3 . Then, with ϑ1 = 0 as reference, the t invariant of Eq. (2.91) is:  t = t13 = −2P 2 (1 − cos ϑ3 ) = t24 = 2M2 (M2 − E4 ) = −2M2 T4

(2.134)

Therefore, 0 ≤ −t ≤ 4P 2 , which bounds the values of t. As such, in elastic scattering, t is always negative, except at cos ϑ3 = 0, where it is zero. The relationship between E3 and E1 as a function of ϑ3 can be directly obtained by solving Eq. (2.94) after substituting P32 = E32 − M32 . The result is a quadratic equation that will produce two values, one corresponding to ϑ3 and the other to ϑ3 + π.

Neutron elastic scattering Relativistic Using the notation of Table 2.1, with the neutron being particle 1 and the nucleus is the target, elastic scattering requires: M3 = M1 and M2 = M4 , i.e. M1 + M2 = M3 + M4 , and as such Q = 0. For simplification, we will express all masses in terms of the mass of the neutron, setting M1 = M3 = 1 and M2 = M4 = A, where A is the mass number of the target nucleus (which when multiplied by the neutron’s rest-mass energy gives the mass of the target nucleus, assuming that the neutron and the proton are equal in mass). Now we will follow the procedure of Algorithm 1 and supplementary relationships from Section 2.5:  P1 = T12 + 2T1

E 2 = (1 + A)2 + 2AT1

κ = E 2 + 1 − A2 = 2[(1 + A) + AT1 ] The minimum value of κ occurs when A = 1 (with hydrogen being the smallest target nucleus). Therefore, κ|min = 2(2 + T1 ) > 0, even for T1 = 0.

110

Chapter 2 Collision Kinematics

The energy of particle 3 in C r is then: κ (1 + A) + AT1 = 2E (1 + A)2 + 2AT1   The minimum value of E3 at A = 1 is E3 |min = 1 + T21 ≥ 1. Therefore, the condition E3 > M3 is always satisfied in this interaction, except when T1 = 0, then T3 = 0. Now, we can proceed with the analysis to obtain:   T12 + 2T1 P  = P3 = E32 − M32 = A (1 + A)2 + 2AT1   T12 + 2T1 P  β3 =  = A (2.135) E3 (1 + A) + AT1  T12 + 2T1 P1 = β0 = E1 + M2 (1 + A) + T1 E3 =

1 (1 + A) + T1 γ0 =  = (1 + A)2 + 2AT1 1 − β02  A A(1 + A)2 + 2AT1 sin ϑ3 β3 sin ϑ3 = tan ϑ3 = γ0 (β0 + β3 cos ϑ3 ) [(1 + A) + AT1 ] + A[A(1 + A) + T1 ] cos ϑ3 E3 = γ0 (E3 + β0 P3 cos ϑ3 ) (1 + A)2 (1 + T1 ) + AT12 + AT1 (2 + T1 ) cos ϑ3 (1 + A)2 + 2AT1

 (1 + A2 ) + AT1 + A(2 + T1 ) cos ϑ3 T3 = E3 − M3 = T1 (1 + A)2 + 2AT1 =

(2.136)

Note that β3 > β0 when A2 > 1. Then, there are always two possible values for ϑ3 , as Fig. 2.6 shows, i.e. for all target nuclides except for hydrogen (A = 1). For the latter case, β3 = β0 , and one of the values of ϑ3 is equal to π. The minimum possible value for T3 is when cos ϑ3 = −1 for A = 1, i.e. T1 |min = 0. That is, T3 and accordingly E3 are never less than zero, as physically expected. With T3 determined, then P3 and cos ϑ3 can be evaluated. Recall thought that all the above values are determined in units of the neutron’s rest mass. In elastic scattering (Q = 0), the back and forward threshold energies are Tb = 0 and Tf = 0, as Eqs (2.84) and (2.82), respectively, show. Therefore, the

111

2.7 Specific Interactions

elastic scattering interaction is always possible and there are no restrictions on the value that ϑ3 can assume. However, when A = 1, i.e. when a neutron elastically scatters by a hydrogen nucleus (a proton), T3 = 12 T1 [1 + cos ϑ3 ]. Consequently, T3 = 0, when cos ϑ3 = −1. Also, for A = 1, β0 = β3 . Therefore, tan ϑ3 = ∞, i.e. the maximum angle of scattering of a neutron with a proton in L is ϑ3 = π2 , at which angle the neutron loses all its energy. Backscattering of neutrons with protons, in a single collision, is, therefore, forbidden. Non-relativistic approximation When T3 << 1, the non-relativistic approximation of Eq. (2.136) leads to:

T3 =

T1 [(1 + A2 ) + 2A cos ϑ3 ] (1 + A2 )

(2.137)

In practice, the above expression is quite valid for neutron scattering, since the rest-mass energy of a neutron is about 1 GeV, while the maximum kinetic energy observed of neutrons produced in reactors or by neutron generators is about 15 MeV. Therefore, the assumption T1 << M1 is easily satisfied. Newtonian Resorting to Newtonian kinematics, Eq. (2.133), when applied to elastic scattering, i.e. with m1 = m3 , m¯ 2 = m2 = m4 and A = mm12 , leads to:

T3 =

T1 [A2 + 1 ± 2A cos ϑ3 ] (1 + A)2

(2.138)

This equation is identical to Eq. (2.137), except for the ±sign, with the positive sign corresponding to ϑ3 < π2 , and the negative sign to ϑ3 > π2 . Another useful formulation found in nuclear engineering textbooks, such as [20], is: T3 =

T1 [(1 + α) ± (1 − α) cos ϑ3 ] 2

(2.139)

 2 where α = A−1 A+1 . This equation, and its counterparts, show that the minimum energy a neutron can carry after a single collision is αT1 , and a neutron can lose its energy in a single collision only if α = 0 (A = 1), i.e. when colliding with a proton (a hydrogen nucleus). The total energy of the particles in C must remain equal to that of C , i.e., according to Eq. (2.126), 1 m1 m2 2 1 1 v1 = m3 v32 + m4 v42 2 m1 + m 2 2 2 2 Given that m1 = m3 , m2 = m4 , A = m m1 , and that momentum balance in C requires   that m3 v3 = m4 v4 , and with the aid of Eqs (2.23) and (2.10), one can show that

112

Chapter 2 Collision Kinematics

the angle ϑ (in C ) and ϑ in (L) are related by the relationship: cos ϑ3 = 

A cos ϑ3 + 1 A2 + 1 + 2A cos ϑ3

One can also show that:   1 cos ϑ32 − 1 + cos ϑ3 cos ϑ32 − 1 + A2 cos ϑ3 = A

(2.140)

(2.141)

Equations (2.138) and (2.141) can be combined into the following single equation: ! "2  cos ϑ3 ± cos2 ϑ3 + A2 − 1 (2.142) T3 = T1 (A + 1) It is interesting to notice that when A = 1, one of the solutions always result in T3 = 0, indicating that no scattering angle, ϑ3 , is allowed to be greater than π 2 , when a neutron is elastically scattered by a proton, as indicated earlier when discussing the forward threshold energy.

Compton scattering In the two-body interaction 2(1,3)4, if M1 = M3 = 0 and M2 = M4 = Me , where Me is the rest mass of the electron, then one has Compton scattering: e− + γ → e− + γ. From the kinematics point of view, this is an elastic scattering, since the reactants and the products are identical. However, as far as the atom is concerned, its electronic structure is altered by this interaction. Since we are concerned here with kinematics, this interaction is classified as two-body elastic scattering. Traditionally, Compton scattering kinematics are arrived at using conservation of momentum and energy. We will use here the invariants as they are quite convenient. We have for the t invariant: t = t13 = −2E1 E3 (1 − cos ϑ3 ) = t24 = 2Me2 − 2Me E4 Conservation of energy in L requires E1 + Me = E3 + E4 , since in Compton scattering the target electron is assumed to be at rest and free. Therefore, it is straightforward to show that: E3 = P3 =

E1 Me Me + E1 (1 − cos ϑ3 )

(2.143)

which is identical to the well-known Compton scattering relationship (keeping in mind that P3 = E3 = T3 ). It is obvious that a photon cannot lose all its energy 1 Me in one collision with an electron, since E3 |min = MEe + 2E1 . The reader can easily

113

2.7 Specific Interactions

show that in C r (where the total momentum is zero), there is no change in electron energy or momentum before and after scattering. In other words, no matter what the angle of scattering is, the energy and momentum of the two bodies remain constant. As such, one can conclude that Compton scattering in C r is isotropic. 2 2 It interesting to note that for Compton √ scattering, s ≥ Me , t ≤ 0; u ≤ Me , and 4 su ≥ Me . That is, the energy of C r (= s) is greater than the electron’s rest-mass energy of 511 keV. Note that, since photons move at the speed of light, classical treatment is not applicable.

Moller and Bhabha scattering Moller scattering is the e− + e− → e− + e− interaction, while Bhabha scattering is e+ + e− → e+ + e− . Both cases lead to the two-body elastic interaction 2(1,3)4 which has M1 = M2 = M3 = M4 = Me . In both interactions, the target particle is an electron at rest. To determine the angle in L at which particle 3, i.e. the scattered electron/positron, emerges at a certain magnitude of energy, E3  (hence momentum, P3 = E32 − Me2 ), one can make use of the t invariant: t = t13 = 2Me2 − 2E1 E3 + 2P1 P3 cos ϑ3 = t24 = 2Me2 − 2Me E4 = 2Me2 − 2Me (E1 + Me − E3 ) = −2Me (E1 − E3 ) Therefore, E3 (E1 + Me ) − Me2 − Me E1 (2.144) P1 P3 where use was made of the conservation of energy, E1 + M2 = E3 + E4 . Similarly for the u invariant: cos ϑ3 =

u = u14 = 2Me2 − 2E1 E4 + 2P1 P4 cos ϑ4 = u23 = 2Me2 − 2Me E3 = 2Me2 − 2Me (E1 + Me − E4 ) = −2Me (E1 − E4 ) Hence, for a given E4 , and subsequently P4 , particle 4 will emerge at the angle determined by: cos ϑ4 =

E4 (E1 + Me ) − Me2 − Me E1 P1 P4

(2.145)

Both these interactions require that s ≥ 4Me2 , a C r energy greater than twice the electron’s rest-mass energy. Also not that t ≤ 0 and u ≤ 0 in these interactions.

Mott scattering This is the interaction of a light charged particle, typically an electron, with the atomic nucleus. It then reasonable in a 2(1,3)4 interaction to assume that

114

Chapter 2 Collision Kinematics

M2 >> M1 , E1 , P1 . The interaction can then be seen to take place in L and the recoil of the target is negligible. In this elastic scattering, M1 = M3 and M2 = M4 . Then the change in the electron momentum is quite small and the momentum transfer can be determined by: |P3 − P1 |2 = t13 = −P12 − P32 + 2P1 P3 cos ϑ3 ≈ −2P 2 (1 − cos ϑ3 ) = −4P 2 sin2

ϑ3 2

(2.146)

2.7.2 Inelastic scattering Neutron inelastic scattering Relativistic This (n,n γ) interaction is a two-body interaction of a neutron with a target nucleus that leads to three products: a neutron, a photon, and a recoiled (and may be excited) nucleus. The interaction is also endoergeic, i.e. it has a negative Q-value, and as such can only take place if the incident neutron has an energy greater than the magnitude of the Q-value. On the other hand, the mass of the reactants and the products does not change, i.e. M1 = M3 , and M2 = M4 in a 2(1,3)4 interaction, when the neutron is the observed particle (particle of interest). Then according to Eq. (2.81), Q = 0, if calculated as the difference in rest-mass energy, which contradicts the fact that Q is negative in this interaction. In order to overcome this difficulty, we will assume that a fictitious rest-mass energy of [−Q] is added to the target, so that M4 = M2 + [−Q], where [−Q] is the internal (excitation) energy absorbed by the target nucleus. We will then let M4 decay on the fly to a photon and a nucleus of mass M2 . Let us first consider the kinematics of the emerging neutron. Threshold energy With M1 = M3 = Mn , where Mn is the rest-mass energy of the neutron, then according to Eq. (2.88) the threshold energy for this interaction is:

1 [(Mn + M2 + [−Q])2 − (Mn2 + M22 )] 2M2  Mn [−Q] + = Mn + [−Q] 1 + M2 2M2   1 Mn [−Q] ≈ [−Q] 1 + + Tf = [−Q] 1 + M2 2M2 A Ef =

(2.147) (2.148)

2 where A = M Mn . Since Q is negative in this interaction, Tf has to be slightly greater than [−Q]. As indicated in Section 1.7.3, the threshold energy is on the order of a few MeV for light nuclei and only 100s of keV for heavy nuclei. Therefore, Q is on the order of magnitude of Tf and much smaller than M2 (which is in the GeV range); hence the approximation in Eq. (2.148).

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2.7 Specific Interactions

Neutron energy Now with E1 > Ef , we can follow either Algorithm 1 or 2 to determine the kinematic properties of the emerging neutron. It is obvious that, unlike in elastic scattering in C r , E3  = E1 and P3 = P1 , since M4 = M2 . Gamma energy Let particle 4, with M4 = M2 + [−Q], decays to a massless photon such that M5 = 0, a residual nucleus with M6 = M2 . Then in the C r of 4, Eq. (2.103) gives:

Pγ = P  = Eγ =

1 2M4 − [−Q] [M42 − M22 ] = [−Q] ≈ [−Q] 2M4 2M4

(2.149)

Using Eq. (2.104), one can determine that the photon’s momentum in L is: Pγ =

M4 [−Q](P4 cos ϑγ4 + E4 ) [M42 − M22 ](P4 cos ϑγ4 + E4 ) = 2 2 2 2(M4 + P4 sin ϑγ4 ) (M42 + P42 sin2 ϑγ4 )

(2.150)

where ϑγ4 is the angle the emitted photon makes with the direction of particle 4. Given that P is typically in the MeV range and the rest-mass energies are on the order of GeVs, one can state that: Pγ ≈

M4 [−Q] = [−Q] M4

(2.151)

Consequently Eγ = [−Q] is also a very good approximation for the photon energy; recall that Eγ2 = Pγ2 . In this approximation, Pγ has no angular dependence, hence its emission can be assumed to be isotropic. The momentum of the residual particle is then:  [M42 + M22 ]P4 cos ϑ64 ± 2E4 M42 P 2 − M22 P42 sin2 ϑr4 (2.152) P6 = 2(M42 + P42 sin2 ϑr4 ) Newtonian As shown in Section 2.6, an inelastic scattering can be made equivalent to an elastic one, by adopting a “hard’’ mass for the target nucleus, using 2 ¯ m¯2 Eq. (2.130). In this equation setting, A = m m1 , and A = m1 , one obtains:

 [−Q] (1 + A) ¯ 2 = A2 1 − A T1 A

(2.153)

keeping in mind that Q is negative. Now, applying Eq. (2.133), with m1 = m3 , m2 = m4 , the kinetic energy of the inelastically scattered neutron emerging at angle of ϑ3 with respect to the incident neutron in C is given by: T3 =

T1 ¯ 2 + 1 + 2A ¯ cos ϑ3 ] [A (1 + A)2

(2.154)

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Chapter 2 Collision Kinematics

¯

¯ and v1 by A+1 v1 in Eq. (2.138) leads also to Eq. (2.154). This Replacing A by A A+1 reflects in essence that the incident neutron appears to the modified “hardened’’ ¯ nucleus to be approaching by a reduced velocity of A+1 A+1 v1 .The angles of scattering in L and C are related in analogy with Eqs (2.140) and (2.141) by: cos ϑ3 = 

¯ cos ϑ3 + 1 A

¯ 2 + 1 + 2A ¯ cos ϑ3 A   2 ¯ ¯2 A cos ϑ3 = cos ϑ3 − 1 + cos ϑ3 cos ϑ32 − 1 + A

(2.155)

(2.156)

2.7.3 Non-elastic collisions Positron annihilation This interaction kinematically is an inelastic scattering, since the product particles are different from the reactant particles. As a two-body 2(1,3)4 interaction, the reaction e+ + e− → γ + γ, has M1 = M2 = Me and M3 = M4 = 0. Due to the involvement of photons, Newtonian kinematics are not suited here.The invariants in this interaction are such that s ≥ 4Me2 , t ≤ Me2 , u ≤ Me2 .The C r energy, E , is such that E > 2M42 , i.e. ≥ 1.022 MeV. Assuming that the electron is the target and is at rest, then the angle in L at which one of the photons emerges with a certain energy is such that: t = t13 = Me2 − 2E1 E3 + 2P1 P3 cos ϑ3 = t24 = Me2 − 2Me E4 = −2Me (E1 − E3 ) − Me2 where use is made of the fact that E4 = E1 + Me − E3 . Then, taking advantage of the fact that E3 = P3 , E3 = P3 =

Me2 + Me E1 E1 + Me − P1 cos ϑ3

(2.157)

Similar use of the u invariant produces the energy and momentum of the other photon: E4 = P4 =

Me2 + Me E1 E1 + Me − P1 cos ϑ4

(2.158)

If the positron has a zero momentum, or more practically if its kinetic energy is much less than Me = 0.511 MeV,then one can set E1 = M1 . Equations (2.157) and (2.157) show clearly that E3 = E4 = P3 = P4 , when E1 = M1 . Angular dependence then disappears from the expressions for the t and u invariants in L. However, the s invariant when E1 = M1 is s = 4Me2 = 2Me2 − 2Me2 cos(ϑ4 − ϑ3 ). That is, ϑ4 − ϑ3 = π, indicating that the two photons emerge at two opposite angles.

2.7 Specific Interactions

117

Since then photon energy is not angular dependent, photon emission is isotropic and photons are emitted at any two opposite directions. Note that this is the essence of positron emission tomography (known as PET), which measures the coincident emission of 511 keV photons to determine the location of a positron source that causes annihilation as the positron comes to rest in the surrounding medium.

Photoelectric absorption Kinematic analysis of the photoelectric effect is not quite straightforward because it involves a bound electron as a target, but the atom as a whole recoils. The photoelectric effect cannot take place in free space, in the absence of the atom. On the other hand, one cannot precisely define the target of the interaction, and has to rely on the probabilistic arguments of quantum mechanics (see Section 3.5.5). Nevertheless, one can represent the photoelectric effect by the two-body interaction 2(1,3)4, with M1 = 0, the product particle is an electron, M3 = Me , and M4 = M2 − Me + Be , i.e. the target loses an electron in the process and gains some excitation energy, Be . Then the threshold energy for the interaction, according to Eq. (2.88) is: (M2 + Be )2 − M22 B2 = Be + e ≈ Be (2.159) 2M2 2M2 The approximation is made possible by the fact that Be , the electron’s binding energy, is at most in the keV range, while the mass of an atom is at least in the GeV range. It is also reasonable to assume that M2 and M4 are much larger than both Me and E1 . Given this, it is also reasonable to assume that C r coincides with L. Since photoelectric absorption takes place only in the field of the atom, it is not unreasonable to reduce the C r energy by the mass of the target atom, M4 , which includes the atomic field potential, Be . The remaining energy is then assumed by the electron. Therefore, √ E3 ≈ E3 = s − M4 ≈ M2 + E1 − (M2 − Me + Be ) = E1 − Be + Me (2.160)   √ E1 . Keeping in mind that E3 = where s = M22 + 2M2 E1 and s ≈ M2 1 + M 2 M3 + T3 = Me + M3 , the above is the well-known Einstein’s (1905) photoelectric equation4 , in which Be is called the work function and is equal to the binding energy of the atomic electron liberated in the interaction (typically a K-shell electron). Note that under this approximation, E4 = M4 + Be , i.e. the momentum given to the residual atom is Be . The momentum of the electron is then: Ef =

P32 = (E1 − Be + Me )2 − Me2 = E12 + 2Me E1 + Be2 − 2Be (Me + E1 ) (2.161) Given that Be << Me , since the maximum value of Be is for the K shell is at most on the order of 100 keV (see Section 1.2.3) Me = 511 keV, then P3 > P1 (=E1 ). 4 1921

Nobel Prize in physics.

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Chapter 2 Collision Kinematics

When E1 >> Be , then P32 = E12 + 2Me E1 , and the momentum given to the recoil atom can be neglected.The momentum balance (P3 sin ϑ3 = P4 sin ϑ4 ) shows that as P4 → 0, sin ϑ3 → 0 and the electron will tend to emerge in the same direction as the incident photon, i.e. with a small scattering angle. Then the residual atom will recoil backwards, since ϑ3 + ϑ4 = 0 given that C r and L almost coincide. When E1 , on the other hand, is close in value to Be , the momentum of the electron becomes almost zero, and ϑ4 → 0. Then, the electron and the residual atom travel in opposite directions, i.e. ϑ3 → π at low energies. As Section 3.5.5, Eq. (3.166) shows, at low photon energy electron emission tends to be in a direction normal to that of the incident photon.

Pair and triplet production This interaction occurs only in the presence of the electric field of an atom or an electron. It can be considered as a three-body interaction: 2(1,3 and 5)4, with M1 = 0, M2 = M4 = M and M3 = M5 = Me , where M is the mass of the target and Me is that of an electron or a positron. However, for the purpose of finding the threshold energy we will combine particles 3 and 5 into one particle of mass equal to 2Me . Then according to Eq. (2.88): Ef =

(2Me + M2 )2 − M22 2Me (Me + M2 ) = 2M2 M2

(2.162)

For pair production in the field of the atom, M2 >> Me , and Ef = 2Me = 1.022 MeV. On the other hand for pair production in the field of the electron, M2 = Me , and Ef = 4M4 = 2.044 MeV.The latter process is the triplet production process, since the target electron also acquires a considerable momentum. Now to examine the kinematical behavior of one of the particles (say the positron) let us consider the two-body inclusive scattering that lumps the other particle, (the electron) into one equivalent particle, i.e. M4 = M2 + Me , with M3 = Me . Given that M1 = 0, one can employ either schemes ofAlgorithms 1 or 2. Since the positron and the electron have an identical mass, once the energy of one particle (the positron) is determined, the energy of the other (the electron) is obtained from energy conservation, such that Eγ = E− + E+ , where the subscripts γ, e−, and e+ are used to refer, respectively, to photon, electron, and positron. The same algorithms can be used to determine the polar angle of scattering for the other particle.

Absorption When the incident particle is absorbed in the target, one can assume in the 2(1,3)4 interaction that E3 = 0, i.e. M3 = P3 = 0. The target 4 then must recoil in the same direction as the incident particle, since there is no momentum in any other direction. With the target being at rest, the s invariant dictates that s = M42 . This in turn results in P4 = 0. Now with ϑ4 = 0, P4 can be shown,

2.7 Specific Interactions

119

using an equation analogous to Eq. (2.96), to be equal to P1 . Notice also that E42 = P42 + M42 is consistent with the conservation of energy, E4 = E1 + M2 , when P4 = P1 and M42 = s = M12 + 2M2 E1 + M22 . Since E1 ≥ M1 , then M4 ≥ M1 + M2 . The threshold energy for this interaction, according to Eq. (2.88), is Ef = E1 . If the formed particle is not stable or quantum-mechanically allowable, as it is mostly the case, since M4 is not likely to correspond to a stable particle, then M4 is simply a compound nucleus that is most likely to decay by the emission of a photon or a charged particle. However, at Ef = E1 = 0, a particle may be formed with a mass M4 = M1 + M2 , which is the case with the absorption of thermal neutrons that have practically a zero energy.

Neutron radiative capture Let us consider the two-body interaction 2(1,3)4, in which particle 1 is a neutron with M1 = Mn , a photon is emitted, M3 = 0, and the target nucleus increased in mass by Mn , i.e. M4 = M2 + Mn . Then according to Algorithm 2: 4sP32 = (s − M4 )2 . The momentum of the emitted γ-ray, according to Eq. (2.96) is: P1 cos ϑγ + (E1 + M2 ) (2.163) Pγ = (s − M42 ) 2(s + P12 sin2 ϑ3 ) In the common case of thermal-neutron absorption, T1 = P1 ≈ 0, and Pγ = Eγ is given by:  Q 2 (M1 + M2 ) =Q 1− ≈Q (2.164) Pγ = Eγ = (s − M4 ) 2s 2(M1 + M2 ) Obviously this is an exoergeic interaction in which Q is positive. The lack of dependence of Pγ on the angle of emission indicates that the interaction is isotropic. The approximation is enabled by the fact that Q << M1 , since M1 is about 1 GeV. Indeed, T1 << E1 , since neutron energy does not exceed 20 MeV in practice, and the above approximation is valid for neutron radiative capture at any energy.

Particle emission and transmutation Nuclear interactions involving charged particles result in the transmutation of the nucleus, while those involving neutron emission produce isotopic transformation. In either case, the general kinematics of the interaction is the same. We will consider here, as examples, a number of common interactions.

Alpha particle production Consider the interaction: 10 B(n,α)7 Li. This interaction is used for detecting thermal neutrons. It has two Q-values: 2.792 MeV when 7 Li is left in the ground state,

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Chapter 2 Collision Kinematics

and 2.310 MeV when it is an excited state, with the latter condition much more likely. Since the Q-value is positive, the threshold energy is negative, as Eq. (2.88) indicates. That is, this interaction can take place at any energy. The kinematics of this interaction at any incident neutron energy can be evaluated using Algorithms 1 or 2 for the analogous 2(1,3)4 interaction. We will consider here, however, the case of thermal neutrons, i.e. at T1 = P1 ≈ 0. Using Algorithm 2, along with Eq. (2.96) and the definition of Q in Eq. (2.81), one has: !  "  1 M3 − M4 2 2 2 P3 = P3 = Q(M1 + M2 + M3 + M4 ) 1 − (2.165) 4 M1 + M2 M3 M4 2 2 With M M1 = 10, M1 = 4, M1 = 7, then P3 = 5.09M1 Q. The lack of angular dependence of P3 , is indicative of the isotropy of the emerging particles. For Q = 2.31 MeV, assuming M1 = 1 GeV, P32 = 11.76 × 103 MeV2 , T3 = E3 − M3 = 1.47 MeV. Similarly, T4 = 0.84 MeV. Notice that T3 + T4 = Q, as would be expected in this case, since the incident particle is assumed to have no kinetic energy. The reader can repeat these calculations using more accurate values for mass, and should arrive at a very similar result.

Proton production The interaction 3 He(n,p)3 H is also used in neutron detection, and has a Q-value of 0.764 MeV.The interaction is useful for detecting both slow and fast neutrons, it is most sensitive to the former. However, since the neutron energy does not exceed in practice 20 MeV, and the neutron’s rest-mass energy is about 1 GeV, T1 << M1 , one can set E1 = M1 , when describing this interaction by the 2(1,3)4 twoM3 M4 2 body kinematics. Now consider, M M1 = 3, M1 = 1, M1 = 3, then using Eq. (2.165), P32 = 1.5M1 Q. With Q = 0.764 MeV, and M1 = 1 GeV, P32 = 1.15 × 103 MeV2 , T3 = 0.57 MeV. Similarly, T4 = 0.19 MeV. These values are very close to those obtained with more exact mass values and satisfy the relation: T3 + T4 = Q, as expected.

Neutron production Neutron generators exploit the interactions 2 H(2 H,n)3 He and 3 H(2 H,n)4 He, with the aid of accelerated deuteron beams in the energy range of 100–300 keV. Since the energy of the incident particles are quite small compared to their rest masses, a quick estimate of the momentum of the emitted neutrons can be obtained with the aid of Eq. (2.165). For the 2 H(2 H,n)3 He reaction, the Q-value is 3.26 MeV, which leads to a value of P32 = 4.89 × 103 MeV2 , assuming a 1 GeV neutron mass, which in turn gives a neutron kinetic energy of about 2.4 MeV. For the 3 H(2 H,n)4 He reaction, the Q-value is 17.6 MeV, leading to a neutron kinetic energy of 14 MeV. In both cases, the incident particle energy is much

121

2.7 Specific Interactions

smaller than the Q-value of the interaction, the energy of the produced neutron stays at about the same energy at all angles. Photoneutron production Let us now consider a photoneutron interaction such as 2 H(γ,n)1 H (encountered in CANDU reactors, which utilize heavy water as a moderator), or the 9 Be(γ,n)8 Be reaction. The former reaction has a negative Qvalue of Q = −2.226 MeV, while the latter’s has Q = −1.666 MeV.With M1 = 0, T1 = E1 , Eq. (2.97) gives:

s + M32 − M42 = 2[M2 E1 + (Q + M3 )M2 − M3 Q] − Q 2

(2.166)

Equation (2.96) then gives the neutron’s momentum at various angles. Note that, since the incident photon has no mass, no further approximations are useful, in terms of neglecting some terms in the equations of the relativistic Algorithms 1 and 2. However, one may use Newtonian kinematics to obtain an approximate solution. This can be achieved by setting the speed, v0 of C equal to mp12 ; notice the change to lower-case variables which indicates the use of classical mechanics. 4 Now, with m1 = 0, T3 can be expressed using Eq. (2.122) as: T3 = m3m+m (T1 +Q). 4 Since Q is quite small, then according to Eq. (2.130), one can set m¯2 = m2 , and m2 , = m3 + m4 , with p1 replacing m1 v1 in Eq. (2.132) with the above value for T3 , one obtains: T3 =

=

m4 1 m3 p12 (T1 + Q) + m3 + m4 2 (m3 + m4 )2 √ p1 2m4 m3 (m3 + m4 )(T1 + Q) + cos ϑ3 2 (m3 + m4 ) M4 1 M3 E12 (E1 + Q) + M3 + M4 2 (M3 + M4 )2 √ E1 2M4 M3 (M3 + M4 )(E1 + Q) + cos ϑ3 (M3 + M4 )2

(2.167)

The second term in the right-hand side of the above equation is usually neglected, when E1 is much less than M3 or M4 , which is commonly the case. Then Eq. (2.167) becomes identical to that reported for instance in [5], except for the use of the Ls angle, ϑ3 , instead of the C s angle, ϑ3 . However, according to T 2M

1 3 → 0. Given that T1 is Eq. (2.23) ϑ3 = ϑ3 , when vv0 → 0, i.e. when 2(T1 +Q)M 2 M4 3 usually in the MeV range, and same is Q, while the rest masses are in the GeV order of magnitude, this condition is typically satisfied, unless T1 is very close to |Q| in value but then the probability of this interaction becomes quite small.

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Chapter 2 Collision Kinematics

Fission This is a drastic and catastrophic interaction, as far as the target nucleus is concerned. Nevertheless, we will use kinematical principles to demonstrate some of its features. Let us consider the specific example of the fission of a 235 U nucleus by a thermal neutron to one of its typical fragments, 95 Sr and 139 Xe. Balance of nucleons necessitates that two neutrons be emitted as well. Let us attempt to determine the energy of one of these neutrons using the concept of inclusive collisions, that is lumping all the other products into one mass, M4 , such that M4 = 235 M3 , with M1 = M3 being the neutron’s rest-mass energy, and M2 = 235 M1 , being the mass of the target nucleus. Since the incident neutron energy is essentially zero, one may use Eq. (2.165) to determine the neutron energy, assuming an approximate Q-value of 200 MeV. This will lead to T3 = 182 MeV, with M1 = 1000 MeV, which is obviously an excessively high energy that is not observed in practice. The mistake that was made in the analysis just presented is that the neutron is treated as a direct product of fission. Neutron production is an after-effect of fission emitted promptly by the fission fragments. Therefore, the energy of the emitted neutrons depends on the excitation energy levels of the emitting nucleus, and can vary widely, with a most probable energy of 0.72 MeV for the fission of 235 U. In addition, some heavier fission fragments may also promptly emit more than one neutron, leading to on average the emission of 2.5 neutrons per 235 U fission. The other forms of energy released in fission, γ and β radiation and neutrinos, are also the result of the decay of fission fragments. Nevertheless, applying the kinematics of Eq. (2.165) to the emission of the above fission products, with M3 referring to 96 Sr and M4 to 140 Xe as the precursors for the emitted neutrons, the obtained kinetic energies for these nuclei are respectively, about 119 and 81 MeV; which are reasonable approximations for the kinetic energy of the fission fragments.

Spallation The spallation process involves the collision of a high energy (typically a proton in the GeV range) light particle with a heavy target, shattering the latter into many small particles and leaving behind a lighter remnant nucleus. Therefore, relativistic kinematics must be applied. The kinematics of spallation is typically divided into two separate steps. In the first step, the energy of the projectile is distributed by collisions among the many nucleons of the target nucleus. This process is known as the intra nuclear cascade, and can result in the emission of some nucleons. The target nucleus is left in an excited state and decays by emitting many low energy charged particles and neutrons; in some cases fission takes place.The de-excitation process is often referred to as “evaporation’’. The kinematics of the intra nuclear cascade process is quite complex as it involves the transport of nucleons via many collisions within the confined potential well of the nucleus. A simplified approach is provided by the abrasion–ablation model5 . In this model, it is assumed that the 5 J.

J. Gaimard and K. H. Schmidt, A reexamination of the abrasion–ablation model for the description of the nuclear fragmentation reaction. Nuclear Physics,Vol. A 531, 1991, pp. 709–745.

2.8 Electromagnetic Interactions

123

projectile strips off a portion of the target (abrasion), leaving the rest of the target as an unaffected (“unheated’’) spectator. The “hot’’ sheared-off portion (called the fireball region) decays by the release of nucleons and bigger fragments, while the remaining portion of the target acquires an additional surface area. The additional surface area, accordingly introduces an excitation energy as the nucleons near the new surface lose their neighbors, see the section on Binding energy (page 22). The subsequent decay (ablation) of the remnant nucleus gives rise to the emission of another set of nucleons and fragments. Experimentally, spallation is studied using reverse kinematics; a process in which the target and projectile are reversed, with heavy ions beams ( ≈ 1 GeV) used to bombard a target of a liquid hydrogen (protons).Then the product particles emerge in the forward direction, due to the forward momentum of the incident ion beam. It becomes then practical to detect the emitted charged particles, by directing them using magnets toward certain detectors.

2.8 Electromagnetic Interactions Electromagnetism affects radiation interactions in a number of ways. First, charged particles in motion (projectiles) produce electromagnetic fields. Targets (atomic electrons, nuclei) also produce their own electric fields. Moreover, photons possess the characteristics of electromagnetic waves that can be affected and affect charged particles as targets. Complete analysis using electromagnetic theory is beyond the scope of this book, but it is dealt with in electrodynamics textbooks such as that of [21]. However, simplified analysis is given here to provide a basic appreciation of the fundamentals.

2.8.1 Coulomb scattering Classical elastic Coulomb scattering The presence of the electrostatic field of atomic electrons produces a repulsive or an attractive effect, depending on the electric charge of the incident particle. This prevents the incident particle from having a “head-on’’ collision, with the target nucleus, and keeps the two particles at some distance apart. On the other hand, an incident particle traveling at the proximity of the target can feel its presence. The latter effect is particularly important given the very long (essentially infinite) range of the Coulomb field. In practice, however, there is some distance, beyond which the mutual effect of the Coulomb force between the two particles is negligible. Within the field of impact, the incident particle will be accelerated by the repulsion or attraction force, hence gain energy and momentum. Some energy will also be transferred to the target and the direction of the incident particle will change. Our interest here is, therefore, to determine this deflection (or scattering) angle and the amount of energy lost by the incident particle, hence that gained by the

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Chapter 2 Collision Kinematics

m2, z2

R C

χ

r

m1, z1 b

Figure 2.9 A schematic showing two charged particles approaching each other.

target. Guided by Appendix B of [1], we will determine this angle in C , where the total momentum is zero, by finding the trajectory of a charged particle, of mass m1 and charge ±z1 e, as it approaches a target of mass m2 and charge ±z2 e, where e is the electronic charge, and the ± sign indicates that the charge can be positive or negative. We will consider as usual the target particle to be initially at rest. Figure 2.9 shows a schematic of an incident particle approaching a target at an impact distance b, with distance r + R between them. In the absence of the electrostatic field between the projectile and the target, the projectile would bypass the target at the distance, b. As the field becomes effective, the incident particle 1 will tend to be repulsed or attracted to the target, depending on whether the charges are alike or are opposite. The incident particle will then follow a curved trajectory that we will describe in terms of polar coordinates (r, χ) with the origin at the center of mass, C , of the two particles. Then r will be the distance from the position of particle 1 to C , and χ being some angle measured with respected to an arbitrary axis. Now in C , we can define the following radial and tangential components of the instantaneous velocity, u, and acceleration, a, for particle 1:  2 d2 r dχ ar = 2 − r dt dt   dχ 1 d 2 dχ uχ = r aχ = r dt r dt dt dr ur = dt

(2.168)

(2.169)

The instantaneous velocity under the influence of the Coulomb forces is designated here by u, in order to distinguish it from the velocity v away from the field effect. Since there is no force in the tangential directions, aχ = 0, and subsequently r 2 dχ dt = constant. The value of this constant can be obtained by taking advantage of the fact that dχ dt is the angular velocity around C of the line connecting the two particles. The angular velocity times the moment of inertia is equal to the angular

125

2.8 Electromagnetic Interactions

momentum. Therefore, using Eqs (2.24) and (2.25) with v2 = 0, since the target is assumed to be initially at rest, one has: dχ m12 2 r = μ12 v1 b dt μ12 r2

μ2 dχ = 12 v1 b = B dt m12

(2.170)

where μ12 is the reduced mass of particles 1 and 2, and B is a constant at a given impact parameter (distance) b and incident particle velocities v1 in L. The radial acceleration, ar , is related to the Coulomb force by:   ±z1 z2 e 2 μ12 2 ±z1 z2 e 2 m1 ar = = 4πε0 (r 2 + R 2 ) 4πε0 r 2 m1   ±z1 z2 e 2 μ212 1 ±A ar = = 2 (2.171) 2 4πε0 m1 r r where ε0 is the permittivity of free space ( ≈ 8.85 × 0−12 C2 /Jm), A is a constant for a given projectile and target, and use was made of the fact that the definition of C is such that m1 r = μ12 (r + R). The sign of A is positive if the two charges are of the same type, indicating a repulsive force, and is negative in the case of attraction, i.e. opposite charges. Now, making use of Eq. (2.170): dr dr dχ dr B = = dt dχ dt dχ r 2 d2 r B 2 d2 r B d2 r dχ = = dt 2 r 2 dχ2 dt dχ2 r 4 which from the definition of ar , Eq. (2.168), and its value (2.171) leads to: 1 d2 r ±A 1 − = 2 2 2 r dχ r B

(2.172)

This differential equation has the solution6 : r=

B2 A FB 2 A

cos χ − 1

(2.173)

  dr 1 dy d2 r 1 d2 1 is used to obtain a directly solvable equation: =− 2 , = = y dχ y dχ dχ2 dχ2 y 1 d2 y d2 y ∓A ∓A − 2 , so that: + y = 2 . This differential equation has the solution: y = 2 + F cos (χ − χ0 ), with y dχ2 dχ2 B B F and χ0 being constants of integration. Measuring the angle χ from a direction away from the vertex for an attractive force and toward the vertex for a repulsive force, one can set χ0 = 0 as the initial value of χ (see Fig. 2.11). 1 Transforming back with y = , leads to Eq. (2.173). r

6 The

transformation r =

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Chapter 2 Collision Kinematics

where F is a constant of integration, which we will determine below by energy conservation. Equation (2.173) is an equation of a hyperbola. Recall that a hyperbola is the locus of points whose distances from two fixed points (foci) have a constant difference, and in polar coordinates (r, χ) with the origin at a focus, the general form of its equation is: r=

±a(2 − 1) ( cos χ − 1)

(2.174)

with the negative sign for the origin of coordinates at the focus enclosed by the two branches of the hyperbola and the positive sign for the origin outside the conic surface of the hyperbola, 2a is the distance between the two vertices of the hyperbola; see Fig. 2.10, and  (called the eccentricity) is the ratio of the distance between the two foci and the two vertices. Therefore, by setting 2 AB 2  = FB A and a = F 2 B 4 −A2 in Eq. (2.174), one obtains the hyperbolic equation of (2.174). When A is positive, i.e. when the force is repulsive, C is a focal point outside the conic section of the hyperbola, while when A is negative (attractive force) C is within the conic section. That is, one branch of the hyperbola will be followed, depending on the nature of the electrostatic force. The two situations are illustrated in Fig. 2.11. Note that in the case of repulsion, r is smallest when

a a

a a

r χ (0,0)

Figure 2.10 A general shape of a hyperbola, with 2a being the distance between its foci. C*

C

χ

a '

' a

a q C

χ

Attractive

q C* Repulsive

Figure 2.11 Hyperbolic orbits for a repulsive Coulomb field (right-hand side) and attractive field (left-hand side) in C.

127

2.8 Electromagnetic Interactions

cos χ = 1; therefore, χ is measured in a direction toward the vertex, while in the case of attraction χ is measured in a direction away from the vertex, since r is minimum at cos χ = −1. Let us determine the integration constant, F. With the equation for r now determined, we can combine the radial and polar components of the instantaneous velocity, u1 of Eq. (2.168)7 , to obtain: u12 = F 2 B 2 sin2 χ +

B2 r2

(2.175)

This enables determining the instantaneous kinetic energy of particle 1, which when added to the potential energy, should be equal to the initial kinetic energy of particle 1. The instantaneous kinetic energy in C is given by:   1 m1 1 2 1 1 2 2 2 T = m1 u1 + mu2 = μ12 (u1 + u2 ) = m1 u1 (2.176) 2 2 2 2 μ12 where u2 is the instantaneous velocity of the target particle in C . The potential 2 2 2e 1 z2 e μ12 energy is 4πεz10z(r+R) = z4πε . Adding this potential energy to T , one obtains 0 r m1 the total instantaneous energy, which must be equal to the initial kinetic energy. Therefore,   1 m1 z1 z2 e 2 μ12 1 2 m1 u1 + = μ12 v12 (2.177) 2 μ12 4πε0 r m1 2 Now using Eqs (2.175) and (2.173), for u1 and r, respectively, one obtains:  F = 2

v1 μ12 B m1

2

 +

A B2

2 (2.178)

With F known, using Eqs (2.170) and (2.171) for B and A, respectively, one obtains an expression for the two parameters controlling the hyperbola:   2   2 4πε μ v b μ12 b 0 12 1 2  = 1 + v1 = 1+ a = b⊥ (2.179) 2 z1 z2 e b⊥ m1 The parameter b⊥ in Eq. (2.179) is such that: b⊥ = 7 Use

is made of Eq. (2.170) and

|z1 z2 e 2 | 1 4πε0 μ12 v12

dr dy 1 ∓A dr = , y = = 2 + F cos χ. dχ dy dχ r B

(2.180)

128

Chapter 2 Collision Kinematics

Note that the eccentricity is the same for attractive and repulsive forces, for the same magnitude of particle charge, and  ≥ 1 always, which excludes the possibility of an elliptical projectile. The eccentricity is also related to the value of χ at r = ∞, χ∞ , by  cos χ∞ = 1, according to Eq. (2.174). In turn, 2χ∞ = π − ϑ . Therefore, the angle of deflection of the incident particle in C , which is also the angle between the asymptotes of the hyperbola, is such that: sin

ϑ1 1 = 2 

tan

ϑ1 b⊥ = 2 b

(2.181)

Therefore, it is now possible to determine the angle of deflection ϑ1 in C , and subsequently the corresponding energy in L, using Eqs (2.179) and (2.181). From Eq. (2.179), one can see that at ϑ1 = π2 , b⊥ = b, i.e. b⊥ is the impact parameter at which the scattering angle in C is π2 . The parameter 2b⊥ is called the collision diameter, since at the impact distance corresponding to this diameter the incoming particle is deflected by 90◦ in C . The minimum separation between the projectile and the target occurs, as indicated by Eq. (2.174) and Fig. 2.11, at r = a( ± 1), depending on whether the force between the particles is repulsive (+) or attractive (−ve). Since the definition of C requires rm1 = Rm2 = (r + R)μ12 , the minimum separation distance between the two particles can be expressed as:  m1 m1 2 + b2 r|min = a( ± 1) = ±b⊥ + b⊥ (r + R)|min = μ12 μ12 # $ 1 = ±b⊥ +1 (2.182) ϑ sin 23 where use was made of Eqs (2.179) and (2.181). The above equation shows that the collision diameter, 2b⊥ , is the closest permissible distance of approach in a head-on collision (ϑ3 = π) in a repulsive field, i.e. when b = 0 between the interacting particles. In an attractive field, the closest distance of approach in a head-on collision is zero, as Eq. (2.182) shows, then the short-range nuclear forces can cause further effects, such as annihilation. One last quantity we will determine here is the energy of the recoiled target, since it is the energy lost by the incident particle in an interaction. Taking advantage of the fact that when the target’s velocity is zero, one has: v 2 = − v0 , as Eq. (2.8) shows. Since the magnitude of velocities does not change in C by elastic scattering, then v4 = v  2 = v0 , where v4 is the velocity of the target in C after collision. Then as the target scatters by an angle ϑ4 in C , the triangle of the vector diagram of Fig. 2.1 applied after collision to particle 2 becomes isosceles, leading to ϑ4 = 2ϑ4 . Since in C (see Fig. 2.2) ϑ3 + ϑ4 = π, then 2 v4 = π − ϑ3 . The triangle of Fig. 2.9 with v4 = v0 also gives ϑ v42 = 2v02 (1 − cos ϑ4 ) = 2v02 (1 − cos ϑ3 ) = 4v02 sin2 23 . The kinetic energy of the

129

2.8 Electromagnetic Interactions

target particle after collision in L is T4 = 12 m2 v42 . Now with Eqs (2.10), (2.179), and (2.181), one obtains: 2 2μ212 v12 b⊥ 2 = T4 = 2 2 m2 b + b⊥ m2 v12



z1 z2 e 2 4πε0

2

1 2 = 2 2 b + b⊥ m2 v12



z1 z2 e 2 4πε0

2

sin2

ϑ3 2

2 b⊥

(2.183) Note that the amount of energy transferred to the target is inversely proportional to its mass. Therefore, energy loss by Coulomb elastic scattering is predominant in the interaction of charged particles with light particles, mainly electrons. The kinetic energy of the incident particle after collision is T3 = T1 − T4 , where T1 is the initial kinetic energy. Interactions in which the energy transfer is large are referred to as hard collisions, while those soft collisions involve a small energy transfer. The maximum energy transfer occurs when b = 0, i.e. at ϑ3 = π. On the other hand, at very large values of b, the amount of energy transfer becomes quite small, and T4 becomes inversely proportional to b2 . Then the scattering angle also becomes small as Eq. (2.181) indicates. It it is, therefore, customarily to set a limit, bmax , on the value of b. In the case of an atomic electron being the target, bmax is chosen as the value at which the energy transfer is about equal to the binding energy, since then the interaction ceases in effect to be an elastic scattering. In scattering with heavy charged particles, even though the energy loss is not great, as Eq. (2.183) indicates, the projectile particle can experience considerable deflection, as is the case with Rutherford scattering. It is also straightforward to show, using Eq. (2.183), that upon head-on collision (b = 0) in a repulsive field z1 z2 e 2 the closest distance of approach occurs when: 4πε = 12 μ12 v12 , i.e., when the 0 (2b⊥ ) Coulomb field’s potential energy balances with the total kinetic energy of the particles in C (see Eq. (2.126)).

Relativistic elastic Coulomb scattering At high velocities, the eccentricity of the incident particle trajectory, Eq. (2.179), is quite high and the angle of scattering, given by Eq. (2.181), is quite small. Therefore, relativistic elastic scattering is extremely fowardly directed that a straight-line trajectory can be assumed. Then the target hardly recoils, and one can assume that it remains at rest. Due to the small angle of scattering, one can also assume that the change in the magnitude of the momentum of the incident particle is quite small. Then, one √can consider the change, qc, in the momentum of the projectile to be equal to −t13 , where t13 is the t invariant. Therefore, using the notation of Table 2.1, one obtains: q2 c 2 = −t13 = −[(E1 − E3 )2 − P12 − P32 + 2P1 P3 cos θ3 ] ≈ 4P22 sin2

θ3 2 (2.184)

130

Chapter 2 Collision Kinematics

where we assumed √ E1 ≈ E3 , and P1 ≈ P3 . The equivalent classical momentum transfer is equal to 2m2 T4 , where T4 is the kinetic energy transfer given by Eq. (2.183), since in the case under consideration the target does not significantly move. Comparing 2m2 T4 of Eq. (2.183) to q2 from Eq. (2.184), one has: P1 =

z1 z2 e 2 c z1 z2 e 2 1 z1 z2 e 2 M1 γ = = 4πε0 b⊥ v1 4πε0 b⊥ β 4πε0 b⊥ P1

P12 =

z1 z2 e 2 M1 γ 4πε0 b⊥

(2.185)

where M1 γ is the total energy (excluding potential energy) (see Table 2.1). Note ϑ that, since the momentum hardly changes, ϑ3 = π − ϑ3 , hence sin ϑ23 = sin 23 . Using Eq. (2.183), one can determine b⊥ , half the collision diameter, as: b⊥ =

z1 z2 e 2 E1 4πε0 P12

(2.186)

Therefore, for a given kinetic energy of the projectile, T1 , one can determine the collision diameter, 2b⊥ , since E1 = M1 + T1 and P12 = T12 + 2T1 M2 . Since the target hardly moves, the scattering angle of the projectile in C r and the corresponding one in L are about equal. Moreover, for the small angle of deflection, which occurs in this type of interaction, tan ϑ ≈ ϑ. Then, in analogy with Eq. (2.181), the small deflection angle of the projectile in L for a given impact parameter, b, can be estimated as: ϑ1 = 2

z1 z2 e 2 E1 4πε0 b P12

(2.187)

Since the energy transfer in this interaction is small, one can equate it to the momentum transfer, qc, determined by Eq. (2.184). That is, T4 = qc. The above analysis is obviously simplified, but enables direct use of the relativistic relationships, when needed.

Quantum effects As Eq. (2.24) indicates, the value of the impact parameter, b, can be determined from the angular momentum, J , around the center-of-mass of the projectile and the target, since J = mv1 b. The angular momentum can be written in units of h  ( = 2π ), where h is the Planck constant, so that J = l , with l being a scaling factor. Classically, there are no restrictions on the value of l. However, the uncertainty principle imposes some constraints on the value of l, since the uncertainty in determining the position of a particle, hence the uncertainty in b, is such that b p ≈ , where p is the particle’s momentum. Therefore, there is

131

2.8 Electromagnetic Interactions

a minimum value of b for a given momentum such that: bq = p , below which the quantum mechanical effects must be taken into consideration. Therefore, the classical analysis which neglects quantum effects is valid when: c e2 z1 z2 e 2 c b⊥ ≈ = z1 z2 = z1 z2 α > 1 bq 4πε0 v1 4πε0 c v1 v1

(2.188)

2

1 where use was made of Eq. (2.180), and α = 4πεe 0 c ≈ 137 is known as the fine structure constant, for the obvious reason that it determines the level at which fine (quantum mechanical) details are relevant. If the condition of Eq. (2.188) is not satisfied, the impact parameter, b, would not have a definite value, since the uncertainty in it would be large. The condition of Eq. (2.188) is easily satisfied when the particles are slow and heavy (large charge). Then the collision diameter, 2b⊥ , is much greater than the radius of a particle, as is the case with Rutherford scattering.

Rutherford scattering As indicated in Section 1.7.3, Rutherford scattering is the scattering of charged particles by heavy nuclei. Under these conditions, the collision diameter, 2b⊥ , is considerably larger than the radius of the target nucleus. The nuclear potential of the target does not then affect the interaction, neither do the electrons surrounding the nucleus. The charge distribution of the interacting particles do not overlap in this interaction. Therefore, Rutherford scattering is a Coulomb scattering for which classical analysis is applicable. Then, the energy loss is not great, but considerable deflection takes place. At the high speeds encountered in accelerators, Rutherford scattering is also used to describe distant nuclear collisions that do not involve direct contact. The interaction is then a small-angle scattering, for which relativistic treatment can be applied.

Electron scattering For incident electrons to be scattered by a nucleus, they must have sufficient energy to overcome the repulsive electrostatic potential of the atomic electrons. That is, the impact parameter of the projectile electron needs to be less than the size of the atom as defined by its orbiting electrons. Electrons subjected to Coulomb scattering by the nucleus experience a large scattering angle, while hardly losing any kinetic energy. This is evident from the fact that the reduced mass, μ12 , of the electron and any nucleus is quite small (about equal to the mass of the electron). Therefore, as Eq. (2.179) indicates,  ≈ 1, and in turn the scattering angle in C is ϑ3 ≈ π, as Eq. (2.181) shows. Given the small mass of the electron compared to that of the nucleus, the C system coincides with L; consequently the scattering angles in L, ϑ3 , is also approximately equal to π. Therefore, the interaction is a backscattering process. The energy acquired by the target, according to Eq. (2.183), is quite small, given the large mass of the

132

Chapter 2 Collision Kinematics

nucleus. The target hardly gains energy and the incident electrons in turn retain their kinetic energy in spite of the large deflection angle. At high velocities, one needs not only to consider the relativistic effects but also the fact that, as Eq. (2.180) indicates, b⊥ decreases with increased projectile velocity. If the velocity is sufficiently high that b⊥ is comparable or lower than the size of the nucleus, the effect of nuclear forces must be taken into account and a nuclear interaction can take place. The electron energy needed for nuclear effects to take place is on the Ze 2 order of 4πε , where Z is the atomic number of the nucleus and R0 is its radius 0 R0 −15 (on order 10 m). This energy is about 1.44Z MeV, which is a considerable energy for a high Z target.

Scattering by atomic electrons Heavy charged particles are hardly affected by the atomic electrons. They do not lose much energy nor they change direction significantly. As in the case of the scattering of an electron by the nucleus, the reduced mass is quite small, b⊥ as Eq. (2.180) shows is large, and the scattering angle in C according to Eq. (2.181) is ϑ3 ≈ π. However, in this case, C coincides with the incident particle. Then according to Eq. (2.4), the scattering angle in L, ϑ3 ≈ 0. Therefore, for all practical purposes, the Coulomb scattering of heavy charged particles can be neglected, as it does not noticeably affect the incident particles. However, the energy gained by the target electrons is quite significant, due to their small mass, as Eq. (2.183) shows.Therefore, heavy charged particles in their wake cause significant ionization of the target atoms. On the other hand, light charge particles, like electrons, are affected considerably by the target electrons. If the energy transferred to the atomic electrons is sufficiently larger than their binding energy, the interaction can be considered as an elastic one. Then, the collision kinematics of Section 2.8.1 can be applied, provided that the incident particle’s energy is not so large that the particle comes close to, or penetrates, the nucleus. In the case of electron–electron scattering, one cannot distinguish between the scattered electron and the recoiled one. Nevertheless, it can be easily shown, using the kinematics of Coulomb scattering for any two particles of equal mass and charge, that the energy of one of the emerging ϑ

electrons is such that T3,4 = T1 cos2 23,4 , where 1 refers to the incident electron and 3 and 4 to the emerging electrons. Due to the equal mass of the interacting particles, the angle of scattering in C , ϑ = 2ϑ, where ϑ is the scattering angle in L. Note that if the energy transfer is greater than 255.5 keV, i.e. greater than half the rest-mass energy of the electron, the electron–electron elastic scattering becomes a Moller scattering and must be treated relativistically.

Soft collisions When the energy transferred by a charged particle is comparable to the binding energy of its target, the reaction becomes an inelastic one, as far as reaction

133

2.8 Electromagnetic Interactions

kinematics is concerned. Soft collisions, which do not impart much energy, are typically encountered when swift charged particles approach a target at a large impact parameter, resulting in small scattering angles. The trajectory of the incident particle can then be approximated as a straight line, rather than a hyperbola, as is the case in elastic scattering. The amount of energy transfer, i.e. the kinetic energy of the target particle then becomes: 2 2μ212 v12 b⊥ 2 T4 = = 2 m2 b m2 v12 b2



z1 z2 e 2 4πε0

2 (2.189)

This was obtained from Eq. (2.183) by setting b >> b⊥ , and using Eq. (2.180).

2.8.2 Radiative collisions An electric charge ze, at position r0 , creates an electric potential, φ( r ), at all positions, r , around it, such that8 : ze φ( r ) = 4πε0



δ[ r  − r0 ] 3  d r | r  − r |

(2.190)

where δ[ r − r0 ] is the Dirac delta function, which is zero everywhere except at r = r0 . A moving charged particle changes its position continuously, i.e. r0 becomes a function of time. The electric field produced becomes then time-varying and can be expressed as: ze φ( r , t) = 4πε0



δ[ r  − r0 (t  )] 3  d r | r  − r |

(2.191)

r − r  c

(2.192)

The time t  is such that: t = t −

where t  is the time it takes for the effect of a charge at r  to reach a point a r , with the electromagnetic effect traveling at the speed of light, c. This time is called the retarded time, as the influence of the charge is delayed by the time it takes to propagate its effect. A moving charge also creates a current, which in turn 8 The

, is such that E = ∇φ, where ∇ is a vector gradient operator. Coulomb’s law for electrostatics: electric field, E ρe , where ρe is the electric charge per unit volume (Coulomb/m3 ). ε0

= ∇ ·E

134

Chapter 2 Collision Kinematics

, the potential of which is expressed as9 : introduces a magnetic field, B μ ze r , t) = 0 A( 4π



v1 δ[ r  − r0 (t  )] 3  d r | r  − r |

(2.193)

where μ0 = 4π × 10−7 Henry/m is the permeability of free space, v1 = ∂r∂t0 , is the speed of the moving charged particle (ε0 μ0 c 2 = 1). Expression (2.191) and (2.193) are known as the Liénard (1898) – Wiechert (1990) potentials of a moving point charge [22]. These two equations are evaluated at the retarded time, t  . Therefore, the integration of these two equations requires relating r to those at r  , but also t to t  . The latter times can be related to each other using Eq. (2.192) as: dt v1 · ( r − r  ) = 1 − dt  c| r − r  |

(2.194)

Given that | r − r  | = c(t − t  ), then t  is also a function of position, which leads to: r − r  ∇t  = −  r − r  ) c 1 − v1c| ·r( − r |

(2.195)

=∇ ×A and E = −∇φ − ∂A , along Now using the electromagnetic equations: B ∂t with Eqs (2.191), (2.193)–(2.195), one obtains [22]: ⎧ ⎫ 2

⎪ ⎪ v R R ⎨ 1 + (1 − 2 ) R − v1 ] ⎬ − v1 × R [ a × R ze c c c = E (2.196)  3 ⎪ 4πε0 c 2 ⎪ R ⎩ ⎭ R − v1 · c

= {R} ˆ × B

E c

(2.197)

= r − r  , where {. . .} indicates that the enclosed function is evaluated at t  , R R = |R|, i.e. the distance between the charge and the point at which the fields 9B =∇

i.e. the magnetic field is the curl of the vector potential A. Ampere’s law: ∇ × B = μ0 J , where i is the × A, electric current density (Coulomb/(s m2 )).

135

2.8 Electromagnetic Interactions

R and a = ∂ v1 is the are estimated at this time, Rˆ = R , is a unit vector along R, ∂t particle’s acceleration10 .  The electromagnetic energy per unit volume (energy density) is given by: 1 B2 2 B 2 μ0 + ε0 E , while E × μ0 is the energy flux density (across a surface). The latter quantity is also known as Poynting vector. The Poynting vector of the fields of Eqs (2.196) and (2.197) is [23]:

× E

1 B × ({R} ˆ ×E ) = E μ0 μ0 c 1 ˆ − (E · {R}) ˆ E ] [E2 {R} = μ0 c 1 ˆ [E2 {R}] = μ0 c

(2.198)

is normal to R, ˆ so that scalar product where use was made of the fact that E · Rˆ = 0. E The electromagnetic field of Eqs (2.196) and (2.197) contains two components: (1) an acceleration-proportional field which decreases with R1 at large distance, and (2) a velocity-proportional field which decreases with R12 and is equal to the electrostatic field at zero velocity. The velocity-dependent field has a Poynting vector, as Eq. (2.198) shows, that decreases with R14 , and as such when integrating over the area of a spherical surface its power decreases with 1 , vanishing quickly with increasing distance. Therefore, the field created by the R2 velocity-dependent term is often referred to as the near field. The acceleration It is also the dominant field is such that its E and B components are normal to R. term at large distance and is the term that accounts for radiation emission by moving charged particles. The Poynting vector of the acceleration field, according to Eq. (2.198), decreases with R12 , which upon integration with a spherical surface area gives a constant total energy. Therefore, the radiation term gives a finite total power even at infinity in vacuum. It is, therefore, referred to as the radiation field, and it affirms the fact that a charged particle radiates (literally sends out energy) only upon acceleration (a particle cannot radiate if moving at a uniform speed in vacuum). Equation (2.198) indicates that the radiation energy propagates in the direction Therefore, radiation energy crosses a spherical surfaces normal R at a rate of R. E2 (energy per unit area per unit time) of μ0 c . The radiation rate per unit solid angle is

R 2 E2 μ0 c , since

the solid angle is the area divided by R 2 . Since E for the radiation v2

in the above discussion the appearance of the term 1 − c 21 and the retarded time, which also appeared when discussing the Lorentz transform and the special theory of relativity in Section 2.3. The electromagnetic theory led, therefore, to the theory of relativity. In fact the Maxwell’s equations of electromagnetism and the associated equations discussed here are already fully relativistic, and there is no need for relativistic corrections.

10 Notice

136

Chapter 2 Collision Kinematics

term of Eq. (2.196) is inversely proportional to R, the energy rate per unit solid angle is independent of R, and is given by [22]:   v1 B dPwr 2 ˆ · × = {R } 1 − {R} E d c μ0 ⎡



 ˆ · a a · v1 {R} c ⎢ = ⎣ 3 + 2  4 2 3 16π ε0 c ˆ · v1 ˆ · v1 1 − {R} 1 − {R} c c ⎤  

2 v2 ˆ 1 − c 21 {R} · a ⎥ −  5 ⎦ ˆ · v1 1 − {R} c z2 e 2

a2



(2.199)

where Pwr is the power (energyper unit time) and  is the solid angle. Note that  n ˆ · v1 in the denominators of the terms in Eq. (2.199), we have the term 1 − {R} c with n = 3, 4, or 5. This term, therefore, influences greatly the angular distribution ˆ · v1 = −β cos ϑ, where ϑ is the angle between of the emitted radiation. Since {R} c ˆ v1 → 0. Then, according to and v1 , with β = v1 , when v1 → c, one has: 1 − {R}· R c c Eq. (2.199), the amount of radiation in the direction along the particle’s velocity (forward direction) is greatly enhanced; a phenomenon known as the relativistic headlight effect. Integration over the solid angle gives [23]: Pwr =

 z2 e 2 2γ 4 2 a − ( a · v1 )2 3 4πε0 3c

(2.200)

 v2 where γ1 = 1 − c 21 · Note here that regardless of relativistic considerations, a charge cannot be accelerated to a velocity equal to c, because then γ → ∞, and an infinite amount of radiation would be emitted; which is impossible. Equation (2.200) is presented in terms of the time, t, which is the time at the point where the field is determined. The power in terms of the time at the particle, t  , is useful when one is interested in determining the energy emission rate per unit volume at the particle, so that it can be added to that of other particles in the volume. This power at t  can be obtained by multiplying the power at t by dtdt of Eq. (2.194).

Bremsstrahlung As a charged particle of charge z1 e approaches a target (atomic electron or a nucleus) of charge z2 e, the Coulomb field between the two, E, subjects the

137

2.8 Electromagnetic Interactions

hν

b

b

z1e

z1e z2e

Figure 2.12 A schematic representation of bremsstrahlung.

projectile to an acceleration, a, given by [23]: a =

E z1 z2 e 2 1 = m1 4πε0 r 2 m1 γ

(2.201)

where r is the distance between the two charges, assuming the target is at rest and ignoring its recoil, and m1 γ is the relativistic mass of the projectile. This acceleration lasts for a short period of time; approximately the time the projectile takes to travel a distance b, where b is the impact parameter. We will assume that the curvature of the projectile’s trajectory during this impulse is small, i.e. the deflection angle is small, and the projectile during acceleration is simply a straight line, as schematically shown in Fig. 2.12. The acceleration can be seen as an impulse, since the acceleration is only significant during that short time period; otherwise the acceleration is negligible when the projectile and the target are far apart. On average, the impulse is perpendicular to the projectile velocity. Let us take the impulse period as being equal to t = vb1 , where b is the impact parameter. In other words, we are assuming that the projectile is repelled, or attracted, by the target over a straight-line distance equivalent to the impact parameter, and beyond that distance the effect of the acceleration becomes negligible. We will also assume that the impulse is in a direction perpendicular to the trajectory of the projectile during the acceleration period, as it tends to throw the particle off its path. Then z12 e 2 2a2 γ 4 according to Eq. (2.200): Pwr = 4πε 3 . Multiplying this power by t gives us 0 3c an approximate estimate of the energy emitted during this impulse. Therefore, using Eq. (2.201) with r = b, one estimates the emitted energy, W , as [23]: z2 e 2 2γ 4 W = 1 4πε0 3c 3



z1 z2 e 2 1 4πε0 b2 m1 γ

2

b z4 z2 e 6 2γ 2 1 = 1 2 3 3 2 3 v1 (4πε0 ) 3c m1 v1 b

(2.202)

This interaction is referred to as a radiative collision, as radiation is emitted in the process. As Eq. (2.202) shows, since W is proportional to z22 , an electron as a target (z2 = 1) is much less effective than a nucleus (z2 = Z, where Z is the atomic number), particularly at high Z values. Eq. (2.202) also demonstrates that, at the same

138

Chapter 2 Collision Kinematics

velocity, electrons emit much more radiation than protons, with the same target, even though they have the same absolute charge value, because of the inverses proportionality with m2 .Therefore, in general, bremsstrahlung is usually neglected in the case of heavy charged particles in the field of other heavy charged particles. However, when a heavy charged particle is accelerated by a free (unbound) electron, the recoil electron experiences acceleration and emits radiation at an energy given also by Eq. (2.202), with z1 replacing z2 , and vice versa, and the mass of the projectile, m1 , replaced by the mass of the target electron. Note that an electron incident on an atom emits radiation by its acceleration by the electrostatic field of the nucleus not by atomic electrons.This is because the incident electron and a target electron would accelerate each other by an equal amount, as Eq. (2.201) shows, but at opposite directions. That is, their electric fields are equal and opposite and produce radiation waves in the far field of bremsstrahlung that are out of phase by an angle π, and in turn cancel each other. However, the motion of the projectile (assuming the target is initially at rest), can cause this phase shift to deviate from the π value. But, as long as this deviation remains small (much smaller than the wavelength of either waves), the electric fields produced by the two electrons will tend to cancel each other. The shift in position of the incident electron during 2π 1 the time of acceleration causes a phase change of at most c t = 2πv cb . Therefore, as long as v1 << c, the two emitted waves would stay at a phase difference of almost π and would tend to cancel each other. At high (relativistic) speeds, the phase difference considerably deviates from π and electron–electron bremsstrahlung becomes relevant. It should be kept in mind that the radiation waves produced in electron–positron bremsstrahlung do not cancel each other out. The bremsstrahlung electromagnetic energy is emitted in the form of a photon, 1 the energy (hence frequency) varies over a frequency range of approximately t , so that: 2  2  dW z12 e 2 2γ 4 z1 z2 e 2 1 b z14 z22 e 6 2γ 2 1 ≈ W t = = dν 4πε0 3c 3 4πε0 b2 m1 γ v1 (4πε0 )3 3c 3 m12 v¯12 b2 (2.203) The average velocity, v¯1 , is used here to account for the energy and momentum carried by the emitted photon, which changes the velocity of the incident particle. Equation (2.203) indicates that for a single particle collision at a certain incident particle energy (hence velocity) and impact parameter, the energy distribution has approximately a uniform spectrum up to the maximum frequency. In order for a photon to be emitted at an energy of hν, the kinetic energy of the incident particle should have a value greater than hν. Therefore, for a given photon frequency, ν, t = 1ν = vb1 , which gives the maximum impact parameter for the impulse producing this photon. The emitted photon carries some momentum, changing in the process the kinetic energy of the incident particle. Therefore, the emitted energy comes at the expense of reduced kinetic energy, causing the particle to suddenly slow down.

139

2.8 Electromagnetic Interactions

It should be kept in mind that the above analysis is approximate, guided by [22] and [23], and is limited to particle speeds much less than c, i.e. for γ ≈ 1. Electrons and positrons, being light particles, can acquire easily relativistic speeds. Then the emitted bremsstrahlung photons can have an energy comparable in value to the rest-mass energy of the electron, and one must deal with the total relativistic energy of the incident electron, not only its kinetic energy. Such analysis is facilitated by considering the electromagnetic field as consisting of “virtual photons’’ which scatter with the incoming electron (i.e. the reverse of Compton scattering). These virtual photons facilitate the energy and momentum transfer process. Bremsstrahlung is then viewed as a scattering process between an electron, or a positron, and a virtual photon, creating a free photon (bremsstrahlung) and a recoil electron, see section 3.4 which deals with quantum electrodynamics. The word “quantum’’ reflects the use of plane waves representing the photons, and the fact that a photon itself is a quantum of energy. The use of quantum mechanics enables the calculation of a wave amplitude for each interaction, the square of which, with proper normalization, determines the interaction probability; see Chapter 3. Therefore, the probability distribution (spectrum) of the bremsstrahlung photons are determined by quantum electrodynamics (see Section 3.6.4). It is worth noting that in spite of these relativistic effects, the flatness of the energy distribution of Eq. (2.203) stays largely valid. Note also that the maximum possible photon energy in relativistic analysis is E = m1 c 2 + T1 , where T1 is the particle’s kinetic energy.

Thomson scattering The scattering of a photon by a free electron is a form of radiative collisions, in which the electron is subjected to the electromagnetic field of the incident photon. In this process, the target electron is accelerated, and begins to oscillate at the same frequency as that of the incident electromagnetic wave. This oscillation produces in turn an oscillating electric field that gives rise to a “retarded’’ (delayed) wave of the same frequency as that of the oscillation. In essence, if the target electron is initially at rest, then the scattered photon would retain its energy. The initial movement of the electron, if present, will cause a Doppler shift between the incident photon and the electron, as well as between the scattered radiation and the observer (due to the motion of the emitter). The result is a scattering frequency determined by: ν3 = ν1 +

1 (k3 − k 1 ) · v3 2π

(2.204)

where the subscripts 1, 2, and 3 designate, respectively, the incident radiation, the target electrons and the scattered radiation, and k is the wave vector of a wave of ˆ a wave number of 2πν c in the direction defined by the unit vector k. The above analysis is applicable to atomic electrons as long as their binding energy is much less than the incident photon energy, so that the electrons can be considered to

140

Chapter 2 Collision Kinematics

be free. It also ignores the change in the momentum of the incident photon and the target electron. This electromagnetic wave analysis also fails at high photon energy, approaching that of the rest-mass energy of the electron, 511 keV, not only because of the large momentum acquired by the target electron, compared to its rest-mass energy, but also because of the quantum nature of the photons (being packets of energy with particle-like behavior). Then, Compton scattering kinematics (see Section 2.7.1), must be applied. Returning to Thomson scattering and viewing the interaction as in effect a Coulomb scattering of the target electron in the electromagnetic field of the incident electron, and assuming non-relativistic conditions, the emitted power (by the electron) per unit solid angle according to Eq. (2.199) is: dPwr e2 = a2 sin2 ψ = d 16π2 ε20 c 3



e2 4πε0 me c 2

2 ε0 cE2 sin2 ψ = re2 si sin2 ψ (2.205)

where the angle ψ here is the angle between the direction of the emitted radiation and the direction of acceleration, and the electron’s acceleration, a, is expressed , with me being the mass of the electron. in terms of the electric field as a = mee E 2

The parameter re2 = 4πεe0 me c 2 is a constant (=2.82 fm), known as the classical electron radius, and si is the magnitude of the Poynting vector, i.e. the instantaneous power per unit area of the incident wave, si = ε0 cE2 . Thomson scattering of photons can also occur with the nucleus. The power emitted per solid angle by the target nucleus can be obtained using Eq. (2.205) by replacing the electron charge, e, by the charge of the nucleus, Ze, and the electron mass by the mass of the nucleus. Obviously, because of the large mass of the nucleus, its acceleration and the amount of power it emits are quite small. Therefore, Thomson scattering with the nucleus is not a very significant interaction. It also combines coherently with the nucleus, making it difficult to distinguish between the two.

Rayleigh scattering While Thomson scattering is an interaction with a free electron, Rayleigh scattering is with bound electrons. When the photon energy is lower than the binding energy of atomic electrons, i.e. when the photon’s wavelength is large, photons are affected by the collective charge distribution of bound atomic electrons. Scattering, therefore, takes place from different parts of the atom, resulting in wave interferences, i.e. a coherent scattering process. This interaction can also be viewed as a process in which the charged particles (atomic electrons) absorb the energy of incident electromagnetic wave, then re-emit it as an electromagnetic radiation. In this interaction, the photon hardly loses energy, but changes its direction slightly by a small angle ϑ. The atom then as a whole recoils, acquiring a momentum, P, without absorbing any energy. The scattering angle is small,

141

2.8 Electromagnetic Interactions

since the recoil imparted to the atom does not produce excitation or ionization. The momentum transfer, P, can then be calculated using the t invariant, as: ( P)2 = −t = 2P 2 − 2P 2 cos ϑ = 4P 2 sin2

ϑ 2

(2.206)

where P is the momentum of the incident photon. Note that at the small angles of Rayleigh scattering, P ≈ Pϑ.

Cerenkov radiation This type of emission does not result in a significant energy loss and is caused by the mismatch of the speed of the electromagnetic field associated with the motion of a charged particle in a dielectric medium and the phase velocity of light in the medium. The electromagnetic properties of a medium are defined in terms of the dielectric constant (permittivity), ε, and the magnetic permeability, μ. In vacuum, εμc 2 = ε0 μ0 c 2 = 1, but in a dielectric medium εμc 2 > 1. In most materials, aside from ferromagnetic materials, μ = μ0 ; therefore, εε0 > 1. An electromagnetic wave √ c ε

travels at the speed of light in vacuum, but its speed in a dielectric medium is √ε0 , which is less than unity (<1).The ratio εr = εε0 is known as the relative permittivity, √

while n = √εε0 is the index of refraction of the medium (with μ = μ0 ). If a charged particle is moving at a speed v higher than the speed of its electromagnetic wave in the medium, the particle will appear as if it is “running’’ away from its field. This is not a physically permissible situation, as the particle has to resonate with its electromagnetic wave. This mismatch causes the production of an optical “shock’’ wave, very similar to the sonic boom produced when the velocity of an object moving in air exceeds the speed of sound in air. For a particle moving with a speed vector v to resonant with an electromagnetic wave: exp (k · x − ωt), where k is the wave number, x is the direction of motion of the wave, and ω is its angular frequency, one has to satisfy the following √ conditions: (1) k = ωc εr , and (2) ω = k · v . The first condition reflects the value of the phase velocity in the medium, while the second condition is the condition required for the particle to resonate with the wave. These conditions are simultaneously met, as graphically shown in Fig. 2.13, for field waves confined within a cone of a half-angle, ϑ, such that: ω c cos ϑ = √ = v εr kv

(2.207)

Note that k · v = kv at ϑ = 0. Field waves outside this cone destructively interference with each other, while constructive interference takes place √ within ϑ. The threshold velocity, as Eq. (2.207) indicates, is equal to √cεr . If vc εr < 1, no physically acceptable value for ϑ can be arrived at and the resonance conditions will not be satisfied, as there will be destructive interference in all directions.

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Chapter 2 Collision Kinematics

k

Shock wave front k· nv

ε k =v r c



ν

Figure 2.13 A schematic for electromagnetic waves associated with the movement of a charged particle in a dielectric medium. 2

Electromagnetic radiation will be emitted at frequencies at which εr > vc 2 , i.e. the dielectric constant of the medium determines the frequency at which Cerenkov radiation is emitted. The emitted electromagnetic wave is propagated · k = 0, and v · E ∝ sin ϑ. The in a direction normal to its electric field; i.e., E latter scalar product is the component of the electric field along the trajectory of the wave. The amount of radiated energy, W , is proportional to the square of that component multiplied by the particle’s charge, as it is proportional to z2 e 2 sin ϑ, since W is proportional to the squared value of the electric field. I. M. Frank and I. G. Tamm, (1937) using classical electromagnetic theory showed that the radiated energy with a short particle path ds is given by:    z2 e 2 c2 c2 dW = 1 − νdν ds for ε > (2.208) r 4πε0 c 2 v 2 εr v2 where use was made of Eq. (2.207), and ν is radiation frequency. It should be kept in mind that in optically transparent materials, εr tends to increase with ν; reaching an infinite value at some frequent νr (a resonance value different from that for Cerenkov radiation). The integral in Eq. (2.208) has a lower limit ν1 at which the conditions for Cerenkov radiation are satisfied, and an upper limit of νr . In optical materials, νr corresponds to a wavelength, λr , of a few hundred manometers (λr νr = c). Equation (2.208) also reflects that the energy distribution 2 (i.e. frequency distribution) of Cerenkov radiation is proportional to (1 − c v2 εr )ν, and leads to a broad and smooth distribution that increases with both ν and εr . This may indicate that Cerenkov radiation has a very high frequency, except for the limitation of νr which restricts it in optical materials (εr = n2 ≈ 1.52 = 2.25) to the visually observable frequency range of the bluish-white light. With each photon having an energy hν, the number of photons, dN , emitted in ds per unit energy, is given, using Eq. (2.208), by:   d2 N αz2 c2 (2.209) = 1− 2 dEds c v εr

2.8 Electromagnetic Interactions

143

2

1 where α = 4εe0 c = 137 is the fine structure constant. In the optical range, 5–20 photons/mm per electron are produced. As can be deduced for Eq. (2.208), the energy loss is quite small by comparison to that caused by other mechanisms, but Cerenkov radiation is easily detectable. It is the bright bluish glow observed in the water of research reactors. As Eq. (2.208), and consequently Eq. (2.209), show, Cerenkov radiation is not dependent on the mass of the charged particle, but only on its charge.

Transition radiation As indicated in Section 2.82, a charged particle moving in a medium with a velocity lower than the phase velocity of light in that medium will not emit radiation. This premise remains valid as long as the phase velocity along the path of the particle does not change. However, change in phase velocity occurs at the interface between two media of different dielectric constants. Then the electromagnetic fields produced by the same charge will be different in the two materials because of the change in the electromagnetic properties of the two materials at the two sides of the interface. This sudden change in the fields as the charge approaches the interface is accommodated by the release of transition radiation. One can imagine that the approaching charged particle on one side of the interface is balanced by a mirror image particle of opposite charge at the other side of the interface, stopping instantaneously when they encounter each other at the interface. Qualitative analysis of this phenomenon involves considerable analysis of the electric field (see [24]). However, some of the basic findings of the analysis are summarized here to demonstrate the relevant aspects of this process. The power of the emitted transition radiation is proportional to the total energy (γm0 , with γ being the usual relativistic factor and m0 is the rest mass) of the charged particle crossing the interface where there is a change in permittivity. Therefore, this interaction is mainly relevant for relativistic particles.This radiation is emitted primarily in the forward direction, i.e. in the direction of motion of the particle (with a maximum at an angle ϑ = γ1 ). The total radiated energy depends on the squared difference of the electric fields at the interface. The number of 1 photons emitted is on the order of αγ, where α = 137 , the fine structure constant. The photon energy typically peaks in the eV range (i.e. in the range of soft X-rays). The process is, therefore, called sometimes X-ray transition radiation.

Delbruck scattering The electromagnetic characteristics of a photon in the presence of the Coulomb field of a nucleus can be examined by viewing the photon as being composed of a virtual electron–positron pair. The virtual electron–positron pair can then be deflected by the Coulomb field, appearing after scattering again as a photon with the same energy as the initial photon. The process is then an elastic interaction.

144

Chapter 2 Collision Kinematics

The momentum transfer in this process can also be calculated using the same equation for Rayleigh scattering (Eq. (2.206)).

2.8.3 Diffraction When a wave is scattered by an obstacle (or an aperture) with a dimension greater than the wavelength, the process is termed diffraction. Since atomic and nuclear radiation (in the form of X- or γ-radiation, respectively) has a relatively very short wavelength (due to its high energy), the obstacles we are concerned with are at the atomic scale, i.e. those formed in a crystallized material by the spacing between atoms in the periodic atomic formation of the crystal. Since radiation behaves then like a wave, we can apply simple optical geometric principles and wave interferences. Let us consider one line in a crystal with a spacing distance between any two spaced atoms (lattice constant) a, subjected to a broad but collimated beam of radiation making an angle χ0 with the crystal line. The beam will diffract by the various atoms on the line. The beam rays diffracted by two adjacent atoms will be in phase, i.e. interfere constructively, when they are phase-shifted from each other by integer multiples of 2π. It can be easily deduced from Fig. 2.14 that the phase shift between the two adjacent rays is 2πa λ ( cos χ − cos χ0 ). Therefore, the condition for complete interference (strongest signal) is such that λa (α − α0 ) = n1 , where n1 is an integer and the αs designate the direction cosine (cosine angle with the array line). The use of the direction cosines enables one to generalize the above condition to the three mutually perpendicular crystal planes. Therefore, the general condition for maximum intensity (complete constructive interference) that can be achieved in a rhombic crystal structure with the lattice constants a, b, and c in each of its three orthogonal axes is: ⎧ ⎫ ⎧ ⎫ ⎧ n1 ⎫ ⎪ ⎪ ⎨α⎪ ⎬ ⎨a⎪ ⎬ ⎬ ⎪ ⎨α0 ⎪ β − β0 = λ nb2 (2.210) ⎪ ⎪ ⎩ ⎪ ⎭ ⎩ n3 ⎪ ⎭ ⎭ ⎪ ⎩ ⎪ γ0 γ c where the ns have integer values and α, β, and γ designate the direction cosines with the two crystal planes (α2 + β2 + γ 2 = 1). The above are known as Laue

χ0 a

χ1

Figure 2.14 A schematic showing diffraction along a crystal line.

145

2.8 Electromagnetic Interactions

diffraction equations. Therefore, there is only one value of λ that will produce maximum interference in the direction defined by the direction cosines (α, β, γ) for a wave incident at the direction defined by (α0 , β0 , γ0 ). In other words, if the incident radiation has a wide spectrum (distribution of frequencies, hence wavelength), a monochromatic (single frequency) radiation is produced in the direction (α, β, γ), provided that the conditions of Eq. (2.210) are satisfied. The angle of deflection, ϑ, between the incident and diffracted rays can be shown to be:   2 2 n1 n22 n32 λ (2.211) + 2 + 2 cos ϑ = 1 − 2 a2 b c with α0 n1

λ = −2 a2 n1 a

+ β0bn2 + γ0cn3 2 2 + nb2 + nc3

(2.212)

Obviously the integers and the direction cosines have to be such that λ > 0. The diffraction process is schematically shown in Fig. 2.15, for a radiation beam incident on a crystal with direction cosines (α0 , β0 , γ0 ) defined with respect to its three mutually perpendicular crystal line axes. The radiation is diffracted to the direction cosines (α, β, γ), after satisfying the conditions of Eq. (2.210), so that the angle of deflection ϑ is such that cos ϑ = α0 α + β0 β + γ0 γ, as determined by Eq. (2.212). The figure also shows a median plane between the incident and reflected rays. This plane is called the net plane, as it is the plane that crosses the sites of an infinite number of atoms forming the crystal. With respect to this plane, the rays appear to be reflected by an angle equal to the angle of incidence (both measured with respect to the net plane). The diffraction process is then seen by an observer as a specular reflection process. This is the process of Bragg diffraction, the conditions for which are typically derived by considering two consecutive net planes. The derivations are obtained here from the Laue conditions (Eq. (2.210)), to show that Bragg reflection is actually a diffraction process and not a scattering process. Let us first remind ourselves of the Miller indices. We have already defined a, b, and c as the distances between adjacent atoms on each of the three lattice axes. Any plane passing through a crystal will intersect with the three lattice axes. The reciprocals of these intersection values are used to determine the lattice indices α, β, γ  χ 2 α0, β0, γ0

 2

 χ 2

Figure 2.15 A schematic showing how diffraction, by an angle ϑ, is equivalent to specular (Bragg) reflection, by an angle χ, on a net plane.

146

Chapter 2 Collision Kinematics

(h, k, and l) defined as h = 1a , k = 1b , and l = 1c . With these indices the distance between two successive net planes is: 1 (2.213) d = 

l 2 h 2 k 2 + + a b c The Miller indices are related to the integers of the Laue conditions, Eq. (2.210) as n1 = nh, n1 = nh, n1 = nh, where n is some common divisor. Such a common divisor exists because the equation of the net plane, with respect to the crystal axes, is n1 x + n2 y + n3 z = 0, so that the integers n1 , n2 , and n3 have a common integer divisor of at least one. With this definition and using Eqs (2.211) and (2.213), one obtains:    2  2 nλ h 2 nλ k l ϑ + + = sin = 2 2 a b c 2d 2d sin χ = nλ (2.214) where in the last step the diffraction angle ϑ is replaced by the reflection angle χ, with ϑ = 2χ. Equation (2.214) is the well-known Bragg’s law of diffraction, derived by W. H. Bragg and W. L. Bragg in 1913 (father and son). It is valid in various crystal structures, since the crystal constants do not appear explicitly in its formulation. From Eq. (2.214), one can conclude that the divisor integer n is the difference in the number of wavelengths between reflections by two neighboring net planes.

2.9 Problems Section 2.2 2.1 For single scattering between a neutron and a nucleus of a mass number A, show that: 1. The magnitude of the relative velocity before scattering is equal to that after scattering. 2. The angle of scattering in L, ϑ, is related to that in C , ϑ , by: tan ϑ =

A sin ϑ A cos ϑ + 1

(2.215)

3. Prove that:

 A cos ϑ = cos2 ϑ − 1 + cos ϑ cos2 ϑ − 1 + A2

4. Prove that: A+1 cos ϑ = 2



T3 A − 1 − T1 2

(2.216)

 T1 T3

(2.217)

147

2.9 Problems 

5. In the special case of scattering with hydrogen, show that ϑ = ϑ2 . 6. In the special case of scattering with hydrogen, show that backscattering in L following a single collision is not possible. 7. In analyzing inelastic scattering, the energy of the one-body equivalent system (reduced mass and relative velocity) is reduced by the excitation energy of the reaction. The new energy is then used in the analysis for the interaction as if it were an elastic scattering. Comment on the logic behind this approach.

Section 2.3 2.2 A particle is moving in the x-direction at a speed u with respect to a frameof-reference, K  , but is moving at a speed faster than u with respect to another frame-of-reference, K . What is the speed of the particle with respect to K , if K  is moving at a speed v in the x-direction with respect to K . 2.3 Show that a relative velocity greater than that of light cannot be attained by one physical system with respect to another, no matter how high the velocity is; keep in mind that no individual speed exceeds the speed of light. 2.4 It is often convenient to express the total particle energy in terms of its restmass energy, m0 c 2 , the momentum relative to the natural units of momentum, 2 m0 c, and the velocity in relation to the speed of light c. Let W = mmc0 c 2 , η = mp0 c and β = vc , prove the following relationships: W 2 = η2 + 1

β2 =

(2.218)

 m 2 η2 1 0 = 1 − = 1 − 1 + η2 W2 m

β2 = η2 = 2 1−β



T m0 c 2

2 

2m0 c 2 1+ T

(2.219)

 (2.220)

2.5 Calculate the error in evaluating non-relativistically the momentum for the following particles at 2 MeV energy: electrons, protons, deuterons, and alpha particles. 2.6 Newton’s second law states that the rate of change in the momentum of an accelerated object is equal to the force causing the acceleration, i.e. F = dmv dt . 1. Express Newton’s second law taking into account relativistic (Einsteinian) mechanics. Hint: take into account that m changes with t. 2. Show that work is still relativistically equal to kinetic energy.

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Chapter 2 Collision Kinematics

Section 2.5 2.7 For a two-body interaction, prove that the energy and momentum in Cr of particles 1 and 2 colliding with each other (with 2 being at rest) are related (using the notation of Table 2.1) to those in L by:   1 2 (2.221) E − (M1 + M2 )2 E 2 − (M1 − M2 )2 P12 = P22 = 2E P1 M2 P1 = P2 = (2.222) E M 2 + M2 E1 E1 = 1 (2.223) E M 2 + M2 E1 E2 = 2 (2.224) E 2.8 Prove the relationships used in Algorithm 2. 2.9 For interacting bodies 1 and 2, with 2 at rest, show that the energy and momentum in L and Cr are related by: P1 M2 P1 = √ (2.225) s E1 =

M12 + M2 E1 √ s

(2.226)

E2 =

M22 + M2 E1 √ s

(2.227)

2.10 For the case of elastic scattering, M1 = M3 , M2 = M4 , determine the values of the Mandelstam invariants in L and Cr and prove that they are equal.

Section 2.6 2.11 Using classical kinematics show that the maximum energy a particle of kinetic energy T1 and mass m1 can lose when elastically scattered by another particle of mass m2 is given by: 4m1 m2 T1 (2.228) Qmax = (m1 + m2 )2 Show with relativistic mechanics that the equivalent expression is:  ⎤  ⎡ 1 1 + 2M T1 ⎦ (2.229) Qmax = T1 ⎣ (M1 +M2 )2 1 + 2M2 T1 Under which conditions do the classical and relativistic expressions agree? Under which conditions would the particle of mass m1 lose all its energy, so that Qmax = T1 ?

149

2.9 Problems

Section 2.7 2.12 For a neutron colliding with a nucleus of mass A: 1. Draw a velocity diagram relating the velocity of the neutron after collision in C to that in L. 2. Calculate the neutron velocity before and after collision in C . 3. Relate the calculated velocities in C to those in L. 4. Prove the relationships: 1 + A2 + 2A cos ϑ3 T3 = T1 (1 + A)2 and cos ϑ3 =

(2.230)

 1 cos ϑ32 − 1 + cos ϑ3 cos2 ϑ3 − 1 + A2 A

(2.231)

where the usual notation of two-body kinematics is used. You will need first to show that: cos ϑ3 = 

A cos ϑ3 + 1

(2.232)

A2 + 2A cos ϑ3 + 1

5. Show that for inelastic scattering, conservation of energy and momentum leads, under some conditions, to the relationships (using the usual two-body interaction notation):  2 (1 − Q ) + 2A cos ϑ  (1 − Q ) 1 + A 3 T3 T1 T1 = (2.233) T1 (A + 1)2 and

 A cos ϑ3 1 −

Q T1

+1  1 + A2 (1 − TQ1 ) + 2A cos ϑ3 1 −

cos ϑ3 = 

(2.234) Q T1

Also show, for Q > 0 and for A large, that T3 = T1 − Q and cos ϑ3 = cos ϑ3 . 2.13 Prove that relativistically the kinematics of elastic scattering between a neutron and a nucleus leads to:  4(μ2 + A − 1)rT1 + 2(2μ2 + A2 − 1)r 2 + 4rμ(T1 + r) μ2 − 1 + A2 T3 = T1 2(A + 1)2 r 2 + 8(1 − μ2 )T12 + 8(1 − μ2 + A)rT1 (2.235) where the usual two-body collision terminology is used, with μ = cos ϑ3 , the rest-mass energy of the neutron is set equal to 12 r and the rest-mass

150

Chapter 2 Collision Kinematics

energy of the nucleus is 12 Ar, where A is the rest mass of the nucleus in units of neutron mass. For 20 MeV neutrons elastically scattered with a target with A = 14, show that the effect of neglecting the relativistic change is insignificant. 2.14 1. For Compton scattering, using the usual notation for two-body collisions, in which α = mhνe c12 , where me c 2 is the electron rest-mass energy, h is Planck constant, and λ refers to the photon’s wavelength, prove the following relationship: λ3 − λ1 =

h (1 − cos ϑ3 ) me c

(2.236)

This expresses the change in photon wavelength as a result of Compton scattering, and is known as the Compton shift. Note that the shift in the wavelength is independent of the incident wave energy. 2. The length mhe c is called the Compton wavelength. Find its value. 3. Show that: Qmax =

hν1 1 1 + ( 2α )

(2.237)

where Qmax is the maximum energy transferred to the electron as a result of photon scattering. 4. Prove that: cot ϑ4 = (1 + α) tan

ϑ3 2

(2.238)

5. Show that the kinetic energy of the emerging electron is given by: T4 = hν1

2α cos2 ϑ4 (1 + α)2 − α2 cos2 ϑ4

(2.239)

6. Using the relationship between ϑ3 and ϑ4 show that: d3 sin ϑ3 dϑ3 −(1 + cos ϑ3 ) sin ϑ3 = = d4 sin ϑ4 dϑ4 (1 + α) sin3 ϑ4 =

−4(1 + α)2 cos ϑ4 [(1 + α)2 − α(2 + α) cos2 ϑ4 ]2

(2.240)

3 and  4 are directions of the incident and scattered photons. where  2.15 Show that the approximate form of the threshold energy for neutron inelastic scattering, Eq. (2.148), corresponds to the excitation energy in C . 2.16 Using conservation principles of energy and momentum, prove that pair production is impossible in vacuum.

2.9 Problems

151

2.17 1. For a neutron-producing photonuclear reaction, show that the kinetic energy of the neutron can be expressed in terms of the mass number of the target nucleus as:  + 1 A−1 2 Tn ≈ Eγ + Q + Eγ A 2A(A − 1)Mn  Eγ 2(A − 1) (Eγ + Q) cos ϑn (2.241) + A AMn where Mn is the neutron’s rest-mass energy, Q is the Q-value of the reaction, and ϑn is the angle between the incident γ radiation and the emitted neutron. 2. For the 2 H(γ,n)1 H reaction, Q = −2.226 MeV, Hanson11 states that for 2.62 MeV gammas, Tn = 196 + 27 cos ϑn keV. Examine the suitability of this expression. 3. For the above expression, determine the spread in the energy distribution of the emitted neutrons. 2.18 For the neutron-producing reaction, 6 Li(p,n)7 Be, Gibbons and Newson12 give:  ! "2 Mp Mn MBe MLi (Tp −Tth ) Tn = Tp cos ϑp ± −sin2 ϑp (2.242) (MLi + Mn )2 Mp Mn Tp 1. Derive this expression. 2. The same reference gives a threshold value for the reaction of Tth = 1.8811 MeV, determine the corresponding Q-value.

Section 2.8 2.19 Apply relativistic kinematics to the elastic scattering of electrons with the nucleus to show that in a heavy target (i.e. with a rest-mass energy greater than that of the incident electron), the electron energy hardly changes even at large scattering angles. dP , demonstrate that a freely 2.20 Using the momentum-energy relativistic rate, dE moving charged particle cannot radiate even in the presence of an electromagnetic field. In other words, external forces have to be exerted on the particle for it to radiate. Hint: the electromagnetic field can be represented by a photon. 2.21 Prove Eqs (2.211) and (2.212) for Laue diffraction for a cubic crystal. 2.22 Prove that a free electron cannot emit or absorb photons. 11 J.

B. Marion and J. L. Fowler, Eds., Fast Neutron Physics, Part II: Experiments and Theory, Interscience Publishers, New York, p. 29, 1963. 12 Ibid, p. 135.

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C H A P T E R

T H R E E

Cross Sections

Contents 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8

Introduction Nuclear Cross-Section Models Neutron Cross Sections Electrodynamics Photon Cross Sections Charged-Particle Cross Sections Data Libraries and Processing Problems

153 156 168 183 194 215 229 238

3.1 Introduction The radiation cross section is the effective area a target presents to a projectile. It determines in turn the probability of the projectile interacting with the target. This cross section is not necessarily identical to the geometric cross section area of the target. For instance, in the case of a charged particle interacting with a target nucleus, the Coulomb field of the latter extends well beyond its geometric boundary, leading to an “effective’’ cross section greater than its area. On the other hand, the same target nucleus would have a smaller cross section for neutrons, which are not affected by the Coulomb field. A cross section not only depends on the type of the target and the projectile interacting with it, but also on the nature of the interaction, i.e. whether it leads to particle scattering, absorption, emission of a secondary particle, etc. The cross section also depends on the energy of the projectile. Roughly speaking, a slower projectile spends more time in the proximity of the target, hence has a higher probability of interaction, while a swift particle can sweep through the target or its potential field without being affected much. The cross section for scattering can also vary depending on the angle of scattering, along with the energy of the projectile. Therefore, one can define a differential cross section, i.e. a cross section per unit solid angle, which varies from one angle to another. In order to further appreciate how the cross section is related to the interaction probability, we will first consider a wide beam of radiation scattered by a large target, and then reduce this to the small target of atoms and nuclei. Radiation Mechanics: Principles & Practice ISBN-13: 978-0-08-045053-7

© 2007 Elsevier Ltd. All rights reserved.

153

154

Chapter 3 Cross Sections

The concept of cross sections is widely associated with atomic and nuclear radiation interactions, but it has its roots in electrodynamics. The surface of a dielectric material, when exposed to an incident electromagnetic field, removes power from the incident wave which is re-emitted as radiation. The amount of power re-radiated (or removed from the incident radiation) per unit incident flux (power per unit area) has dimensions of area, and is called the scattering cross section, σ. This quantity reflects the area of the target intercepted by incident radiation and the ability of the dielectric material to scatter a particular type of incident radiation, but is not a function of the intensity of the power source. In the case of a target of a given surface area, e.g. an electron or a nucleus in the case of Thomson scattering, the cross section directly indicates the ability of the target to scatter the incident radiation. The cross section can also be viewed as the equivalent area of the incident wavefront needed to produce the same amount of power as that re-radiated by the target. Since radiation may scatter in different directions (as measured from the direction of the incident radiation) at different dσ powers, the cross section is determined per unit solid angle as d , where  refers to the solid angle.This cross section is called the differential, or angular, cross section. Consider for a moment that the probability of an incident particle hitting a target is proportional to the geometric area projected by the target to the incoming particle, πR 2 , where R is the radius of the target. The radius of a nucleus, R, 1 as indicated by Eq. (1.15), is R = R0 A 3 with R0 = 1.2 fm. Therefore, R is on the order of a few 10−15 m or so, and the cross-sectional area of a nucleus, πR 2 , is on the order of 10−28 m2 = 1 b, where b refers to cross section’s unit of barn. At low particle energy, the de Borgile wavelength, Eq. (1.9), is typically larger than the radius of the target. Then the wavelength of the incident particle becomes the dominant factor in determining the cross section of interaction (as discussed in Section 3.2). In general, at high energy, the absorption and scattering cross sections, σa and σs , can both be expressed by a combination of the projection area of the incident particle and the geometric cross section of the target nucleus; with the λ former represented by a radius equal to its rational wavelength ( 2π = 1k ). That is, 2 2   1 1 σa = π +R +R σs = π k k  2 1 σt = σa + σs = 2π (3.1) +R k where σt is the total cross section and k = 2π λ is the wave number of an incident particle of wavelength λ. This equation indicates that the cross section is not only a property of the target, but is also a property of the incident radiation. At low energy, the wavelength is much larger than the target’s radius, i.e. λ >> R, and consequently σ >> πR 2 . That is, the interaction cross section is much larger than that of the geometric cross section, and a target appears to an incident particle as a “barn’’ appears to a bullet; hence the use of barn as a cross-section unit. It should be kept in mind though that low-energy interactions with the nucleus occur

3.1 Introduction

155

only with neutrons, since a charged particle needs a sufficiently high energy to overcome the Coulomb barrier of the nucleus. The total cross section is the cross sections of all possible interactions. For each type of interaction, or for a group of interactions, one can define a separate cross section. The classification of these cross sections is given in more detail in Section 3.7. The squared amplitude of a wave equals the power conveyed by the wave. Power is used to express the intensity of electromagnetic waves, and in the definition of the cross section, when dealing with particle radiation, power is replaced with intensity (number of particles per unit time), or flux (intensity per unit area). For a beam of monoenergetic particles, the beam power is simply the intensity times the particle energy. Nevertheless, keeping in mind that power and intensity are related, and that a particle can be represented by an equivalent wave (see Section 1.3), the squared amplitude of the particle–wave also represents the “power’’ of the particle, or more strictly speaking the strength of that particle–wave at a given moment of time and/or space. Upon proper normalization, this intensity is equal to the probability of finding the particle. We can then determine the cross section in terms of the squared amplitude of the wave associated with a particular interaction, with respect to that of the incident radiation. The cross section which indicates the probability of interaction has dimensions of area, while the probability is a dimensionless quantity. In order to reconcile these dimensional differences, let us consider one radiation particle encountering many targets, each with a cross section, σ, in an infinitesimal slab of thickness, dx. If there are N targets per unit volume, then N dx is the number of targets per unit area, and σN dx is the probability of this single radiation particle interacting with the targets in a unit area. In other words, σ is the probability of interaction of a single radiation particle in an infinitesimally small slab containing one target per unit area. The quantity  = σN has dimensions of inverse distance and is known as the macroscopic cross section as it represents the overall cross section for a medium containing N targets per unit volume. In turn, σ is the microscopic cross section. For a flux of φ particles per unit time per unit area, φ, is the interaction rate per unit volume. Since the cross section defines in essence the interaction probability, its value will depend on the field effect of the target on the projectile. The target is represented by its potential field, while a projectile is depicted as a wave, with a wavelength given by the de Broglie wavelength of Eq. (1.9). The interaction of particles with the nuclear potential is studied by quantum mechanics and various nucleus models, while interactions affected by the electromagnetic field of the atomic electrons or the nucleus are governed by electrodynamics. As indicated above, the cross section is the amount of power of electromagnetic radiation removed from an incident beam (and re-radiated by a target) per unit incident flux (power per unit area). Equations (2.199) and (2.200) gave, respectively, the power emitted per solid angle and the total power (over all solid angles), by a moving electric charge. Normalizing these powers by the energy flux gives the cross section as shown in Sections 3.5 and 3.6 for, respectively, photons and charged particles, interacting with the electromagnetic fields of atomic electrons of the nucleus.

156

Chapter 3 Cross Sections

However, the process of determining the cross section for electromagnetic interactions, and for that matter for interactions involving weak nuclear forces, is facilitated by the Feynman diagrams, discussed in Section 3.4. We first address (in Section 3.2), the cross sections of interactions involving the strong forces of nuclear fields. Cross sections for various specific interactions are then discussed. It should be kept in mind that the ensuing sections are intended to provide the theory behind the cross sections and their general behavior. Accurate values for the cross sections should be obtained from the cross section datasets outlined in Section 3.7. That same section shows how cross sections can be manipulated to provide cross sections for compounds and mixtures, and to average them over radiation energy.

3.2 Nuclear Cross-Section Models The effect of the nuclear forces is presented by a nuclear potential. This potential can be a simple square-well, with a constant potential energy over the entire radius of the nucleus, which is assumed to be spherical. However, in order to accommodate changes in projectile energy in scattering and allow absorption of the projectile particle, a complex potential well (with real and imaginary components) is typically assumed. This allows the use of the scattering model (discussed below in Section 3.2.1). More sophisticated models incorporate rotational (spin), vibrational, and dispersive energy effects. The absorption of a particle in a nucleus can result in the formation of a compound nucleus in an excited state, which can further decay by emitting an α particle, a β particle, γ-rays, etc. The cross section for the formation of the compound nucleus facilitates, therefore, the determination of reaction cross sections (as shown in Section 3.2.2). At high particle energies, the excitation levels of the compound nucleus overlap each other and appear as a continuum. This led to the continuum model of cross sections discussed in Section 3.2.3. At even higher particle energies, the excitation levels of the compound nucleus become so densely overlapped that particle emission from the compound nucleus resembles the release of vapor from a boiling liquid, hence the evaporation model of Section 3.2.4. Finally, we discuss the stripping and nucleonic collision models in Sections 3.2.5 and 3.2.7, respectively. The treatment below is mainly relevant to neutrons, which can reach the nucleus easily without being affected by the Coulomb field outside the nuclear potential well. However, the cross-section models are also applicable to high energy charged particles that can overcome the Coulomb barrier.

3.2.1 Optical model The potential model, also called the optical model, of the cross section assumes that the target is represented as a single entity by a potential well, and the incident particle is represented by a plane wave. The effect of the potential field on the incident wave is determined by quantum mechanics (Schrödinger equation). This

3.2 Nuclear Cross-Section Models

157

approach provides applicable models for elastic scattering and absorption, but not for reactions affected by the internal structure of the target, such as the inelastic scattering of neutrons. The potential model is discussed in detail in quantum mechanics textbooks. We will summarize here its basic aspects. The Schrödinger equation was introduced in Section 1.4. We will focus here on the steady-state equation, assuming that the target and its potential as well as the projectile do not change their internal structure during the time of observation. We can then write Schrödinger equation as:   2μ1,2 2 [T − U (r)] = 0 ∇ + (3.2) 2 where  is the wave function of the projectile, U (r) is the potential field of the target, r is the distance from the center of the target, T is the particle’s kinetic energy, and μ1,2 is the reduced mass of the incident particle and the target. The latter parameter is introduced here so that the analysis can be performed in the center-of-mass system (C ) (see Section 2.2). Non-relativistic mechanics is used here for simplification. Note that Eq. (1.19) when rewritten as Eq. (3.2) was modified to include both the particle’s kinetic energy, and the target’s potential to enable examination of the particle outside the potential field, U (r). The potential field is only dependent on the radial distance r, but neither on the axial distance nor the angle. This is due to conservation laws which dictate that the work done to move a particle from one point to another is dependent only on the position of the particle, r. The solution of Eq. (3.2) dictates, as discussed in Section 1.4, some specific discrete quantum states for the presence of a particle within a potential field. Here our interest is to determine the cross section of an interaction. We will do this by first considering a beam of free particles (i.e. away from the potential field), then examine the effect of that potential on the wave function of the free particles as they approach the field. A beam of free particles of wavelength λ, or wave number k = 2π λ , forms a λ = kl , plane wave that can be divided into a set of co-cylinders whose radii are l 2π where l is the angular momentum quantum number. That is, particles with an angular momentum quantum number l occupy the annulus confined between the cylinders of radii kl and l+1 k , which has a cross-sectional area of: π(l + 1)2 πl 2 π(2l + 1) − 2 = 2 k k k2

(3.3)

This also defines the number of free particles that can be found between angular momenta l and l + 1. However, there are no restrictions on the value of l as long as the particles are free from the effect of a potential field, i.e. l can be a continuous function. When the beam is subjected to the field of a nucleus, these particles will occupy orbits defined by l, l + 1, etc. The maximum number of particles that can be absorbed by the potential field provides an upper limit on the value of the

158

Chapter 3 Cross Sections

absorption cross section, represented by the area of the annular zones. For the lth zone, this upper limit is: π(2l + 1) σa,l |max = (3.4) k2 The incident wave satisfies the wave equation, Eq. (3.2), when the potential is zero or at r → ∞, i.e. when the effect of the potential is naught. Therefore, to obtain the wave function for free particles in the direction z, one must solve the equation: ∇ 2  + k2  = 0

(3.5)

, with k2 = 2mT 2

the reduced mass replaced by the particle mass, since this where √ is a free beam. For the plane wave under consideration, exp(ikz), with i = −1, would be an acceptable solution for Eq. (3.5), if the wavefront is propagating in the z direction. The number of particles per unit volume per unit time in this incident wave is |exp(ikz)|2 , since integrating the square of the function over volume gives the total beam intensity, or the number of particles per unit time. The solution exp(ikz) does not reflect the radial behavior of the wave that facilitates the determination of the cross section. It is, however, possible to expand the axial solution exp(ikz) in terms of the radial (r) and azimuthal (ϑ) components using, respectively, the spherical Bessel functions, jl (kr), and the Legendre polynomials, Pl (cos ϑ). The spherical Bessel functions are defined such that: sin kr sin kr cos kr j1 (kr) = − 2 kr (kr) kr  3 1 3 j2 (kr) = − − cos (kr) 3 (kr) kr (kr)2 2l + 1 jl (kr) − jl−1 (kr) jl+1 (kr) = kr j0 (kr) =

(3.6)

The Legendre polynomials are: P0 (cos ϑ) P1 (cos ϑ) P2 (cos ϑ) P3 (cos ϑ)

= = = =

1 cos ϑ 1 2 2 (3 cos ϑ − 1) 1 3 2 (5 cos ϑ − 3 cos ϑ)

(3.7)

The plane wave (z) can then be expressed as [1]: (z) = exp(ikz) = exp(ikr cos ϑ) =

l=∞ 

(2l + 1)i l jl (kr)Pl (cos ϑ)

(3.8)

l=0

The lth component in this expression is called the partial wave. In other words, the plane incident waves is composed of an infinite number of partial waves.

159

3.2 Nuclear Cross-Section Models

Potential scattering We will now examine the scattering of the plane wave represented by Eq. (3.8) by the radial potential field, U (r), called potential scattering. Recall that the analysis here is done in the center-of-mass system, C . The square of the amplitude of the scattered wave at an angle ϑ gives the probability of scattering at that angle. In the far field, i.e. away from wave interference effects near the potential field, the scattered wave, s , can be expressed by a plane wave propagating radially from the center of the field, but declines in amplitude with increasing distance. That is, s = f (ϑ) exp(ikr) , where f (ϑ) is a function that depends on the wave number, k, r of the incoming wave, as well as on U (r). The term exp(ikr) indicates that the scattering wave is propagating outward, in positive r (away from the center of U (r)), in the same fashion exp(ikz) propagates in the positive direction of z. The 1r term serves to conserve the number of scattered particles, which has to maintain the same value over the surface of a sphere of radius, r. Since the intensity of particles is proportional to the squared value of the magnitude of its wave, then the scattering wave has to be proportional to 1r . The scattered wave combined with the incident wave gives the total wave, which can be represented away from U (r), i.e. at kr >> l, by: f (ϑ) exp(ikr) (3.9) r That is, subtracting the incident wave, exp(ikz), from the total wave gives the scattered wave at kr >> l. We will try then to determine the total wave by solving the Schrödinger equation, Eq. (3.2), in the presence of the potential, then subtract the wave for free particles, exp(ikz). A solution for the total wave function, t , must include the effect of U (r). Analogous to Eq. (3.8), a solution for t can be expressed as: t = exp(ikz) +

t =

l=∞ 

(2l + 1)i l exp(iϕl )Rl (kr)Pl (cos ϑ)

(3.10)

l=0

where Rl is a radial function similar to jl (kr), but incorporates the effect of U (r), and ϕl is the phase shift of the lth partial wave caused by the scattering potential U (r). This phase shift is due to the presence of the potential U (r); independent of ϑ, but depends on the value of k. The value of ϕl can be numerically evaluated for given potential fields. The scattering function, f (ϑ), can be expressed as: f (ϑ) =

l=∞ 1  (2l + 1)[exp(2iϕl ) − 1]Pl (cos ϑ) 2ik l=0

=

1 k

l=∞  l=0

(2l + 1)exp(iϕl ) sin ϕl Pl (cos ϑ)

(3.11)

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Chapter 3 Cross Sections

Note that f (ϑ) is a complex function, with real and imaginary components, but the square of its absolute value can be written as [1]: ⎧! "2 ⎨ l=∞  2l + 1  2  f (ϑ) = f (ϑ)f ∗ (ϑ) = (sin 2ϕl )Pl (cosϑ) ⎩ 2k l=0 !l=∞ "2 ⎫ ⎬  2l + 1 (cos 2ϕl − 1)Pl (cos ϑ) + (3.12) ⎭ 2k l=0

where f ∗ is the conjugate of f . Recalling that the differential scattering cross section is the intensity of the scattered radiation per unit solid angle per unit flux, one can readily evaluate its value, given that we have expressions for the incident and scattered waves. The incident flux is |exp(ikz)|2 v particles per unit area per unit time, where v is the speed of the incident particles (assuming that they all have the same speed), since as indicated above the number of particles per unit volume per  2   unit time wave is |exp(ikz)|2 . Similarly, the scattered flux is  f (ϑ) exp(ikr)  v, r if in potential scattering the particle changes its direction, but not its velocity. The number of scattered particles crossing an element of area, dS, at some solid (defining a surface area dS = r 2 d), is therefore, angle, d, around a direction,   2  f (ϑ)   r exp(ikr) vr 2 d. Consequently, the differential scattering cross section can be expressed as: dσs = | f (ϑ)|2 d

(3.13)

The elastic scattering cross section, σes , is the integral of the differential cross section over 4π, with d = 2π sin ϑdϑ:  π σes = 2π | f (ϑ)|2 sin ϑdϑ (3.14) 0

Using Eq. (3.12), the above integral yields: σes =

l=∞ l=∞ l=∞     π  2 exp(2iϕl ) − 12 = 4π (2l + 1) (2l + 1) sin ϕ = σes,l l 2 2 k k l=0

l=0

l=0

(3.15) The same result can be arrived at by dividing the scattered current by the incident flux for each partial wave. The formulation of Eq. (3.15) for the cross section assumes energy is conserved in C , or in other words for particles remaining in

161

3.2 Nuclear Cross-Section Models

the total wave, t . It is, therefore, referred to as shape elastic scattering (hence the “es’’ subscript), in contrast to that caused by elastic scattering that occurs via, for example, compound nucleus formation. The maximum possible scattering cross section occurs when the scattered wave is phase-shifted by an angle π with the incident partial wave, since the partial wave is doubled in amplitude. Therefore, the maximum partial scattering cross section occurs when sin ϕl = 1, then: σes,l |max =

4π(2l + 1) k2

(3.16)

The absorption cross section can be obtained by dividing the absorbed (incoming–outgoing) current by the incident flux for each partial wave, to obtain [19, Sections V. C. & V. J.]: σa =

l=∞ l=∞    π  exp(2iϕl )2 − 1] = (2l + 1)[ σa,l k2 l=0

(3.17)

l=0

Notice that σa,l |max occurs when exp(2iϕl ) = 0, and is identical to the value obtained using geometric arguments (Eq. (3.4)). This is also four times the maximum scattering cross section (Eq. (3.16)). It should be kept in mind that this absorption cross section refers to particles “optically’’ removed from the incident beam, i.e. those that are not shape-scattered. Equations (3.15) and (3.17) enable formulating the cross section in terms of partial waves. These partial waves are designated, according to the terminology of Table 1.4, as s, p, d, . . . , for l = 0, 1, 2, . . . , respectively. For l = 0, the differential 2 dσ cross section for elastic s-wave scattering, according to Eq. (3.11), is d = sink2ϕ0 , which is independent of the angle of scattering, the scattering of the s-wave is, therefore, isotropic in the center-of-mass system, since the above analysis is performed in C . For s-scattering, the maximum scattering cross section, as Eq. (3.16) indicates, is 4π . For an s-wave, the scattering cross section is purely geometric, and, k2 therefore, depends on the size of the target nucleus, as well as the wave number of the incoming particle. This dependence can be expressed as: σes,0 =

4π 2 sin kR k2

(3.18)

where R is the radius of the target. At low particle energy, kR << 1 and Eq. (3.18) is reduced to: σes,0  4πR 2

kR << 1

(3.19)

which is the geometric cross section of the target. Then the scattered wave is not affected by the internal nature of the potential, but merely by its rigid (impenetrable) shape, which is assumed to be spherical.

162

Chapter 3 Cross Sections

Differential cross section dσ Equation (3.12) together with Eq. (3.13), define the differential cross section, d . The latter equation shows that the cross section is expanded in a series of the squares of the Legendre polynomials, P(cos ϑ). Since the squares of a polynomial are also function of the polynomial itself, dσ d is customarily expressed as an expansion of P(cos ϑ), so that:

 2l + 1 dσ = Bl (k)Pl (cos ϑ) d 4π l

(3.20)

l=0

where Bl s are known as the Legendre coefficients, and are dependent on k and ϕl : the particle’s energy and the nature of the target (its potential well).

3.2.2 Compound nucleus The absorption cross section, σa of Eq. (3.17), includes all interactions that involve the disappearance of the incident wave into the target. These interactions can be incorporated into the formation of a compound target, and σa becomes then the cross section for compound system formation. The fate of this compound system is determined by considerations other than the potential-field effects of the optical model. If the target is a nucleus, a number of reactions can take place as discussed in Section 1.7.3. One of the possibilities is the formation of a compound nucleus, which then decays in a number of ways. The compound nucleus can decay to the ground state of the parent target, emitting a particle of the same type as that of the projectile. Then, the total kinetic energy of the incident particle and the nucleus are conserved, and the reaction is in effect an elastic scattering. The cross section of this mechanism, σce , is added to that of the shape elastic scattering, σse of Eq. (3.15), leading to the elastic scattering cross section: σes = σse + σce

(3.21)

The compound nucleus may also decay by inelastic scattering, by emitting a particle identical to the incident particle while leaving the nucleus in an excited state. The cross section for this process is designated as σin . Naturally, the magnitude of a compound-nucleus cross section depends on the probability of forming the compound nucleus itself and the probability of its decay in the manner corresponding to the cross section. When an incident particle is trapped in the potential field of the target, i.e. captured by a nucleus, it resides in one of the excited states of the compound system. Therefore, the probability of forming a compound nucleus is largest when the kinetic energy of the incident particle, added to the potential energy released as the particle drops into the potential well, is equal to one of the excitation energy levels of the compound system. Therefore, the cross section for the formation of the compound nucleus, σC , is equal to the maximum absorption cross section,

163

3.2 Nuclear Cross-Section Models

σa |max of Eq. (3.4) multiplied by the penetrability of the potential well and the nuclear surface, also called the transmission coefficient, as it represents the fraction of the incident wave which is not reflected, i.e. scattered. That is, for the lth partial wave: π(2l + 1) σC,l |max = Tl (3.22) k2 where Tl is a transmission coefficient (0 ≤ Tl ≤ 1). If σa of the optical model is taken as the cross section for the formation of the compound nucleus, then comparing Eqs (3.17) and (3.22) shows that Tl = 1 − |exp(2iϕl )|2 . Therefore, exp(2iϕl ) is referred to as the reflection factor. At low particle energy, Tl = 0 for √ l > 0 and T0 is proportional to E, where E is the particle energy. Then σC is proportional to E1 , since k ∝ E. This is the so-called 1v behavior. When the energy of the incident particle coincides with that of an excitation level, a resonance occurs which increases the probability of forming a compound nucleus. Then the resonance effect is superimposed upon the transmission coefficient, which is a smoothly varying function with particle energy. Therefore, the cross section for the formation of the compound nucleus, σC,l , for a partial wave l around a resonance energy E0 for an incident particle of an energy E in the neighborhood of E0 , is formulated as: σC,l =

z,l  π(2l + 1) 2 2 k (E − E0 )2 + 

(3.23)

2

where z,l is the energy width of the resonance (full width at half-maximum) for the lth partial wave for the absorption of the incident particle, z, (in order for the compound nucleus to be formed), and  is the total width (see Fig. 3.1). This is known as the Breit–Wigner single level formulae, and is applicable when  << D, i.e. when two adjacent resonances do not overlap; the so-called resolved resonances. The energy width, , for a given resonance level in the compound nucleus is related to the mean lifetime, τ, of that level in the formed metastable compound nucleus, by the uncertainty principle (see Section 1.33), so that τ = . The optical model of Section 3.2.1 does not reflect the resonance effect caused by the intrinsic nature of the target nucleus, but gives the overall effect.Therefore,the sC

D E0

Figure 3.1 A schematic of cross-section resonances.

E

164

Chapter 3 Cross Sections

average of the resonance cross section can be related to that obtained from the optical model. For an average energy spacing D, with  << D, the average of Eq. (3.23) can be approximately expressed as: π(2l + 1) 2πz,l  σC,l  ∼ = k2 D

(3.24)

where · designates the average value. Comparing this average value to the expression of Eq. (3.23) for the same cross section, one gets: Tl =

2πz,l  D

(3.25)

The compound nucleus can decay by emitting a number of particles and by releasing γ-rays. Each permitted mode of emission is represented by a partial width b (not related to the partial wave), so that b is the fraction of transitions (decay channels) that results in the release of particle i. The cross section σb,l for the release of particle b is expressed as: σb,l = σC,l

π(2l + 1) b z,l b = 2 2  k (E − E0 )2 + 

(3.26)

2

Therefore, the cross section for resonance scattering, i.e. emission of the same type as the incident particle, is given by the cross section σ(z,z) obtained by setting b = z in Eq. (3.26). There are, however, two channels for this (z, z) interaction: elastic and inelastic. The elastic cross section provided by Eq. (3.26) must be added to that of the corresponding partial l shape elastic scattering of Eq. (3.15) to provide the total elastic scattering cross section. This addition of the amplitudes of the waves corresponding to resonance shape scattering can cause interferences that may lead to maxima and minima as the cross section changes with particle energy. The cross section for all possible channels of decay except that for resonance scattering is the reaction, or absorption, cross section, σa . The cross section for a reaction (n, b), including all partial waves, is given by: σb =

∞ 

σb,l

(3.27)

l=0

In the above expressions describing the cross section resonances, the intrinsic angular momentum (spin) of the target nucleus and the intrinsic spins of the colliding particle, were ignored. The effect of these factors can be complicated when applied to the derivation of the cross sections. It is, however, incorporated into a parameter called the statistical weight factor or coefficient, g. For a projectile of spin i, a target nucleus of spin J and a compound nucleus of spin JC , 2JC +1 g g = (2I +1)(2J +1) . The correction factor introduced into Eq. (3.26) is 2l+1 , which eliminates the 2l + 1 term from the relationship.

165

3.2 Nuclear Cross-Section Models

The resonance width is temperature dependent, since the thermal motion of the target nucleus affects the relative velocity of the incoming neutron with respect to the target nucleus. As the temperature is elevated, the width of the cross section resonance tends to widen while its peak value is reduced. This is called Doppler-broadening. Resolved resonances are encountered when the incident particle energy is in the eV to the tens eV range, where the spacing between nucleus excitation levels is on the order of eV. At particle energies above a few hundred eV, the resonances in heavy nuclei tend to overlap with each other, and become unresolved (indistinguishable as individual resonances).Then  is much larger than D, and the cross section is evaluated at some discrete energies and averaged over contiguous energy intervals.

3.2.3 Continuum theory At incident particle energies above about 1 MeV, the resonances of the compound nucleus become broader and more closely spaced, in effect forming a continuum. Also at high energy, the wavelength at the incident particle becomes small, and the geometric cross section of the target nucleus, πR 2 , begins to affect the cross section. In other words, the cross section will depend on both the inner structure of the target nucleus, as well its radius R. Then, theoretically, the compound nucleus cross section approaches an asymptotic value of: 

1 σC = π R + k

2 

B 1− T

(3.28)

where T is the kinetic energy of the incident particle in C , B is the Coulomb potential at a separation distance of R + 1k . Equation (3.28) shows a smooth behavior of the cross section with k, hence with incident particle energy. However, interferences between the incident wave and outgoing particle wave can occur, causing some maxima and minima in the cross section variation with energy. For neutrons, B = 0, and the absorption and the elastic scattering cross sections both

2 asymptotically approach the value of π R + 1k . Since k is quite large at high energy, σ approaches the value of the geometric cross section, πR 2 , which is in the range of a few barns.

3.2.4 Evaporation At high incident particle energy above about 15 MeV, or when  >> D, many levels of the compound nucleus overlap. Particle emission from the compound nucleus can then be seen as a continuous process, similar to liquid evaporation. The energy of the emitted particle will then be emitted with an energy equal at most to the excitation energy of the compound nucleus minus the separation energy of the particle (less the Coulomb barrier energy in the case of charged

166

Chapter 3 Cross Sections

particles). For charged particles, the cross section for particle emission is zero at incident particle energy below the threshold energy, the Q-value of the reaction. The compound nucleus can be left in an excited state with the particle energy emerging at a lower energy. The cross section for this evaporation process increases with incident particle energy, since the cross section for the compound nucleus increases with energy.When the excitation energy for the formation of compound nucleus is sufficient for the release of two particles, the cross section for the release of one particle decreases, while that for the emission of two particles increases. This manner of behavior repeats itself when three particles are emitted and so on. The type of particle emission (neutrons, protons, or a combination of thereof) depends on the inverse cross section of the emitted particle: the cross section with which the emitted particle would be absorbed if it were incident on the product nucleus resulting from the decay of the compound nucleus. The value of the inverse cross section is the same as that of the compound nucleus cross section (in the inverse reaction). The cross section for the emission of more than one particle then depends on whether the energy of the incident particle is sufficiently high to bring the compound nucleus to the excitation level needed to produce a number of particles. However, the cross section for neutron emission dominates over that for the emission of charged particles, since the latter particles face the Coulomb barrier. Among charged particles, proton emission is more likely than α-particle emission, due to the smaller charge of the former and the subsequently lower resistance it encounters through the Coulomb barrier.

3.2.5 Stripping The stripping process discussed in Section 1.7.3 causes an incident particle composed of more than one nucleon, such as a deuteron or a helium ion, to be stripped of one of its nucleons. The emitted particle then does not give an energy that correspond to any of the excitation levels of the compound nucleus, and is emitted forwardly with respect to the incident particle. In fact, the compound nucleus may be transformed to a product nucleus in the ground state. This will then increase the reaction cross section for the emitted particle, showing a peak beyond those corresponding to the resonances.

3.2.6 Photonuclear reactions Although photons usually interact with atomic electrons or are affected by the nuclear field without penetrating it, highly energetic photons can penetrate the nucleus and result in the emission of nucleons, α particles, or other particles. Noticeable at a relatively low photon energy are the photoneutron reactions: 2 H(γ,n)1 H and 9 Be(γ,n)8 Be; with threshold energies of 2.226 and 1.666 MeV, respectively, since these two nuclides have the lowest neutron separation energy among all nuclides. The cross section for photonuclear reactions exhibits the so-called giant resonances, or more preciously giant dipole resonances. This gamma-absorption

3.2 Nuclear Cross-Section Models

167

process is in essence the opposite of the electric dipole (E1) gamma-decay process described in Section 1.6.5. These resonances occur at a photon energy of 1 about 15 MeV in 208 Pb, and the resonance energy is proportional to R − 2 ,1 and 1 consequently, in accordance to Eq. (1.15), is also proportional to A− 6 , where R is the radius of nucleus and A is its mass number. It reaches about 25 MeV in 16 O. These resonances are attributed to the huge vibration of the neutrons in the nucleus as a collective versus that of protons, like two interpenetrating fluids oscillating toward, then away, from each other. This vibration process has a resonance frequency at which the absorbed photon excites the nucleus, causing it to emit a neutron, a proton, etc. The reason that 2 H is vulnerable to disintegration by photons is that it consists of one neutron and one proton, which in the ground state have parallel spin (with I = l + 12 , where l is the angular momentum quantum number and 12 is the spin). The anti-parallel level (I = l − 12 ) is an excited level which is unstable by about 65 keV [1]. The electromagnetic field of a photon incident on a deuteron causes a change in the total nuclear-angular momentum number by I = 1, resulting in a spin-flip. A parity change of the target nucleus can also be caused by the electric vectors of the incident photon (a dielectric dipole transition, E1), then the proton-neutron pair disintegrates with an angular distribution of sin ϑ, where ϑ is the angle with the Poynting vector of the photon. On the other hand, the magnetic vector causes no parity change (a magnetic dipole transition, M 1) and the disintegration becomes isotropic in C . The latter photomagnetic disintegration has a large cross section just above 2.2 MeV, while the cross section for photoelectric disintegration peaks above about 2.5 MeV [1]. The cross sections for the E1 and M 1 disintegration modalities are given by [1]: "3 ! B(Eγ − B) ρ2 (3.29) 3 Eγ (1 − ρr0 ) ⎡ ⎤ 1a 2   2 [1 − ] B(Eγ − B)  2πα ρ ⎣ (μp − μn )2 σ(M 1) = 2 ⎦ (3.30) 3 2mc Eγ 1 + 1 ak  dσ 3 1 (σ(M 1) + σ(E1)sin2 ϑ (3.31) = 4π 2 d 8πα σ(E1) = 3

where Eγ is the incident photon energy, α is the fine structure constant, B is the deuteron’s binding energy (2.225 MeV), 3 r0 is the effective range of the nuclear force for the triplet (2S + 1 = 3)2 state, 1 a is the scattering length of the nuclear 1 M.

Goldhaber and E. Teller, On nuclear dipole vibrations, Physical Review, Vol. 74, pp. 1046–1049, 1948. in a nucleus are strongly bound to those of the same angular momentum, same l. For l = 0, the nuclear forces result in S states of even parity. Singlet states have anti-parallel spin, total spin S = 0, while triplet states have parallel spin, S = 1. The total number of states is 2S + 1. Since each state has the same probability of occurrence, 2S + 1 is called the statistical weight.

2 Nucleons

168

Chapter 3 Cross Sections

potential (amplitude of neutron scattering wave at low neutron energy) for singlet states (S = 0, 2S + 1 = 1), μp and μn are the magnetic dipole moments3 of the proton and neutron, respectively, and m is the reduced mass of the neutron and 2m(Eγ − B) 2 the proton, ρ2 = 2mB , k2 = . The photoneutron emission from 9 Be can 2 be explained with similar arguments, given its low neutron separation energy.

3.2.7 Nucleonic collisions At particle energy above about 50 MeV, reaction cross sections do not vary much with energy, and emitted particles emerge in the forward direction with respect to the incident particle. This is attributed to the direct collision of the incident particle with individual nucleons in the target nucleus. The struck nucleons can themselves collide with other nucleons.

3.3 Neutron Cross Sections Neutron interactions in practical applications have an upper energy limit of about 15 MeV, corresponding to the maximum energy of fission neutrons and those produced in the fusion 2 H(t,n)4 He reaction (also used in neutron generators). Therefore, its kinematics (as indicated in Section 2.7), are dealt with using Newtonian mechanics. Nevertheless, the variable E is used here to express kinetic energy (not the total energy), to comply with the convention of neutron crosssection libraries. The dependence of cross sections on energy is often described by the excitation function (also called) the transmutation function. This excitation function, as the name indicates, reflects the excitation state of the target nucleus. Nevertheless, even for elastic scattering an excitation function exists, since elastic scattering can be the result of the absorption and subsequent re-emission of a neutron. The excitation function is a continuous function of energy when the neutron energy is sufficiently high to correspond to closely spaced, broad, and partially overlapped excitation levels. The resulting cross sections are then smooth functions of energy. At low energy, in the keV and eV range, discrete widely spaced nuclear excitation levels lead to resonances (peaks) in the neutron cross sections when the neutron energy coincides with one of the excitation levels. The resonance peaks at lower energy tend to be distinguishable from each others (resolved). At higher energies (above a few hundreds keV in heavy elements) the resonance peaks become less sharp and broader, and the resonances take a smoother appearance (become unresolved). At even higher neutron energy, the overlapping of resonances becomes complete and the continuum behavior of the cross section with energy dominates. The behavior of various types of cross sections with energy is discussed below. We begin first by presenting the overall = 2.793 μN , μn = −1.913 μN , μN = 3.152 × 10−8 eV/tesla = 1 nuclear magneton. These are due to the internal structure of the neutron and the proton.

3μ p

169

3.3 Neutron Cross Sections

reaction cross sections, i.e. the cross section that includes all product energies and directions. The differential cross sections with respect to angle or energy are then presented for separate sets of products in Section 3.3.6.

3.3.1 Elastic scattering Elastic scattering can be the result of potential scattering, in which the neutron does not penetrate the nucleus, or by resonance scattering which leaves the nucleus in the ground state. The potential scattering cross section, as Eq. (3.18) indicates, is purely geometric (for s-wave scattering), and, therefore, tends to be constant with energy. Potential scattering also is the dominant mode of scattering for light nuclei in most of the energy range, but eventually resonance elastic scattering takes place as the energy increases. In heavy nuclei, resonance scattering prevails at much lower energy than in light nuclides. The neutron cross section for elastic scattering at the kinetic energy E expressed as a summation of the partial wave cross sections in the lab frame of reference (L) [25] is: σ=

L 

σl (E)

(3.32)

l

where l is the angular momentum for incident neutron partial wave l, and L is the total number of l states, Nr (l,J ) 4π 2 π   σl = (2l + 1) 2 sin ϕl + 2 gJ σr,l k k J r=1 ⎡

⎢ σr,l = ⎣

2nr (E − Er )2 +

 2 − r 2

(3.33) ⎤

2[nr r sin2 ϕl − (E − Er )nr sin 2ϕl ] ⎥ ⎦  2 (E − Er )2 + 2r (3.34)

where ϕl is the phase shift of the partial wave l; k is the neutron wave number in C : √   2mn E A (3.35) k=  A+1 with A being the target’s mass number, or more accurately the ratio of the mass of the target nucleus to mn , the neutron’s mass; gJ is the statistical weight factor: gJ =

2J + 1 2J + 1 = 2(2I + 1) (2 × + 1)(2I + 1) 1 2

(3.36)

and represents the sum over projections of the spin of the compound nucleus resonance state, J , and an averaging over projections of the neutron spin, 12 , and

170

Chapter 3 Cross Sections

the spin of the target nucleus I ; nr is the neutron resonance width: nr =

Pl (E)nr (|Er |) Pl (|Er |)

(3.37)

with Pl being the penetration factor; Er is the resonance-peak energy: Er = Er +

Sl (|Er |) − Sl (E) nr (|Er |)) 2Pl (|Er |)

(3.38)

with Sl being a shift factor; r is resonance widths for resonance r: r (E) = nr + γr + fr + xr

(3.39)

with γr , fr , and xr , being the resonance widths for, respectively, the gamma, fission and any other competitive interactions, with the first summation carried over all possible values of angular momentum of the resonance states of the compound nucleus excited by a partial wave l; and Nr (l, J ) is the number of compound nucleus resonances produced by the incident partial wave l which has a spin J . The energy, |Er |, is expressed as an absolute value so that the same expressions can be used in inelastic scattering, since then the resonances below the threshold energy are at a “negative’’ energy in the inelastic reaction channel. The resemblance between the first term in Eq. (3.33) and that of Eq. (3.18) for the potential scattering cross section indicates that the first term in the former equation is for potential scattering summed over all possible partial waves. The phase shift (negative of that of a hard sphere), ϕl , is defined, in analogy with Eq. (3.18) in terms of ρ = ka, where k is the neutron’s wave number in L and a is called the channel radius at which the neutron wave function vanishes for the reaction channel. A neutron channel is defined by the quantum numbers, l, s, and J , where s is the channel spin, which is the vector sum of the neutron spin ( 12 ) and I . The second term in Eq. (3.33) relates to resonance elastic scattering; note that gJ replaces the orbital momentum terms in Eq. (3.26). In the resonance partial cross section Eq. (3.34), the first term is the direct resonance cross section (see Eq. (3.23)) the other two terms include interference effects between the potential scattering waves and the resonant waves (those causing excitation, hence compound nucleus formation) [19, Section V. E.]. The penetration factor (Pl ) is equal to ρυ, where υ is the probability of the neutron passing the potential barrier, considering a centrifugal barrier. Equation (3.37) indicates that the neutron width, nr , is inversely proportional to Pl , i.e. the higher the penetration factor the smaller the neutron width (hence the lower the chance of the compound nucleus formation). This enables evaluation of the cross section at energies around the energy at which the resonance peaks. The resonance energy of Eq. (3.38) is adjusted by an energy shift factor, to reflect that the intrinsic (energy dependent) resonance width of the target nucleus can be shifted due to the formation of the compound nucleus. Table 3.1 gives the values of ϕl , Pl , and Sl as function of ρ for the first few l-values. For resonance

171

3.3 Neutron Cross Sections

Table 3.1 Phase shift (ϕl ), penetration factor (P l ), and (energy) shift factor (Sl ) as function of ρ = ka, for first four values of l [25, 19] l

ϕl

Pl

Sl

0 (s-wave)

ρ

ρ

0

1 (p-wave)

ρ − tan−1 ρ

ρ3 1 + ρ2

−

1 1 + ρ2

2 (d-wave)

ρ − tan−1

3ρ 3 − ρ2

ρ5 9 + 3ρ2 + ρ4

−

18 + 3ρ2 9 + 3ρ2 + ρ4

3 (f-wave)

ρ − tan−1

15ρ − ρ3 15 − 6ρ2

225 +

−

675 + 90ρ2 + 6ρ4 225 + 45ρ2 + 6ρ4 + ρ6

ρ7 + 6ρ4 + ρ6

45ρ2

scattering, the summation over l extends up to l = 2 [25]. Note that Sl is zero for s-waves, and is negative for higher order partial waves. Figure 3.2 shows the resonance cross sections for 16 O. Notice the dip in the cross section at 2.37 MeV neutron energy. This dip is called an inverted resonance (or a cross-section window) and its presence can be deduced from Eq. (3.34). When the phase shift is π2 , and assuming that rn ≈ r , then for s-wave scattering and for gJ = 1, the resonance term in the cross section at E = Er is equal to − 4π , k2 which is equal in magnitude and opposite in sign to the potential scattering term. The negative resonance term is due to the interference effect. The fact that the dip in the cross section did not reach a zero value is due to the contribution of higher order partial waves. Equation (3.32) does not include interference between neighboring resonances. This is done by adding to Eq. (3.33) the term [25]:

 Nr (l,J ) r−1 2nr ns (E − Er )(E − Es ) + 14 r s π     gJ  2   2 k2 J  2 r=1 s=1 (E − E  )2 + r (E − Es ) + 2s r 2

(3.40)

where the summation over s includes all the r resonances. This is the multilevel formulation of the resonance cross section. When the resonances are unresolved, average values for the neutron widths are used, along with the energy spacing between resonances. Then the summation over r in Eq. (3.33) is replaced by [25]: ! " , 2π2  gJ n n 2 (3.41) − 2¯ nl,J sin ϕl ¯ l,J k2 J  l,J D

172

Chapter 3 Cross Sections

20

Cross section (b)

15

10

5

0

2

4

Figure 3.2 Elastic scattering for endfplot.shtml).

6

16 O,

8 Energy (MeV)

10

12

14

plotted using ENDFPLOT (http://atom.kaeri.re.kr/

where · designates an average over the number of sets of resolved resonance parameters (each set with its own J -value), and the subscript r is dropped, since ¯ is the average level spacing. Widths all resonances are considered at once, and D are determined at the neutron energy for which the cross section is evaluated. Energy averaging is assumed to eliminate the asymmetric term: E − Er .

3.3.2 Inelastic scattering Inelastic scattering is one of the possible outcomes of the de-excitation of the compound nucleus. Its resonance width is included in xr of Eq. (3.39). The single-level resonance scattering for a competitive reaction, x (i.e., any reaction other than elastic scattering, radiative capture or fission) is given by: σx =

L 

σx,l (E)

(3.42)

l

σx,l

Nr (l,J ) π   = 2 gJ k J r=1

nr xr (E − Er )2 +

 2

(3.43)

r 2

For inelastic scattering resulting from the excitation of the pth level in the compound nucleus, σn ,l is evaluated with [25]: xr (E) = n r

Pl  (E − Ep∗ ) Pl  (|Er − Ep∗ |)

(3.44)

173

3.3 Neutron Cross Sections

where l  is the orbital momentum of the scattered neutron (which can be different from l of the incident neutron) and Ep∗ is the threshold energy for level p in L, determined from the Q-value of the reaction as: (A + 1)Q (3.45) A where A is the ratio of the mass of the target nucleus to that of the neutron. For inelastic scattering to occur, the neutron energy, E, has to exceed the threshold energy, Ep . Therefore, the penetration function Pl  in Eq. (3.44) is set equal to zero, if its argument is negative (i.e. when E < Ep∗ ). Inelastic scattering is usually assumed to be isotropic in L, since there is no particular preference for emission from the excited nucleus. Ep∗ = −

3.3.3 Radiative capture In the resonance region, the capture cross section is expressed by the formulation: σγ =

L 

σγ,l (E)

(3.46)

l

where for a single level resonance: σγ,l =

Nr (l,J ) π   g J k2 J r=1

nr γr (E

− Er )2

+

 2

(3.47)

r 2

and in the unresolved region: σγ,l

, 2π2  gJ n γ = 2 ¯ l,J k  l,J D J

(3.48)

with the radiative capture width, γ , begin energy independent [25]. For lowenergy neutrons, l = 0, and the radiative capture cross section becomes: σγ =

π k2

nr γr (E − Er )2 +

 2

(3.49)

r 2

The maximum value of σγ occurs at E = Er . Let √ this maximum value be designated as√σ0 , and given that k is proportional to E, nr is proportional to k, hence to E, and rγ is constant with energy, Eq. (3.49) can be expressed as:  2  r 2 Er σγ = σ0 (3.50)  2 E r  2 (E − Er ) + 2

174

Chapter 3 Cross Sections

Thermal neutrons usually have energies well below the resonance energy for most nuclei, i.e. E << Er , then as Eq. (3.50) indicates, σγ is proportional to √1 , and E

consequently to 1v . This is the 1v behavior observed for the neutron absorption cross section for most nuclides. Note that this 1v behavior is also observed when the resonance is broad, i.e. when r >> (E − Er ).

3.3.4 Fission The resonance fission cross section is give by: σf =

L 

σf ,l (E)

(3.51)

l

For a single-level resonance: σf ,l =

Nr (l,J ) π   g J k2 J r=1

nr fr (E

− Er )2

+

 2

(3.52)

r 2

where the fission width, nf , is independent of energy. In the unresolved region: , 2π2  gJ n f σf ,l = 2 (3.53) ¯ l,J k  l,J D J The average total number of neutrons per fission at a given energy, ν(E), is typically expressed as a polynomial expansion of E [25]: ν¯ =

I 

ci E i−1

(3.54)

i

where ci s are the polynomial’s coefficients.

3.3.5 Charged-particle production The cross section for the production of a charged particle following a neutron absorption is also expressed by relationships (3.42) and (3.43), except that the resonance width for the charged particle takes into account that the produced particle can have different quantum numbers from those of the neutron or the target nucleus. Therefore, the width for a charged particle (z) at neutron energy E is expressed as: Pz l  s J  (E) xr (E) = zr (3.55) Pz l  s J  (|Er )

175

3.3 Neutron Cross Sections

where the primed quantum numbers indicate that the emitted charged particle has angular momentum and spin values that are different from those of the incident neutron.

3.3.6 Energy and angular distribution Elastic scattering The differential cross section for elastic scattering in the interaction 2(1,3)4 from an energy E1 and direction  1 to energy E3 and direction  3 is reported in terms of the Legendre coefficients in C to decouple the energy–angle dependence dictated by the interaction’s kinematics (see Eq. (3.20)). Therefore, σ(E1 )  dσ(E1 → E3 ; μ) = (2j + 1)Bj (E1 )Pj (η) d 4π j=0  1−α 1−α E1 − E1 η ×δ E3 − E1 + 2 2 J

(3.56)

where σ(E1 ) is the total scattering cross section at energy E1 , μ =  1 ·  3 , η =  1 ·  3 = A1 [μ (A2 − 1 + μ2 ) − 1 + μ2 ] with the primed values being in C 2 (as per Eq. (2.141)), α = ( A−1 A+1 ) , A is the ratio of the target mass to the neutron’s mass, and δ is the Dirac delta function expressing energy–momentum conservation for elastic scattering, Pj is an ordinary Legendre polynomial of order j , and Bj is jth Legendre coefficient in C . Equation (3.56) shows that the differential cross section is an inseparable function of energy and angle; i.e. it cannot be expressed as the multiplication of two functions, one for energy and the other for angle. The choice of the number of coefficients, J , in the Legendre expansion depends on the mass of the target nucleus and the neutron energy. However, some rough criteria are available to estimate the required order of expansion in C . Neutron scattering can be assumed to be isotropic ( J = 0), spherically symmetric (s-wave), in C , if the neutron de Broglie wavelength is greater than its radius [2], or equivalently, if: E1 < 10 A− 3

2

(3.57)

where E1 is in MeV. This p-wave scattering approximation ( J = 1), cosine distribution in the C , is valid when [2]: 10 A− 3 < E1 < 40 A− 3 2

2

(3.58)

This is true for the potential scattering component of the cross section, i.e. without considering resonance scattering.

176

Chapter 3 Cross Sections

In general, the required number of coefficients may be chosen such that it satisfies the inequality [2]: E1 < 10(J + 1)2 A− 3 2

(3.59)

The use of the s-scattering or p-scattering approximations in C is adequate for a large number of nuclides and over a wide range of incident-neutron energies; as relationships (3.57) and (3.58) indicate. For hydrogen, the assumption of isotropic scattering in C is valid up to 10 MeV neutron energy, as relationship (3.57) indicates. Since hydrogen is an effective slowing-down element which brings high energy neutrons to below the above energy limit after a few collisions, isotropic scattering in C can be applied to hydrogen-containing materials without much loss of accuracy. However, this assumption is not valid for thermal neutrons where chemical binding becomes important (see Section 3.3.7). The p-scattering approximation accommodates angular distribution bias toward either forward or backward directions. Then, the coefficient B1 is negative when backscattering is predominant, and assumes a positive value when forward scattering dominates. The bias in the forward 2 direction is on the order of 0.07 A 3 E1 , with E1 measured in MeV [26, XVIII]. The differential cross section of scattering in L is expressed in terms of the angle cos−1 μ. For large mass number materials, μ ≈ η, as L and C almost coincide, the Legendre coefficients of Eq. (3.56) can be directly used. However, for light nuclides, the transformation of the Legendre coefficients from L to C is not as straightforward. In L, the differential cross section of scattering can be expressed as: L dσ(E1 → E3 ; μ) 1  Sl (E1 → E)Pl (μ) = d 4π

(3.60)

l=0

where Sl is the lth Legendre coefficient in L, which is related to that in C , Bj in Eq. (3.56), by [26]: J  (2l + 1)σ(E1 ) Sl (E1 → E3 ) = Pl [μ0 (E1 → E3 )] (2j + 1)Bj (E1 )Pj (η) (3.61) (1 − α)E1 j=0

   E3 A−1 E1 where μ0 (E1 → E3 ) = A+1 − 2 E1 2 E3 , Eq. (2.217), expresses the energy–momentum conservation in L. For isotropic scattering in C , only the first term in the summation of Eq. (3.56) is required, i.e. J = 0. Equations (3.60) and (3.61) then give: L  dσ(E1 → E3 ; μ) σ(E1 ) = (2l + 1)Pl (μ)Pl [μ0 (E1 → E3 )] (3.62) d 4π(1 − α)E1 l=0

177

3.3 Neutron Cross Sections

In C :  1−α 1−α dσ(E1 → E3 ; μ) 1 δ E1 − E3 + E1 − E1 η = 4π 2 2 d

(3.63)

The right-hand sides of Eqs (3.62) and (3.63) can only be equal to each other when L approaches infinity. Then the summation in Eq. (3.62) approaches a delta function, δ(μ − μ0 ) [27]. A very large number of coefficients in L is, therefore, needed to approach isotropic scattering in C . Therefore, the use of a limited number of coefficients in L, for low mass number nuclides, leads always to an approximate treatment of the differential cross section. For heavy nuclides, L and C almost coincide with each other, and η and μ become nearly equal. Then Sl can be expressed using Eqs (3.56) and (3.60) as:  1−α E1 Sl (E1 → E3 ) = σ(E1 )(2l + 1)Bl (E1 → E3 )δ E3 − E1 + 2 1−α E1 η − (3.64) 2

Neutron-producing-reactions Neutron production includes inelastic scattering, prompt neutron emission from fission, delayed fission neutrons, (n,2n) and (n,3n) reactions. In these interactions, neutrons are assumed to be emitted isotropically, since the energy of the emitted neutrons is decoupled from that of the incident one. The differential cross section is then expressed as [25]: dσ(E1 , E3 ) = νσ(E1 )p(E1 → E3 ) d3

(3.65)

where σ(E1 ) is the interaction cross section, ν is the number of neutrons generated in the interaction (1 for inelastic scattering, 2 for (2,2n), or ν¯ (E1 ) of Eq. (3.54) for fission), and p(E1 → E3 ) is a normalized distribution function (per unit energy). The distribution p(E1 → E3 ) can be composed of the combination of several analytical formulations, each described by a function, fk (E1 → E3 ). Then, at a particular incident energy, p(E1 → E3 ) can be expressed as: p(E1 → E3 ) =

K 

pk (E1 )fk (E1 → E3 )

(3.66)

k

where pk (E1 ) is the weight given to the function fk (E1 → E3 ). The function f (E1 → E3 ) can assume a number of analytical formulations, ranging from an arbitrary tabulated function to a Maxwellian or a Watt spectrum [25]. The average energy of the emitted neutrons must be lower than the available energy for the reaction. The latter energy is equal to the incident energy plus or minus the

178

Chapter 3 Cross Sections

reaction’s Q-value (depending respectively on whether the reaction is exoergeic or endoergeic). That is: Eavailable = E1 +

1+A Q A

(3.67)

with the Q-value adjusted to the L system. The available energy is in turn larger than νE¯3 , where E¯3 is the average energy of produced neutrons.

Gamma production The energy distribution of γ-rays produced by neutron capture or elastic scattering is determined by the yield of the decaying level in the compound nucleus. The differential cross section for the production of γ-rays of energy Eγ by the inelastic scattering of a neutron of energy E, can be expressed as [25]:  dσ(E1 , Eγ )    = δ(Eγ − {εj − εi })Aij σm (E1 ) Rmjα dEγ m α i j α−j

(3.68)

where σm (E1 ) is the cross section for excitation level m in the compound nucleus by a neutron of energy E1 , the delta function excludes the energy levels which do not produce γ-rays of energy Eγ by transition from level j of energy εj to level i of energy εi , and Aij is the γ-ray branching ratio of transition from j to i, and Rmjα is the probability a nucleus excited at level m will de-excite to level j in α transitions, which is the sum of the products of the α branching ratios from level m via intermediate levels to level j. For gamma production by neutron capture, the photon production cross section is given in the form of tables of the neutron energy, E1 , and multiplicity, yk (E1 ), where k refers to a particular photon energy, and yk (E1 ) is the number of photons produced (yield) given by [25]: yk (E1 ) =

σγ,k (E1 ) σγ (E1 )

(3.69)

where σγ is the radiative capture cross section. Equation (3.69) is applicable to discrete energy gammas, and is integrated over energy for a continuum photon spectrum. Alternatively, a differential cross section is given by [25]: dσγ,k (E1 , Eγ ) = σγ,k (E1 )fk (E1 → Eγ ) d

(3.70)

where σk is the photon production cross section of a particular discrete photon or a photon continuum, and f (E1 → Eγ ) is a normalized distribution function.

179

3.3 Neutron Cross Sections

Charged-particle production When the energy–angle distribution are coupled by interaction kinematics, as is the case in the production of charged particles or when considering the residual nucleus left after a neutron reaction, the energy and angular distribution is described by the production cross section [25]:   1 [σi (E1 )y(E1 )fi (μ, E1 , E3 )] (3.71) σ(E1 , E3 , μ) = 2π where i defines a certain reaction product, σi is the interaction cross section at the incident energy, yi , is the product yield or multiplicity, fi is the normalized angle– energy distribution of the product per unit energy per unit angle cosine, and 1 the factor 2π indicates that the products are assumed to be emitted isotropically in the azimuthal direction (a reasonable assumption for the unpolarized neutrons encountered in most applications). Similar to neutron elastic scattering, Eq. (3.56), fi (E1 , E3 , μ) for a two-body interaction involving incident neutrons can be simply expressed as pi (E1 , μ), since μ and E3 are related to each other by the interaction’s kinematics. Then Legendre expansion can be used to decouple energy and angle so that [25]:  p(E1 , μ) =

fi (E1 , E3 , μ)dE3 =

L  2l + 1 l=0

2

al Pl (μ)

(3.72)

where al is the lth Legendre coefficient associated with the lth Legendre polynomial, Pl (μ). Note that μ extends in value from −1 to 1 with a total range of 2, and for isotropic scattering L = 0, and p(E1 , μ) = 12 , a0 = 1, and P0 (μ) = 1. Gamma emission is generally considered to be isotropic. However, the photon angular distribution can be expressed as [25]: pk (E1 , μ) =

L  2l + 1 l=0

2

al,k Pl (μ)

(3.73)

where k designates a certain discrete photon or photon energy distribution and al,k is the lth Legendre coefficient.

3.3.7 Thermal neutrons All the neutron cross sections discussed above are for a single nucleus, i.e. chemical molecular bond effects are not considered. This is quite valid at neutron energies above the chemical binding energy (a few eV). For thermal neutrons, the thermal motion of the target nuclei and the molecular structure should be accounted for. The lattice structure of a solid can lead to coherent scattering of the neutrons, the cross section for which are discussed in Section 3.3.7. By definition, thermal

180

Chapter 3 Cross Sections

neutrons are in “thermal equilibrium’’ with the medium in which they travel. Therefore, thermal neutrons can gain or lose energy as they traverse a medium. The state of the target atom, as being bound in a crystal lattice or in a chemical compound, affects the thermal neutron’s interaction probability, i.e. cross sections. For example, for water a cross section obtained by adding the contribution of its individual elements (H and O) will result in values different from those measured at the thermal neutron energy at a given medium temperature. The motion of thermal neutrons resembles that of a dilute gas, and the distribution of the kinetic energy of thermal neutrons can be described by the Maxwell–Boltzmann distribution, derived from the kinetic theory of gases. The probability of neutrons having an energy E is expressed in the form [20]: √   n(E)dE 2π E E = dE (3.74) 3 exp − n kT (πkT ) 2 where n(E)dE is the number of neutrons of energy from E to E + dE, n is the total number of neutrons in the system, k is the Boltzmann constant, and T is the absolute temperature. This distribution gives an energy kT corresponding to the most probable velocity, or 0.025 eV at 300 K.The temperature, T , associated with a Maxwellian distribution of a neutron population, is also called the“neutron temperature’’.

Incoherent inelastic scattering Chemical binding increases the scattering cross section for thermal neutrons, but barely affects the absorption cross section [27]. Scattering of thermal neutrons causes the molecule to recoil as one unit. Since the mass of a molecule is typically larger than that of the neutron, C and L coincide. This type of scattering is considered to be incoherent inelastic scattering, when the neutron is viewed as a wave, since neutron waves do not interfere with each other. The interaction is inelastic, since the neutron waves exchange energy with the molecules. Scattering by bound atoms becomes isotropic in L, if it was isotropic in C [27]. For these two isotropic situations, the differential scattering cross section for bound and free atoms is expressed as: dσs,f (η) σsf = (3.75)  d 4π σsb dσs,b (μ) = (3.76) d 4π dσs,f (η) dσs,b (μ) dμ (3.77) dη =  d d where4 σfs and σbs are the scattering cross sections for free and bound atoms, respectively, which are respectively isotropic in C and L, the subscript s refers to 4σ fs

is also expressed, similar to Eq. (3.19), as σfs = 4πb2 , with b referred to as the neutron scattering length.

181

3.3 Neutron Cross Sections

scattering at all angles, and η and μ are the cosines of scattering angles in C and L, respectively. However, when η = μ = 1, the scattering angle in both C and L is dσfs (1) dσbs (1) equal to zero, and no change in neutron energy would occur.Then d  = d , and using Eqs (3.75) to (3.77):   dσfs (1) dη  dσbs (1) dη  σbs = 4π = σfs = 4π d d dμ η=μ=1 dμ η=μ=1 Now using the relationship between η and μ given by Eq. (2.141):  η = A1 (A2 − 1 + μ2 )μ − 1 + μ2 , one obtains: 

1 σsb = 1 + A

2 σsf

(3.78)

The bound cross section of scattering for thermal neutrons is, therefore, higher than that for free atoms, and the difference between them is highest at lowest A values. Therefore, in low mass-number nuclides, particularly those used in neutron moderation, attention must be given to the evaluation of the scattering cross section for thermal neutrons. Using the free-gas model, the differential cross section for thermal neutrons is described by the so-called S(α, β) treatment. With the usual assumption of isotropic scattering in the azimuthal direction, the double differential cross section with an atom of mass number A is given by [25]:     Mn σbs,n Ef d2 σ β (Ei → Ef , μ, T ) = Sn (α, β) (3.79) exp − d dE 4πkT Ei 2 n with

 Ei + Ef − 2μ Ei Ef

Ef − Ei AkT kT where the summation is over the atoms in the compound, Ei and Ef are, respectively, the initial and final energies of the neutron at a thermal temperature T , Mn is the number of atoms of type n in the chemical compound, σbs,n is the bound cross section for atoms of type n. In Eq. (3.80), β is a normalized change in the energy of the neutron, while α is proportional to the square of the change in momentum. Recall that σbs,n can be obtained from those for free atoms using Eq. (3.78). Tabulated values of S(α, β) are available for some neutron moderating materials, such as light water, heavy water, graphite, and polyethylene, at some selected temperatures [25]. However, in its simplest form for a free gas, S(α, β) is expressed as:   2 1 α + β2 S(α, β) = √ (3.80) exp − 4α 4πα Obviously for bound compounds, this scattering function is more complex. α=

and

β=

182

Chapter 3 Cross Sections

Under thermal equilibrium, the cross section for energy gain is related to that of energy loss by:   dσ(Ei → Ef , μ, T ) dσ(Ef → Ei , μ, T ) −Ef −Ei Ei exp = Ef exp d dE kT d dE kT (3.81) That is, S(α, β) is invariant under the transformations Ei → Ef and β → −β.

Incoherent elastic scattering In solids, the atoms are sufficiently close to each other that a low-energy neutron, behaving as a wave of a large wavelength, would be influenced by more than one atom at a time. Neutron waves can then be elastically scattered, without any change in energy. In non-crystallized solids with lattice structure, this elastic scattering is incoherent, i.e. there is no interference between neutron waves, since the atoms are fully well ordered.The same effect is observed in hydrogenous solids, such as polyethylene, water ice, and zirconium hydride. The double differential cross section for elastic incoherent scattering of neutrons by a nucleus of mass number A is expressed as [25]:  d2 σs,inc (E1 → E3 , μ, T ) σsb W (T ) exp −E1 (1 − μ) δ(E1 − E3 ) (3.82) = d3 dE3 4π A where use is made of the usual terminology, W is a temperature-dependent factor known as the Debye–Waller integral and accounts for the reduction in scattering by motion of atoms around their equilibrium positions, and the delta function reflects the fact that there is no change in the neutron energy. This scattering is isotropic, since the scattered neutron waves are randomly distributed in phases. The total cross section for this interaction integrated over energy and angle is:  1 − exp − 4WA(T ) (3.83) σs,inc (E) = σsb 4W (T ) A

The Debye–Waller integral is such that:  W (T ) = 0

ωmax

ρ(ω) ω

!

ω 1 + exp(− kT ) ω 1 − exp(− kT )

" (3.84)

where  ωmax ρ(ω) is the frequency (vibration) spectrum of the target atoms, ρ(ω) = 1, where ωmax is the maximum frequency. At zero absolute temper0 ature, W = 0 and σs,inc (E) = 0. The incoherent elastic cross section is smaller than the bound cross section for incoherent inelastic scattering given by Eq. (3.78).

183

3.4 Electrodynamics

Diffraction In a crystalline solid, such as graphite, beryllium, or a ceramic, Bragg neutron diffraction can take place in a manner similar to that of X-rays (see Section 2.8.3). Diffraction is a coherent scattering process, which favors particular directions, at which the scattering cross section peaks. The scattering cross section is expressed as [25]:  d2 σcoh (E1 → E3 , μ, T ) σsb  W (T ) = fi exp −4E1 d3 dE3 E1 E <E A i

1

× δ(μ − μi )δ(E1 − E3 )

(3.85)

where fi is related to the structure factor of the crystal (see Section 3.5.4), μi = μ − 1 − EEi , and Ei is the Bragg peak (or edge), and the other parameters are as defined in Eq. (3.82). The δ(μ − μi ) function ensures the satisfaction of Bragg’s law of diffraction, Eq. (2.214), expressed in equivalent energies, and the fi factors are calculated from the properties of the lattice structure and its constituent atoms. The Debye–Waller integral accounts for the disturbance of an idealized lattice structure, introduced by atomic thermal motion. The cross section integrated over all angles and energies is:  W (T ) σsb  σcoh = (3.86) fi exp −4E1 E1 E <E A i

1

3.4 Electrodynamics 3.4.1 Quantum electrodynamics The Schrödinger equation, Eq. (1.19), was given for a free particle, approaching a potential field. We will generalize this equation to the form: i

∂ = H ∂t

(3.87)

where H is the Hamiltonian operator.This operator has an eigenvalue equal to the total energy. For states consisting of photons only, where there is no interaction between the  photons, H is the summation of the Hamiltonian for each state, Hλ , i.e. H = Hλ . The eigenvalues of Hλ are Eλ = nλ hνλ , where νλ is the photon’s frequency and nλ is the number of photons of wave function λ . An interaction that causes a photon absorption obviously increases the number of photon states by 1, while a photon emission decreases the number by 1. The quantum state of a particle includes negative energy eigenstates (holes), since the energy of a

184

Chapter 3 Cross Sections

 free particle E = ± p2 c 2 + mc 2 . This is known as the Dirac electron theory. Since it is not physically possible to observe an electron with a negative energy, all negative-energy electron states are assumed to be fully filled with electrons, and transitions into these states are not permitted. Consequently, the state in which all the negative-energy states are occupied, and no positive-energy state exist, is a zero-energy, zero-momentum, and zero-charge state. As such, electrons in the negative-energy state neither contribute to the physical field, nor to the system’s charge, energy and momentum. Transition between states of positive and negative energies allows for the process of electron–positron production and the mutual annihilation of particles by anti-particles. For instance, the creation of a positron and a negative electron from a photon in pair production is attributed to the liberation of an electron from the virtual sea of negative-energy states, with the hole left in this sea becoming the positron. On the other hand, in positronelectron annihilation, the hole in the sea of negative-energy states is filled with an electron in a positive–energy state, annihilating both the positron and the electron. Comparing Eqs (3.87) and (1.19), it is obvious that for a free particle (i.e. in a 2 2 zero potential field), H = − 2m ∇ . The Hamiltonian for a charged particle in an electromagnetic field is defined as follows: H ≡ zeφ +

 2 (mc 2 )2 + | pc 2 − ze A|

(3.88)

are, respectively, the electric and vector magnetic potentials, where φ( r ) and A m is the particle’s rest mass, and ze is its charge. Equation (3.88) includes the potential energies of the electric and magnetic fields, the kinetic energy (via the momentum p) and the rest-mass energy. If the kinetic energy is small compare to 2 , the Hamiltonian can the rest-mass energy, or in other words, mc 2 >> ( pc 2 − ze A) 2 2 − ze A) + mc 2 . Since the Hamiltonian is an operator be approximated by zeφ + ( pc 2mc 2 on the wave function , the constant value in the operator has no effect on φ, and for non-relativistic conditions H is simply reduced to: H ≡ zeφ +

2 | pc 2 − ze A| 2mc 2

(3.89)

For a photon, the relativistic Hamiltonian of Eq. (3.88) must be used, but obviously with mc 2 = 0. The Hamiltonian of Eq. (3.88) represents one particle. For an interaction involving more than one particle, each particle is represented by its Hamiltonian, and the total Hamiltonian for all particles will be simply the summation of the Hamiltonian of individual particles. It is convenient to express the Hamiltonian into three components: He corresponding to charged particles, Hγ to photons, and Hint to the electromagnetic field in which they interact. The Hamiltonian H0 corresponds to the free (non-interacting) photons and electrons. Therefore,

185

3.4 Electrodynamics

the Hamiltonian H0 satisfies the Schrödinger equation: ∂ (3.90) = H0  ∂t

The transformation:  = exp iH0 t , is used to arrive at a time-dependent

description of transfer interactions. The solution for exp iH0 t gives the energy quantum number, n. It is straightforward to show that:     iH0 t ∂ iH0 t ∂ i = i  exp − iH0 exp  (3.91) ∂t  ∂t  i

Therefore,  is time dependent. However, the interaction Hamiltonian, Hint , alters  and  , as the Schrödinger equation becomes: ∂ = (H0 + Hint ) ∂t

Now the representation:  = exp iH0 t , gives:     ∂ iH0 t iH0 t = (H0 + Hint ) exp  − H0 exp  i ∂t     iH0 t Hint  = exp  i

(3.92)

(3.93)

where use was made of Eq. (3.92). In essence, one needs to solve for: i

∂   = Hint ∂t

(3.94)

with:

    iH0 t iH0 t Hint exp − (3.95) = exp   The focus here is on solving for  , since the squared value of its magnitude, properly normalized, determines the cross section of the interaction, and Eq. (3.94) enables the evaluation of the transition probability between the initial and final states of an interaction. The solution of Eq. (3.94) can be expressed as a series expansion of the eigenstates of H0 , i.e. the states of the unperturbed (non-interacting) system entities. If an eigenstate of H0 is n , then:  bn (t)n (3.96)  (t) =  Hint

n

Recalling that the squared value of the amplitude of the wave function is a measure of probability, then |bn (t)|2 is the probability that the system is still in the

186

Chapter 3 Cross Sections

unperturbed state at time t. The discrete nature of Eq. (3.96) enables the expres as a matrix in which an element H  sion of the Hamiltonian Hint int nm represents a transition from state m to state n. Equation (3.96) in Eq. (3.94) gives: ∂bn (t)   i = Hint nm bm (t) (3.97) ∂t m The Hamiltonian H0 represents free states; therefore, for an eigenstate n its eigenvalue is equal to En . Consequently:  i(En − Em )t  Hint nm = Hnm exp (3.98)  where Hnm is a matrix element in Hint . Then, Eq. (3.97) becomes:  ∂bn (t)  i(En − Em )t i = bm (t) Hnm exp ∂t  m

(3.99)

For weak interactions, such as those of electromagnetic interactions, it is sufficient to arrive at an approximate solution for Eq. (3.97). We will consider the case of one entity in the initial state (t = 0), then two entities, and so on. If at time t = 0 there is only one entity (say an atomic electron in an excited  can move to one of many n state) in state n = 0, which upon perturbation by Hint states (say moves to another lower-excitation state), then b0 (0) = 1, and bn (0) = 0 for n = 0. To a first order approximation involving only transition into and from state 0, Eq. (3.99) gives:  ∂bn (t) i(En − E0 )t i = Hn0 exp b0 (t) (3.100) ∂t  and

 ∂b0 (t)  i(E0 − En )t i = bn (t) H0n exp ∂t  n

(3.101)

Integrating Eq. (3.100) over time to obtain bn (t) is facilitated if we assume that the interaction is so immediate within the interaction time that b0 does not change, i.e. remains equal to unity. Then:  i(En −E0 )t   t 1 − exp  Hn0 i(En − E0 )t bn (t) = (3.102) dt = Hn0 exp i  t=0 En − E0  The probability of transition from state 0 to n is then |bn (t)|2 . In order to conserve energy between the initial and final states, En must be equal to E0 , which gives the transition rate [28]: n0 =

1 − cos[(En − E0 ) t ] |bn (t)|2 = 2|Hn0 |2 t(E0 − En )2 t

(3.103)

187

3.4 Electrodynamics

On the limit, as t → ∞ (or when t >> E0 ), n0 = 0 except when En = E0 . How ever, the integral of n0 over all energies, including E0 , is equal to 2π Hn0 dEn .  If the states have a continuous spectrum, so that there are ρn dEn states with an energy interval dE around En , then one can express the transition rate as: 2π |Hn0 |2 ρn (3.104)  Note that according to Eq. (3.88), Hn0 is proportional to ze; i.e. in the case of an electron Hn0 is proportional to e. Therefore, n0 is on the order of e 2 , which is a first order approximation (first order perturbation) for evaluating the transition rate (probability per unit time) that does not allow for the change of two initial states. Let us consider an initial state that contains two entities, say a photon and an electron, leading to a final state of also two entities, e.g. a scattered photon and a scattered electron. In order to allow the final states to be different from the initial state, we have to allow the photon and the electron to go through a temporary intermediate state that permits the disappearance of the incident photon and the re-emergence of another photon. We can do this with photons, since as indicated earlier the absorption and emission of a photon is accomplished between states by simply changing the number of photons from one state to another by one. This intermediate state is a virtual state, let us call it state n . Then, state n has one more photon than state 0, while state n has one less photon than state n . Analogous to Eqs (3.100) and (3.101), one has:  ∂bn (t) i(En − E0 )t i = Hn 0 exp b0 (t) (3.105) ∂t  n0 =

and

 ∂bn (t)  i(En − En )t i = bn (t) Hnn exp ∂t  

(3.106)

n

At the initial state, state 0, one still has b0 (0) = 1 and bn (0) = 0 when n  = 0. Similar to Eq. (3.102), one has:  i(E −E )t exp n  0 bn (t) = Hn 0 (3.107) E0 − En Notice that the initial condition bn (0) = 0 is not satisfied here, since it is not necessarily an intermediate state. Another way of looking at this is that Hn 0 does not exist at t = 0, as it is only temporarily meaningful. With Eq. (3.107), Eq. (3.106) becomes:  i(En −E0 )t exp   ∂bn (t) Hnn Hn 0 (3.108) i = ∂t E0 − En  n

188

Chapter 3 Cross Sections

However, by comparing Eqs (3.108) and (3.104), one can see that these two operations can be combined into one compound operation with a Hamiltonian matrix element:  Hnn Hn 0 Kn0 = (3.109) E0 − En  n

Since each Hamiltonian element is on the order of ze, or e for an electron, the compound operator is on the order of e 4 . Now similar to Eq. (3.104), the transition probability per unit time becomes:  2 2π  Hnn Hn 0  n0 = (3.110)   ρn    E0 − En  n

This second order perturbation can be extended to many orders by considering multiple intermediate states. Obviously this is only possible when E0 − En  = 0, where now n refers to any of the many intermediate states considered. The Hamiltonian of Eq. (3.95) has no diagonal elements, which implies a change in photon’s number from the initial to the final states by ±1 (emission or absorption). If there is no change in the number of photon states, but photons lose energy, a second order perturbation needs to be introduced. To accommodate a change of state energy from En to En + En , while still satisfying Eq. (3.106), an intermediate state, n , is introduced so that:  ∂bn (t) i(En − En )t i = Hn n exp bn (t) ∂t   i(E −E − En )t exp n n (3.111) = cHn n En − En + En  −i En t (3.112) bn (t) = c exp  Notice the analogy with Eq. (3.107) and the fact that if En = 0, bn (t) = c, a constant, which is when equal unity restores the problem back to that of Eq. (3.108):  ∂bn (t)  i(En − En )t i = bn (t) Hnn exp (3.113) ∂t   n

Now, assuming n << En − En (notice that E includes the rest-mass energy):  Hnn Hn n

En = (3.114) En − En  n

This is a second order perturbation of energy, which introduces the diagonal elements of Eq. (3.109).

3.4 Electrodynamics

189

The cross section is obtained by normalizing the transition probability to the flux of incident projectiles. For a two-body interaction, the incident flux is simply the relative velocity between the two particles, v1 + v2 , per unit volume.Therefore, the cross section for transition from an initial state i to a final state f is expressed in general as:  2   if V 2π  Hfj Hji  ρf (3.115) σif = v1 +v2 = v1 + v2   j Ei − Ej  V where j and k represent intermediate states and V is the space volume.These intermediate states can be represented by diagrams, as described in Section 3.4.2 below. However, formulation of the cross section requires explicit determination of the matrix elements of the interaction Hamiltonian, Hint . These are given in quantum electrodynamics textbooks (see e.g. [28, 29]). We will not address the direct determination of these Hamiltonians here to avoid being side tracked to detailed quantum-mechanics calculations, which are beyond the scope of this book. Note also in addition to the interaction Hamiltonian, a field Hamiltonian, Hfield , is also evaluated for the interaction of a charged particle in the Coulomb field.

3.4.2 Feynman diagrams The concept of intermediate states, described above involves the introduction of virtual, or mediating particles (also described in Section 1.4), which enable proceeding from an initial state to final ones in stages. This process is quite useful in weak interactions, and in strong-field interactions at very high particle energy, where projectile-target coupling is weak. One can view a 2(1,3)4 interaction as consisting of three states: initial, intermediate, and final. In the initial state, the projectile 1 strikes the target 2. In the final state, particles 3 and 4 emerge. The intermediate state involves the mediating particle, also called the virtual or coupling particle. While conservation of energy and momentum is required at the initial and final states, it is not necessary in the intermediate state. Although this intermediate state is not physically observable, it is permitted by quantum mechanics; though energy is temporarily not conserved (as such it is a virtual process). One may think of the free space as being full with virtual particles and that, during the intermediate state of an interaction, a virtual particle is borrowed to facilitate the interaction and then given back to the free space. The above three interaction processes are described by Feynman diagrams [13, 30]. These are space–time diagrams in which time points upward and space to the right-hand side (or vice versa); though spatial spacing does not indicate the distance between particles. An electron is represented by a solid arrow that points along the direction of time, while a positron (an anti-particle) points in the direction of reverse time. A photon on the other hand is represented by a wavy line. Since the photon does not have an anti-particle (it is its own anti-particle), no direction is associated with it. Note though these diagrams are not trajectories, but

190

Chapter 3 Cross Sections

Final

g

e Initial (b)

(a)

Figure 3.3

Feynman diagrams for Compton scattering.

are merely indicative of interaction progression. Therefore, an interaction begins at the bottom of the diagram (initial state) and ends at its top (final state). A vertex is an intersection of three lines and represents an interaction, at which energy and momentum are conserved. Particles entering or leaving the diagram must represent real particles, for which the total energy, E, the momentum, P, and rest mass, M , as defined in Table 2.1, are such that E 2 = P 2 + M 2 . This relationship, on the other hand, does not hold for virtual particles, which are not observable in reality. To illustrate this concept, let us consider the Feynman diagrams for Compton scattering, the scattering of a photon with a free electron. There is more than one diagram, since this collision can take place by a number of interaction successions as shown in Fig. 3.3. In Fig. 3.3(a), the target electron absorbs the incoming photon, forming a virtual electron (energy is not conserved), the latter electron then emits a photon. The second possibility, Fig. 3.3(b), is that the target electron emits a photon, creating a virtual electron (with negative energy), which recovers its energy by absorbing the incoming photon. That is, in the first case the photon is absorbed at the point of electron entry, while in the second case the photon is absorbed as the electron exits.

The S-matrix Feynman diagrams were introduced in quantum electrodynamics to facilitate the calculation of the S-matrix approximately. This matrix describes the relationship between two collision, or decay, states. An element Sf i of the S-matrix is the expected (average) value of a matrix element of the quantum-mechanical operator S (a transfer function) that transforms the system from an initial (i) to a final ( f ) state. That is: Sf i =  f |S|i (3.116) where · defines the expected value. The S operator is defined in terms of the so-called Dyson expansion of the S-matrix: S=

∞  n=0

S (n)

(3.117)

191

3.4 Electrodynamics

where S 0 = 1 and corresponds to no interaction at all. All other higher order terms, are defined in terms of an operator, H (corresponding to the wave function of the incident particle and the field of the target) for 1st, 2nd, and higher order perturbations of the field. We will introduce here a brief description of the Smatrix and its relationship to the cross section to familiarize the reader with the terminology. For a more detailed analysis consult textbooks on relativistic quantum mechanics, such as, [28, 29, 31]. Natural units are used typically in relativistic quantum mechanics for convenience (see Table page 318). In these units, the fundamental constants c and h  = 2π are taken as being equal to unity, where c is the speed of light and h is the Planck constant.Then the natural unit for length remains as the meter (m), but the natural unit for mass becomes m−1 c −1 , and that for time is mc −1 . Consequently, energy and momentum as defined in Table 2.1 are divided by c. Note that the momentum of a wave hcλ becomes equal to 2π λ in natural units, hence the use of the wave number, k, to designate the momentum of a wave. Since the four-vector energy-momentum is conserved in between the initial and final states of an interaction (see Section 2.5), then Sf i can be expressed as: Sf i = δf i + i(2π)4 δ4 (Pf − Pi )Tf i

(3.118)

where δf i is the Kronecker delta function (δf i = 0, if f  = i, and equal to unity when i = f ), while δ4 (Pf − Pi ) is the Dirac delta function (which is equal to zero if the argument is not zero), and Tf i is the scattering amplitude. The raising of the Dirac delta function and 2π to the fourth power ensures conservation of the four energy-momentum vector, in order to accommodate√the four-dimensional Fourier transform of the electromagnetic field5 , while i = −1 simply designates a phase change. The transition probability, if , from the initial to the final state is given by the square of the S-matrix element, multiplied by Vt, where V is the space volume and t is the observation time. When the states i and f are different6 , and when i = f : if = (Vt)(2π)4 δ4 (Pf − Pi )|Tf i |2

(3.119)

Then, the transition rate (probability per unit time) is: if = V (2π)4 δ4 (Pf − Pi )|Tf i |2 Df

(3.120)

where the Df factor, called the density of final state, takes into account transition to a continuum of many final states. This final state density is given by: # $ # $ . V d3 P . V d3 k Df = (3.121) (2π)3 (2π)3 f

5 Some

f

textbooks use the square root of this quantity in accordance with the definition of symmetry in Fourier transforms. 6 δ4 (P ) is the Fourier transform (over the four-dimensional space) of unity. For a large space, large Vt value, δ4 (P )2 = tV (2π)4 δ4 (P ).

192

Chapter 3 Cross Sections

where P is the three-momentum vector for a particle with a finite mass, and k is that for a massless particle (a photon). The scattering amplitude, Tf i in Eq. (3.120), is often replaced with the normalized amplitudes Mf i , defined as:

Tf i =

! . i

⎤ "⎡ . √1 √ 1 ⎦ Mf i ⎣ a(k) a(k) 2EV 2EV f

(3.122)

where E is the total energy of a particle (with a finite mass) in either the initial for a photon of energy Eγ in either state is given (in or the final state, and a(k) natural units) by: 2= |a(k)|

R(k) ε0 Eγ V

(3.123)

where ε0 is the permittivity of free space and R is the fraction of the total field energy that is attributable to the electrical energy (the latter is equal to 12 ε0 |E|2 , where E is the electric field of the wave). Eq. (3.122) in (3.120) gives: 2   . .   1 1 4 4  if = V (2π) δ (Pf − Pi )  a(k) √ a(k) √ Mf i  Df 2EV f 2EV   i

(3.124)

Note that in the products of Eqs (3.120) and (3.124), only the present particles are included. For example, in photon annihilation where there is only one photon in the initial state and two particles (an electron and a positron) in the final state, one would have: . i

√1 a(k) = a(k γ ) 2EV

and

. f

1 1 √1 a(k) =√ √ 2E− V 2E+ V 2EV

where the γ, − and + subscripts refer to a photon, an electron, and a positron, respectively. The differential cross section for a given interaction is obtained by dividing the transition probability, Eqs (3.120) and (3.124), by the incident flux. This is a differential cross section, since it involves a portion of the three-vector momentum domain as indicated by Eq. (3.121). Let us consider the two-body interaction 2(1,3)4. In the center of mass system (C ), where the total momentum is zero, the incident particle flux is equal to the relative velocity of the incident particles divided by the volume V , i.e. the number of tracks in volume V per unit time; see Eq. (1.45) for definition of flux.Therefore,

193

3.4 Electrodynamics

in natural units [29]: φincident =

=

P1 P2 P β1 + β2 E + E2 = 1 = V V V  (E1 E2 + P 2 )2 − M12 M22

VE1 E2



 1 1 + E1 E2   1 2 2 2 2 2 2 (s − M1 − M2 ) − M1 M2 = VE1 E2 (3.125)

where the usual notations are used with the primed values being in C . The general expression for the differential cross section in C for a two-body collision is then given by dividing Eq. (3.124) by Eq. (3.125): E1 E2 dσ = (2π)4  δ4 (Pf − Pi )|Mf i |2 |Si Sf |2 Df  2 2 2 2 (E1 E2 + P ) − M1 M2 with Si = Sf =

.

(3.126)

(3.127)

i

√1 a(k) 2EV

(3.128)

f

√1 a(k) 2EV

.

The S-matrix is evaluated in terms of expansion coupling parameters determined with the aid of the Feynman diagram for a particular reaction. The order of the expansion series depends on the number of mediating particles introduced. One can add as many mediating particles as needed to achieve the desired accuracy. However, usually an expansion of the second order suffices for electromagnetic interactions, which has a coupling constant at each vertex of the diagram equal √ 1 ). This process is viewed to α, with α being the fine structure constant (= 137 as a perturbation of the potential field, and obviously is only useful when the coupling constant is small (as in the case of weak nuclear interactions). Each intermediate line in a Feynman diagram contributes a factor (a complex number) to the amplitude of the process. The amplitude factors for all possible diagrams give the total amplitude of the process. Note that the amplitudes are added in the same manner the amplitudes of multiple waves are combined to obtain the resultant wave. Using the absolute value of the total amplitude squared, gives the expected rate of the interaction. This rate with proper normalization with respect to the flux of the incident radiation gives the angular cross section of the interaction. The process of constructing the S-matrix using Feynman diagrams, though simple, leads to complicated mathematical expressions. Feynman diagrams produce terms corresponding to the amplitude of the scattered wave in the quantum mechanics treatment discussed in Section 3.2.

194

Chapter 3 Cross Sections

3.5 Photon Cross Sections We will start by the simplest process of Thomson scattering, in which the electromagnetic field of a photon affects a single free electron, then generalize the discussion to Compton scattering and Rayleigh scattering. We begin first with a reminder of the definition of radiation polarization.

Polarization An electromagnetic wave is the result of two perpendicular electric and magnetic fields, and propagates in a direction mutually perpendicular to these fields, as shown in Fig. 3.4. A transverse wave propagates in a direction normal to the fields inducing it. The direction of the electric field, as given by the unit vector ˆ is normal to that of the magnetic field B; ˆ it is, therefore, sufficient to describe E, ˆ A wave such as that shown on Fig. 3.4 the direction of the wave by kˆ and E. ˆ In general, an electric is called linearly polarized, with a polarization vector E. field can be represented by two mutually perpendicular directions, Eˆ 1 and Eˆ 2 , with each having a magnetic field in the direction kˆ × Eˆ i , with i = 1 or 2. If the fields in Eˆ 1 and Eˆ 2 are in phase, one still has a linearly polarized wave at an angle tan−1 EE21 with Eˆ 1 , where Ei refers to the strength of the electric field in the direction of Eˆ i . If Eˆ 1 and Eˆ 2 are not in phase, the wave resulting from their combined effect is called elliptically polarized, or circularly polarized in the special case of E1 = E2 . Polarization in photon interactions with atomic and nuclear fields is not generally important, except at low photon energy, because scattered photons are then linearly polarized. Therefore, at low photon energy, one can no longer assume isotropic scattering in the azimuthal direction, due to the anisotropy caused by linearly polarized Compton or Rayleigh scattered photons. kˆ

ˆ B

Eˆ

Figure 3.4 Transverse wave propagation unit vector (kˆ ), with respect to those of the electric ˆ ). field (Eˆ ) and the magnetic field (B

195

3.5 Photon Cross Sections

kˆ3 c

z kˆ3 q y

Eˆ w kˆ1

x

Figure 3.5 A schematic showing radiation and field directions in Thomson scattering.

3.5.1 Thomson scattering As discussed in Section 2.8.2,Thomson scattering can be viewed as a Coulomb scattering of the target electron in the electromagnetic field of the incident photon. Then, the power emitted by the electron per unit solid angle is given by Eq. (2.205). This power is indicative of the intensity, hence, probability of interaction. However, since we are interested in the cross section of the incident photons, the power given by Eq. (2.205) has to be normalized with respect to the energy flux of the incident photon, which is the si term in the same equation. Therefore, the differential scattering cross section for Thomson scattering is given by: dσ = re2 sin2 ψ d

(3.129)

where re is the classical electron radius (= 4πεe0 me c 2 = 2.818 × 10−15 m), and ψ is the angle between the direction of the emitted radiation and the direction of acceleration. Since the latter angle is not directly the scattering angle of the incident radiation, we will pay attention to each definition with the aid of Fig. (3.5). In this figure, the direction of the incident radiation is defined by the unit vector, kˆ 1 , and the scattered wave is defined by the unit vector, k 3 , while the unit vector Eˆ 1 defines the direction of the electric field associated with the incident magnetic wave such that kˆ 1 · Eˆ 1 = 0. Since the direction of acceleration of the electron is in the same direction as Eˆ 1 , then cos ψ = kˆ 3 · Eˆ 1 . Now let kˆ 3 be defined in terms of the (x, y, z) coordinate system and the angles ϕ and ϑ, also shown in a Fig. 3.5, then: 2

kˆ 3 = (cos ϕ sin ϑ)ˆx + (sin ϕ sin ϑ)ˆy + (cos ϑ)ˆz

(3.130)

where xˆ , yˆ , and zˆ are unit vectors in x, y, and z, respectively. Then the angle of scattering of the photons is ϑ, since kˆ 1 · kˆ 3 = cos ϑ, with kˆ 1 ≡ xˆ . At ϑ = π2 , kˆ 3 is perpendicular to kˆ 1 , leading to ψ = 0 and ϑ = π2 . Then according to Eq. (3.129), the angular cross section will be zero, i.e. there is no Thomson scattering at a π2 scattering angle. This is true if the radiation was polarized in a particular direction Eˆ 1 . In general, however, any direction in the plane x–y is perpendicular to kˆ i ;

196

Chapter 3 Cross Sections

any vector emanating from the origin in this plane, i.e. at any angle ϕ, can funcˆ It is reasonable then to use an average value tion as an acceptable vector for E. of cos2 ϕ in the expression for sin2 ψ to obtain an average value for the latter. With cos ψ = cos ϕ sin ϑ and sin2 ψ = 1 − cos2 ϕ sin2 ϑ, for unpolarized incident radiation, the average value is: sin2 ψ = 1 − cos2 ϕ sin2 ϑ = 1 −

1 1 sin ϑ2 = (1 + cos2 ϑ) 2 2

(3.131)

The direction of polarization is not important in most applications. Therefore, the average value of Eq. (3.131) can be used for sin2 ψ in Eq. (3.129) to give:  dσ  re2 (1 + cos2 ϑ) = (3.132) d unpolarized 2 Integrating Eq. (3.129) over all solid angles yields the overall Thomson scattering cross section: σ=

8π 2 r = 6.6524 × 10−29 m2 3 e

(3.133)

The Thomson scattering of a photon is independent of its energy, and occurs only at photon energies much less than the rest-mass energy of the electron, but at an energy higher than its binding energy. Therefore, Thomson scattering is frequency-coherent. However, the movement of the target electron away and toward the incident radiation results in Doppler shifting in frequency (see Eq. (2.204)). The Thomson scattering by a nucleus is obtained by replacing the charge and the mass in the definition of re with the corresponding charge and mass of the target nucleus. That is, for a particle of charge ze and mass m:  2 dσ (ze)2 = sin2 ψ (3.134) d 4πε0 mc 2 2  (ze)2 8π (3.135) σ = 3 4πε0 mc 2 Therefore, the Thomson cross section for scattering with protons and heavy charged particles is quite small in comparison to that with the lighter electron, and is, therefore, negligible.

Anomalous scattering When an incident photon has an energy near that of the binding energy of an electron in the target atom, the conditions for Thomson scattering are no longer applicable. However, the Thomson scattering cross section can be adjusted by

197

3.5 Photon Cross Sections

multiplying it by a form factor. This scattering is then referred to as anomalous scattering, a form of “resonance’’ scattering. The anomalous scattering factors are given for various elements in the elastic photon-atom scattering RTAB database (http://www-phys.llnl.gov/Research/scattering/).

3.5.2 Compton scattering The main characteristic of Thomson scattering is that it is frequency-coherent, i.e. both the scattered photon and the incident photon have the same frequency (hence energy). This implies that the target electron is not affected in the process, neither gaining momentum nor energy. Although this classical treatment is acceptable at large wavelength (low frequency), where optical considerations are applicable, it is not physically realistic at high frequency due to the large momentum carried by the incident photon that can no longer be ignored. Then, we are in the domain of Compton scattering and quantum effects. The Feynman diagram for Compton scattering 2(1,3)4 was shown in Fig. 3.3. The cross section in C (P1 = P2 and P3 = P4 ; with the usual primed notation removed for convenience) for this interaction can be written, with the aid of Eq. (3.126) along with Eq. (3.121), as: E1 E2 4 δ (P3 + P4 − P1 − P2 )|Mf i |2 s − M22   2 # 3 $ # 3 $   V d P 3 V d P 4 1 1  ×  a(P 1 ) √ (3.136) a(P 3 ) √  (2π)3 (2π)3 2M2 V 2E4 V

dσ = (2π)4

=

1 e4X 2E1 E2 (s − M22 ) (2E1 E2 E3 E4 ) ε20

#

V d3 P 3 × (2π)4 δ4 (P3 + P4 − P1 − P2 ) (2π)3

$#

V d3 P 4 (2π)3

$ (3.137)

Recall that the above expression is in natural units, and notice that use was made of 2P1 P2 = s − M22 , with M1 = 0, since the photon is massless with s being the invariant of Eq. (2.87), and V disappears as it is also included in the definition of by Eq. (3.123). The factor X in Eq. (3.137) is given by [29]: a(k)    1 1 P1 P3 − 1 + 2 Eˆ 1 · Eˆ 3 + (Eˆ 1 · P 1 )(Eˆ 3 · P 3 ) + X= 2 P3 P1 P1 2 1 ˆ ˆ (3.138) − (E1 · P3 )(E3 · P1 ) P3 where Eˆ is a unit vector in the direction of the electric field polarization. This factor X incorporates |Mf i |2 . In order to obtain the angular

198

Chapter 3 Cross Sections

differential cross section for the product photon, we use d3 P 3 = P32 dP3 d7 , where  refers to the solid angle within which photons are scattered. Since for photons E = p, then p32 dp3 d = E32 dE3 d. Due to the conservation of momentum, one has P 4 = P 1 + P 2 − P 3 , while conservation of energy requires E42 = M42 + P42 = M42 + (P 1 + P 2 − P 3 )2 . Now integrating Eq. (3.137) over d3 P 3 and d4 P 4 , one obtains [29]: dσ 1 = d 2E22



e2 4π

2 

E3 E1

2

X (1 − β cos ϑ3 )2

(3.139)

where ϑ3 is as usual the angle the scattered photon makes with the incident photon, and β is the velocity of the target electron in C (measured in natural units). The introduction of the classical electron radius enables easy conversion 2 between natural and normal units. In natural units, re = 4πεe0 me c 2 . With E2 = γme , where γ = √ 1 2 . Equation (3.139) becomes: 1−β

dσ re2 X = 2 d 2γ (1 − β cos ϑ3 )2



E3 E1

2 (3.140)

c = Mαe (in natural units), where α is the fine structure Keeping in mind that re = α M e 1 constant and Me is the electron’s rest-mass energy, one can see how α = 137 appears 2 in the cross section of Eq. (3.140) as a coupling constant. Recall that α = 4πεe 0 c 2

e . or in natural units α = 4πε 0 Now we can set re = 2.818 × 10−15 m in Eq. (3.139) and return the entire equation to the normal units. Recalling that E3 is uniquely related to E1 by the angle of scattering via Eq. (2.143), then the latter equation and Eq. (3.140) indicate that the Compton scattering cross section is in general inversely proportional to E1 . That is, it is dominant at lower photon energies, provided that they are higher than those of the binding energies of atomic electrons to avoid photoabsorption. Then one can assume the target electron to be at rest, i.e. β = 0 and γ = 1. In the electron rest-mass frame of reference, L, P2 = 0, and the factor X of Eq. (3.138) becomes:   1 P1 P3 − 1 + 2(Eˆ 1 · Eˆ 3 )2 + (3.141) X= 2 P3 P1

Since in most applications, polarization is not important, then an averaging of X over the initial states of photon polarization and a summation over the final states of polarization can be performed. Then (Eˆ 1 · Eˆ 3 )2 can be replaced, as was 7 This

is similar to having an infinitesimal volume of r 2 dr d, which upon integration over all solid angles gives the volume of a sphere of radius r.

199

3.5 Photon Cross Sections

done in Thomson scattering, Eq. (3.131), by 21 (1 + cos2 ϑ3 ). However, the terms that are independent of Eˆ 1 and Eˆ 3 in Eq. (3.141) need to be doubled (multiplied by 2). This along with Eq. (3.141) in (3.140) gives the well-known Klein–Nishina expression for the differential cross section in Compton scattering: dσ r2 = e d 2



E3 E1

2 

E1 E3 + − sin2 ϑ3 E3 E1

 (3.142)

Note that at very low photon energy, E3 << M2 , Eq. (2.143) shows that E1 and E3 become nearly equal. Eq. (3.142) becomes identical to the differential cross section of Thomson scattering (Eq. (3.129)). In essence, Thomson scattering is the classical (low-energy, non-relativistic, no quantum effects) form of Compton scattering. In other words, the sin2 ϑ3 term of Eq. (3.142) can be viewed as the term corresponding to the Coulomb scattering of the target electron in the elec 2 tromagnetic field of the incident electron. The term EE31 in Eq. (3.142), which is always less than the one due to the loss in photon energy in this interaction, signifies a decrease in interaction probability due to the motion of the target electron during the interaction. The terms EE13 and EE31 do not include dependence on the scattering angle, due to the relativistic quantum-mechanical effects, not taken into account in Thomson scattering. At very high energy, the third term in Eq. (3.142) becomes equal to EE13 and   r2 the differential cross section approaches a value of 2e EE31 . Also at high energy, collision kinematics (Eq. (2.143)) indicates that E3 approaches a value of Therefore, the Klein–Nishina at high energy approaches:      re2 M3 1 re2 E3 dσ  = = d E1 >>M3 4 E1 4 E1 sin2 ϑ23

M3 1 − cos ϑ3 .

(3.143)

3 which is valid when ϑ32 >> 2M E1 . This indicates that small-angle (forward) scattering dominates at high energy. The overall Compton scattering cross section (at all angles), as a function of the incident energy, is obtained by integrating Eq. (3.142), to get [1, 29]:

   2E1 M22 (M2 + E1 ) 2E1 (M2 + E1 ) σ(E1 ) = − ln 1 + E1 3 M2 (M2 + 2E1 ) M2  +  M2 (M2 + 3E1 ) 2E1 M2 − (3.144) ln 1 + + 2E1 M2 (M2 + 2E1 )2 

2πre2

200

Chapter 3 Cross Sections

Recall that M2 is the rest-mass energy of the electron (= 511 keV). The following approximations can come handy:   ⎧ 8πre2 2E1 ⎪ ⎪ 1− ⎪ ⎪ 3 M2 ⎪ ⎪ ⎪ ⎨ 2 2 8πre M2 + 2M2 E1 + 1.2E12 σ(E1 ) ≈ ⎪ 3 (M2 + 2E1 )2 ⎪ ⎪   ⎪ ⎪ ⎪ M2 2E1 1 ⎪ 2 ⎩ πre ln + E1 M2 2

E1 << M 2 (3.145) E1 >> M2

Notice that at intermediate and high energy, the Compton scattering cross section is roughly inversely proportional to E1 . The Klein–Nishina cross section of Eq. (3.142) assumes that the target electron was free and at rest before the collision. This assumption is valid when the incident photon energy is much higher than the binding energy of a target’s atomic electron. Otherwise, some of the incident photon momentum is utilized to liberate the target electron. A momentum q (in relativistic terms) such that: q=

 E12 + E32 − 2E1 E3 cos ϑ3

(3.146)

impacts the atomic electron. The probability of scattering (differential cross section) decreases as the value of q increases, and is, therefore, lowest at small scattering angles. A correction is then introduced to the Klein–Nishina cross section of Eq. (3.142), by accounting for binding energy via a multiplicative correction factor, S(q,Z) Z , where Z is the atomic number. The function S(q, Z) is called the incoherent scattering function, and represents the probability of liberating an atomic electron (by excitation or ionizing), by the sudden impulsive action of a recoil momentum q to the atom. At a given Z, S(q, Z) varies from zero (complete photon absorption) to a maximum of Z.Tabulated values for S(q, Z) are given in [32]. The kinematics of Compton scattering as given by Eq. (2.143) dictates a unique relationship between the energy and angle of scattering. Therefore, the dσ differential cross sections, d , defines also the scattering cross section from E1 to E3 , since the polar scattering angle, ϑ3 , defines E3 and scattering in the azimuthal direction is isotropic. This unique energy–angle relationship is, however, distorted if the target electron is not initially at rest, due to the Doppler effect associated with the relative motion of the incoming photon and the moving electron. The extent of this Doppler broadening depends on both the incident photon energy and the structure of the atom with which the photon interacts. In order to accommodate this Doppler effect, one must define a double-differential cross section that takes into account both the change in the photon energy and the scattering angle.

201

3.5 Photon Cross Sections

An expression for this cross section is given by8 : r 2 E3 M2 d2 σ = e dE3 d 2 E1 q 

YJ (Pz )  2 12 Pz 1+ M 2

(3.147)

where Pz is the projection of the initial momentum of the electron on the direction of scattering:   1 [E1 E3 (1 − cos ϑ3 ) − M2 (E1 − E3 )] (3.148) Pz = q and the parameter Y is given by:     1 1 R 1 1 2 R −2 + − − Y = + R R R R R R ⎧! ⎫  2 " 12 ⎨ E1 E1 − E3 cos ϑ3 Pz ⎬ Pz R = + 1+ M2 ⎩ M2 q M2 ⎭   E1 E3 E3 E1  (1 − cos ϑ3 ) = R − −1 R = R− M2 E3(0) M22

(3.149)

(3.150)

(3.151)

where E3(0) is the energy of the photon scattered by an electron at rest. The photon scattering energy, E3 , is given by:  Pz q  + 1 E3(0) (3.152) E3 = E1 M2 keeping in mind that E3(0) is determined by Compton kinematics, Eq. (2.143). The function J (Pz ) is known as the Compton profile, i.e. the distribution of energy around  ∞ E3(0) . The quantity J (Pz )dPz is dimensionless, and is normalized so that ∞ J (Pz )dPz = 1. The function J (Pz ) is bell-shaped and symmetrical about Pz = 0; tabulated values of which are given in [33] for individual atomic orbits and for the entire atom, for elements of atomic number from 1 to 102. Incorporating both Doppler broadening and binding-energy effects leads to the following double-differential scattering cross section8 : r 2 E3 M2 d2 σ = e dE3 d 2 E1 q 

YJ (Pz )  2 12 Pz 1+ M 2

8 D.

(E1 − E3 − Be )

(3.153)

Brusa, G. Stutz, J. A. Riveros, J. M. Fernández-Varea, and F. Salvat, Fast sampling algorithm for the simulation of photon Compton scattering. Nuclear Instruments and Methods A, Vol. 379. pp. 167–175, 1996.

202

Chapter 3 Cross Sections

where Be is the electron’s binding energy and the function step function:  1 if x > 0 (x) = 0 otherwise

(x) is the Heaviside

which reflects the fact that Compton scattering can only take place when the energy deposited to the target atomic electron, E1 − E3 , is greater than Be . Integrating the double-differential scattering cross section of Eq. (3.153) over E3 leads to the single-differential cross section, with S(q, Z) included. Therefore, the incoherent scattering function, S(q, Z), is equivalent to the integration of the Compton profile J (Pz ) over all possible photon scattering energies. It is quantum mechanically possible for the final state in the scattering of a photon by an electron to contain two quanta (photons). The transition matrix element in this double Compton scattering process is one order higher than that for single Compton scattering. Therefore, the probability of double scattering is lower than that of single scattering by α (the fine-structure, coupling, constant), the probability of emission of two pairs of photons is lower than that of single photo emission by α2 , and so on. The relevance of multiple emissions is mainly in double scattering. The double scattering cross section has the limits [28]: ⎧  2 E1 ⎪ 2 ⎪ ⎨ αre Me σ≈ ⎪ M ⎪ ⎩ αre2 e E1

E1 << M2 (3.154) E1 >> M2

where Me refers to the electron’s rest-mass energy. The two photons are emitted mostly into small angles (i.e. in the forward direction). Moreover, it is more probable to have one photon emitted at a small angle and the other at a larger angle, than to have both photons emerge at large angles. When one of the emitted photons has a very small energy, the process becomes for all practical purposes equivalent to that of single scattering. In practice, a low threshold energy is imposed in double scattering, so that if one of the photons has an energy lower than this threshold, the interaction is considered to be a single Compton scattering. Integrating over the energy, E5 , of the emitted second (lower energy) photon from the threshold limit, Eth , up to an energy Em (<<E1 , an upper limit for the second photon to be considered of low energy), one obtains the differential cross section for the contribution of double Compton scattering [28]:  2 dσD E1 Em 2 2 re α (1 + cos2 ϑ3 )(1 − cos ϑ3 ) ln Eth < Em << E1 = 3π Me Eth d (3.155) where ϑ3 is the scattering angle of the main (higher energy) photon. As Eth → 0, the cross section diverges. This is known as the “infrared divergence’’ problem, discussed also in Section 3.6.4.

3.5 Photon Cross Sections

203

A second order correction to the Compton scattering process can be included, in the Feynman diagram, by introducing photon absorption and emission (or emission followed by absorption) processes.This introduces the so-called radiative correction. At low photon energy, the correction for this effect is given by [28]:  2   Me

2 2 E1 dσRad = − re α −3(1 + cos ϑ3 + cos2 ϑ3 ) + cos3 ϑ3 ln d 3π Me E1 + Me + (1 + cos2 ϑ3 )(1 − cos ϑ3 ) ln Eth <<E1 << Me (3.156) Eth where Eth is a low threshold energy imposed to avoid convergence of the logarithmic term when E5 approaches zero. Since this is a negative correction to the cross section, the infrared conversion term in Eq. (3.156) cancels that of the double ME Em E Compton scattering correction in Eq. (3.155), since ln M Eth = ln Em + ln Eth . This eliminates the infrared conversion problem altogether, as the two effects occur concurrently; and demonstrates that infrared divergence appears only when higher order corrections to the cross section are not accounted for. Equation (3.156) also shows that the correction for the radiative cross section at low energy is of the same order of magnitude, but with an opposite sign, as that of the double Compton scattering in the same energy range, Eq. (3.155). Therefore, the two effects counteract each other. Both the radiative correction (Eq. (1.156)) and the double Compton cross section (Eq. (3.155)) are insignificant, since both contain E1 1 the factors: α = 137 and M << 1, which are not included in the Klein–Nishina e Compton cross section, Eq. (3.142). The combined contribution of the two effects of double Compton scattering and the radiative correction is about 0.25% at 4 MeV photon energy, increasing to about 1% at 100 MeV [34].

3.5.3 Rayleigh scattering Rayleigh scattering, like Thomson scattering, is a coherent scattering process. However, in Rayleigh scattering the photon is scattered by bound atomic electrons, without causing atomic excitation or ionization. In essence, Rayleigh scattering is a scattering by the atom as a whole. Therefore, the charge distribution of all electrons in an atom must be simultaneously considered. This is done by introducing the atomic form factor, F(q, Z), to the Thomson cross section. This factor, also called the scattering factor, is a function of E1 , Z, and q. Here, the recoil momentum acquired by the atom, q, causes no loss of photon energy, i.e. E1 = E3 . Using Eq. (3.146), then:  ϑ3 (3.157) q = 2E12 − 2E12 cos ϑ3 = 2E1 sin 2 Now for unpolarized photons, the differential cross section for Rayleigh scattering becomes equal to theThomson scattering of Eq. (3.132) multiplied by [F(q, Z)]2 ,

204

Chapter 3 Cross Sections

with the cross section now given per atom. Therefore, dσ r2 = e (1 + cos2 ϑ3 )[F(q, Z)]2 d 2

per atom (unploarized)

(3.158)

The form factor values are given in [32]. This factor is nearly independent of the scattering angle for photon energies below about 2 keV, but at higher energy it sharply decreases with the scattering angle, becoming predominantly a small-angle forward scattering interaction. At least 75% of the scattering angles are confined to scattering angles within the interval [34]: # $ 1 3 0.0133Z 0 ≤ ϑ3 ≤ 2 sin−1 (3.159) with E1 in MeV E1 For example, for Fe (Z = 26), at E3 = 1 MeV, 75% of Rayleigh scattering occurs within angles less than 4.5◦ . This small angle of scattering at higher energy is dictated by the fact that the recoil energy imparted into the atom must not produce excitation or ionization. The same fact makes also this scattering cross section much smaller than that of Compton scattering at high photon energy. The overall (integrated at all scattering angle) cross section for Rayleigh scattering is given in [35].

3.5.4 Diffraction Diffraction is in effect a Rayleigh scattering process that involves more than one atom, with the wave scattered by one atom coherently added to those scattered by other atoms. For a unit cell of N atoms9 and volume Vc whose plans are aligned to satisfy the Bragg scattering conditions (see Section 2.8.3), the coherent addition of the scattered waves for incident radiation of wavelength λ leads to the cross section [36]: 2   1 + cos2 ϑ 2 λ 2 σ|Bragg/Lau = re (3.160) mF d 2NVc 2 hkl hkl

where hkl represents a lattice plane (defined by the Miller indices) from which diffraction occurs, d is the spacing in the plane and m is its multiplicity (the number of atoms generated by the symmetry around the position of a single atom). Notice the similarity between the expression of Eq. (3.160) and the differential cross section for Rayleigh scattering, Eq. (3.158), and that the former is much larger than the latter. The F|hkl factor in Eq. (3.160) is known as the structure factor and consists of the sum of the atomic form factors for individual atoms in the diffracting plane. The cross section of Eq. (3.160) is applicable when the Bragg scattering conditions, Eq. (2.214), are satisfied. However, thermal motion distorts 9 The

atoms in a unit cell are those described by the stoichiometry of a compound chemical.

205

3.5 Photon Cross Sections

the alignment of atoms in a crystal, resulting in the thermal diffuse scattering (TDS) approximation cross section [36]:

σ|TDS =

re2

 λ2  1 + cos2 ϑ 2 m F d × [1 − exp(−2W )] 2NVc 2 hkl

(3.161)

hkl

where W is a thermal diffusion parameter, which depends on temperature, as well as on the type of diffracting atom and its location in the cell. The TDS cross section vanishes at zero absolute temperature, when W = 1, otherwise it is is always present. The similarity between Eqs (3.161) and (3.160) indicate that the distortion introduced by σ|TDS , though broadly distributed, peaks when Bragg conditions are satisfied, but the peak value is much smaller than the Bragg scattering cross section. Other distortions to Bragg scattering are introduced by inelastic scattering, scattering in the air surrounding the source and detector, etc.

3.5.5 Photoelectric effect Both the Pauli exclusion principle and kinematical considerations do not permit a free electron to absorb or emit a photon, because it cannot accommodate the recoil energy and momentum; though a free electron can coherently scatter (as in Thompson scattering). On the other hand, the more bound an electron, the more likely it will absorb a photon, since the atom can recoil as a whole, allowing the electron to move to an excited state or even leave the atom altogether. Therefore, the photoelectric effect is most probable for K-shell electrons.Typically, about 80% of photoabsorption is caused by K-shell electrons. For the photoelectric effect to occur, the energy of the incident photon must be greater than the binding energy of the target electron. Therefore, once the incident energy of the photon is below the binding energy of the K shell, the cross section for the photoelectric effect drops sharply, since electrons from that shell can no longer be ejected. The process is repeated as the photon energy drops below each subsequent (L1, L2, etc.) shell. Therefore, the photoelectric cross section exhibits the so-called absorption edges; a sawtooth-like change in the cross section behavior that reflects the fact that the incident photon energy has exceeded the binding energy of various atomic shells. Since the binding energy of the K atomic shell is much less than the rest mass of the electron (13.6 eV for hydrogen’s and 115.606 keV for uranium10 ), it is reasonable to assume that relativistic effects are negligible in the photoelectric absorption process. Assuming also for now that the energy of the incident photon is larger than the ionization energy of the atom in which the target electron resides, i.e. the photon energy is higher than the K absorption edge of the atom, the matrix element for the Hamiltonian interaction for this process from the initial state i to 10 See

http://www.webelements.com/webelements/elements/text/periodic-table/bind.html

206

Chapter 3 Cross Sections

the final state, f , is given by [28]:  e Pe Hf i = − √ M 4πε0 V 3

2χ3 8πχ3 c 3 cP1 [χ2 + (P 1 − P 3 )2 ]2

(3.162)

where V is the space volume, Pe , is the component of the ejected electron’s momentum, P3 , in the direction of polarization of the incident photon, 1 χ = ZαM2 with α being the fine structure constant (= 137 ) and Z is the atomic number of the target nucleus, and all the parameters have their usual meaning. Now Eq. (3.115) for the cross section dσ within dE, with Eq. (3.162) and with the relative velocity v1 + v2 = c, gives:  + e 2 Pe2 2χ3 64π2 χ2 6 c 6 2πV ρf dσ = c 4πε0 M32 V 2 cP1 [χ2 + (P 1 − P 3 )2 ]4  2 2 + ρf e Pe 2χ3 64π2 χ2 6 c 6 2π (3.163) = 2 c 4πε0 M3 cP1 [χ2 + (P 1 − P 3 )2 ]4 V ρ

The final state density per unit volume Vf can be evaluated using Eq. (3.121) after restoring to normal units from natural units and dividing by V dE3 , since ρf is a number per unit energy: ρf d3 P 3 P32 dP3 d P3 E3 dE3 d P3 M3 d = = = = 3 3 3 V (2πc) dE3 (2πc) dE3 (2πc) dE3 (2πc)3

(3.164)

with d being a solid angle around the direction of emission of the electron, and it is assumed that the kinetic energy of the ejected electron is much smaller than its rest-mass energy, so that E3 ≈ M3 . Now the differential cross section becomes:   e4 M3 Pe2 P3 χ5 4πε0 c  dσ = 32 d e2 (4πε0 )2 M32 P1 [χ2 + (P 1 − P 3 )2 ]4 =

M3 Pe2 P3 χ5 32re2 α P1 [χ2 + |P 1 − P 3 |2 ]4 2

(3.165) 2

1 Recall that re = 4πεe0 Me and the fine structure constant = 4πεe 0 c = 137 . This is the angular cross section for the scattering of the electron. If ϑ3 is the scattering angle and ϕ3 is the angle of polarization (angle between the plane of the incident photon and target electron and the direction of polarization of the former), then Pe = P3 sin ϑ3 cos ϕ3 . The momentum transfer term according to the kinematics of the reaction (see Section 2.7.3), is equal to (P 1 − P 3 )2 = P12 + P32 − 2P1 P3 cos ϑ3 . Energy conservation also requires that E3 = E1 − I + M3 , assuming here that full ionization occurs. The ionization energy can be estimated as I = Z 2 Me α2 , with Me = M3 being the

207

3.5 Photon Cross Sections

electron’s rest-mass energy. Still with non-relativistic conditions where one P2 χ2 + P 2 can set E3 = 2M3 e , it can be shown that E1 = 2Me 3 << Me , χ2 + (P 1 − P 3 )2 = E1 (2Me + E1 − 2P3 cos ϑ3 ) ≈ 2Me E1 (1 − β cos ϑ3 ), with β = vc . One can also show that under these conditions P 2 ≈ 2E1 M1 . With these approximations, the cross section of Eq. (3.165) becomes [28]:  3.5 √ 4 2 sin2 ϑ3 cos2 ϕ3 dσ 2 5 4 Me = re Z α d3 E1 (1 − β cos ϑ3 )4

(3.166)

This expression of the cross section shows that photoelectrons are predominately emitted when ϕ ≈ 0 and ϑ ≈ π2 , i.e. close to the direction of polarization of the incident radiation. However, as the denominator of Eq. (3.166) shows, emission in the forward direction around ϑ = 0 becomes significant when β approaches 1, but then relativistic conditions must be addressed as shown below. We first examine the overall cross section which is obtained by integrating Eq. (3.166) over all angles. To obtain a cross section for all K-shell electrons, the obtained integral must also be multiplied by the number of electrons in this shell, which is two. Therefore, the K-shell photoelectric cross section is [28]:  3.5 8π 2 5 4 √ Me σK = re Z α (4 2) 3 E1

(3.167)

2 Recall that 8π 3 re is theThomson cross section of Eq. (3.133) introduced here only for convenience. It is obvious that the photoelectric cross section is dominant only at low energy and in dense (high Z) materials. The cross section of Eq. (3.166) needs, however, to be adjusted at low photon energies comparable to the electron’s binding energy, and at very high energy where relativistic effects are important. At low energy, the Hamiltonian of Eq. (3.162) needs to be corrected to take into account the absorption edges in the cross section which occur when the photon energy is close to the binding energy of the shell. The correction for the Hamiltonian leads  to the  introduction of the approximate multiplicative Be correction function f E1 − Be given by [28]:

 f (ξ) = 2π

⎡



− tan4ξ−1 ξ

⎤

Be ⎣ exp ⎦ E1 1 − exp(−2πξ)

ξ2 =

Be E1 − Be

(3.168)

where Be is the electron’s binding energy (= I in the case of the K shell). Obviously at E1 = Be , ξ approaches ∞, and f (ξ) approaches 2π, causing an increase in σK by 2π. On the other hand, when E1 is the neighborhood of Be , f (ξ) causes σK to decrease by a factor of 2π exp(−4) = 0.115.

208

Chapter 3 Cross Sections

At high photon energies, where relativistic effects are relevant, the relativistic Hamiltonian leads to the cross section [28]:   5  3 8π 2 5 4 2 1.5 Me σK = r Z α (γ − 1) 2 3 e E1 ! $" #  4 γ(γ − 2) γ + γ2 − 1 1 × ln (3.169) + 1−   3 γ +1 2γ γ 2 − 1 γ − γ 2 − 1 where γ = √

1 1 − β2

E1 =1+ M and E1 >> I . At very high energies, E1 >> Me , e

γ ≈ 1, leading to:

3 σK = 2



 8π 2 Me re Z 5 α4 3 E1

(3.170)

showing a much lower rate of decrease with energy; only with E11 in comparison to E13.5 for the low energy cross section, Eq. (3.166). At these extremely relativistic 1 energies, the angular distribution is given by the Sauter [37] distribution for the polar angle of electron emission from the K shell, ϑ3 , as: f (cos ϑ3 ) =

γ 1 − cos ϑ32  1 + (γ − 1)(γ − 2)(1 − β cos ϑ3 ) 4 (1 − β cos ϑ3 ) 2

(3.171)

where f (cos ϑ3 ) is a distribution function. Although this function indicates that no electrons are emitted at ϑ3 , at theses extremely relativistic conditions with β approaching unity, a considerable number of electrons are emitted at a small angle.

3.5.6 Pair production The creation of a positron/electron pair is due to the transition of an electron from the virtual sea of electrons of negative energy to a state of positive energy. The hole left creates the positron. A minimum energy of 2Me , twice the electron’s rest-mass energy, is required. Therefore, all cross sections for this interactions exist only at a photon energy greater than 1.022 MeV. Pair production can be represented by the Feynman diagram of Fig. 3.6, which shows the scattering of a positron first by a virtual photon from the nucleus, then by the incoming photon. Since in Feynman’s representation, a positron is viewed as an electron traveling backward in time, the product electron is the result of the scattering of the positron by one of the two photons. Triplet production is similar except that it is an interaction with one of the atomic electrons, which acquires energy sufficient to liberate it from the nucleus. Therefore, in pair production or triplet production, the Hamiltonian is a combination of the Hamiltonian, Hfield , of the positron interacting with the field of the target (nucleus in case of pair production and atomic electron in case of triplet production) plus the Hamiltonian of the scattering of a positron by a photon, Hint . This greatly facilitates the

209

3.5 Photon Cross Sections

e e × Nucleus



Figure 3.6

Feynman diagram for pair production.

evaluation of the total Hamiltonian of the interaction, Hfi = Hfield + Hint , from the initial to the final state, since Hfield is the Hamiltonian of an electron (albeit here with a positive charge) in an electromagnetic field, in which the momentum of the electron changes. The Hamiltonian Hint is that of the interaction of a free particle (a positron here) with a photon (or vice versa), a process in which momentum is conserved. The Hamiltonians for these two fundamental processes can be determined (see e.g. [28]). Assuming Hf i is given, to determine the cross section, as Eq. (3.115) shows, ρ one also needs to determine the final state density per unit volume Vf . This is evaluated using Eq. (3.121), similar to Eq. (3.164): ρf d3 P + P+ P− Me2 d+ d− d3 P − = = V (2πc)3 dE+ (2πc)3 dE− (2πc)6

(3.172)

where the positive sign refers to the positron and the negative sign to the electron, and the kinetic energies of the positron and the electron are assumed to be much smaller than their rest-mass energy. The relative velocity (v1 + v2 ) in Eq. (3.115) is here the speed of the incident photon, c. Now proceeding in a manner similar to that followed in deriving other cross sections, one obtains [29]: αZ 2 2 Me2 P+ P− dE+ dσ = r d+ d− (2π)2 e q4 E1 ⎡ # $2 ˆ ˆ P P · E · E − + 1 1 × 2 ⎣q 2 − P − · P 1 P + · P 1 # − # −

P − · Eˆ 1 P + · Eˆ 1 2E+ + 2E− P − · P 1 P + · P 1

$2

$⎤ q2 E12 P + · P 1 P − · P 1 ⎦ (3.173) 2− + + (P + · P 1 )(P − · P 1 ) P − · P 1 P + · P 1

210

Chapter 3 Cross Sections

where as usual the subscript 1 refers to the incident radiation, and q2 = |P 1 − P + − P − |2 ; recall that P1 = E1 , since the incident projectile is a photon and α is the fine structure constant. Note that the cross section presented here is per nucleus, since this is an interaction with the entire atom. For unpolarized radiation, one obtains the Bethe–Heitler pair-production formulae [28]: dσ αZ 2 2 Me2 P+ P− dE+ = r d+ d− (2π)2 e q4 E13  2 2) 2 sin2 ϑ (q2 − 4E 2 ) β+ sin2 ϑ+ (q2 − 4E− β− − + × + (1 − β+ cos ϑ+ )2 (1 − β− cos ϑ− )2 2β+ β− sin ϑ+ sin ϑ− cos ϕ(4E+ E− + q2 − 2E12 ) (1 − β+ cos ϑ+ )(1 − β− cos ϑ− ) 2 2 2 2 2 (γ+ − 1) sin ϑ+ + (γ− − 1) sin ϑ− + 2E1 (3.174) γ+ γ− (1 − β+ cos ϑ+ )(1 − β− cos ϑ− ) −

where the β and γ terms have their usual Lorentz’definition, the scalar products of Eq. (3.173) are expressed in terms of the polar angles the positron and the electron make with the incident photon, ϑ+ and ϑ− , respectively, and the azimuthal angle ϕ of the emitted pair (which has to be in the same plane to conserve momentum). When E+ >> 2Me and E− >> 2Me , the polar angular distribution of both the positron and the electron takes the form [28]:   2 M2  ϑ± Aϑ± e +B  2 2 ln 1 + E 2 1 Me 2 ϑ± + E1

dσ = dϑ±

(3.175)

where A and B are constants. Therefore, at extremely relativistic conditions, the angular distribution of the created pair peaks in the small-angle (forward) direc2Me e tion. The average angle of emission is then equal to about M E1 or T± [1, 28], where T refers to the kinetic energy. Near the threshold energy of the interaction, the bias to the forward direction is not as pronounced. Integrating Eq. (3.174) over all scattering angles leads to [28, 29]:    dσ 2 2 4 γ+ γ− 2 2 P+ P− = αZ re + − − − 3 (β+ β− )2 γ− γ+ γ+ γ− dE+ E13    2  l − γ+ 8 E1 l+ γ− + 2 − l+ l− + L − γ+ γ− + + 2 3 Me γ− − 1 γ+ − 1

211

3.5 Photon Cross Sections



1 (β+ β− )2



1 E1 2 Me /"  +

×

1+

− 2

1 E1 2 2γ Me β+ γ+ β− −

 l+ + l− −

l+ γ− l − γ+ − 2 2 β+ γ+ β− γ− (3.176)

where l± =

1 1 + β± ln β± γ± 1 − β±

L=

2 γ+ γ− (1 + β+ β− ) + 1 ln E1 β+ γ+ β− γ− M e

More detailed expressions can be found in [38]. At very high energies, extreme relativistic conditions, the cross section of Eq. (3.176) is simplified to [28]:   2 2 2 2E+ E− 1 dσ 2 2 E+ + E− + 3 E+ E− ln = 4αZ re − (3.177) dE+ E1 Me 2 E13 Integrating Eq. (3.176) gives the overall cross section for pair production, but the result is too complicated to be represented by a simple function. However, two extremes for this cross section can be analytically represented, one at an energy close to the reaction’s threshold energy and the other at highly relativistic conditions. For these two conditions, one has [29, 28]: ⎧  3 ⎪ π ⎪ 2 r 2 E1 − 2 ⎪ αZ E1 − 2Me << Me ⎪ e ⎪ ⎪ 12 Me ⎪ ⎪ ⎪   ⎨ 28 2E1 218 2 2 σ = αZ re ln E1 >> 2Me : negligible screening − ⎪ 9 Me 27 ⎪ ⎪ ⎪   ⎪ ⎪ ⎪ 28 183 2 ⎪ 2 2 ⎪ ln 1 − E1 >> 2Me : complete screening ⎩ αZ re 9 27 Z3 (3.178) The pair production cross section is, therefore, approximately proportional to Z 2 , but increases rapidly with photon energy, eventually reaching a constant value at very high energies. The above expressions for the cross section for pair production, though quite complicated, are based on the Born approximation, which is a first order approximation of the effect of the field of the nucleus as being proportional to the strength of the potential and the amplitude of the incident wave. In essence, the incident particle is considered as a plane wave incident on a potential field. This approximation is valid when αZ << 1 and when the charged particle is quite fast, i.e. when the charged particle is not strongly repelled or attracted by the nucleus. This approximation ignores a number of effects: (1) screening of the nuclear field by atomic electrons, (2) the repulsive/attractive effect of the Coulomb field on the

212

Chapter 3 Cross Sections

emitted electrons and positrons, (3) pair production by the atomic electrons, and (4) higher order corrections to the Feynman diagram of the interaction. The latter is the radiative correction, and is typically neglected, since its effect is quite small. Pair production with the atomic electrons results in triple production, which is discussed in Section 3.5.7. The other two effects are discussed below, beginning with the screening effect. For atoms of high atomic number, the negative charge of the atomic electrons weakens the positive charge of the atomic nucleus, a process known as screening. This effect is important at high photon energies where pair production is produced + E− at large impact parameters. This effect is important when 2E E1 Me is not much less than 1 1 , i.e. when the energies of both the positron and the electron are αZ 3 sufficiently high to allow contributions from high impact parameters [28, 29]. The cross section is then corrected by a screening factor. For complete screening, 1 + E− i.e. when 2E 1 , the arguments of the logarithms in Eq. (3.177) and E1 Me >> αZ 3

the second relationship in Eq. (3.178) are replaced by 1831 . This results in an Z3 asymptotic distribution that favors giving almost the entire available kinetic energy (= E1 − 2Me ) to either the positron or the electron, i.e. screening reduces the chance of the electron and the positron having equal energies. The charge of the nucleus repels the positron and attracts the electron. This Coulomb effect is a departure from the Born approximation and creates asymmetry in the distribution of the emitted particles, causing the probability of pair production to be small for large P+ and large for small P− .To correct for this effect under non-relativistic conditions, the cross section is multiplied by the following factor [29]: ξ+ ξ− ; [exp(ξ+ ) − 1][1 − exp(−ξ− )] 2παZ and β+ , β− << 1 ξ± = β±

f (β+ , β− ) =

(3.179)

This correction is needed when the condition ξ± << 1 is not satisfied, i.e. at low positron and electron energies and large Z-values. It distorts the chance that the electron and positron acquire the same energy. A small positron momentum, P+ , tends to lower the value of f (β+ , β− ), reducing the pair production cross section, while the opposite is true for small P− . In the extremely relativistic case, the following additive correction is introduced to the cross section of Eq. (3.177) [28]: 2 + 2E E  E 2 + E− dσ  3 + − 2.414(Zα)2 = −2αZ 2 re2 + 3 dE+ E1

β+ , β− >> 1 (3.180)

This correction is quite significant when E1 is equal to a few Me s, decreasing in relative effect with increasing energy, and eventually flipping sign to become an additive effect.

213

3.5 Photon Cross Sections

3.5.7 Triplet production This is an interaction with the field of the atomic electrons. Triple production, as shown in Section 2.7.3, takes place only at a photon energy greater than 4Me . It can, therefore, be considered to be a relativistic interaction. A common practice is to add the cross section for the triplet production to that of pair production, producing a cross section, κ, for the entire neutral atom (including both its electrons and nucleus). This is done by replacing Z 2 in the cross section of pair production by Z(Z + η), so that [34]: Z(Z + η) σpp Z2 E1 E1 3 + Zα − 0.00635 ln3 ln η ≈ 9 2Me 2Me κ = σpp + σtp =

(3.181) E1 > 4Me

(3.182)

where pp and tp refer, respectively, to pair and triple production. The value of η varies from a small fraction to about 1. Therefore, the addition of η to elements of large Z produces a κ-value close to that of σpp . As such, triple production is important in low Z elements.

3.5.8 Delbruck scattering Classically, Delbruck scattering approaches that of Thomson scattering with the nucleus, if the nucleus’ field is considered to be equivalent to that of a point charge of a mass equal to that of the target nucleus, M and a charge, Ze, where Z is the atomic number of the nucleus. Then, the Delbruck scattering cross section approaches the Thomson nuclear cross section, given by Eq. (3.135). In essence, while Thomson scattering is the classical counterpart of Compton scattering, Delbruck scattering is the non-relativistic limit of the photon Compton scattering by the nucleus. This approximation is valid under non-relativistic conditions, Me ≥ E, where E is the photon energy. LikeThompson scattering, Delbruck scattering is dominant at small scattering angles. However, its effect is quite negligible in comparison to Thomson scattering with the electrons, but the two combine coherently. Delbruck scattering is described by the Feynman diagram of Fig. 3.7, which shows the formation and subsequent recombination of a virtual electron/positron pair in the static (non-recoiling) field of the nucleus. Delbruck scattering can, therefore, be viewed as photon-by-photon scattering, with the incoming and outgoing photons being real ones and the “target’’ photons being virtual. At E > 2Me , Delbruck scattering can be seen as a pair production in the nuclear field followed by positron–electron annihilation. In other words, pair production is the absorptive process in Delbruck scattering, while pair annihilation is its radiative process. Alternatively, Delbruck scattering can be seen as radiative correction to Compton scattering with the nucleus. In other words, the Feynman diagram of Fig. 3.7 can be added as an extension of that of Compton scattering without

214

Chapter 3 Cross Sections

Photon × Nucleus

× Nucleus

Photon

Figure 3.7

Feynman diagram for Delbruck scattering.

affecting the outcome of the Compton interaction, since the photon energy does not change in Delbruck scattering. Given these analogies, the differential cross section for Delbruck scattering is expressed in two parts: dispersive and absorptive. The first part is the real component of the scattering amplitude, and the absorptive part is its imaginary component. Therefore, at a scattering angle, ϑ = 0, the scattering cross section is expressed as [29]:  dσ  = |a1 + ia2 |2 = a12 + a22 (3.183) d ϑ=0 ⎧ E2 73 1 ⎪ ⎪ (αZ)2 re 2 E << Me ⎨ 72 32 Me (3.184) a1 = ⎪ 7 E ⎪ ⎩ (αZ)2 re E >> 2Me 18 Me ⎧ 0; E ≤ 2Me ⎪ ⎪ ⎪  3 ⎪ ⎪ E ⎨ 1 2 (αZ) re −2 E − 2Me << Me (3.185) a2 = 24 Me ⎪    ⎪ ⎪ 7 E 2E 109 ⎪ ⎪ ⎩ (αZ)2 re ln − E >> 2Me 9π Me Me 42 The values of a1 and a2 at energy in between the above low and high energy limits can be found in [29]. The a2 factor being an absorptive term is only present when the pair production limit is exceeded. Therefore, for E >> 2Me , a2 should be corrected by the screening and atomic electron effects as in pair production (discussed in Section 3.5.6). However, the dispersive part of the cross section, a1 , is dominant from low photon energy to about 20 Me , and the absorptive component of the cross section does not have much physical significance. Notice

215

3.6 Charged-Particle Cross Sections

that the cross section for Delbruck scattering is on the order of (αZ)4 re2 , while the pair production cross section is on the order of αZ 2 re2 . Therefore, the pair production cross section is much larger than that for Delbruck scattering even at high Z-value. Therefore, Delbruck scattering is usually ignored, though it still has a noticeable effect in heavy nuclei. The Delbruck scattering cross section at ϑ > 0 is quite complicated, and depends on the momentum transferred to the nucleus, q = 2E sin ϑ2 (in relativistic terms). The cross section decreases sharply with increasing q [29]. Obviously q = 0 at ϑ = 0, where the cross section is maximum. Finally, it should be noted that Delbruck scattering combines coherently with other elastic photon scattering processes; namely with Thomson, Rayleigh, and nuclear resonance scattering.

3.6 Charged-Particle Cross Sections As with neutrons and photons, the cross sections for charged particles are dσ , which upon formulated in terms of the angular differential cross section, d integration over all solid angles gives the overall interaction cross section, σ. It dσ , is often, however, desirable to obtain the energy differential cross section, dE where E is the energy of the incident radiation or that of the scattered or emitted radiation. The latter cross section can be obtained from the angular cross section using interaction kinematics, which relate energy and direction to each other.

3.6.1 Coulomb scattering This interaction is handled through classical treatment of Coulomb scattering (no quantum effects), which is valid when the collision diameter, 2b⊥ (see Eq. (2.181)) is much larger than the radius of the target particle. Then the effect of the potential of the target is negligible. Let us consider the case of a particle of mass m1 and charge z1 e interacting with another charged particle of mass m2 and charge z2 e. The angular momentum, J , of the two particles around their center of mass (C ) is J = mm2 1+mm2 2 vr b = μ12 v1 b = l , where b is the impact parameter (distance of approach), μ12 is the reduced mass, and vr is the relative velocity between the two particles (v1 in L when particle 2 is at rest), l is a proportionality parameter not confined to integer values, since quantum effects are not considered. Then, b = l μ12 v1 . The smallest observable change in the impact parameter, b, is governed by the Heisenberg’s uncertainty principle (Section 1.3.3), i.e. b p ≥ , where p is the momentum transfer in Coulomb scattering. An estimate of p can be obtained z1 z2 e 2 by taking the product of the Coulomb force 4πε 2 and the effective time of inter0b action, which is about

2b v1 , i.e.

2z1 z2 e  0  v1 b

p ≈ 4πε . Accordingly, b|min = p = 4πε . 2z1 z2 e 2 0 v1 b 2

2

β v1 In terms of the fine structure constant, α = 4πεe 0 c , b b |min ≈ 2z1 z2 α , where β = c .

216

Chapter 3 Cross Sections

That is, the minimum relative uncertainty in the impact parameter, b b |min , is small 2z1 z2 α only when β >> 1. This inequality defines the domain of validity of the classical collision theory, and is expressed in terms of the collision diameter, 2b⊥ , Eq. (2.180), as: β=

v1 << 2z1 z2 α c

αz1 z2 = kb⊥ >> 1 β

or equivalently

(3.186)

λ where k = 2π = μ12 v1 , with λ being the Broglie wavelength. The satisfaction of the above condition makes it possible to measure b as a definite value, with a small uncertainty, without the need to resort to quantum mechanics where only the probability distribution of b can be considered. The domain of the classical theory is the extreme opposite of the domain of the Born approximation, which assumes full wave properties. As Eq. (3.186) indicates the classical theory is applicable at large values of z1 z2 and not for slow particles, and the opposite is true for the Born approximation. Within the domain of classical theory, one can define the differential cross section for an impact parameter within the small interval between b and b + db as the area of the annulus between the radii b and b + db, which is 2πb dx. Then using Eqs (2.179) and (2.181), one has b = b⊥ϑ , where ϑ3 is the scattering angle tan

3 2

in C . The differential cross section in C is expressed then as: dσ|C = 2πb db = π # =

2 cos b⊥

sin3

ϑ3 2

ϑ3 2

z1 z2 e 2 ϑ 8πε0 μ12 v12 sin2 23

dϑ3 =

$2

2 b⊥

ϑ 4 sin4 23

d

d

(3.187)

where use was made of Eq. (2.180) to arrive at the last expression. The cross section in L can be evaluated as: dσ|L =

dσ|C d d d

tan ϑ3 =

sin ϑ3 cos ϑ3 +

with

and

(3.188)

m1 m2

d 2πd cos ϑ3 m2 sin3 ϑ3 = = d 2πd cos ϑ3 (m2 + m1 cos ϑ3 )sin3 ϑ3

(3.189)

(3.190)

where as usual the primed angles are in C . Equation (3.189) is obtained from Eq. (2.23), along with the fact that in C the total momentum remains zero so

217

3.6 Charged-Particle Cross Sections

that v3 m1 = v0 m2 , with particle 2 being initially at rest, while Eq. (3.190) can be derived from Eq. (3.189). Comparing the expression of Eq. (3.187) to the general formulation of the cross section based purely on quantum mechanical principles, one can see that the two would be identical if the scattering function in Eq. (3.13) is such that 2 f (ϑ3 ) = 8πε μz1 zv22esin2 ϑ . This is a scattering function that reflects the scattering 0 12 1

3

2

1 z2 e , where r is the distance between effect of the Coulomb potential: U (r) = z4πε 0r the two interacting particles. The classical approach is valid when 2z1βz2 α >> 1, i.e. when the effect of the nucleus potential-well is not significant. However, the Coulomb field extends to r = 0, traversing by the inner potential well. Therefore, for small values of 2z1βz2 α , the influence of the potential field of the nucleus becomes considerable, and the classical theory ceases to be valid. Eliminating ϑ3 from Eq. (3.187) using the kinematic relations of Section 2.8.1, one obtains the cross section for energy transfer:

2 2 2πz12 z22 e 4 1 dσ 2 πv1 μ12 1 = = 2πb db = 2b⊥ dQ m2 Q 2 m2 v12 Q 2

(3.191)

where Q is the energy transferred to the target during the interaction. It is obvious that the energy transfer cross section is high when the target mass is small. The cross section also favors small energy transfers. It is interesting to notice in Eq. (3.191) that the mass m1 of the incident particle does not affect the energy transfer cross section. If the two interacting particles are identical, one would not be able to distinguish between them after the interaction. Their energy-transfer cross section will then correspond to an energy transfer Q to one particle, plus an energy transfer T − Q to the other particle, where T is the kinetic energy of the incident particle, assuming the target was initially at rest. The addition of these two cross sections then gives:  dσ 2πz12 z22 e 4 1 1 = m = m1 = m2 + dQ Q 2 (T − Q)2 mv12 2 !  2 "  2Q Q T 2πz12 z22 e 4 1− (3.192) +2 = 2 Q(T − Q) T T mv1 This cross section is applicable when T is selected to be such that Q ≤ (T − Q), since the case for Q ≥ 0.5 T is already included in the cross section via the (T − Q) term.

3.6.2 Rutherford scattering The classical treatment discussed in Section 3.6.1 is applicable to Rutherford scattering, which is the scattering of slow particles by heavy nuclei (large atomic

218

Chapter 3 Cross Sections

number). A number of interesting physical aspects of Rutherford scattering are presented below, using its cross section. According to Eq. (2.181), impact distances from zero to some value b produce scattering angles in C from ϑ3 to π. Integrating Eq. (3.187) over this range of angles gives the cross section: σ(≥ϑ3 )

=

ϑ3 2 cos 2 πb⊥ ϑ sin 23

= πb2

(3.193)

2 , which is the area of a disk with a It is interesting to notice that σ(≥ π2 ) = πb⊥ diameter equal to the impact diameter (2b⊥ ). This reflects well the geometry of the problem since b⊥ corresponds to the impact diameter at a π2 scattering angle, 2 area corresponds to the area that causes back deflection from π to and the πb⊥ 2 π. Using Eq. (3.189), the cross section for backscattering (ϑ3 ≥ π2 ) in L is:

σ(ϑ3 ≥

 2   z1 z2 e 2 m12 π )=π 1 − 2 4πε0 m1 v12 m22

(3.194)

Equation (3.193) shows that σ(≥ 0) = ∞, though b also becomes infinite. This indicates that there will always be scattering, even if the incident particle is approaching at an infinite distance; which is a reflection of the fact that the Coulomb field has an infinite range. However, in reality the Coulomb field of the nucleus is neutralized by the field of the atomic electrons when the distance of approach becomes large. Nevertheless, Eqs (3.193) and (3.187) indicate that the cross section is highest at small angles of scattering, i.e. forward scattering is dominant in C . At relativistic conditions, the collision radius, b⊥ in Eq. (3.187) can be replaced by the relativistic one given by Eq. (2.186). Under these conditions, as indicated in Section 2.8.1, the angle of scattering is smaller than that for non-relativistic conditions for the same impact parameter, which enhances the cross section for relativistic conditions at the same value of b⊥ . The scattering of electrons by nuclei is also subjected to a Coulomb scattering component governed by Rutherford scattering. Here, m1 << m2 , and using Eq. (3.189), ϑ3 = ϑ3 . Therefore, the cross sections given in Section 3.6.1, as well as Eq. (3.193), are equally applicable in the L system for the Rutherford scattering of electrons by nuclei. Since electrons tend to be relativistic, the differential cross section takes the form: dσ = d



Ze 2 4πε0 γme c 2 β2

2

1 4 sin4 ϑ23

 =

Zre γβ2

2

1 4 sin4

ϑ3 2

(3.195)

with Z being the atomic number of the target nucleus, v1 = βc and m1 = me γ, where me is the electron’s rest-mass energy, β and γ are the familiar Lorentz

219

3.6 Charged-Particle Cross Sections

factors, and re is the classical electron radius. The backscattering cross section is then:  2 Ze 2 π (3.196) σ(ϑ3 ≥ ) = π 2 4πε0 γme c 2 β2 Note that 2  e2 ≈ 0.25 b π 4πε0 me c 2

3.6.3 Mott scattering As indicated by relationship (3.186), classical collision theory is valid when 2z1 z2 α >> 1.This condition was imposed by the uncertainty principle to allow the β measurement of the impact parameter, b, as a definite value. When this condition is not satisfied, one must obtain probabilistic estimates of b via quantum mechanics, taking into account the potential field of the nucleus as this happens when the incident charged particle is close to the nucleus. Light charged particles moving at high speed do not also meet the conditions of the classical theory. Mott scattering deals with the scattering of electrons under the conditions of quantum mechanics. Figure 3.8 shows the Feynman diagram of this interaction. The transition probability per unit time between the initial and final states is given by [29]:  if =

Ze 2 4πε0

2

2πδ(Ef − Ei ) V2

#

$2

4π 4P 2 sin2

ϑ 2

 γ

2

ϑ 1 − β sin 2 2

2

 (3.197)

where Z is the charge of the target nucleus, ϑ is the scattering angle, β and γ are the Lorentz parameters and P is the momentum and energy of the incident and deflected electron, which do not change as implied by the delta function. The first term in Eq. (3.197) is obviously due to the Coulomb field, the denominator of the third term contains the momentum transfer as given by Eq. (2.146), and the last term is the result of averaging over the initial states and the summation over all possible polarization in the final state. Multiplying by the density of the final state

e × Nucleus e

Figure 3.8

Feynman diagram for Mott scattering.

220

Chapter 3 Cross Sections

(one electron) given by Eq. (3.121) as

V d3 P 3 , integrating over P3 , and normalizing (2π)3

with the incident flux, Vβ in natural units, gives the differential cross section [29]: dσ = d



Z 2 re2 4



1 − β2 sin2

ϑ 2 (β2 γ)2 sin4 ϑ2

At the non-relativistic limit, the Mott cross section is reduced to:  2 2 dσ Z re 1 = d 4 β4 sin4 ϑ2

(3.198)

(3.199)

This expression is identical to the Rutherford scattering cross section, Eq. (3.195) with γ = 1, which shows that the Rutherford scattering cross section is an approximation of that obtained with quantum mechanics. The relativistic form of the Mott cross section [29] is: dσ Z 2 re2 1 = 2 2 d 4 (β γ) sin4 ϑ2    ϑ ϑ ϑ 1 − sin × 1 − β2 sin2 + αZβπ sin 2 2 2

(3.200)

where α is the fine structure constant, with the added terms, compared to Eq. (3.198), represent in essence a second order Born approximation that allows the term αZ to contribute to the cross section. Notice that the cross sections of Eqs (3.198)–(3.200) approach infinity as ϑ → 0 (i.e. when the momentum transfer is small), due to the infinite range of the Coulomb field. This situation, which occurs at large impact parameters, is in practice corrected for by the screening effect of the atomic electrons.

3.6.4 Bremsstrahlung The Feynman diagram for bremsstrahlung is shown in Fig. 3.9. It involves the scattering of an electron twice, by a virtual photon from the nucleus and by the free Photon

e × Nucleus e

Figure 3.9

Feynman diagram for bremsstrahlung.

221

3.6 Charged-Particle Cross Sections

photon created in the interaction. This is the opposite of pair production, shown in Fig. 3.6, with the only difference being that the latter involves the scattering of a positron, while bremsstrahlung is typically the scattering of an electron; though the process is also applicable to positrons (recall that bremsstrahlung is negligible for heavy charged particles). Because of the similarity of bremsstrahlung and pair production, the Hamiltonians used in developing their cross sections are identical, and as such many of the relationships and trends described for pair production in Section 3.5.6 are equally applicable to bremsstrahlung. However, the two interactions differ in the density of the final state and in the definition of the flux of the projectile particle. The final state in bremsstrahlung contains an electron and a photon, designated here, respectively, by 3 and 4, with 1 identifying the incident electron, as usual. ρ The final state density per unit volume Vf , using Eqs (3.121) and (3.164), is: ρf d3 P 3 d3 P 4 P3 E3 E42 3 d4 = = V (2πc)3 dE3 (2πc)3 dE4 (2πc)6

(3.201)

Recall here that energy conservation requires that E1 = E3 + E4 and that P4 = E4 . The relative velocity (v1 + v3 ) in Eq. (3.115) is equal to the velocity of the 1 incident electron, cP E1 , where again the terminology of Table 2.1 is used. Analogous to Section 3.5.6, the following is the differential cross sections for bremsstrahlung [29]: αZ 2 2 Me2 P3 dE4 d2 σ = r d3 d4 (2π)2 e q4 P1 E4 ⎡# $2 3 · Eˆ 4 3 · Eˆ 4 P P × ⎣ 2E1 E4 − 2E3 E4 P 3 · P 4 P 1 · P 4 # −

E42 q2 #

+ E42

P 3 · Eˆ 4 P 1 ·Eˆ 4 − P 3 · P 4 P 1 · P 4

$2

P 1 · P 4 P 3 · P 4 q2 E42 − 2+ − (P 3 · P 4 )(P 1 · P 4 ) P 3 · P 4 P 1 · P 4

$" (3.202)

where q2 = |P 1 − P 3 − P 4 |2 and Eˆ 4 is a unit vector in the direction of the photon’s electric field, i.e. the direction of polarization. Relationship Eq. (3.202) is valid for both positrons and electrons. When the momentum transfer, q, is small, the cross section will tend be quite high, due to the dependence on q4 in Eq. (3.202). The momentum transfer is almost always small in the extreme relativistic case, where most of the emission occurs in the forward direction with an average angle

222

Chapter 3 Cross Sections

on the order of ϑ4 = γ11 [29]. Summation over all polarization directions gives the cross section for unpolarized radiation, expressed as [29]: d2 σ αZ 2 2 Me2 P3 dE4 = r d3 d4 (2π)2 e q4 P1 E4  2 2 β3 sin (ϑ3 − ϑ4 )(4E12 − q2 ) β12 sin2 ϑ4 (4E12 − q2 ) × + [1 − β3 cos(ϑ3 − ϑ4 )]2 (1 − β1 cos ϑ4 )2 2β1 β2 sin (ϑ3 − ϑ4 ) sin ϑ4 cos ϕ(4E3 E1 − q2 + 2E42 ) [1 − β3 cos(ϑ3 − ϑ4 )](1 − β1 cos ϑ4 ) γ32 β32 sin2 (ϑ3 − ϑ4 ) + γ12 β12 sin2 ϑ4 2 (3.203) + 2E4 γ3 γ1 (1 − β3 cos(ϑ3 − ϑ4 )(1 − β1 cos ϑ4 )

−

where, as our usual notation, the angles are with respect to the projectile particle; although most textbooks present this expression in terms of the angle with the emitted photon. The azimuthal angle, ϕ, is the angle between plane 3–1 (of the deflected electron and the incident electron) and plane 4–1 (of the photon and incident electron). For given polar angles of emission ϑ3 and ϑ4 , Eq. (3.203) indicates that maximum emission occurs at ϕ = π2 , i.e. with the photon in a direction perpendicular to the 4–1 plane. Eq. (3.203) is known as the Bethe– Heitler formulae. Integrating Eq. (3.203) over all directions of photon emission and electron deflection yields [29]: !   dσ 2 γ1 γ3 4 P 2 3 2 2 − = αZ re + − dE4 P1 E4 3 (β3 β1 )2 γ3 γ1 γ3 γ1    2  l 3 γ1 8 E4 l1 γ3 + − l + L l γ + γ + 13 3 1 2 2 3 Me γ1 − 1 γ3 − 1    1 E1 1 l1 γ3 l 3 γ1 + l1 − l 3 + 2 − 2 × 1+ 2 (β3 β1 ) 2 Me β1 γ1 β3 γ3 /"  1 E1 (3.204) +2 Me β12 γ1 β32 γ3 where l1 =

1 1 + β1 1 1 + β3 ln l3 = ln β1 γ1 1 − β1 β3 γ3 1 − β3 2 γ1 γ3 (1 + β1 β3 ) − 1 L= ln E1 β1 γ1 β3 γ3 M e

with β and γ having their usual Lorentz’ definition.

223

3.6 Charged-Particle Cross Sections

The cross section has the following limits: ⎧ 16 2 2 Me2 |P1 | + |P3 | ⎪ ⎪ ln αZ re |P1 | << Me ⎪ ⎨ 2 3 E4 P1 |P1 | − |P3 | dσ =    dE4 ⎪ γ32 2 γ3 2Me γ3 γ1 1 1 ⎪ 2 2 ⎪ E1 >> Me 1+ 2 − ln − ⎩4αZ re 3 γ1 E4 2 E4 γ1 (3.205) The maximum photon energy is equal to E1 − Me , then P3 = 0 and the cross section becomes equal to zero. Coulomb scattering of the electrons does, however, lead to a finite cross section, as discussed below. The relationships of Eq. (3.205) shows that the probability of a photon emission with an energy E4 is approximately proportional to E14 . Therefore, more low-energy photons are produced than higher-energy ones. However, the cross section of Eq. (3.205) indicates that as E4 approaches zero, the cross section diverges logarithmically since P3 then approaches P1 . This divergence is known as the infrared divergence problem, as it refers to the large wavelength near the infrared frequency range of electromagnetic radiation. This divergence is avoided by setting a low-energy threshold limit below which the energy of the emitted photon is considered to be too low to be of significance and photon emission is neglected altogether. Then the electron interaction with the nuclear field of the nucleus is simply considered to be Coulomb scattering. Screening of the nuclear field by atomic electrons limits the exceedingly large cross section obtained by Eq. (3.205) as γ1 approaches infinity, i.e. at very high incident electron energies as shown below. As in the case of pair production (Section 3.5.6) the above cross sections for bremsstrahlung are based on the Born approximation. The effect of the attractive (repulsive in case of positrons) force of the Coulomb field of the nucleus are corrected for under non-relativistic conditions, by multiplying the cross section by the Sommerfeld factor [29]:  f (β3 , β1 ) =

   1 − exp − 2παZ β1 β1   β3 1 − exp − 2παZ β3

β+ , β− << 1

(3.206)

This factor is always greater than unity, because β1 > β3 , and it does not change the angular distribution of the emitted photon. When β3 approaches 0 (i.e. P3 → 0), this factor becomes infinite, while the cross section, as Eq. (3.205) shows, converges toward zero.Therefore, the corrected cross section at the maximum photon energy E4 = E1 − Me (when P3 = 0) has a non-zero value. In effect, the field accelerates the incoming charged particle, increasing its kinetic energy beyond the incident energy, hence allowing bremsstrahlung at the apparent nominal incident particle energy. The correction for departure from the Born approximation

224

Chapter 3 Cross Sections

for relativistic energies is similar to that of Eq. (3.180), and is given by [28]:   E3 E12 + E32 2 dσ  [2.414(Zα)2 ] β+ , β− >> 1 (3.207) = −2αZ 2 re2 − E4 E1 E3 3 dE4 The electrons of high Z elements also introduce a screening effect which counters the field of the nucleus at high impact parameter (high energy). Again as in pair production, the effect of complete screening is approximately accounted for by replacing the argument of the logarithm in the second term of equation in (3.205) by 1831 . This also eliminates the logarithmic divergence in the cross Z3 section at low photon energy (since E4 no longer appears in the argument of the logarithm). The same correction makes the cross section approach a finite value as γ1 approaches infinity. Bremsstrahlung can also take place in the field of the atomic electron. As in the case of triple production (Section 3.5.7) the atomic electron can recoil with a large momentum. The increase in the bremsstrahlung cross section due to this process can be accounted for, as in the case of triple production, by replacing Z 2 in the above expression by Z(Z + η), with η as defined in Eq. (3.182) for pair production. At extremely high energies, η = 1 is a reasonable approximation for the value of η.

3.6.5 Moller scattering The Feynman diagram for this electron–electron process is shown in Fig. 3.10. The differential cross section for this interaction is given in C by [29]: ! "  (2γ 2 − 1)2 2γ 4 − γ 2 − 14 re2 (γ 2 − 1)2 dσ  = 2 2 − + d C γ (γ − 1)2 sin4 ϑ sin2 ϑ 4 (3.208) where ϑ is the scattering angle, and γ  is the Lorentz parameter, with the prime indicating that the variables are in C . The first term in the expression for Eq. (3.208) is similar to that of Rutherford scattering, Eq. (3.195), and is

e

Figure 3.10

Feynman diagram for Moller scattering.

e

225

3.6 Charged-Particle Cross Sections

attributable to the Coulomb field between of the two charges. The second term of Eq. (3.208) is an exchange term. In L, the Moller cross section is [29]:  2  dσ  cos ϑ 2 4(γ + 1) = re  2 d L β γ [2 + (γ − 1)sin2 ϑ]2 !   "  4 4 3 γ −1 2 1+ 2  (3.209) × − + sin4 ϑ sin2 ϑ 2γ sin ϑ with all variables in L, except ϑ which is in C with: γ = 2γ 2 − 1 2 − (γ + 3)sin2 ϑ 2 + (γ − 1)sin2 ϑ 8(γ + 1)cos ϑ d d = [2 + (γ − 1)sin2 ϑ]2

cos ϑ =

(3.210)

Under non-relativistic conditions, β << 1 and γ < 1, as Eq. (3.210) shows, ϑ = 2ϑ. The cross section in L then becomes:   dσ  1 1 1 2 4 cos ϑ − + β << 1 (3.211) = re d L β4 sin4 ϑ sin2 ϑ cos2 ϑ cos4 ϑ and in C : !  dσ  1 re2 = −   4 d C 16β sin4 ϑ2 sin2

1 ϑ 2

 cos2 ϑ2

+

1  cos4 ϑ2

" β << 1

(3.212)

The first term in the square brackets is the Rutherford scattering cross section of Eq. (3.195) with Z = 1 and γ = 1. The third term is present because the incident and target particles are indistinguishable, and as such while one electron scatters by an angle ϑ , the other would scatter by π − ϑ to conserve momentum. The middle term in the square brackets of Eqs (3.211) and (3.212) is not a classical term and arises from the interference between the two symmetric wave functions of the identical interacting particles. Due to the interference term, the cross section of Eqs (3.211) and (3.212) is referred to as Mott scattering, because it takes into consideration the wave mechanics of the intersecting particles, which is an essential feature of Mott scattering as discussed in Section 3.6.3. Therefore, if the energy loss per collision is less than half the rest-mass energy of the electron, the interaction is considered to be Mott scattering, and the cross section of Eqs (3.211) and (3.212) can be employed; otherwise the scattering is a Moller scattering.

226

Chapter 3 Cross Sections

Integrating any of the differential cross sections of Moller scattering over the solid angle leads to a value equal to twice the overall cross section for the interaction, due to the symmetry of this scattering interaction caused by the presence of two identical particles. The integral also diverges at zero and π scattering angles, due to the infrared divergence problem discussed in Section 3.6.4, which is overcome by threshold limits in the integral.

3.6.6 Bhabha scattering This interaction is similar to Moller scattering, as can be seen by comparing the Feynman diagrams of the two processes (Figs 3.10 and 3.11). However, in Bhabha scattering, virtual annihilation and subsequent re-generation can take place. The cross sections for the two reactions are similar except for the term for the “exchange’’ cross section for Moller scattering, which is replaced by an annihilation term in Bhabha scattering.The differential cross section in C is given by [29]: " !      re2 ϑ 2 dσ  1     4 4 ϑ = + 2(β γ ) 1 + cos 1 + 2β γ cos d C 16γ 2 [β γ  sin ϑ ]4 2 2 2 !    "  1 ϑ 2 1 ϑ 4     − + 2β γ cos 3 + 2 2β γ cos  2 4 2 γ 2 [β γ  sin2 ϑ2 ]2 +

1 [3 + 4(β γ  )2 + (β γ  )4 (1 + cos2 ϑ )] γ 4

(3.213)

where ϑ is the electron’s angle of scattering in C . Once again the divergence of the cross section at ϑ = 0 is present, but unlike Moller scattering their is no divergence at ϑ = π, since the interacting particles are not distinguishable from each other and as such the π angle of scattering for one particle does not correspond to a zero scattering angle for another. Because of the lack of symmetry between the two particles, integration over all solid angles (avoiding the zero scattering angle) provides an overall cross section for the interaction which is not twice the total value as in Moller scattering. One more difference between Bhabha and Moller scattering appears when considering the non-relativistic cross section limits of

e

Figure 3.11

Feynman diagram for Bhabha scattering.

e

227

3.6 Charged-Particle Cross Sections

e

Figure 3.12

e

Feynman diagram for pair annihilation.

both. For Bhabha scattering:  dσ  re2 = d C 16β4 sin4

ϑ 2

β << 1

(3.214)

which when compared to that of Moller scattering, Eq. (3.212), shows that the Bhabha scattering cross section, in the non-relativistic limit, coincides with that of Rutherford scattering, without the interference and symmetry terms observed in Moller scattering.

3.6.7 Pair annihilation A free electron and a positron can annihilate each other in flight as indicated by the Feynman diagram of Fig. 3.12. Typically a pair of photons is produced, but the annihilation process can result in the production of three photons; the probability of the latter is lower by an order of α (the fine structure constant) than the former. The differential cross section for annihilation expressed as the interaction 2(1,3)4 ≡ e+ (e− , γ)γ, is given by [29]:  dσ  Me2 X re2 E3 = (3.215)  d3 C 4 E4 (P 1 · P 2 )2 − M 4 (1 − cos χ ) where χ is the angle between the two emerging photons 3 and 4, with X for all polarization directions given by: 

P3 P4 X =2 + P4 P3



 + 4Me2

1 1 + 2 2 P3 P4



 − 2Me4

1 1 + 2 2 P3 P4

When the electron is at rest, the cross section in L is then [29]:    dσ  re2 1 P3 1 + cos2 χ 1 + γ1 − = 2 β1 γ1 P4 1 − cos χ d3 L

2 (3.216)

(3.217)

228

Chapter 3 Cross Sections

The overall annihilation cross section is [29]:      πre2 1 2 σ= 2 γ1 + 4 + ln γ1 + γ1 − 1 − β1 (γ1 + 3) γ1 β1 γ1 (1 + γ1 ) (3.218) The non-relativistic limit is simply: σ=

πre2 β1

β1 << 1

(3.219)

Although this cross section appears to approach zero as β1 → 0, the current of incident positrons eβ1 also approaches zero, leading to a finite interaction rate. When E1 << Me and the electron is at rest, reaction kinematics dictates that the two emitted photons have each an energy Me and are emitted in opposite directions, i.e. with χ = π. Then the differential cross section, as Eq. (3.217) indicates, r2 dσ  is equal to d  = 2βe 1 , and shows no directional preference, i.e. isotropic emis3 L sion. Integrating this cross section over all solid angles, and dividing by 2, since over 4π one of the photons would be counted twice, leads to the same cross section as that of Eq. (3.219). When a slow positron is captured by an electron, the pair momentarily exists in a bound state similar to that of the hydrogen atom. The bound entity is called a positronium. This is, however, an unstable state and  annihilates itself within about α52M = 10−10 s = 0.1 ns [29]. The annihilation of e the positron–electron is then accomplished by positronium decay. At very high energies, the cross section becomes [28]: σ=

πre2 (ln 2γ1 − 1) γ1

β1 >> 1

(3.220)

The cross section for annihilation of fast positrons is quite small, but increases with decreasing energy. Therefore, a positron tends to lose all its energy by slowingdown (bremsstrahlung) before being annihilated. A positron can also be absorbed, hence annihilated by a bound electron, in a manner similar to that of the photoelectric effect. This process is also known as single-photon annihilation, since only a single photon is emitted, with the atom recoiling to provide the needed momentum balance. The cross section per atom for this reaction with the K-shell electrons is given by [28]:  1 2 4 γ1 + 2 2 4 5 2 σK = 4πre α Z γ + γ1 + − ln (γ1 + β1 ) (3.221) β1 (γ1 + 1)2 1 3 3 β1 with the following limits:

⎧ 4πre2 4 5 ⎪ ⎨ α Z β1 3 σK = 1 ⎪ ⎩ 4πre2 α4 Z 5 γ1

β1 << 1 β1 >> 1

(3.222)

3.7 Data Libraries and Processing

229

Dividing the above cross sections by the atomic number Z, one obtains the cross section per electron. Comparing the latter with the cross section for two-photon annihilation, it is obvious that the cross section for single-photon annihilation is much lower than that for two photons (notice the high power of α). The singlephoton cross section becomes significant only at very high energies in a target with large Z.

3.7 Data Libraries and Processing 3.7.1 Libraries Cross-section data has been compiled in the form of electronic libraries, many of which are available on the Internet. The ENDF (Evaluated Nuclear Data File) format is the internationally recognized form for nuclear data. The word “evaluation’’ indicates that the cross-section values are determined by analyzing experimental data in conjunction with calculations based on appropriate models to obtain the best possible estimate of a cross section. ENDF data sets are updated only when more accurate measurements or models become available. Two different databases originally emerged: ENDF/A and ENDF/B. The former was intended for tentative partial evaluations that upon satisfactory evaluation was to be incorporated in the more complete evaluation of ENDF/B, which was made available to users. At the time of writing this manuscript, the latest version of the ENDF/B was ENDF/B-vi [25]; documentation for which was available online at http://www.nndc.bnl.gov/nndcscr/documents/endf/endf102/ The ENDF format gives each nuclide (including isomers) a unique material number (MAT). Each file stores a certain type of data and is given a label MF, e.g. descriptive and miscellaneous data are given in file MF = 1, resonance parameter data are contained in file MF = 2, etc. Reaction cross sections are designated with MT labels for different incident particles, as shown in Table 3.2; each referring to an ENDF section. Cross sections for neutron, photon, and charged particles can be found in the ENDF library. In addition to the ENDF library, the following databases are available: JEFF (Joint Evaluated Fission and Fusion File), JENDL (Japan Evaluated Nuclear Data Library), CENDL (China Evaluated Nuclear Data Library), and BROND (Library of Recommended Evaluated Neutron Data). These libraries are maintained by different centers around the World, coordinated via the Nuclear Reaction Data Centres Network (NRDC) of the International Atomic Energy Agency (http://www-nds.iaea.org/nrdc.html). Besides the above general cross-section libraries, a number of specialized databases are available. For example, XCOM is a photon cross sections database available at http://physics.nist.gov/PhysRefData/Xcom/Text/XCOM.html. Electron elastic scattering differential cross sections, and inelastic scattering properties in solids, are posted on http://www.ioffe.rssi.ru/ES/. The RTAB database

230

Chapter 3 Cross Sections

Table 3.2 MT numbers used in the ENDF format with z designating an incident particle. For more details see http://www.nea.fr/html/dbdata/ data/nds_eval_mfmt.htm MT

Reaction

1

(n,total)

4 5 27

Total neutron = 2, 4, 5, 11, 16–18, 22–25, 28–37, 41, 42, 44, 45, 102–117 (z,z0 ) Elastic scattering (z,non-elastic) = 4, 5, 11, 16–18, 22–25, 28–37, 41, 42, 44, 45, 102–117 (z,n) = 50–91 (z,anything) Sum of all reactions not given an MT number (z,abs) Absorption = 18, 102–117

101

(z,disap)

11 16 17 18 19 20 21 22 23 24 25 28 29 30 32 33 34 35 36 37 38 41 42 44 45

(z,2nd) (z,2n) (z,3n) (z,fission) (z,f ) (z,nf ) (z,2nf ) (z,nα) (z,n3α) (z,2nα) (z,3nα) (z,np) (z,n2α) (z,2n2α) (z,nd) (z,nt) (z,n3 He) (z,nd2α) (z,nt2α) (z,4n) (z,3nf) (z,2np) (z,3np) (z,n2p) (z,npα)

50

(z,n0 )

51 ... 90 91

(z,n1 ) ... (z,n40 ) (z,nc )

2 3

Disappearance = 102–117

= 19, 20, 21, 38 1st chance fission 2nd chance fission 3rd chance fission

4th chance fission

Residual nucleus in the ground state, z  = n; for z = n, MT = 2 is used Residual nucleus in 1st excited state Residual nucleus in 40th excited state Continuum neutron production (Continued)

231

3.7 Data Libraries and Processing

Table 3.2

(Cont.)

MT

Reaction

102 103 104 105 106 107 108 109 111 112 113 114 115 116 117

(z,γ) (z,p) (z,d) (z,t) (z,3 He) (z,α) (z,2α) (z,3α) (z,2p) (z,pα) (z,t2α) (z,d2α) (z,pd) (z,pt) (z,dα)

500 501 502 504 505 506 515 516 517 522 534

Radiative capture = 600–649; inelastic scattering for z = p = 650–699; inelastic scattering for z = d = 700–749; inelastic scattering for z = t = 750–799; inelastic scattering for z = 3 He = 800–849; inelastic scattering for z = α

Total charged particle stopping power Total photon interaction cross section Photon coherent scattering Photon incoherent scattering Imaginary scattering factor Real scattering factor Pair production, electron field = 515 + 517; pair production, total Pair production, nuclear field Photoelectric absorption K (1s1/2) subshell photoelectric cross section ... Q3 (7p3/2) subshell photoelectric cross section

... 572 600

(z,p0 )

601 ... 648 649 650

(z,p1 ) ... (z,p48 ) (z,pc ) (z,d0 )

651 ... 698 699

(z,d1 ) ... (z,d48 ) (z,dc )

Residual nucleus in the ground state, z  = p; for z = p, MT = 2 is used Residual nucleus in 1st excited state Residual nucleus in 48th excited state Continuum proton production Residual nucleus in the ground state, z  = d; for z = d, MT = 2 is used Residual nucleus in 1st excited state Residual nucleus in 48th excited state Continuum deuteron production (continued)

232

Chapter 3 Cross Sections

Table 3.2

(Cont.)

MT

Reaction

700

(z,t0 )

701 ... 748 749 750

(z,t1 ) ... (z,t48 ) (z,tc ) (z,3 He0 )

751 ... 798 799

(z,3 He1 ) ... (z,3 He48 ) (z,3 Hec )

800

(z,α0 )

801 ... 848 849

(z,α1 ) ... (z,α48 ) (z,αc )

Residual nucleus in the ground state, z  = t; for z = t, MT = 2 is used Residual nucleus in 1st excited state Residual nucleus in 48th excited state Continuum triton production Residual nucleus in the ground state z  = 3 He; for z = 3 He, MT = 2 is used Residual nucleus in 1st excited state Residual nucleus in 48th excited state Continuum 3 He production Residual nucleus in the ground state z  = α; for z = α, MT = 2 is used Residual nucleus in 1st excited state Residual nucleus in 48th excited state Continuum alpha production

(http://www-phys.llnl.gov/Research/scattering/) provides cross sections for Rayleigh scattering, and anomalous scattering factors. The scattering lengths and bound scattering cross sections for thermal neutrons can be found at http:// www.ncnr.nist.gov/resources/n-lengths/.

3.7.2 Processing and manipulation It is often useful to pre-process cross-section data from cross-section libraries into a form suitable for use in certain applications. For instance, it is often desirable to obtain the cross section for a mixture or a compound of nuclides; collapse the cross sections into a number of energy groups; adjust the resonance cross sections for Doppler broadening at a certain medium temperature; and so on; or simply to plot the cross sections for display or examination. A number of computer codes have been developed for this purpose, and can be acquired through the Radiation Safety Information Computational Center (RSICC) (http://wwwrsicc.ornl.gov/), and the Data Bank of the the Nuclear Energy Agency (NEA) (http://www.nea.fr/html/databank/welcome.html). Of particular interest is the NJOY Nuclear Data Processing System (http://t2.lanl.gov/codes/codes.html), a modular computer code described in [39]. The Nuclear Data Section of the International Atomic Energy Agency has also made available another modular

3.7 Data Libraries and Processing

233

set of independent pre-processing computer codes [40]. These codes can be downloaded via the Internet: (http://www-nds.iaea.org/ndspub/endf/prepro/). Although a number of codes are available for processing cross sections, the mathematical and physical aspects of some of these procedures are described below.

Energy grouping The cross section, σ(E), is a function of energy, E, but an average value over an energy range, called a group, can be obtained as:  σ(E)W (E)dE σ¯ = (3.223) W (E)dE where W(E) is a weight function, typically taken as a function that describes the variation of the flux with energy in the medium for which the cross section is to be applied. This flux variation may not be known, but in reactor applications, the reactor flux is typically taken to be inversely proportional to E, i.e. W(E) = E1 , since neutrons slow-down from higher energy to low energy, increasing neutron population in the latter. In a shielding problem, one may take, W (E) to be equal to the energy spectrum of the radiation source impinging on the shielding material. Obviously, transport calculations (see Chapter 4) need to be performed to find the appropriate weight function.This can be done iteratively, i.e. one can start with an educated guess for W (E) and utilize the average cross sections in transport calculations to obtain the flux distribution with energy, repeat the cross-section averaging process with the obtained flux energy distribution, and redo the transport calculations with the revised cross sections, and so on, until an acceptable weight function is arrived at. However, this is not usually necessary in most applications. Equation (3.223) is adequate when the cross section varies smoothly with energy, as it is the case for most cross sections, except in the resonance range (or near the absorption edges in case of photons). In such strongly energy-varying behavior, the cross sections can be evaluated one point energy at a time, at a number of contiguous points that reasonably cover the resonance region. The cross sections obtained at these points are then used to obtain an average value over the resonance. When averaging resonances over spin states, the distribution of the spin states should be included along with the weight function. Averaging a resonance would naturally result in a value lower than the maximum cross section at the resonance energy, and the more points taken for averaging the less is the underestimation of the maximum value. In averaging the differential cross sections with energy for neutrons, care must be taken to accommodate the variation of their Legendre coefficient with energy. As Eq. (3.60) shows, the differential cross section is expressed in L in terms of Legendre coefficients, Sl (E1 → E3 ), for energy transfer from E1 to E3 . This coefficient is obtained by weighting the cross section with the Legendre

234

Chapter 3 Cross Sections

polynomial, Pl , and integrating over all angles, i.e.:    1  d    2l + 1 Sl (E1 → E3 ) = 4π  dμ Pl (μ)σ(E1 → E3 ; μ)   dμ  2 −1

(3.224)

where the notation of Eq. (3.60) is utilized. However, the cross section in ENDF libraries is reported in C , as Eq. (3.56) indicates. Then substituting Eq. (3.56) in Eq. (3.224) gives [26]:  J 2l + 1 1 σ(E1 )  dμ Pl (μ) (2j + 1)Bj (E1 )Pj (η) 2 4π j=0 −1     dη  1−α 1−α   E1 − E1 η (3.225) ×   δ E3 − E1 + dμ 2 2

Sl (E1 → E3 ) = 4π

J  2l + 1 σ(E1 )Pl (μ) (2j + 1)Bj (E1 )Pj (η) = (1 − α)E1 j=0

(3.226)

where α = A−1 A+1 , with A being the ratio of the mass of the target nucleus to that of the neutron. Now, one can define a transfer matrix from one energy group, g, to another group, g  , and from an angular segment, d, to another segment, d  , by integrating Eq. (3.226) over the energy interval and the angular segments of interest, so that an element of this matrix is given by:   L    1  g  dE g dE W(E)Sl (E → E )  Tg,d;g  ,d  = 4π g dE W(E) l /    d W()P l ( ·   ) d   d (3.227) × d  W() d d

are appropriate weight functions with energy and angle, where W (E) and W () respectively. In arriving at Eq. (3.227), the weights are assumed to be separable, i.e. if the flux is used as a weight function, the change in flux with energy is considered to be independent from the change with direction. For inelastic neutron scattering, isotropy is assumed, and the matrix becomes simply an energy-to-energy operator in which an element relating an energy group g to an energy group g  is given by:      g dE g  dE W(E )σin (E → E )   (3.228) σin ( g → g ) = g dE W (E) The energy of the scattered photon, E  , emerging from the de-excitation of the ith level of the target nucleus is related to the incident neutron energy by conservation

235

3.7 Data Libraries and Processing

of energy in C so that: 

E =E



A−1 A+1



 − Ei

A A+1

 (3.229)

where Ei is the excitation energy for the ith level. When the excitation levels are too closely spaced to allow the determination of discrete energy levels, as typically the case with heavy nuclides and at higher neutron energies, expression Eq. (3.228) is no longer applicable. Then, the evaporation model discussed in Section 3.2.4 is used to describe this continuous inelastic scattering process and provide a probability for its occurrence. Some continuous distribution, f (E  ), for the energy spectrum of the scattered neutron can be assumed within the energy band at which inelastic scattering takes place. This distribution resembles that of the Maxwell spectrum evaluated at some effective temperature, but a fission-like spectrum can also be used. An energy mesh can then be defined over the energy range, and the probability for evaporation scattering, pg , is determined for each group. The fraction ofneutrons scattered into a particular energy group g  is fg  = g  f (E  )dE  , with f (E  )dE  = 1 for all groups. Then pg fg gives the transfer probability for an incident neutron of energy in group g. The same approach can be used to generate the transfer probabilities for (n,2n) reactions.

Thermal averaging The thermal energy at room temperature is 0.025 eV. However, this is the most probable energy, and a cross section evaluated at this energy alone would not be reflective of the distribution of thermal neutrons depicted by the MaxwellBoltzmann distribution of Eq. (3.74). It is, therefore, conventional to determine a cross section weighted by the Maxwell–Boltzmann distribution by the integral:       E E E I = σ(E) exp − d (3.230) kT kT kT where E is the neutron energy, T is the medium’s absolute temperature and k is the Boltzmann constant. If σ(E) is proportional to the inverse of the neutron velocity,  1 kT v , then σ(E) = σkT E . The integral of Eq. (3.230) becomes √ π 1 2 σ(kT ). Since not all cross sections have a v behavior, the factor:

i.e. σ(E) ∝

I=

2 I g=√ π σ(kT )

(3.231)

is used as a measure of the deviation of the cross section from the 1v behavior. The procedures of Eqs (3.230) and (3.231) are used to calculate integrals and g factors for the thermal capture and fission cross sections and for the resonance cross sections of the same two interactions.

236

Chapter 3 Cross Sections

In reactor physics calculations, lethargy is often used to designate the energy groups over which the cross sections are averaged. Lethargy is defined as: u = ln

Eup E

(3.232)

where E is the neutron energy and Eup is some upper energy usually taken as 10 MeV. Since neutron slowing-down typically results in a flux that is proportional to E1 , groups with the same lethargy width will have an equal portion of the overall flux.

Doppler broadening The thermal motion of the target nucleus affects the relative velocity of an incoming neutron, so that the neutron as seen by the nucleus would have a larger velocity when the neutron is approaching it, and lower velocity when it is moving away from it. This affects the cross-section resonance which would appear to broaden as the temperature increases. This is because neutrons which had an energy that exactly matched a resonance peak would no longer have an energy that coincides with the peak energy (hence would have a lower cross-section magnitude), while neutrons that had energies outside the resonance domain would be brought into the resonance’s region of influence. The thermal motion of the target nuclei can be described by the Maxwell–Boltzmann distribution of Eq. (3.74), so that: √   N (E2 )dE2 2π E2 E2 = dE2 3 exp − N kT (πkT ) 2

(3.233)

where N (E)dE2 is the number of target nuclei per unit volume of energy between E2 and E2 + dE2 , N is the total number of nuclei per unit volume, k is the Boltzmann constant, and T is the medium’s absolute temperature. The singlelevel Breit–Wigner formula for neutrons, Eq. (3.26), can be expressed for the absorption cross section at l = 0 (zero-angular momentum) as:  σ(E1 − E2 ) = σ0

E0 [(E0 )]2 E1 − E2 [(E1 − E2 )]2 + 4[(E1 − E2 ) − E0 ]2

(3.234)

where E1 is the neutron energy in L, E1 − E2 is the relative energy between the neutron and the target nucleus, E0 is the relative energy at which the resonance peaks reaches a value of σ0 , and (E) is the total neutron width at energy E. Averaging over the thermal motion of the target nucleus using the distribution Eq. (3.233) gives the resonance cross section at temperature T for a neutron of

237

3.7 Data Libraries and Processing

energy E as [4]: 1  σ0 AE0 2 σa (E1 , T ) = 2E1 πkT  ∞ [(E0 )]2 × d(E1 − E2 ) [(E1 − E2 )]2 + 4[(E1 − E2 ) − E0 ]2 0 √ √ √ √      A( E1 − E2 + E1 )2 A( E1 − E2 − E1 )2 − exp − × exp − kT kT (3.235) where A is the ratio of the mass of the target nucleus to that of the neutron, and appears to ensure that both the neutrons and the target nuclei have the same energy of kT corresponding to the most probable velocity. This averaging process reduces the peak value of a resonance as temperature increases, and broadens the energy range of its influence.

3.7.3 Compound and mixture cross sections The macroscopic cross section defined in Section 1.8.4 can be calculated for a compound or a mixture using either the atomic density or the mass density of the constituent nuclides. In terms of atomic density, the macroscopic cross section for a compound is calculated from the  microscopic cross sections, σi s, of the constituents according to Eq. (1.58):  = i Ni σi , where Ni is the atomic density of atoms of type i given by: Ni =

ρ ρ ni Ai ρ ni = = wi Mu M Ai u Ai u

(3.236)

where ρ is the material density, u is the atomic mass unit, M is the molecular weight of the compound, ni is the number of atoms of type i in the molecule (e.g. for H2 O, nH = 2, and nO = 1), Ai is the mass number of element i in i Ai the molecule, and wi = nM is the weight fraction of element i. Therefore, using Eqs (1.58) and (3.236): =

 i

wi

 ρ σi = wi i Ai u i

(3.237)

where i is the macroscopic cross section of element i, if it had the density of the compound. Equation (3.237) enables the calculation of the macroscopic cross section in terms of the mass (or weight) fractions, wi s, and is also applicable to mixtures.

238

Chapter 3 Cross Sections

The macroscopic cross section of a compound can also be obtained by viewing the compound as forming a mixture of materials each with a density, ρi , and i Vi , where Vi is the volume occupied by coma weight fraction, wi , with wi = ρρV ponent i in the mixture and V is the total volume of the mixture. Therefore, wi = ρρi αi , where αi = VVi is the volume fraction occupied by component i. Then, using Eq. (1.60), the macroscopic cross section of the compound (or its equivalent mixture) becomes: =

 ρi αi σi i

(3.238)

Ai u

This expression is obviously applicable to mixtures. However, it should be kept in mind that the macroscopic cross section of a compound should not be calculated from the microscopic cross sections of its constituent elements when molecular vibration can affect the radiation interaction, as in the case of thermalneutron scattering; then measured values for the compound should be used (see Section 3.3.7).

3.8 Problems Section 3.2 3.1 In Problem 1.10, it was shown that the electron’s magnetic moment, M , is given by Mm = − 2me e ωr , where ωr is the angular moment of the electron, e is its charge and me its mass. Taking into account the quantization of angular momentum of atomic electrons, show that the z-component of the electron magnetic moment is given by Mz = −m" μB , where z is the direction of an applied external magnetic field, m" is the magnetic quantum number and e μB = 2m (known as the Bohr magneton). The splitting of spectral lines by e the application of the external magnetic field is known as the Zeeman effect. What is the value of μB ?

Section 3.3 3.2 The Breit–Wigner formula expresses the radiative capture cross section around a resonance at a neutron energy E0 as:  n γ λ20 E0 (3.239) σ(n,γ) =

4π E (E − E0 )2 +  2 2

3.8 Problems

239

where λ0 is the neutron wavelength at energy E = E0 , g is a constant known as the statistical factor, n is the neutron resonance width, γ is the radiativecapture width and  = n + γ is the total width. Show that for a broad resonance,  >> (E − E0 ), the capture cross section becomes proportional to 1v , where v is the velocity of the incident neutron. Show also that the same 1 v behavior is obtained at low neutron energy, E << E0 . 3.3 The 1v behavior of the neutron cross section at low energy, though describes the general behavior of the cross section, is not strictly adhered to. Using the Online Cross Section Graphs of the Table of Nuclides of the Korea Atomic Energy Research Institute (http://atom.kaeri.re.kr/endfplot.shtml), or a similar library: 1. Identify the energy range within which the capture (σc ) and fission (σf ) cross sections of 235 U, 238 U, and 239 Pu can all be described by the 1v dependence. √ 2. Within this energy range, show that σc E for U238 assumes a reasonably constant value. 3. Within the above energy range, normalize the total absorption cross section (σf + σc ) of 235 U and 239 Pu to that of 238 U, and plot the normalized cross sections versus the neutron energy. With these plots examine the validity of the 1v proportionality of 235 U and 239 Pu compared to that of 238 U. σ 4. Within the same energy range, plot the ratio σf +f σc versus neutron energy for 235 U and 239 Pu, and using these plots, examine the validity of the 1v proportionality for σf and σc for these two nuclides. 5. In a thermal neutron reactor employing natural or enriched uranium, both 235 U and 238 U are present, and 239 Pu is produced. What is the mechanism that causes this production? 6. In the above described reactor, U and Pu contribute to fission. If the fuel temperature of this reactor increases, will the amount of fission increase or decrease? Base your answer on the change in the neutron velocity as a result of this temperature increase, and its effect on σf and σa given the plots obtained above.

Section 3.4 3.4 The SI system of units uses length (L) in meters, mass (M ) in kilograms and time (T ) in seconds, to measure kinematical and physical attributes. These are common quantities of everyday life. However, any three linearly independent combinations of these quantities can be employed. In relativistic quantum mechanics, the fundamental constants of c and  are often encountered, and the natural units are chosen so that c and  are both equal to unity and the energy is expressed in GeV. Therefore, kinematical and physical attributes are expressed in GeV. Determine in natural units, the values

240

Chapter 3 Cross Sections

of a (1) meter, (2) kilogram, (3) seconds, (4) barn, (5) Joule, and (6) the fine structure constant, α. Hint: Equate the dimensions of a given quantity to the units of (GeV)x c y z , find the values of x, y, and z, then the conversion factor. 3.5 Feynman diagrams are used to facilitate the calculation of the cross sections of electromagnetic interactions and weak interactions of particles. These diagrams represent the intermediate states, the amplitudes of which are summed to obtain the cross section (which is proportional to the absolute value of the total amplitude squared). The figure below shows a number of single vertex and complete Feynman diagrams. Explain what each of the graphs in the figure signify.

(a)

(f)

(b)

(c)

(g)

(d)

(h)

(e)

(i)

Section 3.5 3.6 Using xcom: Photon Cross Sections Database of the National Institute of Standards and Technology (http://physics.nist.gov/PhysRefData/Xcom/Text/XCOM.html), or a similar library: 1. Plot the microscopic cross sections for all photon absorption and scattering processes for O,Al, Fe, and Pb in the energy range from 1 keV to 10 MeV. 2. For each of the above plots, explain the behavior of the cross section with energy. 3. Comparing the above plots with each other, explain the change in their behavior with energy, from one material to another. 3.7 Show that the classical Thomson cross section: 8π 2 σT = r = 0.6653 b/electron 3 e can be obtained from the differential Thomson cross section:

(3.240)

241

3.8 Problems

3σT r2 dσe (E, ) = e cos2 ϑ + 1 = (cos2 ϑ + 1) d 2 16π

(3.241)

8πr 2

where σT = 3 e . 3.8 Plot the ratio between the Compton and the photoelectric cross sections for H and Si. Determine whether this ratio can be used to detect moisture in sand. 3.9 The Klein–Nishina relationship gives the number of photons per unit solid 1 ,ϑ3 ) , where the usual two-body notation is used. Determine: angle, dσc (hν d3 1. The number of photons scattered into a unit solid angle corresponding (hν,ϑ3 ) . Show how that relationship differs to a scattering angle ϑ3 , i.e. dσc dϑ 3

1 ,ϑ3 ) from dσc (hν . d3 dσc , and per 2. The probability of Compton recoil electrons per solid angle, d 4 dσc unit scattering angle, dϑ4 . 3. That the probability of photon Compton scattering per electron energy, dσc dT4 is:

 2 (1 + α)2 − α2 cos2 ϑ4 dσc 2π dσc = dT4 d3 α2 me c 2 (1 + α)2 − α(2 + α)cos2 ϑ4

(3.242)

4. The energy range within which Compton scattering can be assumed to be approximately isotropic and describe the angular bias outside this energy dσc (hν1 ,0) dσc (hν1 ,0) and dσ range, using the ratios dσ π . c (hν1 ,π) c (hν1 , ) 2

3.10 Plot the Rayleigh differential scattering cross section as a function of angle for photons at 0.1, 1, and 10 MeV incident on iron using“RTAB:the Rayleigh scattering database’’ at (http://www-phys.llnl.gov/Research/scattering/elastic. html), or a similar library. Compare the results and examine the validity of the relationship that estimates the largest angle of Rayleigh scattering as:

1

ϑmax = 2 sin

with E in MeV.

−1

0.0133Z 3 E

(3.243)

242

Chapter 3 Cross Sections

Section 3.6 3.11 The angular cross section of Moller scattering in L is given by Eq. (3.209). Show that when expressed in terms of energy loss, the cross section becomes: −

dσ 2πre2 me c 2 = dw β2 T !   "   γ −1 2 1 1 2γ − 1 1 dw + + − × w 2 (1 − w)2 γ2 w(1 − w) γ (3.244)

where w is the relative energy transfer, w = W T , T is the kinetic energy of the incident electron and W is the energy lost by this electron following the collision. Hint: Use reaction kinematics to show that cos ϑ = 1 − 2w. This reaction is symmetric, i.e. the two reactant particles and product particles are indistinguishable from each other. Show how this symmetry is accommodated in the above expression. 3.12 The angular distribution of the power irradiated by a charged particle, observed at distance R n from the initial position which is far away from the particle, after being accelerated only for a short time, (during which the ˙ and the velocity, β c, do not change much in direction and acceleration, βc, magnitude), can be expressed as: ˙ 2 dP(t  ) e 2 n × |{( n − β ) × β}| = d 4πc (1 − n · β )5

(3.245)

where time t  is the retarded time. ˙ and β are parallel 1. Simplify the above relationship for the cases in which β to each other with cos ϑ = n · β. 2. The non-relativistic case of β << 1 is known as the Larmor formula for radiation power, derive this result and plot the angular distribution as a polar plot in ϑ. 3. For the extreme relativistic case of β → 1, show that the angle at which the radiation intensity is maximum is given by:  / 2 − 1 β→1 1 1 + 15β ϑ = cos−1 → 3β 2γ where γ = √

1 . 1 − β2

(3.246)

243

3.8 Problems

4. For very small angles, show that the angular power distribution is simplified to dP(t  ) 8e 2 β˙ 2 γ 8 (γϑ)2 ≈ d πc (1 + γ 2 ϑ2 )5

(3.247)

5. Plot the angular distribution for a slow particle with β = 0.01 and a fast particle with β = 1. Compare the relative strength and angle with maximum intensity. Discuss the physical significance of the above relationship.

Section 3.7 3.13 Using the Online Cross Section Graphs of the Table of Nuclides of the Korea Atomic Energy Research Institute (http://atom.kaeri.re.kr/cgi-bin/ endfplot.pl), or a similar library: 1. Plot the neutron total microscopic cross section in the energy range of 1 to 3 MeV for 1 H, 12 C, 14 N, and 16 O. 2. For each of the above plots, explain the behavior of the variation of the cross section with energy. 3.14 Using one of the neutron cross section databases available on the Internet, examine and explain the validity of the statement: “The neutron elastic scattering cross section is constant over a wide energy range for light nuclei such as 2 H, 11 Be, C, and 16 O’’. 3.15 Multigroup cross sections are obtained by averaging the cross sections over energy groups of pre-determined widths, assuming a certain distribution of the particle flux with energy. Assume a E1 flux variation between groups, and a linear variation of σs (E) within each energy group and in energy between group boundaries, so that:  Eg+1 σs (E) = ln (ag + bg E) Eg 

(3.248)

where Eg+1 and Eg are the energy limits for group g, Eg ≤ E ≤ Eg+1 and ag and bg are group constants. 1. Devise an energy group structure of equal lethargy intervals for 10 energy groups for 0 ≤ E ≤ 10 MeV, with u = ln EE0 , E0 = 10 MeV. 2. Derive expressions for the group averaged neutron cross sections for hydrogen (1 H) in terms of lethargy.

244

Chapter 3 Cross Sections

3. Given that the partial cross section in L for the neutron elastic scattering from energy group g to energy g  is expressed as:  2l + 1 l σg,g  = dE  (1 − α) ug  J  dE × σs (E)Pl (μ) (2j + 1)fj (E)Pj (η) (3.249) E2 j=0 2  where E designates neutron energy, α = A−1 A+1 , ug is the lethargy width of group g, the integrals are over the energy bounds of each group, and the Legendre coefficients are in C . Derive expressions for the group and group-to-group differential scattering cross sections. Note that neutron scattering with hydrogen can be assumed to be isotropic in C , up to an energy of about 10 MeV. For the Legendre poly l nomial assume the expansion Pl (μ) = ls=0 Als μBs , where Als and Bsl are pre-determined constants. 3.16 Eq. (3.235) for resonance broadening of the neutron absorption cross section can be simplified as:  E0 σa (E1 , T ) = σ0 ψ(x, θ) (3.250) E1   ∞ dy (x − y)2 1 exp − (3.251) ψ(x, θ) = √ 4θ 1 + y2 4πθ −∞ 0 kT where x = 2 (E1 − E0 ) and θ = 4EA 2 .

1. By comparing Eq. (3.235) to the above expressions, state the conditions under which the simplified expression is valid. 2. Provide an expression for the cross section at the low- and high-energy ends (tails) of the resonance. 3. Reduce the simplified expression further to the unbroadened natural form. 4. The broadened cross section for neutron scattering is given by: σ0 n (E0 ) ψ(x, θ) γ √ σ0 (E0 ) 2μ1,2 E0 Rχ(x, θ) + 4πR 2 + γ   ∞ ydy (x − y)2 1 exp − χ(x, θ) = √ 4θ 1 + y2 πθ −∞

σs (E1 , T ) =

(3.252) (3.253)

3.8 Problems

245

where μ1,2 is the reduced mass of the neutron and the target nucleus, and R is the radius of the target. Explain the physical significance of each term in this expression in comparison to the analogous expression for the absorption cross section. 3.17 Express the cross section (for any interaction) of boiling water under saturation conditions in terms of the void fraction (volume fraction of the vapor phase).

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C H A P T E R

F O U R

Transport

Contents 4.1 4.2 4.3 4.4 4.5 4.6

Boltzmann Transport Equation Modal Solution Methods Nodal Solution Methods Stochastic Methods Transport of Charged Particles Problems

247 259 266 284 306 308

In Chapter 1 various radiation interaction mechanisms were introduced. Chapter 2 dealt with the interaction kinematics, while Chapter 3 examined the interaction probability as quantified by the cross sections. In each of these chapters, the discussion focused on a single interaction event. In reality, however, single events only occur in thin and/or not-too-dense media. Otherwise, one interaction follows another, and many particles are transported at the same time. The transported particles can also generate different types of particles, e.g. neutrons can produce γ photons, photons liberate electrons, electrons give rise to X-rays, and the latter can liberate electrons, etc. These processes are described collectively by the transport equation, which is introduced in Section 4.1. This is an integrodifferential equation that can be solved: (1) in terms of a polynomial expansion of various modes; (2) by a discretization of the solution function at different points (nodes) in the transport space; or (3) stochastically using probability distributions derived from the transport equation. These modal, nodal, and stochastic solution methods are discussed in Sections 4.3, 4.2 and 4.4, respectively.The special features of the transport of charged particles in matter are treated separately in Section 4.5.

4.1 Boltzmann Transport Equation 4.1.1 Basics Assumptions A generalized equation describing the transport of neutral particles is developed from simple bookkeeping principles. We focus first on neutrons, then incorporate Radiation Mechanics: Principles & Practice ISBN-13: 978-0-08-045053-7

© 2007 Elsevier Ltd. All rights reserved.

247

248

Chapter 4 Transport

photons.The derivations here are guided by [41].The obtained equation is known as the Boltzmann transport equation and relies on two fundamental assumptions: I. Particles travel in straight lines, without changing direction, maintaining the same velocity, until they interact with atomic nuclei. II. The transport process is linear. Assumption I is consistent with the physics of neutrons, which dictate that they are affected only by the short-range nuclear forces. Consequently, external fields, such as gravity or electromagnetic fields, have no effect on neutrons. The implication of Assumption I is that the behavior of neutrons in the proximity of the nucleus is not taken into account. This is a reasonable assumption, since in the lab system (L) an observer can only record events away from the nucleus. That is, quantum mechanics is not considered to influence the transport process directly; though the cross sections of the medium in which neutrons travel are based on quantum mechanics. The linearity of the transport process (Assumption II) requires that (i) neutrons do not interact with each other, (ii) statistical variation in the number of neutrons be ignored, and (iii) the medium with which neutrons interact maintains the same physical properties while the transport process is taking place. Typically, the number of neutrons is sufficiently large that statistical variations are negligible, but their number is small enough so that neutron–neutron collisions can be ignored. This is due to the fact that the density of neutron populations for which transport analysis is performed is usually large, but is small compared to the number of target nuclei. In addition, the medium with which neutrons interact does not usually change at a rate faster than the interaction rate of the neutrons with the medium, consequently the physical attributes of the transport system can be considered to be constant. Therefore, the macroscopic cross sections can be calculated independently, without knowing the density of the transported particles.

Neutron gas With the above two assumptions, one can view a neutron population as a “gas’’ (or a fluid) moving freely in straight lines until it encounters an obstacle, a nucleus or a number of nuclei. There is, however, a fundamental difference between a neutron gas and a classical fluid, in that neutrons do not abide by the principle of impenetrability. This principle allows fluids to form interfaces between each other, but does not permit them to penetrate solid barriers. This in turn limits the movement of fluids to within directions prescribed by solid boundaries. Neutrons, on the other hand, do not respect boundaries and can move in all directions. As a consequence, neutrons do not form clouds, droplets, clusters, and the likes. The mathematical implication is that neutrons can have a full velocity directional distribution in all possible directions. Classical fluids are described by scalars, vectors, and tensors which are functions of position only, with velocity being a dependent variable (related to momentum). With neutrons, one must use a function of position and velocity (direction and value) to describe their transport.

249

4.1 Boltzmann Transport Equation

In other words, in addition to the four independent variables used in classical fluids (three for space, one for time), one must have three more independent variables to describe the transport of neutrons: the magnitude of velocity and the direction of motion1 . Therefore, a seven-dimensional space is used for describing neutron transport: r , v , t, where the vector r designates the spatial location of a point in the space (with three coordinates), the vector v designates the velocity vector (another three coordinates: direction and magnitude), and t refers to time.

Boltzmann equation A systematic cataloging of neutrons within a finite volume ( r v t) in the seven-dimensional phase space leads to the transport equation. Let n( r , v , t) be the number of neutrons per unit volume, or neutron density, at the space location designated by the arguments in parenthesis. The rate of change of n( r , v , t) within an infinitesimal volume element d r d v dt around ( r , v , t) can be determined by bookkeeping, which considers all processes via which neutrons enter, leave, are created or destroyed. This results in the Boltzmann (1872) transport equation2 , which is a self-contained relationship that does not need a proof, but can be justified on physical grounds. It is designated, therefore, as a fundamental equation, given by: ∂n( r , v , t) = ∂t − v · ∇n( r , v , t) − v  total ( r , v , t) n( r , v , t) +  v  scatter ( r , v  → v , t) n( r , v  , t)d v  + v  νfission ( r , v  → v , t) n( r , v  , t)d v  + Q( r , v , t)

(Streaming) (Removal) (In scattering) (Fission) (Source) (Fundamental equation)

(4.1)

The terms in the right-hand side of Eq. (4.1) describe the following processes: Streaming through volume: − v · ∇n( r , v , t) is the net number per unit time of neutrons leaving minus those entering the volume element at r , with velocity v , at time t. Even in a vacant (material free) volume element, the number of neutrons leaving the volume will be different than that entering it, due to neutron divergence (spread). Removal: −vtotal ( r , v , t) n( r , v , t) is the number per unit time of neutrons removed from the element by absorption or scattering, where total is the macroscopic total cross section of material within the volume. 1 Recall

that a direction unit vector needs a polar angle and an azimuthal angle, or two direction cosines, to be fully defined; the sum of the squares of direction cosines is equal to unity. 2 This equation was originally derived for the kinetic theory of gases.

250

Chapter 4 Transport

 In scattering: v  scatter ( r , v  → v , t) n( r , v , t) d v  is the number per unit time of neutrons scattered, with velocity v , into the element, where scatter is the macroscopic scattering cross section of material within the volume.  Generation (fission): v  νfission ( r , v  → v , t) n( r , v , t) d v  is the number per unit time of neutrons produced, with velocity v , within the element by fission, where fission is the macroscopic fission cross section of material within the volume and ν is the average number of neutrons produced per fission. Source: Q( r , v , t) is the number per unit time of neutrons generated by an external source, per unit time with velocity v .

In terms of energy and direction Photons move at a constant speed, equal to the speed of light. This may appear to eliminate velocity as one of the seven independent variables of the transport space. However, photons do not necessarily have the same energy, but always possess the same speed. Therefore, to accommodate photon transport, energy usually replaces the scalar magnitude of velocity when dealing with radiation transport. This is also in agreement with the common convention of using neutron energy, rather than neutron velocity. The velocity vector v is, therefore, replaced by kinetic representing the direction of travel (or solid angle). energy3 , E, and a vector , In order to eliminate the explicit appearance of v in Eq. (4.1), the particle flux (see Section 1.8.2), φ = vn, is used, instead of the particle density, n. With: ⎧  2E ⎨ for neutrons v = , (4.2) v = mn ⎩ c , for photons t) dE d dt where mn is the neutron mass and c is the speed of light.With n( r , E, , being the number of particles (neutrons or photons) per unit volume having an about , the transport equation, energy in dE about E and direction in d Eq. (4.1), becomes: t) ∂n( r , E, , · ∇vn( r , E, , t) − total ( r , E, t) vn( r , E, , t) = − ∂t    → , t) + scatter ( r , E  → E;     → , t) + νfission ( r , E → E;   + Q( r , E, , t) × v  n( r , E  , t) dE  d

(4.3)

When dealing with photons, the fission term can be replaced by a secondary photon generation cross section. 3 E,

rather than T , is used to designate kinetic energy, to adhere with the conventional notation of the transport equation.

251

4.1 Boltzmann Transport Equation

In terms of flux With the angular flux density defined as: t) = vn( r , E, , t) φ( r , E, ,

(4.4)

Equation (4.3) gives: t) 1 ∂φ( r , E, , t) · ∇φ( r , E, , t) − total ( r , E, t)φ( r , E, , = − v ∂t    → , t) + scatter ( r , E  → E;    → , t) + νfission ( r , E  → E;   , t) dE  d  + Q( r , E, , t) × φ( r , E  , 

(4.5)

This equation accommodates both neutrons and photons, with v in the right-hand side of the equation replaced by c when dealing with photons. Photon energy can also be replaced by hν, where ν is the photon’s frequency. The seven independent variables involved in the transport equation and its integro-differential nature make it quite difficult to obtain analytical solutions, except in very few special cases. Even numerical solutions are very extensive and consume considerable amount of computational time. However, we will demonstrate below that the conventional Lambert laws of optics and the wellknown Fickian diffusion equation can be deduced by simplifying the Boltzmann transport equation.

4.1.2 Transport in void In void, and in the absence of particle sources , one would expect that the classical law of mass conservation be satisfied. The conservation of mass as seen by an observer moving with the particle “gas’’ at speed v , requires that: dn ∂n = + v · ∇n = 0 dt ∂t

(4.6)

In fluid mechanics, dn dt is known as the “material derivative’’, and is related to the partial derivative at a fixed position in space, by the streaming term, as Eq. (4.6) shows. Applying the transport equation, to void, in the absence of a source, Eq. (4.3) becomes: t) ∂n( r , E, , t) = − v · ∇n( r , E, , (4.7) ∂t It is obvious that Eqs (4.6) and (4.7) are identical, which shows that the Boltzmann particle transport equation, as one would expect, conserves mass in the absence of sources or material that can absorb or generate new particles.

252

Chapter 4 Transport

It is interesting to show that Eq. (4.7) is in effect Newton’s first law of motion, which states that if a body in motion is not acted upon by external forces, its momentum remains unchanged.The particle flux is equivalent to the momentum of particles per unit area per unit mass per unit time. Within a small spatial volume

x y z, the rate of momentum loss must be compensated for by a change in the number of particles. Therefore, for the mass within the volume, one has:

y z[φ(x) − φ(x + x)] + x z[φ(y) − φ(y + y)] ∂n + x y[φ(z) − φ(z + z)] = x y z ∂t

(4.8)

With φ = nv, on the limit:   ∂ ∂ ∂ ∂n =− nvx + nvy + nvz ∂t ∂x ∂y ∂z = −∇ · n v = − v · ∇n − n∇ · v = − v · ∇n

(4.9)

The particle velocity, v, stays constant within a void. Therefore, ∇ · v = 0. Eq. (4.9) then becomes identical to Eq. (4.7).

4.1.3 Divergence law One of the common simple laws of physics is the decrease in the flux of rays emitted from a point source with the inverse of the distance from the square

source. This is the so-called inverse square R12 law, where R is the distance from the source; which is usually derived from the fact that the total intensity of radiation emitted from a source at a certain time is constant. When considering a sphere of radius R around the source, the flux for a unit source becomes equal to 1 ; hence the R12 law of divergence. The same law can be arrived at from the 4πR 2 Boltzmann transport equation. In steady state, Eq. (4.5), in void, is reduced to: · ∇φ( r , E, ) = Q( r , E, ) 

(4.10)

Since velocity and energy do not change as the particles travel in void, we will drop E from the ensuing expressions. The scalar product in Eq. (4.10) can be so that: expressed in terms of the direction in which the particles are traveling, , · ∇φ( r , ) = || 

∂ ∂ = φ( r , ) φ( r , ) ∂R ∂R

(4.11)

Therefore, one can write: where R is a distance along . ∂ = Q( r , ) φ( r , ) ∂R

(4.12)

253

4.1 Boltzmann Transport Equation

Integrating along R, in an infinite space, gives:  ∞  ∞ ∂ dR = dR φ( r , ) Q( r , ) −∞ ∂R −∞ is then: The flux along the direction   ∞ dR Q( r  , ) φ( r , ) = −∞

(4.13)

(4.14)

only at some position r (any other The integral represents the angular flux along , would have a different value of r ). A dummy non-zero value of the flux along  variable r  was introduced in Eq. (4.14) because any source anywhere along r  will contribute to the value of the flux at r , since with particles directed in  these particles will eventually reach r , as Fig. 4.1 schematically demonstrates. The which enables integrating Eq. (4.13) vectors r and r  are related by r  = r − R , resulting in: over all possible values of ,  ∞ ) dR d φ( r ) = Q( r − R , (4.15) −∞ 4π

and dV = dR dA, where A and V refer, respectively, to area With and volume, the above equation can be expressed as:  ) Q( r − R , φ( r ) = dV (4.16) R2 d = dA/R 2

For an isotropic point source at r0 , Eq. (4.16) becomes: = Q( r , )

Q0 δ(r0 ) 4π

(4.17)

→ →

Q (r ,)

→

R →

r

→

r

Figure 4.1

Relationship between r, r  , and R.

254

Chapter 4 Transport

where Q0 is the source strength (number of particles per unit volume) and the delta function is equal to zero everywhere except at r = r0 where it is equal to unity. The flux then becomes: Q0 φ( r ) = 4π



Q0 V δ( r − R0 ) dV = 2 R 4πR02

(4.18)

This is the well-known inverse square law of radiation divergence, for a source of total intensity equal to Q0 V . In optics, this is known as Lambert’s first law.

4.1.4 Attenuation law Lambert’s third law of optics4 states that the intensity of light traveling in an absorbing medium decreases exponentially with distance. This law can also be deduced from the Boltzmann equation, while elucidating the inherent limitations of this law. This attenuation law is only obtainable from Eq. (4.5), when it is reduced to: · ∇φ( r , E, , t) − total ( r , E)φ( r , E, , t) = 0 −

(4.19)

which is arrived at when considering (i) a steady-state domain, within which there is; (ii) no particles entering from outside (i.e. no scattering into domain); (iii) no generation of particles (e.g. fission); (iv) no external sources; and (v) no change of particle energy or (vi) direction. This is typically achieved far away from the source of the particles in the idealistic situation of a steady well-collimated beam of radiation, that is so narrow that particles entering the beam from outside its domain and those generated within are so small in comparison to the radiation considered. In the beam configuration, particles do not change direction. When monitoring only radiation within the beam path, that is radiation not removed or absorbed within the medium traversed by the beam (the so-called uncollided radiation), particle energy does not change. A narrow beam by definition is also and E can be omitted one dimensional. Under all these conditions combined, t, , from Eq. (4.19). For a beam in the x-direction, the divergence term in Eq. (4.19) d d is reduced to −ˆx · dx xˆ = − dx . Equation (4.19) is then reduced to: −

4 Lambert’s

dφ(x) = total (x)φ(x) dx dφ(x) = −total (x)dx φ(x)

(4.20)

second law in optics stipulates that for rays incident on a surface at an angle, the illuminance is proportional the cosine of the angle with the normal to the surface. It applies only to directional flux (i.e. current).

255

4.1 Boltzmann Transport Equation

Integrating from x = 0 to some value x along the beam:  x  dφ(x) total (x)dx =− φ(x) 0  x φ(x) =− total (x)dx ln φ(0) 0   x φ(x) = φ(0) exp − total (x)dx

(4.21)

0

When (vii), the cross section total remains constant along the distance of integration: φ(x) = φ(0) exp[−total x]

(4.22)

Equations (4.21) and (4.22) are two different forms of the attenuation law of radiation, where six restrictive conditions are needed to arrive at the former and seven for the latter. These assumptions are usually hidden when deriving the exponential law of attenuation by observing that the reduction in flux, −dφ, within a distance dx, is proportional to the flux itself, φ, and to the traveled distance dx, with total being the proportionality constant. Notice too when arriving at these relationships, radiation divergence is not accounted for as the beam is assumed to maintain its direction and suffers no divergence. This is only practically valid over a short distance, since radiation always spreads with distance, or when divergence ceases to be significant at distances far away from the position of the source producing the particles.

4.1.5 Point kernel Combining the divergence and attenuation laws of Sections 4.1.3 and 4.1.4, respectively, the flux of uncollided (suffered no scattering) radiation at a point at r due to an isotropic point source at point r0 of strength Q can be calculated as: φuncollided ( r ) = Q( r0 , E)K ( r ; r0 , E) = Q( r0 , E)

exp[−total (E)| r − r0 |] 4π| r − r0 |2

(4.23)

where K ( r ; r0 , E) is called the point kernel, as it defines the response at a point, r , due to a point source at some other point, r0 . Notice that there is no change in energy in Eq. (4.23) since no particle collisions are taken into account along the distance r − r0 . Collisions can be accounted for susing the so-called buildup factor, B(| r − r0 |, E), evaluated from independent more detailed calculations, or experiments. Then the flux at r due to the point source Q at r0 becomes: φ( r ) = B(| r − r0 |, E)Q( r0 , E)

exp[−total (E)| r − r0 |] 4π| r − r0 |2

(4.24)

256

Chapter 4 Transport

The buildup factor can be expressed in general as: B(R, E) = 1 + a1 (E)(total R) + a2 (E)(total R)2 + · · ·

(4.25)

where the a’s are non-negative constants that can be obtained from more detailed radiation calculations. Obviously, the buildup factor, B(R, E), is greater than one. The point-kernel approach can be extended to volumetric radiation sources, which are approximated by a distribution of point sources. The simplicity of this point-kernel approach makes it attractive for use in radiation shielding calculations, particularly gamma-radiation for which buildup factors are readily available [42]. The QAD [43] computer code is one of the programs used for this purpose.

4.1.6 Diffusion theory Another common approximation of the transport equation is that of the diffusion theory, the so-called Fickian diffusion. Fick’s law states that the current of particles (directional flow) is proportional to the gradient of the particle flux (concentration), with the particle flowing from high to low concentration. Mathematically, Fick’s law is expressed as: J = −D∇φ (4.26) where J is the particle current and D is the diffusion coefficient. This law is based on the fact that in regions of high flux there is more particles to scatter away, than in regions of low flux. Therefore, there is more particle flow from high to low flux regions rather than in the opposite direction. Fick’s law considers neither the energy nor the direction of the particles, it only addresses the spatial gradient of the flux. The lack of angular dependence implies that the scattering process is isotropic, while the absence of energy dependence requires that the diffusion theory be applied only to particles of the same energy (or when an average particle energy is assumed). With those assumptions in mind, we will drop energy and direction from Eq. (4.5). In addition, we will combine the last three terms in the right-hand side of Eq. (4.5) into a single source term, S, with the understanding that all the three terms contribute to the increase in the flux. Equation (4.5) can then be expressed as: 1 ∂φ( r , t) · ∇φ( r , t) − total ( r )φ( r , t) + S( r , t) = − v ∂t Considering that: = · ∇φ + φ∇ ·  = · ∇φ ∇ · (φ)

(4.27)

(4.28)

term disappearing since diffusion is an isotropic process. With with the φ∇ ·  and using Fick’s law of Eq. (4.26), Eq. (4.28) becomes: J = φ, · ∇φ = ∇ · (φ) = ∇ · J = −∇ · (D∇φ) 

(4.29)

257

4.1 Boltzmann Transport Equation

Equation (4.29) in Eq. (4.27), for a medium of a constant diffusion coefficient, D, gives the diffusion equation: 1 ∂φ( r , t) = −D∇ 2 φ( r , t) − total ( r )φ( r , t) + S( r , t) v ∂t

(4.30)

This derivation elucidates that the diffusion equation is only applicable when (i) scattering is isotopic (or approximately isotopic). The diffusion can also be used only (ii) away from boundaries, and (iii) away from strong sources or sinks (absorbers), but is (iv) not applicable in a voided medium. The latter assumption is due to the fact that the diffusion coefficient in void is indefinite, while assumptions (ii) and (iii) are due to the fact that near boundaries and in the presence of radiation sources and sinks a strong directional preference of particle flow is established. Such directional bias is inconsistent with assumption (i), and breaks down the linearity of Fick’s laws, Eq. (4.26). The nature of these assumptions become evident in the discussion of Section 4.2.2.

4.1.7 Adjoint transport equation Let us consider a steady-state problem and write the corresponding particle transport equation in the compact form: T φ = −Q

(4.31)

where T is an operator that incorporates all the terms in Eq. (4.5) except for the Equation (4.31) describes the transport of particles external source, Q( r , E, ). within the domain of the problem.This problem has a natural boundary condition of no incoming particles at the free surfaces defining the external boundaries of the problem’s domain. In other words, the domain of the problem is defined by free-surface boundaries where no particles enter into the problem’s domain. Let us turn the problem around and rather than considering a source, we will consider With the detector as a “source’’, a detector with a response function D( r , E, ). one can write an equation “adjoint’’ to Eq. (4.31): T + φ+ = −D

(4.32)

This equation has the boundary condition of no adjoint flux, φ+ , at the free surface adjoint to that of Eq. (4.31). Equations (4.31) and (4.32) are related by:   + r , E, )d r d dE (4.33) Q( r , E, )φ ( r , E, )d r d dE = D( r , E, )φ( The right-hand side of Eq. (4.33) is obviously the overall response of the detector to the flux induced by the source Q, while φ+ in the left-hand side of the equation can be seen as the source component contributing to the detector response.

258

Chapter 4 Transport

In the above analysis, so far, no conditions have been imposed on the source Q. Any source value or distribution will result in a detector response that satisfies 0 , E0 ), Eq. (4.33) gives the Eq. (4.33). For a delta-function unit source, δ( r0 ,  value of the adjoint flux:  + r , E, )d r d dE φ ( r0 , 0 , E0 ) = D( r , E, )φ( (4.34) 0 , E0 ) would have a nil If the adjoint flux is zero, then the unit source δ( r0 ,  contribution to the detector D( r , , E). The value of the adjoint flux is in turn an indication of the “importance’’ of a particle in contributing to a detector. A particle at the free surface boundaries of the problem domain have a flux of zero importance, since the outgoing flux at the boundary is naught and cannot in turn contribute to the problem. We have considered a steady-sate analysis for simplification, but the above analysis can be easily extended to a time-varying problem, leading to the adjoint Boltzmann transport equation: −

t) 1 ∂φ∗ ( r , E, , · ∇φ∗ ( r , E, , t) − total ( r , E, t)φ∗ ( r , E, , t) = − v ∂t   → , t)φ∗ ( r , E  ,  , t)dE  d  + scatter ( r , E  → E;    → , t)φ∗ ( r , E ,  , t)dE  d  + νfission ( r , E  → E;  t) + D( r , E, ,

(4.35)

This equation is a “final-value’’ problem, unlike Eq. (4.5) which is an initial-value problem, in which the flux is determined in subsequent times based on values at a previous time. In Eq. (4.35), the adjoint flux, φ∗ , is given at a final time, tf , and the values at earlier times are determined by integrating in reverse time, as evident by the negative sign of the time derivative in Eq. (4.35) compared to that in Eq. (4.5). That is, an adjoint source (detector) affects the importance of particles at earlier times, but has no effect on particles after they are detected. On the other hand, an ordinary source has no effect on the regular flux at earlier times, but changes its value at later times. Similarly, the kinematics of the adjoint transport equation is backward, i.e. instead of a particle losing energy as it moves from a source to detector, it gains energy in adjoint calculations, since they are directed from a detector to a source. This again is physically plausible, since a detector receives particles that have interacted (losing energy in the process), but when acting as “an adjoint source’’ a detector would “energize’’ particles to bring them back to the energy of the initial source. Adjoint calculations are useful, for example, in determining the worth of a control rod in a reactor, where the rod acts as an absorber of radiation, in

259

4.2 Modal Solution Methods

a manner analogous to that of a detector. However, because of the small volume of a control rod, it is unlikely to receive much radiation, in comparison to the rest of the reactor volume. Another example is in estimating the radiation dose at points around a large distributed source of radiation, such as a cask containing nuclear waste, where adjoint calculations can be used to determine the source regions and energies of most relevance to the magnitude of the recorded dose. In performing Monte Carlo simulations, a rough adjoint calculation can be used to determine the importance-sampling (biasing) parameters that can be used to reduce the variance of the results (see Section 4.4.7).

4.2 Modal Solution Methods One of the difficulties in dealing with the seven independent variables of is that while one can designate a reference point the transport space (t, r , E, ) for the first three variables, i.e. a zero time, a spatial origin point, and a zero has no origin. A vector  can have the same value energy the direction vector, , at different points r , and different values at the same r . This poses difficulty in arriving at a solution for the transport equation. However, interaction kinematics, as shown in Chapter 2, for unpolarized radiation and when spin is neglected, are rotationally invariant; that is the change in momentum depends on μ, the cosine and that of the of the angle between the direction of the incident particle ()  scattered particle ( ), not on the specific orientation of each vector. In other words, the azimuthal direction at the same polar direction does not affect the interaction’s kinematics. The modal method of solution overcomes the above-mentioned difficulties by focusing on the angle dependence of the flux [41, Chapter 1]. It expands a over 2l + 1 subspaces, by expressing it as the summation of 2l + 1 function, g(), modal functions. This is consistent with the decomposition of the scattering cross section into its (2l + 1) partial waves (see Section 3.2). This decomposition affects, in the transport equation, Eq. (4.5), as it has however, the operator acting on  also to be decomposed into 2l + 1 components. Let Il be some operator that such that: generates the lth components of the function g()  2l + 1 ·  ) g( )  Pl ( Il g() = gl = (4.36) d 4π where Pl is a Legendre polynomial of order l (see Eq. (3.7)). The function g() can now be reconstructed by summing all its modes: = g()

N  l=0

gl =

N  l=0

= (I0 + I1 + · · · + IN )g() Il g()

(4.37)

260

Chapter 4 Transport

where the summation is limited here to some finite integer, N , as a reasonable value, since summation to infinity is impractical. Equation (4.37) is a spherical harmonics expansion, since the Legendre polynomials are a special case of the more general spherical harmonics (the angular portion of the solution to Laplace’s equation in spherical coordinates). Spherical harmonics are a set of functions in ·  and the azimuthal angle ϕ. A spherical harmonics the angular variable μ =  of the (l, m) order is given by:  (2l + 1)(l − m)! m m Plm exp(imϕ) (4.38) Yl () = Yl (μ, ϕ) = (l + m)! where Plm (μ) is the associated Legendre polynomial: Plm (μ) = (1 − μ2 )m/2

dm Pl (μ) dμm

(4.39)

For m = 0, the no-spin condition, the associated Legendre coefficients (polynomials) are identical to the Legendre coefficients Pl . Spherical harmonics are attractive for use with the particle transport equation, because they are invariant with respect to the orientation of the axes of coordinates, as they depend only on μ. In order to apply the spherical harmonics decomposition of Eq. (4.37), we will express the transport equation, Eq. (4.5), in a compact form in terms of mathematical operators, as follows: 1 ∂φ · ∇ + Tˆ ]φ + Q = T φ + Q = [− v ∂t

(4.40)

where the operator Tˆ = −Ttotal + Tscatter + νTfission incorporates the total, scattering and fission cross sections and associated integrals, while the operator · ∇ and T , called the transport operator, includes both the streaming operator  the interaction operator Tˆ . To apply the spherical harmonics decomposition of Eqs (4.37)–(4.40), we face the difficulty that the operators T and I0 + · · · + IN are not mutually commutive operators, that is:   (4.41) T (I0 + · · · + IN ) g()d = (I0 + · · · + IN )T g() d This lack of commutation does not permit the use of the rotationally invariant nature of scattering, assumed earlier. The two operators do not commute with each other, because the T operator contains the ∇ operator, being a differential to each other. Therefore, while the operator that links adjacent modes of g() l to g() l+1 , by reducing via differoperation ∇(I0 + · · · + IN )g couples g() entiation a higher order subspace by an order of one, the commutive operation (I0 + · · · + IN )∇g has no effect on the order of the subspaces. One way to get

4.2 Modal Solution Methods

261

these two operations to be commutive is to take an infinite number of modes, since then reducing the number of subspaces by one in a set of infinite size is insignificant. In practice, however, the number of modes, N , is finite, no matter how large its value is. One has, therefore, to resort to some approximation to make the two operations nearly commutive. Instead of the operators T and I0 + · · · + IN , let us consider the operator: (I0 + · · · + IN ) T (I0 + · · · + IN ), which is obviously a self-commutive operator. This in effect creates new subspaces. It also requires coupling all the N subspaces with each other, via a matrix operation of the form: ⎡ ⎤ T00 · · · T0N ⎢ . ··· . ⎥ ⎢ ⎥ ··· . ⎥ (I0 + · · · + IN ) T (I0 + · · · + IN ) = ⎢ . (4.42) ⎣ . ··· . ⎦ TN 0 · · · TNN where Tij = Ii TIj . The modified operator also requires the flux and the source in Eq. (4.40) to have corresponding projections in the new subspaces, due to the introduction of the operator I0 + · · · + IN in front of the T operator. The flux is then replaced by the vector: ⎡ ⎤ ⎡ ⎤ φ0 I0 φ ⎢ . ⎥ ⎢ . ⎥ ⎢ ⎥ ⎢ ⎥ (4.43) ⎢ . ⎥=⎢ . ⎥ ⎣ . ⎦ ⎣ . ⎦ IN φ φN The projections of the source Q are: ⎤ ⎡ ⎤ ⎡ I0 Q Q0 ⎢ . ⎥ ⎢ . ⎥ ⎥ ⎢ ⎥ ⎢ ⎢ . ⎥=⎢ . ⎥ ⎣ . ⎦ ⎣ . ⎦ IN Q QN The lth element of the flux and source vectors are:  2l + 1 ·  )φ( r , E,   , t)  Pl (  φl = d 4π  2l + 1 ·  )Q( r , E,   , t)  Pl (  d Ql = 4π

(4.44)

(4.45) (4.46)

One needs now to solve for N projections of the flux, and sum them to obtain the original flux, according to Eq. (4.37). Let us first examine the impact of the transformation (4.42) on the individual · ∇, is replaced by terms of the transport operator, T . The streaming operator, 

262

Chapter 4 Transport

Ii (·∇)I j = (·∇) ij , and will produce the ijth element of the matrix of Eq. (4.42), such that: · ∇)ij = 0 unless i = j ± 1 ( (4.47) The scattering and fission operators commute only with themselves, since they are rotationally invariant: (Ttotal )ij = (Tscatter )ij = (ν Tfission )ij = 0

unless

i=j

(4.48)

One now has a tri-block-diagonal matrix, with (−Ttotal + Tscatter + νTfission )ll on · ∇)ll±1 one row above and below the diagonal. The lth the diagonal, and −( modal equation is then: 1 ∂φl · ∇)ll−1 φl−1 + (− · ∇)ll+1 φl+1 = (− v ∂t +(−Ttotal + Tscatter + νTfission )ll φl + Ql

(4.49)

When considering n modes, the spherical harmonics solution is known as the Pn approximation. We will now consider the first order approximation to give a physical meaning to the first two modes.

4.2.1 P 1 approximation In order to incorporate the divergence term, at minimum one must consider the zeroth and first terms, otherwise, as Eq. (4.47) indicates, a P0 approximation would not contain the streaming term. With N = 1, only the (I0 + I1 )T (I0 + I1 ) operator needs to be considered, leading to the matrix equation:   1 ∂ φ0 (Tˆ )00 = · ∇)10 (− v ∂t φ1

· ∇)01 (− (Tˆ )11

  φ0 Q0 + φ1 Q1

(4.50)

where Tˆ is the interaction operator of Eq. (4.40). The flux and source terms in Eq. (4.50) are defined according to Eqs (4.45) and (4.46) as follows:  1 φ( r , E, t)  φ( r , E, , t) = φ0 = d 4π 4π  3  ( ·  ) φ( r , E,   , t) d (4.51) φ1 = 4π  1 Q( r , E, t)  Q( r , E, , t) = d (4.52) Q0 = 4π 4π  3  ( ·  ) Q( r , E,   , t) d (4.53) Q1 = 4π

263

4.2 Modal Solution Methods

It is obvious that φ0 and Q0 are scalar quantities (integrated over all directions), being, respectively, the flux and source strength, per steradian. The first mode of the flux, φ1 , is related to the current density by:  3 3  · J ( r , E, t)  , t) =  φ( r , E,   · d  (4.54) φ1 = 4π 4π The operator terms in Eq. (4.50) are resolved with the aid of Eq. (4.36). The streaming terms are expressed as:  3  ( ·  )φ d (− · ∇)01 φ1 = −I0 ( · ∇)I1 φ = −I0  · ∇ 4π  1   .∇φ1 d (4.55) = −I0  · ∇φ1 = − 4π  3 · ∇)I0 φ = −I1  ·  )  · ∇φ0 · ∇φ0 = −  ( · ∇)10 φ0 = −I1 ( d (− 4π  3  · ∇φ0   · ∇φ0 = − d =− (4.56) 4π 3   5 d   . The In the above, use was made of the identity dyadic operator 4π right-hand side of Eq. (4.55) is related to the current density of Eq. (4.54) by:     1 3  1    d  · ∇φ1 = − d  · ∇ J − 4π 4π 4π  3 1 1    ∇ · J = − ∇ · J (4.57) d = − ∇· 4π 4π 4π

Here also use was made of the identity dyadic operator. The transport operator Tˆ 00 , according to Eq. (4.36), is given as: Tˆ 00 = −total + scatter + νfission

(4.58)

The (Ttotal )11 portion of the operator or Tˆ 11 is equal to the total cross section, since  3 ·  )total ( r , E, t) J = total ( r , E, t) J  ( (4.59) d Ttotal J = 4π The above integration is made possible by the use of the identity dyadic and the On the other hand, the fact that the total cross section does not depend on . scattering operator component, (Tscatter )11 , encompasses both elastic and inelastic 5  is the linear transformation of a unit vector Vˆ

directions of Vˆ can be shown to be equal to

 · Vˆ ). The integration of this transformation over all possible to ( the definition of the identity dyadic.

4π 3 , hence

264

Chapter 4 Transport

scattering. However, for neutrons, the elastic scattering portion of this operator can be expressed as:  3 ·  )elastic →  , t) J  ( Telastic J = r , E → E,  d scatter ( 4π  3  μelastic →  , t) J = μ0 elastic d r , E,  = scatter ( removal J (4.60) 4π where μ0 is the mean cosine angle of scattering and removal is called the removal cross section; the introduction of which along with μ0 simplifies the calculation of (Telastic )11 . Note that the second term in Eq. (4.60) is possible due to the unique relationship between angle and energy in elastic neutron scattering. Using Eqs (4.59) and (4.60), it is convenient to define the transport operator: Ttransport = Ttotal − (Tscatter )11 = total − μ0 elastic removal

(4.61)

Now, with all the relevant variables and operators explicitly defined, the P1 approximation equations can be expressed in terms of the scalar flux density, φ = 4πφ1 , and the current density, J , as: 1 ∂φ = Tˆ 00 φ − ∇ · J + 4πQ0 v ∂t 1 1 ∂ J = − ∇φ + Tˆ 11 J + 4πQ1 v ∂t 3

(4.62) (4.63)

4.2.2 Diffusion equation The P1 approximation enables derivation of the diffusion equation, without invoking explicitly Fick’s law as was done in Section 4.1.6, but under the following assumptions: ∂ J ∂t

= 0: The directional flux (or current density) is constant with time. This assumption is reasonable for regions not very close to boundaries or sources or strong localized absorbers. 2. Q1 = 0: The source has no directional dependence, which is also an acceptable assumption when obtaining the flux at regions far away from strong sources or localized absorbers. 3. (νTfission )l,l = 0 for l ≥ 1: Fission and other generation reactions are isotropic. This is reasonable for most particle generating reactions, since emitted particles usually emerge isotropically in the center-of-mass system (C ). When the material inducing the generation term is composed of heavy nuclei, this assumption becomes equally valid in L. −1 4. Ttransport exists: This, as Eq. (4.61) indicates, requires that the total cross section, and consequently all other cross sections, of the material have non-zero values. This assumption, therefore, makes the diffusion theory inapplicable in void. 1.

265

4.2 Modal Solution Methods

elastic 5. inelastic scattering  scattering : This assumption is satisfied in most materials when considering neutron scattering. Inelastic neutron scattering is also isotropic, and as such does not perturb the overall isotropic behavior of the diffusion theory.

The above assumptions are consistent with those outlined in Section 4.1.6 for the diffusion theory when derived from Fick’s law. Applying Assumption 1 eliminates J from Eqs (4.62) and (4.63), allowing the two equations to be combined into a single equation: 1 −1 1 ∂φ = ∇ · Ttransport ∇φ + (Tˆ )00 φ + 4πQ0 (4.64) v ∂t 3 where Ttransport is as defined in Eq. (4.61), and use was made of Assumption 2. Assumption 4 permits the inversion of the transport operator, while Assumption 3 eliminates fission from the T11 operator, allowing the use of Ttransport . Assumption 5 allows only the use of neutron elastic scattering (while ignoring that of inelastic scattering) in defining Ttransport . The use of the removal cross section is adequate for analysis of neutron diffusion in a medium containing heavy elements; the so-called Fermi age diffusion theory. In such systems, neutrons lose their energy gradually in small increments and are scattered isotropically. Then, the diffusion problem becomes continuous in energy and analytical solutions can be found. When dealing with a medium containing light elements, such as hydrogen, the neutron can lose a significant amount of energy in one collision, and the direction of neutron scattering, though isotropic in C , becomes forward biased in L. In this case the calculation of the average cosine angle, μ0 , and consequently the removal cross section of Eq. (4.60), would not be reflective of the abrupt change in energy and the non-isotropic nature of the scattering process. In light materials, analytical continuous solutions for the diffusion equation are not possible. However, multigroup diffusion theory can be utilized. The energy domain is divided into a number of discrete finite energy groups, each group reflecting a single average energy. By convention, the group with the highest energy is designated as group 1, with g = 1, while the last group, group G, has the lowest energy (usually the thermal energy). The diffusion equation becomes a set of G-coupled linear second order equations. For each group, an energy-averaged cross section is calculated. The groups are coupled to each other by scattering and generation terms. For an energy group, g, the diffusion equation for the flux in the group, φg , is expressed as:  1 ∂φg (scatter + νfission )g,g  φg + Q0,g = −∇ · Dg ∇φg − (total )g φg + g ∂t  g

(4.65) In order to ensure the spatial continuity of flux at material interfaces, φg and Dg ∇φg |n are made equal at both sides of the interface, where ∇φg |n is the derivative of the flux normal to the surface. Since the diffusion equation is not valid in

266

Chapter 4 Transport

void (Assumption 4) the diffusion coefficient, Dg , in each group must be finite, and significantly greater than zero. For spatial continuity, Dg must also be continuous. Since the diffusion equation is not applicable near external boundaries (Assumption 1) the physical boundary is artificially extended to a point outside the physical domain of the studied system so that at the extrapolated boundary: ∂φg φg + βg ∂ˆnoutward = 0, where βg is the extrapolated length. For more information on the use of the diffusion theory, the reader can consult reactor physics textbooks, such as [3] and [4]. The diffusion equation can also be used with X- and γ-ray photons provided that the assumptions of the diffusion theory, outlined above, are satisfied. In a medium of a thickness of around one-mean-free path or less, the conditions of the diffusion theory are not satisfied, since only one collision on average would occur, which does not allow the conditions of isotropy and remoteness from boundaries to be established. The diffusion theory is best suited for use within a large system, such as a nuclear reactor.

4.2.3 Numerical solution and computer codes The modal formulation of Eqs (4.36) and (4.37) facilitates the handling of the angular dependence of the flux. However, time, energy, and space dependence still needs to be dealt with. Variation with energy is accommodated by the use of the multigroup approximation, which divides the energy domain into many groups with average cross section values supplied for each energy group. Spatial dependence of flux is handled using conventional finite-difference, or finiteelement, methods, with the latter employed to accommodate irregular geometries. Coupling between different flux modes are dealt with iteratively, starting with an initial guess. In a PN approximation, one obtains N + 1 first order differential equations of the type of Eq. (4.49). When N is odd, the number of equations is even. At a steady state, the odd modes of flux can be eliminated, leading to a system of second order equations, which can also be solved by the finitedifference or the finite-element method, as done in the MARC-PN code [44]. There are also computer codes that solve the diffusion equation with the finitedifference methods (e.g. SNAP-3D [45]), or using the method of finite-elements (e.g. FEM-2D [46]).

4.3 Nodal Solution Methods Nodal methods of solution rely on discritization, where discrete interval are defined between points (nodes). Finite-difference and finite-element methods are examples of such nodal methods, and as mentioned in Section 4.2 are used in the solution of the flux modes. This requires, however, the solution of N + 1 flux modal equations in a PN approximation. A more direct approach is to discretize all

267

4.3 Nodal Solution Methods

the seven independent variables of the transport equation, in a manner similar to that used in the finite-difference method. Then the Boltzmann transport equation is integrated over a finite-difference cell. The integro-differential form of the Boltzmann equation is then replaced by a set of simultaneous difference equations, which are numerically solved. Note that applying difference methods directly to the Boltzmann equation (rather than integrating it over each cell) leads to numerical instability in the solution, due to the difficulty of faithfully coupling the angular flux components. The discretization of the particle direction poses a challenge, since unlike space, energy, and time, the vector representing the direction of a particle has no fixed origin. We, therefore, focus first on the discretization of the angular space before proceeding further to the discretization of the other independent and dependent parameters in the transport equation.

4.3.1 Discretization of directions: discrete ordinates The particle flux at a point, φ, is the integral of its angular-dependent component, that is: f (),   φ = f ()d (4.66) The integral in the above equation can be approximated by weight sums of the at some discrete directions, m, so that: value of f () 

 = 4π f ()d

M 

m) wm f (

(4.67)

m

m ). Implementing the approximation where wm is the weight associated with f ( of Eq. (4.67) into the transport equation, Eq. (4.5), defines the so-called “discrete ordinates’’ approximation of the transport equation. The factor 4π is introduced in Eq. (4.67) so that: M 

wm = 1

(4.68)

m=1

is evaluated. In essence, where M is the number of angular intervals at which f () wm can be seen as the surface area in units of 4π, associated with a solid angle m . The condition of Eq. (4.68) ensures also that an isotropic flux has the magnitude of the scalar flux (flux integrated over all angles). The accuracy of a quadrature set depends on the number of selected discrete m and wm can be selected for the points, M . In principle, an arbitrary set of  quadrature formulae of Eq. (4.67). The Gauss quadrature method provides a “best fit’’ for integrating a function, since an M -point Gauss quadrature integrates

268

Chapter 4 Transport

correctly all polynomials of order 2M or less. However, there are a number of restrictions imposed on the selection of the discrete ordinates that do not make it possible to use Gauss quadratures in all cases. A number of moment conditions need to be satisfied in an angular set.  While Eq. (4.68) resembles the zeroth moment of the direction cosines (i.e. μ0 d = 1 with  in units of 4π), higher order moment conditions can also be defined. In general one can define the following generalized moment condition:  1 μl ηm ξ n d = Constant at given values of l, m, and n (4.69) 4π where the division by 4π is due to the normalization of the solid angle. Equation (4.69) provides the so-called (l, m, n) moment, with Eq. (4.68) giv 1 ing the (0,0,0) moment. The first moment (1,0,0): μd = 2π 4π −1 μdμ = 0, and the other two first moments (0,1,0) and (0,0,1), require the satisfaction of the conditions: M M M    wm μm = 0 wm ηm = 0 w m ξm = 0 (4.70) m=1

m=1

m=1

The above conditions ensure that the current of an isotropic flux is zero, and formulate the so-called “flow condition’’. Satisfaction of the second moment 1 2  1 μ dμ = condition (2,0,0), μ2 d = 2π 4π −1 3 , and the other second moments (0,2,0) and (0,0,3), requires that: M  m=1

wm μ2m =

M M 1  1  1 wm η2m = wm ξm2 = 3 m=1 3 m=1 3

(4.71)

This is known as the “diffusion theorem condition’’, as it is the next order of approximation after isotropic scattering and is related to the current equation (see Section 4.2.2). The larger the number of moment conditions a quadrature set satisfies, the more faithfully such a set reflects the angular space; or in other words the larger the quadrature set the more moment conditions need to be satisfied. In addition, to the moment conditions, some additional physical restrictions m vector is defined by the directional cosines (μm , ηm , ξm ) in the apply. The  three principal directions ( x, y, z ) in a rectangular coordinate system (see Fig. 4.2). By definition, μ2 + η2 + ξ 2 = 1, as such only two direction cosines need to be m is defined known to define a direction. Since radiation can change direction,  locally at each spatial point. However, selected discrete directions must be rotationally invariant, that is, rotating the axes of reference of the directional cosines m defining the solid (μm , ηm , ξm ) should not change the direction of the vector  angle of the segment m. Rotational invariance can be assured by selecting the direction cosines for each discrete angle from a set of predetermined directional cosines.Then rotating the axes of a pair of direction cosines, say μm and ηm , around

269

4.3 Nodal Solution Methods

→

z

→

j

→

V (x, y, z)

→

h →

y

→

m

→

x

Figure 4.2

Directions in a rectangular (x, y, z) coordinate system.

the axis of the third (ξ), will not produce an angular segment that does not belong to the set. Similarly, rotation around the η-axis requires that ξm be chosen from the same set as μm , and the same argument applies for rotation around the μ-axis. Therefore, the μ, η, ξ axes must be discretized in the same manner, in order to guarantee symmetry. However, in one- and two-dimensional problems (where the flux is implicitly integrated over the absent dimension(s)), strict adherence to symmetry can be relaxed in the absent dimension. That is, in a two-dimensional problem, the same  set must be used for selecting μm and ηm , but the resulting third value (ξm = ± 1 − μ2m − η2m ) does not have to belong to the same set. Moreover, in one- and two-dimensional problems, not all points in the unit sphere of the solid angle domain are needed. For example, in a two-dimensional rectangular x-y geometry, only half the ξ range is needed, say the upper hemisphere. Then, the flux becomes symmetric in ξ (which is parallel to the z axis). In one-dimensional cylindrical geometry, the flux is symmetric in ξ and η (which correspond to the axes of the polar and azimuthal angles, respectively), then only a quadrant of the sphere of the solid angle, containing the entire range of μ, is needed. If each axis, in a completely symmetric arrangement of solid-angle intervals, is divided in the same fashion into n points, then the problem is said to be solved to the Sn approximation.The order of n is even for complete symmetry. Note though for an Sn problem, the number of solid-angle segments, M , is not necessarily equal to n, but in a fully symmetric problem M is determined by: M = n(n + 2)

(4.72)

A symmetric S2 quadratic set can be constructed by simply setting μ2√= η2 = ξ 2√= 1, i.e. two points are assigned on each directional cosines axis: m ’s, constructed from − 33 and + 33 . This leads to eight different directions,  1 1 1 1 ± √ , ± √ , ± √ , with an equal weight of 8 . It can be easily shown that all the 3 3 3 above discussed conditions are satisfied, up to the third moment.

270

Chapter 4 Transport

With the direction cosines distributed in the same manner on each axis, the m terminate on the surface of a unit sphere, forming end points of each vectors  latitudes. Level symmetric (constant latitude) quadrature sets are available, generated by the computer code DOQDP [47]. This code provides fully symmetric sets (required in three-dimensional geometries), “half symmetric’’ sets (with rationally symmetry only about one axis), or non-symmetric sets (points on each axis chosen from independent sets). In addition, biased sets, where the number of directions with negative and positive orientations are not equal, are also given. These biased sets are useful when a large directional bias (anisotropy) is expected in the problem. For illustration Table 4.1 gives a symmetric three-dimensional S4 set. Notice that although Eq. (4.72) requires M = 24 when n = 4,Table 4.1 lists 32 discrete directions. The additional eight directions have wm = 0 and ξm = 0. These added dummy directions are called reference or boundary directions; needed when dealing with curved geometries to facilitate coupling between directions,  as discussed later in Section 4.3.5. For these zero-weight directions: μm = − 1 − η2m , since then ξm = 0. Table 4.1 is divided into subsets, called η levels, with each subset starting with a reference direction and ordered with increasing value of μm . The ξm value need not to be explicitly given. since they can be calculated as ξm = ± 1 − μ2 − η2 . The first quadrant for η (that with negative η) is given positive ξm values, the second quadrant is identical to the first one, except that it has negative ξm direction cosines. The third and fourth quadrants are similar to the first and second quadrants, except they have positive ηm s. For a two-dimensional problem, the missing third direction is considered to correspond to the direction of ξ, and the corresponding discrete angles with positive and negative ξm are combined into one direction, with the weights of the two original angles added. Then, in a two-dimensional case, a weight of 0.083333 is used in the non-zero weight directions, and only a total of 16 discrete angles remain (angles 9–16 and 25–32 are no longer needed, and the remaining angles are reordered to give a consecutive set). In a one-dimensional problem, the directions with common ηm values are combined, their weights are added, and the quadrature set is recorded in increasing μm values as shown in Table 4.2. Notice that the reference direction has ηm = 1, due to the absence of the η direction; hence μm = −1 for the only remaining zero-weight direction. One last observation on the quadrature sets: zero does not appear as a directional cosine, since it does not allow rotational invariance.

4.3.2 Discretization of time, energy, and space The hierarchy of the solution process in the discrete ordinates method is of the following order: time, energy, angle, and space. That is, a time interval is first considered, within that interval a solution is found within each energy group, and at each energy group the transport problem is solved in angle and space. The solution proceeds to the next time interval in the same sequences as the previous

271

4.3 Nodal Solution Methods

Table 4.1

m

A symmetric three-dimensional S 4 quadrature [47]

ηm

1, 9

−0.47140

−0.88192

0.00000

0.0000000

2, 10

−0.33333

−0.88192

∓0.33333

0.0416665

3, 11

0.33333

−0.88192

∓0.33333

0.0416665

4, 12

−0.94281

−0.33333

∓0.00000

0.0000000

5, 13

−0.88192

−0.33333

∓0.33333

0.0416665

6, 14

−0.33333

−0.33333

∓0.88192

0.0416665

7, 15

0.33333

−0.33333

∓0.88192

0.0416665

8, 16

0.88192

−0.33333

∓0.33333

0.0416665

17, 25

−0.47140

0.88192

0.00000

0.0000000

18, 26

−0.33333

0.88192

∓0.33333

0.0416665

19, 27

0.33333

0.88192

∓0.33333

0.0416665

20, 28

−0.94281

0.33333

∓0.00000

0.0000000

21, 29

−0.88192

0.33333

∓0.33333

0.0416665

22, 30

−0.33333

0.33333

∓0.88192

0.0416665

23, 31

0.33333

0.33333

∓0.88192

0.0416665

24, 32

0.88192

0.33333

∓0.33333

0.0416665

Table 4.2

ξm

wm

μm

A symmetric one-dimensional S 4 quadrature

m

μm

wm

1

−1.00000

0.00000

2

−0.88192

0.16667

3

−0.33333

0.33333

4

+0.33333

0.33333

5

+0.88192

0.16667

time step. If the problem is a steady-state one, or within each time step, the solution starts with the first energy group. Guided by [41, Chapter 3], we now introduce the discretization of the independent variables of time, energy, and space, since the discretization of angles (directions) was previously discussed in Section 4.3.1.

272

Chapter 4 Transport

The discretization of the various dependent terms of the Boltzmann transport equation is given in Section 4.3.3.

Time Time, t, is discretized into the intervals, s = 1, . . . , S, such that the time ts+ 1 covers 2 the period from ts to ts+1 . That is, non-centered subscripts are used to describe time, with the increasing values of s referring to increasing time.

Energy Energy is divided into energy groups, such that g = 1 designates the highest energy, while g = G corresponds to the lowest energy. Group g covers the energy interval from Eg− 1 to Eg+ 1 . 2

2

Spatial intervals A point i, j, k in the spatial space is given by the rectangular coordinates: xi , yj , zk . One can then define a set of non-overlapping mesh cells each with a volume: Vi+ 1 ,j+ 1 ,k+ 1 , as shown in Fig. 4.3. That is, the cell volume is given by centered 2 2 2 subscripts. However, cell surfaces are given by non-centered subscripts with areas: Ai,j+ 1 ,k+ 1 = Ai+1,j+ 1 ,k+ 1 = (yj+1 − yj )(zk+1 − zk ) 2

2

2

2

Bi+ 1 ,j,k+ 1 = Bi+ 1 ,j+1,k+ 1 = (xi+1 − xi )(zk+1 − zk ) 2

2

2

2

Ci+ 1 ,j+ 1 ,k = Ci+ 1 ,j+ 1 ,k+1 = (xi+1 − xi )(yj+1 − yj ) 2

2

2

2

→

z

i,j,k  1

i,j  1,k 1 i  1,j  1,k  1

i  1,j,k  1

Bi  ½, j  1,k  ½(side) i,j  1,k → y

i,j,k

i,j  1,k  1 i  1,j,k →

x

Ai  1,j  ½, k  ½(front) Ci  ½, j  ½,k (bottom)

Figure 4.3 An elementary volume, Vi+ 1 ,j+ 1 ,k+ 1 , in rectangular coordinates. 2

2

2

273

4.3 Nodal Solution Methods

4.3.3 Multigroup approximation The nodal solution of the transport equation requires averaging of the flux density, and other dependent variable in the transport equation, Eq. (4.5), over discrete time intervals, energy groups, solid angles, and spatial volume elements. We will start first with energy to define the flux-related quantity for which a multigroup solution of the transport equation can be obtained. A mean-value approximation allows expressing the average flux at energy group g by:  Eg− 21 t) = φg ( r , ,

Eg+ 1

t) dE φ(E, r , ,

2

(4.73)

Eg− 1 − Eg+ 1 2

2

t) is the number of particles in group g per unit volume, where φg ( r , , per unit energy, per unit solid angle, per unit time. It is customarily to solve for the quantity:   t) = Eg− 1 − Eg+ 1 φg ( r , , t) (4.74) Ng ( r , , 2

2

which is the number of particles in group g per unit volume, per unit solid angle. Therefore, at time ts (no time averaging) one has the mean value:   ts )dV d Ng ( r , , (4.75) Ng,s,m,i+ 1 ,j+ 1 ,k+ 1 = V   2 2 2 V  dV d When averaging over time, one has: Ng,s+ 1 ,m,i+ 1 ,j+ 1 ,k+ 1 = 2

2

2

2

1 ts+1 − ts



ts+1 ts

Ng,s,m,i+ 1 ,j+ 1 ,k+ 1 dt 2

2

2

(4.76)

For convenience, let us refer to Ng,s+ 1 ,m,i+ 1 ,j+ 1 ,k+ 1 simply by N with the 2 2 2 2 understanding that N is averaged over time, energy group g, solid angle m, and the volume around i + 12 , j + 21 , k + 12 . Any value N at some other discretized point will be given with the explicit superscripts of that point. The same approach will be used with other subscripted parameters, i.e. the absent subscript of a parameter is considered to be by default g, s + 12 , m, i + 12 , j + 12 , or k + 12 , as appropriate.

4.3.4 Discretization of transport equation We will now consider each term in the Boltzmann transport equation of flux, 1 ∂N Eq. (4.5). The time-derivative term, 1v ∂φ ∂t , or equivalently, v ∂t , is discretized as: t) 1 ∂N ( r , E, , 1 Ns+1 − Ns = V wm v ∂t vg ts+1 − ts

(4.77)

274

Chapter 4 Transport

·∇ N .This term is equivalent to The streaming term in Eq. (4.5) becomes: − direction = −(out − in) = in − out. the difference in particle density, along the  direction is given by: The number of particles entering a cell in the  Flux in = wm (μm Ai Ni + ηm Bj Nj + ξm Ck Nk )

(4.78)

The number of particles exiting the same cell is: Flux out = wm (μm Ai+1 Ni+1 + ηm Bj+1 Nj+1 + ξm Ck+1 Nk+1 )

(4.79)

Therefore: · ∇ N = wm [μm (Ai Ni − Ai+1 Ni+1 ) − + ηm (Bj Nj − Bj+1 Nj+1 ) + ξm (Ck Nk − Ck+1 Nk+1 )]

(4.80)

The removal term in Eq. (4.5) becomes −t N : −t N = −wm N V

(4.81)

For simplification, all other remaining terms in Eq. (4.5), the collision, fission, and source terms, are represented by a single variable: Collision, fission, and external sources = wm SV

(4.82)

A more detailed definition of this term is given later in Section 4.3.6. Using Eqs (4.77), (4.80)–(4.82) and dividing by wm t, one obtains the discretized form of the transport equation (4.5): (Ns+1 − Ns )V = μ(Ai Ni − Ai+1 Ni+1 ) vg t + η(Bj Nj − Bj+1 Nj+1 ) + ξ(Ck Nk − Ck+1 Nk+1 ) − N V + SV

(4.83)

This equation is an exact particle conservation over a finite cell in the sevendimensional space, and its limit over an infinitesimally small volume in the sevendimensional space gives6 : 1 ∂N · ∇ N + N = S + v ∂t which is a form of the transport equation (4.5). 6 · ∇N

=μ

∂N ∂N ∂N N. +η +ξ =∇ ·  ∂x ∂y ∂z

(4.84)

275

4.3 Nodal Solution Methods

The space and time domains can be divided into intervals according to the level of details in flux distribution desired in the simulated problem. The energy group structure should be selected to capture important variations with energy in the cross section of the material in the studied problem.

4.3.5 Curved geometries Curved geometries are described by cylindrical and spherical coordinates, shown in Figs 4.4 and 4.5, respectively. These coordinates have some ramifications on the discretization process. First, opposite surfaces in a volume element no longer have the same area, as in the case of the parallelepiped element created in rectangular coordinates (Fig. 4.3). Table 4.3 compares the volume and areas of an element in the three coordinate systems. Although this change in surface area from one edge of an element to another is accommodated for in the streaming term, Eq. (4.80), it affects the behavior of the streaming term. This becomes obvious by examining →

→

j

z

→

V →

(r,q,z )

h →

q

→

q

→

m q →

r

Figure 4.4

Directions in a spherical (r, ϑ, z) coordinate system. →

→

→

V

m

r

→

j q

(r,q,w) →

w

→

h

w

Figure 4.5

→

q

Directions in a spherical (r, ϑ, ϕ) coordinate system.

276

Chapter 4 Transport

Table 4.3 Volume and area of a spatial volume element in different coordinate systems

Coordinates

V

Aj

Bj

Ck

Rectangular (x, y, z)

(xi+1 − xi ) ×(yj+1 − yj ) ×(zk+1 − zk )

(yj+1 − yj ) ×(zk+1 − zk )

(xi+1 − xi ) ×(zk+1 − zk )

(xi+1 − xi ) ×(yj+1 − yj )

Cylindrical (r, ϑ, z)

2 − r 2) π(ri+1 i ×(ϑj+1 − ϑj ) ×(zk+1 − zk )

2πri ×(ϑj+1 − ϑj ) ×(zk+1 − zk )

(ri+1 − ri ) ×(zk+1 − zk )

2 − r 2) π(ri+1 i ×(ϑj+1 − ϑj )

Spherical (r, ϑ, ϕ)

4π 3 3 3 (ri+1 − ri )

πri2 ×Sj ×(ϕk+1 − ϕk )

2 − r 2) π(ri+1 i × sin 2πϑj ×(ϕk+1 − ϕk )

2 − r 2) π(ri+1 i ×(ϑj+1 − ϑj )

×Sj ×(ϕk+1 − ϕk )

Sj = 12 ( cos 2πϑj − cos 2πϑj+1 ). Angles ϑ and ϕ are measured in units of 2π.

the analytical form of the streaming term in the three coordinates systems: ⎧ ∂N ∂N ∂N ⎪ μ +η +ξ ; rectangular ⎪ ⎪ ⎪ ∂x ∂y ∂z ⎪ ⎪ ⎪ μ ∂N r η ∂N ∂N 1 ∂N η ⎪ ⎪ + +ξ − ; cylindrical ⎨ ∂z r ∂ω N = r ∂r 2 r ∂ϑ · ∇N = ∇ ·   μ ∂N r η ∂N sin ϑ ξ ∂N ⎪ ⎪ + + ⎪ ⎪ r 2 ∂r r sin ϑ ∂ϑ r sin ϑ ∂ϕ ⎪ ⎪ ⎪ 2) ⎪ ∂ cos ϑ ∂ N (1 − μ Nξ 1 ⎪ ⎩ + − spherical r ∂μ r sin ϑ ∂ω (4.85) where in  cylindrical coordinates  ω is the angle of revolution about the ξ axis, such 2 ω is the that μ = 1 − ξ cos ω, η = 1 − ξ 2 sin ω, while  in spherical coordinates  2 angle of revolution about the μ-axis with η = 1 − ξ cos ω, ξ = 1 − ξ 2 sin ω, and all the angles on the above equation are in radians. The complicated forms of the divergence term in curved geometries is due to the fact that, as evident from Figs 4.4 and 4.5, the angular directions are not in the same absolute directions at m , would all points. The result is that a particle moving in a certain direction,  have different values for direction cosines (μm , ηm , ξm ) in the curved-geometry, though its direction does not change. Therefore, the discrete ordinate equation for the streaming term must reflect this change in the values of the direction This can cosines without changing the absolute direction of the particle, . be accomplished by a meticulous discretization of the streaming term in the cylindrical and spherical coordinates. However, the practice is to introduce a correction to the steaming term of Eq. (4.80).

277

4.3 Nodal Solution Methods

In order to overcome the problem of the changing orientation of the spatial axes, hence direction cosines, in cylindrical and spherical geometries, coupling between angles is introduced in the streaming term, Eq. (4.80), so that: · ∇ N = wm [μm (Ai Ni − Ai+1 Ni+1 ) + ηm (Bj Nj − Bj+1 Nj+1 ) − +ξm (Ck Nk − Ck+1 Nk+1 )] − (αm+ 1 Nm+ 1 − αm− 1 Nm− 1 ) 2

2

2

2

(4.86)

The parameter α couples m + 12 to m − 12 , within the same element, to provide coupling between angular intervals by introducing a particle flow from the edges of the interval. The parameter α is called the curvature coefficient. The transport equation, Eq. (4.83), at steady state then becomes: −μ(Ai Ni − Ai+1 Ni+1 ) −η(Bj Nj − Bj+1 Nj+1 ) −ξ(Ck Nk − Ck+1 Nk+1 )

(4.87)

1 + (αm+ 1 Nm+ 1 − αm− 1 Nm− 1 ) + N V = SV 2 2 2 2 wm To determine the appropriate values for α, let us consider a problem in which the particle removal rate is exactly equal to its generation rate (i.e. N = S). In a steady-state problem, the value of N should stay constant. Further, let us assume for simplification that curvature occurs only in one direction, i.e. Bj+1 = Bj and Ck+1 = Ck . Then Eq. (4.87) is reduced to: (αm+ 1 Nm+ 1 − αm− 1 Nm− 1 ) = wm μm (Ai Ni − Ai+1 Ni+1 ) 2

2

2

2

(4.88)

This equation provides a relationship between the different values of αs, but does not give them specific values. Acceptable values of α should not result in the destruction or generation of new particles. Therefore, one must have: M 

(αm+ 1 Nm+ 1 − αm− 1 Nm− 1 ) = αM + 1 NM + 1 − α 1 N 1 = 0 2

2

2

2

2

2

2

2

(4.89)

m=1

When neither NM + 1 nor N 1 are equal to zero, then 2

2

α 1 = αM + 1 = 0 2

2

(4.90)

That is, particles are allowed to flow out of the angular segment at m = 1, but not into it; and the opposite is true for segment at m = M . Then, since it is assumed that the value of N does not change: M 

αm+ 1 − αm− 1 = − 2

m=1

M 

2

m=1

wm μm (Ai+1 − Ai ) = 0

(4.91)

278

Chapter 4 Transport

For curvatures with Ai+1 > Ai , the condition: M 

wm μm = 0

(4.92)

m=1

must be satisfied.This is one of the first-moment conditions imposed on the angular quadrature set (Eq. (4.70)). The same argument can be extended to a geometry curved in the other remaining dimensions, which then requires the satisfaction of the three first-moment conditions of Eq. (4.70). These first-moment conditions are easily satisfied by any symmetric quadrature set, but for a non-symmetric set in a curved geometry one must ensure that the employed quadrature set obeys this condition.

4.3.6 Source term The source term, S in Eqs (4.83) and (4.87), consists of external (independent) sources, and internal (flux dependent) sources generated by particle collisions and fission, when present. An external source is expressed simply as Q, or in full indices as Qg,s+ 1 , m,i+ 1 ,j+ 1 ,k+ 1 , which is the number of source particles emitted 2 2 2 2 per unit volume, per unit angle, per unit time, at the considered energy, time, angle, and volume element indicated by its subscripts. A neutron fission source is usually assumed to be isotropic, with a certain energy spectrum. The number of fission neutrons produced in group g is, therefore, expressed as: (F N )g = (νfission )g N¯ g

(4.93)

where N¯ g is the average number, over all angles, of neutrons in group g. However, the fission term in the transport equation represents the contribution of fission in all energy groups to a certain group g. The fission source term is, therefore, given by: (Fission source)g = χg

G 

(F N )h = χg

h=1

G 

νfission N¯ h = χg F h

(4.94)

h=1

where χg represents the fission spectrum, and F is the total number of fission neutrons. The collision source naturally includes angular as well as energy, dependence. Therefore, (Collision source)g,m =

M G   h=1

h,g,m,m Nh,m wm

(4.95)

m =1

where h,g,m,m is a four-dimensional array, which is also a function of position. The summations in Eq. (4.95) allows for both downscattering and upscattering (in

279

4.3 Nodal Solution Methods

the case of thermal neutrons). It requires, however, the angular flux, Nh,m , in all directions, energies, and positions, to be available. However, resorting to the usual expression of the scattering cross section as a Legendre expansion (Section 3.2) this collision term, can be significantly simplified and the computations process (as well as the associated computer data storage and manipulation) becomes much less complicated. With the scattering cross section expressed in the fashion of Eq. (3.20) as:  → E, ) = (E , 

L 

 · ) (2l + 1)l (E  → E) Pl (

(4.96)

l=0

where the 4π factor was dropped, since the angle wm is measured in units of 4π, the collision source can be expressed as:

(Collision source)g,m =

L 

(2l + 1)

l=0

G 

lh,g

h=1

l 

fml,r Nhl,r

(4.97)

r=0

where Nhl,r is the moment of the angular flux: Nhl,r

M 

=

m =1

wm fml,r Nh,m

(4.98)

 · ) in Legendre and associated and fml,r is the result of the expansion of Pl ( l,r Legendre polynomials. The array fm is a two-dimensional array with l + 1 values of r for each l. The definition of the collision source relies now on the Legendre coefficients of the transfer (differential) cross section lh,g .The use of discrete ordinates along with the Legendre expansion of the scattering cross sections produces the so-called Sn Pn method.

4.3.7 Solution of S n equations Difference methods The aim of the solution of Sn equations is obviously to find the values of the flux, N , as defined by Eq. (4.74), at all the discrete elements designated in the sevendimensional transport space. However, the fission and collision sources, Eqs (4.95) and (4.97), depend on values of N . Even without considering these source terms,

280

Chapter 4 Transport

the time-dependent Sn equation takes the form: (Ns+1 − Ns )

V − μ(Ai Ni − Ai+1 Ni+1 ) v t −η(Bj Nj − Bj+1 Nj+1 )

−ξ(Ck Nk − Ck+1 Nk+1 )   1 αm+ 1 Nm+ 1 − αm− 1 Nm− 1 + 2 2 2 2 wm +N V = SV

(4.99)

which requires the evaluation of eleven flux terms, at each cell; namely Ns+1 , Ns , Ni , Ni+1 , Nj , Nj+1 , Nk , Nk+1 , Nm+ 1 , Nm− 1 , and Ns+ 1 ,g,m,i+ 1 ,j+ 1 ,k+ 1 = N . 2 2 2 2 2 2 These seven fluxes are, however, present within the same seven-dimensional cells or at its boundaries. Some relationship can be assumed between these unknowns fluxes. Some of the difference methods for relating the above eleven fluxes to each other are described below. Difference methods relate the value of the flux, N , at one end of a finite interval to those at the middle of the interval and at the other end. Perhaps the most obvious approach is to assume a linear change in the flux throughout the interval. In the seven-dimensional problem at hand, the linear difference method can be seen as relating linearly the fluxes at the tips of a diamond to the flux at its center. This diamond difference method assumes, therefore, that the arithmetic average over the edges of a cell is equal to the average over the entire cell, that is: Ns+1 + Ns = Ni + Ni+1 = Nj + Nj+1 = Nk + Nk+1 = Nm+ 1 + Nm− 1 = 2N 2 2 (4.100)

Then, if it is assumed that Ni , Nj , Nk , Nm− 1 are known, that is, if the calculations 2 proceed in the direction of increasing m, s, i, j, and k, then Eq. (4.100) can be used to remove Ns+1 , Ni+1 , Nj+1 , Nk+1 , Nm+ 1 , from Eq. (4.99), which is then 2 solved for the unknown fluxes, so that: N =

˜ Ns + SV μANi + ηB Nj + ξC Nk + αNm− 1 w1 + V 2

˜ + V 2μAi+1 + 2ηBj+1 + 2ξCk+1 + 2αm+ 1 w1 + V

(4.101)

2

where A = Ai+1 + Ai , B = Bj+1 + Bj , C = Ck+1 + Ck , α = αm+ 1 + αm− 1 , and 2 2 ˜ = 2 . Equation (4.101) indicates that N is a weighted average of the flux  v t sources in the considered cell. The linear change of the flux assumed in the diamond method can lead to negative values of flux, caused by a steep decline in the flux. This can occur if

281

4.3 Nodal Solution Methods

the size of the cell is large, the material’s total cross section is large and/or the particle severely changes its direction as it crosses a cell. In either of these cases, linearity can be adversely affected. Then a step change in the flux across the cell can remedy the negative flux problem. The step difference assumes that the flux N is constant throughout the cell, so that: N = Ns+1 = Ni+1 = Nj+1 = Nk+1 = Nm+ 1 2

(4.102)

Boundary and initial conditions enable the solution for N to advance from one cell to another and from one time interval to the next. Although this step difference method ensures that N is always positive, it provides a very abrupt change in the flux from one cell to another, and as such it is not a very accurate solution method. It is possible, however, to employ the linear diamond model everywhere in the problem, except when it produces a negative flux where it can then be replaced by the step difference method. However, this can come at the expense of flux oscillations between adjacent cells that can destabilize the solution process. The weighted difference attempts to provide a flux variation in between the abrupt change of the step difference and the linear change of the diamond method. The flux across a cell is assumed to be a weighted average of the flux at opposing edges of the cell, that is: ⎧ ai Ni+1 + (1 − ai )Ni + ⎪ ⎪ ⎪ ⎨aj Nj+1 + (1 − aj )Nj + N = ak Nk+1 + (1 − ak )Nk + ⎪ ⎪ ⎪ ⎩a N 1 + (1 − a )N 1 m m+ m m− 2

(4.103)

2

where the a’s are fractions between 12 and unity, since the step difference is produced when all a’s are equal to unity, while all a’s equal to 0.5 give the diamond difference. Equation (4.103) is then used to calculate Ni+1 , Nj+1 , Nk+1 , and Nm+ 1 from the corresponding values in a preceding cell and their averages; start2 ing with known values at boundaries and from initial conditions. The weighted difference method reduces the chance of arriving at negative fluxes. More elaborate difference methods and guidance on their use can be found in [48]. Such methods include various ways for choosing the weighting constants, obtaining solutions within finer cells within the problem’s original cells, the utilization of more elaborate function fits based on semi-analytical solutions of the transport equation within a cell, and the application of the more sophisticated characteristics method.

Boundary conditions The solution of the difference equations discussed above is not possible without boundary conditions, and in transient problems initial conditions. Boundary

282

Chapter 4 Transport

conditions take various forms, the simplest of which is the void or vacuum condition in which all flux terms at the boundary and inward directed to the problem geometry are set equal to zero. This void boundary condition is typically used in place of air at the boundary of a problem geometry. In problems where symmetry exists, the reflected boundary condition can be imposed at the line of symmetry, where the inward flux is set equal to the outward flux in the direction of reflection. Such a reflected boundary condition can be imposed for example at the centerline of a cylinder, the horizontal mid-plan of a homogeneous cylinder and at the outer surfaces of cylindrical lattices representing fuel cells in a reactor. Another way that enables conducting the calculation on a portion of geometry in a larger system is to impose an albedo (also called gray) boundary condition at the boundary of the studied geometry. The albedo boundary condition is such that the incoming flux at the boundary is isotropic and the incoming particle current is only a fraction, α, of the outgoing current. That is, incoming Nm m wm  α = outgoing wm Nm m

(4.104)

Knowing the value of α from previous calculations and or measurements, enables the calculation of Nm at the boundary, but requires iteration to determine the numerator of Eq. (4.104). In the white boundary condition, α = 1. When the flux at the boundary is fully known, e.g. from previous calculations, or when it can be made available to the boundary of the studied portion of the geometry, one has a flux (or a fixed source) boundary condition. This boundary condition can also be used in solving successive portions of geometry, by bootstrapping. In the process, the problem is solved to a certain boundary and the flux at the boundary is used as a flux boundary condition to the adjacent portion of the geometry and so on. On more boundary condition is the periodic boundary condition, which equates the incoming flux in one boundary to the flux in the same direction at the opposite boundary. This boundary condition is useful when dealing with a continuing array of symmetric objects.

Iterative solution The calculation of the fission source (when present) and the collision source, using Eqs (4.97) and (4.95), respectively, requires determining in advance the flux at all other cells in the problem. This is done iteratively starting by an initial guess. We will assume here a steady-state, or a quasi-steady-state, problem. In the latter case, Ns+1 is set equal to Ns , to enable the evaluation of Ns+ 1 = N in Eq. (4.99) 2 at all other energies, angles, and spatial intervals, then the same equation is used to calculate Ns+1 , assuming Ns is known from the solution for the previous time step (with N0 known from the initial conditions). Calculations typically follow the flow of particles, starting from the direction of an outer boundary condition and proceeding progressively to the next most

4.3 Nodal Solution Methods

283

inward direction. An inward spatial sweep is made in all incoming directions, followed by the imposition of inner boundary conditions, then an outward calculation sweep is conducted. With energy, calculations also follow the particle flow, proceeding from the highest energy, g = 1, to lower energies, until the last group G. The collision and fission sources are calculated within this inner iteration using available flux information. The iteration process is repeated until the problem converges to an acceptable level within an energy group. When convergence is assured in one group, calculations proceed to the next group, and so on. In problems involving fission and upscattering, outer iterations are conducted after the inner iterations for all groups are completed. An outer iteration repeats an entire cycle of inner iterations, using previously calculated fluxes to determine these internal sources. Outer iterations also enable the computation of the multiplication factor, or equivalently the eigenvalues in problems involving fission. This is done by comparing the total value (over all cells) of the fission source between outer iterations.

4.3.8 Computer codes A number of computer codes are available for solving the steady-state transport equation in one dimension (ANISN [49]), two dimensions (DORT [50]), and three dimensions (TORT [50]). TDTORT [48] is the time-dependent version of TORT. An Sn code that utilizes the finite-difference method is ONETRAN [51], which uses “linear discontinuous finite-element representation for the angular flux in each spatial mesh cell’’. These codes can be acquired through the Radiation Safety Information Computational Center, Oak Ridge, TN (http://wwwrsicc.ornl.gov), or the OECD Nuclear Energy Agency, France (http://www.nea.fr). The multigroup approximation of the Sn method, and the corresponding multigroup cross sections, facilitate the performance of adjoint calculations. As explained in Section 4.1.7, these calculations require following the particle energy upward, from a detector to a source. The matrix of the multigroup scattering cross section from one group to another can be readily transposed to provide the “upward’’ (in energy), cross sections required for the transport calculations. The main advantage of the Sn method is that it provides a detailed distribution of the flux with time (in a transient problem), energy, angle, and location. The method is well suited for thick deep-penetration shielding problems, as well as reactor calculations. It is amenable also to bootstrapping by solving a problem piece-by-piece. However, convergence of the solution in Sn analysis is not always attained, and even when it occurs it may not be uniform throughout various regions of the problem. If an angular interval is too wide, the so-called rayeffect can take place, in which radiation streaming into a particular direction is underestimated. The choice and size of the angular quadrature, spatial elements, energy groups, time interval (in transient problems), the difference scheme, and the iteration convergence criteria, can affect the accuracy of the solution. Care must also be taken when handling internal voids (or air) to avoid negative fluxes

284

Chapter 4 Transport

and ensure proper directional flow of particles in these regions (within which no significant interactions take place).

4.4 Stochastic Methods 4.4.1 Introduction The stochastic solution of the particle transport problem, better known as the Monte Carlo method7 , relies on the simulation of particle movement and interactions via random variables sampled from probability distributions. The Monte Carlo method in its simplest form involves a process for producing a random event, repeated N times with each trial being independent of the others. At the end of the simulation, the results of all trials are averaged together to provide an estimate of the quantity of interest. This simulation process resembles a laboratory experiment, and is sometimes referred to as the method of stochastic, or statistical, experiments. Obviously the larger the number of trials, the more confidence one has in the estimated quantity. Therefore, Monte Carlo results must be reported along with the statistical variability associated with it. A Monte Carlo result reported without such variability, or statistical “error’’, is simply not worth reporting at all, since one cannot ascertain the credibility of the result. On the other hand, an advantage of the Monte Carlo method is that it provides a measure of confidence in the reported results; or in other words, it quantifies the statistical “doubt’’ that accompanies the calculated result. In general, the error associated with an estimated quantity is inversely proportional to the square root of the number of trials, that is:  1 Error ∝ (4.105) N Therefore, decreasing an error by a factor of two requires quadrupling the value of N . With the advent of computers, it is now possible to track a very large number of particles and simulate complex problems in practically short times and with readily available computers. The Monte Carlo method can handle complex geometries, since unlike other numerical methods it does not involve a direct solution of the integro-differential particle transport equation. As such, the solution process is not restricted to a certain system of coordinates (rectangular, cylindrical, or spherical), needed in other solution methods to formulate the differentials and the integrals involved in the analytical form of the Boltzmann equation, Eq. (4.5). However, for the same reason, the Monte Carlo method does not provide a solution for the distribution of the particle flux at every point in the problem’s seven-dimensional domain; only 7 The

name of the method comes from the city of Monte Carlo in the principality of Monaco, known for its gambling houses.

285

4.4 Stochastic Methods

solutions at some pre-designated specific locations in the space are calculated. Since the Monte Carlo method is a computational process in which random variables are involved, we begin by explaining what it is meant by a random variable, and reviewing some important statistical concepts.

4.4.2 Random variables and statistical basis In ordinary English usage, a random variable is the outcome of any process that proceeds without any discernible aim or direction. Mathematically, a random variable is a variable whose value is not known in any given case, but the values it can assume are known along with the probabilities with which it assumes these values. Therefore, a random variable, ξ, assuming discrete values, is expressed as:   x x2 . . . x n ξ= 1 (4.106) p1 p2 . . . pn where the xi s are possible values of ξ and the pi s are the corresponding probabilities. For the ithe value, one can write P(ξ = xi ) = pi , or pξ (xi ) = pi , to express the probability with which xi occurs. In a continuous process, random variables are defined with the aid of a function f (x), called the probability density function (pdf), or simply the density distribution, so that:  P(ξ ≤ x0 ) =

x0

−∞

f (x)dx

(4.107)

with x0 being a certain value of x, f (x) ≥ 0 and −∞ < x < ∞, but some other finite interval (a, b) could have been assumed. This function is normalized such that:  ∞ f (x) dx = 1 (4.108) −∞

That is, the zeroth moment of the pdf is equal to unity. The first moment gives the expected value of x:  ∞ ξ = ξf (x) dx (4.109) −∞

The second moment, defined centrally with respect to ξ, gives the variance of the distribution of pdf:  σ (ξ) = 2

∞

−∞

[ξ − ξ]2 f (x) dx

(4.110)

286

Chapter 4 Transport

Another important quantity employed in Monte Carlo simulations is the cumulative density function (cdf ), also called the distribution function, defined as:  x0 f (x) dx (4.111) F(x0 ) = P(ξ ≤ x0 ) = −∞

Then, one can define the probability of ξ having values within some range (a, b) by:  b f (x) dx = F(b) − F(a) (4.112) P(a < ξ < b) = a

Equations (4.111) and (4.112) indicate that cdf represents an area under pdf, covering the range of interest.

4.4.3 Abstract analysis It is useful to define random variables in a mathematically abstract fashion, and relate them to the probability theory. This requires a definition for an event and the probability associated with it. We will start first by defining an event as the occurrence of a specified outcome of an experiment. The event that includes all possible events is designated here by the set . An event # contained within  is expressed as subset of : # ⊂ . The probability is a real-valued function of the events of an experiment with properties [52]: P(∅) = 0 P() = 1 and P

#∞ 0 i=1

$ #i

=

0 ≤ P(#) ≤ 1 for all # ⊂ 

∞ 

f (#i )

if #i ∩ #j = ∅ i  = j

(4.113)

(4.114)

i=1

where the union, ∪, indicates that events exist simultaneously, while the intersection, ∩, defines common events. In an experiment repeated n times, with an event # occurring n(#) times, one would expect n(#) n to cluster about a unique number P(#). The abstract notion of probability requires that the function P assigns to every event # a number with the above probability. Random variables are certain real-valued functions of points of . Let a point of  be denoted by ω and let ξ be a real-valued function on the points of . The set of all points ω such that ξ(ω) ≤ x, where x is some real number, is expressed as: #(x) = {ω|ξ(ω) ≤ x}

(4.115)

which states that #(x) is a subset of  that depends on the value of x. If for every x, the set #(x) corresponds to an event, then ξ is a random variable to which one can associate a probability so that: P{#(x)} = P{ω|ξ(ω) ≤ x} = P{ξ ≤ x} defined for every x

(4.116)

287

4.4 Stochastic Methods

F(x) p

x

x0

Figure 4.6 A schematic of a cumulative density function, F(x), with a discontinuity.

This gives the formal definition of random variables. The cumulative density function (cdf ) is defined as a real-valued function of a real variable so that: F(x) = P{ξ ≤ x} (4.117) This function has the following characteristics [52] (see Fig. 4.6): F is a monotonically non-decreasing function. F(−∞) = 0 and F(∞) = 1. F(a) − F(b) = P(a < ξ ≤ b), for a < b. If at x0 F becomes discontinuous by abruptly increasing its value by p, then P{ξ = x0 } = p, and the probability that the random variable ξ takes on the value x0 is a greater than zero. 5. F is continuous on the right at every event x, i.e. any discontinuity in F is included in the value of F(x). 6. If the derivative of F with respect to x exists at point x, then  +

x

x dF lim P x − <ξ ≤x+ =

x = f (x) x (4.118)

x→0 2 2 dx 1. 2. 3. 4.

If the derivative, f (x) exists, it then defines the probability density function (pdf ). In performing an experiment, not all points of  will be encountered, but after conducting N independent trials one may obtain ω1 , . . . , ωN points. The probability space (N ) is used to refer to all N -tuples of these points in . Then, the probability of the occurrence of (N ) is: PN (ω1 , . . . , ωN ) =

N . i=1

P(ωi )

(4.119)

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Chapter 4 Transport

One can define N random variables on (N ) , ξi , i = 1 to N : ξi (ω1 , . . . , ωN ) = ξ(ωi )

1≤i≤N

(4.120)

If ξ (N ) is a discrete random function of N variables, such that: ξ (N ) =

N 

ξi

(4.121)

i=1

it is also a random variable on (N ) . Then ξ (N ) represents the total number of occurrences of the event ωi in N repetitions of the experiment. For any random variable, ξ, of any distribution function with a mean, or expected value, m,Tchebycheff’s theorem [52] states mathematically that the standard deviation, σ, of this random variable is such that: P{|ξ − m| > kσ} ≤

1 k>0 k2

(4.122)

If k relates an error to σ by  = kσ, then for the random variable ξ (N ) , one can state that: 1 P{|ξ (N ) − p| > } ≤ (4.123) 4N 2 (N ) where p = ξ¯ (N )  and ξ¯ (N ) = ξN . This theorem proves the proportionality of error with the inverse of the square root of the number of trials, relationship (4.105). The cornerstone of the Monte Carlo method is the Central Limit Theorem, also known as the law of large numbers. This theorem states that ξ (N ) will be approximately normally distributed even if ξ is not. Formally, the theorem states that if ξ1 , . . . , ξN is a sequence of independent and identically distributed random variables with a common mean m and variance σ 2 , then [52]: ξ¯N =

N 1  ξ (N ) ξi = N i=1 N

(4.124)

is asymptotically normal (m, Nσ ). Therefore,  lim P

N →∞

ξ¯ − m √σ N

/ ≤ x0

1 = √ 2 π



x0

−∞



−x2 exp 2

 dx

(4.125)

The theorem assumes that both m and σ exist, that is they are given converging integrals (or series summations). Applying the Tchebycheff ’s

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4.4 Stochastic Methods

theorem yields then:  /   2  2    ∞  ξ¯ − m  1 1 −x −x   N  dx− √ dx exp exp P  σ < ∼ √   √  2 2 2 π 2 π −∞  N (4.126) In the above equation ∼ reads “asymptotically equal’’. Consequently,  P{|ξ¯N − m| < } ∼

2 π



√  N σ

0



−x2 exp 2

 dx

(4.127)

The central limit theorem is the  backbone of the Monte Carlo method, since the summation of random variables ( ξi ), regardless of their original pdf, will lead to a normal distribution, as long as the random variable ξ is sampled independently (one trial does not depend on the others) from the same distribution and that the summation is taken over a large number of trials. The implications of this theorem is that the well-known confidence levels associated with the normal distribution can be used to determine the degree of variability associated with the mean value of the experiment. That is, the mean value, ξ¯ , can be used as an estimate of the average value of the random variable ξ. This value approaches the “true’’ expected value, m, as the number of trials approaches infinity. The variability estimated by: N 1  2 2 s = ξ − N i=1 i

#

N 1  ξi N i=1

$2 (4.128)

is called the sample variance. It is not directly an estimate of the distribution variance, but it is a good estimate of which, since σ2 =

N s2  N −1

(4.129)

The one-sigma confidence interval indicates that there is a 68% probability that m lies within the interval [ξ¯ + σe , ξ¯ − σe ], where σ σe = √ N

(4.130)

The 95% (two-sigma) confidence interval is [ξ¯ + 2σe , ξ¯ − 2σe ], and higher levels of confidence are defined accordingly. Since σ is not known, the following estimate is used: ⎡ $2 ⎤ # N N 2   1 1 σ 1 ⎣ σe2 = (4.131) = ξ2 − ξi ⎦ N N − 1 N i=1 N i=1

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A useful quantity utilized in Monte Carlo computations is called the fractional standard deviation (fsd) or relative error, defined as relative error = fsd =

σe ξ¯

(4.132)

A fsd of less than 0.05, or 5%, is usually desired in Monte Carlo calculations. It must be kept in mind that the use of the central limit theorem, and the subsequent employment of the normal distribution confidence levels, require the mean and variance values to exist (i.e. they cannot be indefinite). The summation used to evaluate the mean and the variance must also converge toward the correct value. Moreover, sampling of the random variable must not leave “holes’’ within the distribution, i.e. sampling must reasonably cover every region in the distribution. It is difficult though to judge whether the number of trials, N , is sufficiently large (ideally infinity) to assure satisfaction of the asymptotic limit of the central limit theorem, since infinity by definition does not have a quantifiable value. However, a good Monte Carlo code, such as MCNP [53], provides a number of statistical tests to ensure that the conditions of the central limit theorem are met, so that statistical validity can be lent to the results of the sampling process. Below is a list of the criteria tested by the MCNP code to ensure statistical validity of the results: 1. The mean value does not monotonically increase or decrease as a function of the number of trials, N , toward the end of the experiment. Satisfaction of this test ensures a converging solution. 2. The magnitude of the relative error is low (below 5% or so), an indication of the convergence of the mean value. 3. The relative error decreases monotonically with N (with a √1 decrease) toward N the end of the experiment. 4. The variance of the variance (the estimated relative variance of the relative error) is low (less that 10% or so), and monotonically decreases (with N1 decrease) toward the end the experiment. This a further indication of convergence. 5. The figure-of-merit defined as ( fsd )−2 T −1 , where T is the computing time, maintains a statistically constant value and a non-monotonic behavior with N toward the end of the experiment. An erratic behavior of the figure-of-merit indicates that the problem is not converging. Increasing the computing time should result in an increase in the number of trials, hence a reduction in the relative error according to Eq. (4.105). If this does not occur, it is likely an indication that the particle is trapped within a certain trial somewhere within the problem’s domain. 6. The high-score (but low probability) end of the normal distribution of the estimated quantity is adequately sampled. When N is sufficiently large to satisfy the asymptotic condition of the central limit theorem both the two low-probability tail ends of the expected normal distribution would have contributed almost equally to the estimated mean value. However, low scores do

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4.4 Stochastic Methods

not significantly alter the value of the summation in Eq. (4.124), used for estimating the mean value. Therefore, low scores do not much affect the value of the estimated mean. On the other hand, the absence of the large scores would lead to underestimation of the mean value. The Pareto distribution gives the probability that ξ is greater than a certain value. The corresponding pdf for the Pareto distribution can be expressed as [53]: f (x) |Pareto

x − 1+k 1 k 1+k = a a

(4.133)

where a and k are constants. Note that when k = −1 the distribution is uniform, while it resembles an exponential decay distribution when k −→ 0 (which has a low probability for events with a large value). The parameters a and k can be obtained by a functional fit using the scores with the largest value in the experiment. From this functional fit, the value 1k + 1 (the slope of the Pareto distribution) is determined. A zero slope (i.e. k = −1) indicates that the sampling of large score values is uniform, which is likely caused by insufficient sampling of large scores. On the other hand, a very large k value reflects the far tail end of a negative exponential distribution. Therefore, the slope should not exceed some value that is large, but not exceedingly large. The MCNP code uses a value of 10 for a “perfect score’’, and requires a slope greater than three for the original distribution of the random number to be considered “completely’’ sampled. This will also be indicative of the second-moment existence requirement of the central limit theorem. 7. The skewness of the normal distribution around the mean value, caused by the absence of large scores, decreases as the number of trials, N , increases, because more large scores are sampled. The absence of large scores is likely also to lead to underestimation of the variance as given by Eq. (4.128). Consequently, the associated confidence intervals around the estimated mean (which is also underestimated) will not correspond to its nominal confidence level, that is a one-sigma confidence level would then correspond to less than 68% of the area under the“true’’normal distribution dictated by the central limit theorem. It is also not realistic to assume that the confidence interval is symmetric around the mean, since the distribution is not truly normal in the absence of large scores. One then must define an asymmetric confidence interval around the estimated mean. This is done by adding an estimated statistical shift, δ, to the mean to produce the midpoint of the now asymmetric confidence interval about the mean. The value of this shift is based on the central third moment [53]:  (ξi − ξ¯ )3 δ= 2S 2 N Now the 68% confidence interval is [ξ¯ + σe + δ, ξ¯ − σe ]. The shift, δ, should decrease with N1 , reaching zero as N approaches infinity, where the conditions of the central limit theorem are fully met. However, when δ  0.5 σe , the

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conditions for applicability of the central limit theorem can be considered to be “substantially satisfied’’. 8. All intervals of the distribution are sampled and no intervals are over-sampled. The MCNP code provides a plot of an empirical estimate of the distribution function for the estimated quantity based on the frequency of sampling the associated random variable. The plot of this function can then be visually examined for unsampled (holes) or over-sampled (spikes) regions, or any other irregular behavior that may indicate a bias in sampling the random variable. It is important to examine the above criteria at the end of a Monte Carlo simulation to validate the statistical viability of the results.

4.4.4 Random numbers Before discussing the way a random variable is sampled in a Monte Carlo numerical experiment, let us first distinguish between the definition of a random variable and a random number. As indicated in Section 4.4.2, random variables are realvalued functions associated with probabilities. Random numbers are random variables with the same probability of occurrence, i.e. have a constant (uniform) pdf. Conveniently, random numbers are defined within the range [0,1].Therefore, a sequence ρ1 , . . . , ρn , of random numbers, is such that:  b (1)dx = b 0 ≤ ρi ≤ 1 (4.134) P{ρi ≤ b} = 0

and P{ρi1 , . . . , ρin ≤ b} = bn

with i1 , . . . , in , all different

(4.135)

The multiplication of probabilities in the above equation is possible when random numbers ρi1 , . . . , ρin , are mutually independent, with ii1 , . . . , iin , being all different. Consequently, the cdf for a random number ρ is:  ρ (1)dx = ρ (4.136) F(ρ) = 0

4.4.5 Random number generation Random number generators are a standard feature in computing machines. However, such numerically generated numbers are not entirely random, they are pseudo random numbers, i.e. they are approximately random. A numerical scheme is used to generate a long set of random numbers with the same probability of occurrence, while being independent from each other. This is a desirable requirement in numerical computations, so that one can repeat the same sequence of calculations for debugging purposes. Such pseudo random numbers are legitimate for use in Monte Carlo experiments as long as they satisfy the two important criteria of:

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4.4 Stochastic Methods

1. Uniform distribution: Each number generated has the same probability of occurrence as any other number in the sequence. 2. Independence: The occurrence of a number does not rely on previous or subsequent occurrences of any other numbers. The modulus method is one of the methods used to generate random numbers. The random numbers are generated as follows: ρi = a ρi−1

(mod M )

(4.137)

where a is a constant, M = 2k , and k is the number of bits per word used by the computing machine, and “mod’’ is the modulus8 . The first number, ρ0 , in the series generated by Eq. (4.137), functions as a seed, so that a sequence of random numbers can be repeated by choosing the same seed.

4.4.6 Sampling Standard sampling The sampling (generation of trials) of a random variable, ξ, can be accomplished by equating its cdf to a random number, ρ, since the range of ρ (0 ≤ ρ ≤ 1) is the same as that of cdf. Therefore, if ξ has a pdf of f (x) then its sampling is realized by constructing a sequence of numbers ξ1 , . . . , ξn (−∞ ≤ ξi ≤ ∞), such that [52]:  x0 P{ξi ≤ x0 } = f (x)dx (4.138) −∞

and

 P{ξi1 , . . . , ξin ≤ x0 } = P{ξi1 ≤ x0 }, . . . , P{ξin ≤ x0 } =

n

x0 −∞

f (x) dx (4.139)

The random variables ξi1 , . . . , ξin , are mutually independent, if i1 , . . . , in are all different. Now by taking advantage of the fact that both the cdf, F(x), and random numbers ρ span values within the same interval [0,1], one can sample a random number, ρ, and set it equal to the cdf, F(x), so that:  x0 F(x) = P{ξ ≤ x0 } = f (x) dx = ρ (4.140) −∞

With ρ known, one can solve for x to obtain a value that is sampled from the distribution f (x). If direct solution for x is not possible, some other means can be used, such as the rejection method. Assuming a pdf to be such that 0 ≤ f (x) ≤ fmax , 8 Modulus

is a number or quantity that produces the same remainder when divided into each of two quantities, so B that A = B (mod M ) reads A is congruent to B module M and means A is the remainder of M ; for example: 15 (mod 13) = 2.

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Chapter 4 Transport

and a ≤ x ≤ b, in the rejection method, a pair of random numbers, ρ1 , ρ2 , is sampled. The first random number is used to select a value x = a + ρ1 (b − a) with a corresponding pdf of f (x1 ). Then if x < ρ2 fmax , x is accepted as a sampled value; otherwise another pair of random numbers is sampled and the process is repeated until the sampled value is accepted. Reference [54] provides methods for sampling from a variety of analytical solutions. For a discrete distribution, cdf is evaluated as:  F(xi ) = pi (4.141) x≤xi

where pi is the probability of occurrence of xi . The sampling of an xj+1 is then accomplished when: j  i=0

pi < ρ ≤

j+1 

pi

(4.142)

i=0

with p0 = 0, and j = 0, 1, 2, . . .

Importance sampling The above methods of sampling are straightforward sampling methods from probability density functions (pdfs). Sometimes, a portion of the range of a pdf is more relevant to the application at hand and it would be more efficient to sample more from that important range and avoid wasting time sampling from a less relevant range. Such sampling will, however, be biased, but this bias can be corrected for if the expected value for the original and biased distributions stay the same. Using Eq. (4.109), one has:  ∞  ∞  ∞ dF(x) ξ = xf (x)dx = x dF(x) = x ∗ dF ∗ (x) −∞ −∞ −∞ dF (x)  ∞  ∞ f (x) f (x) = (4.143) x ∗ dF ∗ (x) = x∗ dF ∗ (x) with x∗ = x ∗ f (x) −∞ f (x) −∞ Therefore, after sampling x∗ from the biased pdf, f ∗ (x), instead of the original pdf, f (x), the scored random variable is adjusted by the factor ff∗(x) (x) , when estimating the average of Eq. (4.124). Equation (4.143) indicates that any biased pdf can be used for sampling, provided that the scoring process is adequately adjusted to account for the bias. However, it can be easily shown that the second moment of the biased pdf cannot be equal to that of the first. Therefore, the variance estimated for the biased pdf will not be equal to that of the original distribution. Importance sampling should be performed only if it leads to a reduction in the variance of the estimated quantity; otherwise it is a wasteful exercise. Experimenting first with the original distribution to find the range of x that contributes most to the effective value can help in devising a biased function that is likely to reduce the variance.

4.4 Stochastic Methods

295

Splitting and Russian roulette In a complex sampling process that involves many stages of sampling, as in the case of particle transport, it is not convenient or practical to provide a biased pdf for importance sampling at every stage. Instead one can identify regions within the domain of the problem which can be considered as either important, i.e. most likely to contribute to the solution, or unimportant. In important regions, effort should be made to increase the number of samples, and the opposite should occur in unimportant regions. However, some adjustment must be made to the scored quantity to account for this preference in sampling. In important regions, sampling can be performed n times (instead of once), generating n random variables, the score of each variable must, however, be multiplied by n1 to compensate for the bias caused by“splitting’’the sampling process into n branches. Each branch can in turn be split into n other branches if it remains in an important region. The value of n can vary from an important region to another. In unimportant regions, a game of Russian roulette9 can be played. If the probability of reaching an unimportant region is estimated to be equal to q, then sampling is performed with a probability q and no sampling is performed with probability 1 − q, i.e. samples are “killed’’ with a probability 1 − q.To compensate for this bias, the score of surviving samples is multiplied by a factor of 1q . Once more, like importance sampling, the estimate of the expected value remains unbiased, because of the introduced adjustment, but the associated variance will not be equal to that of the original distribution, since the second moment is not conserved. Splitting and Russian roulette should be applied so that they reduce the variance of the estimated quantity. This happens when the adjusted scores are not very different from each other. Therefore, the reduced score of a splitting sample should not be very different in value from the magnified score of a sample that survived the Russian roulette game.

Other sampling methods In order to reduce the variance and/or reduce computational time, some computer codes take advantage of a priori information [54]. For example, when sampling N trials from a pdf distribution, f (x), the range of x can be divided into J subintervals. The probability of sampling from each interval (often called a batch) is then determined as pj from f (x), and each interval, j, is sampled pj N times; an estimate of the random variable, ξ¯j , is then evaluated for this batch. After sampling all intervals, a final estimate of ξ¯ is calculated as: ξ¯ = N1 pj ξ¯j . This is the systematic sampling method, which results in a lower variance than the standard sampling method. The process of importance sampling described above can also be used to give preference to the batches that have most influence on the solution, with the corresponding estimate adjusted to compensate for the introduced 9 This

sampling technique was named by the fathers of the Monte Carlo method, John von Neumann and Stanley Ulam, after a game of chance thought to be popular in the Russian army.

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Chapter 4 Transport

bias. This so-called stratified, or quota, sampling can also reduce the variance of the estimated value, if properly performed. In some situations, the outcome of a trial can be analytically evaluated once the random variable is sampled. One can use this analytical result as an expected value, and estimate the final solution as an average of all sampled analytical estimates.This expected value estimator eliminates the statistical fluctuation in the estimated value and results in a reduced variance, provided of course that none of the analytical values are too extreme (e.g. infinite or out of bounds).

4.4.7 Particle transport In order to apply the Monte Carlo sampling process described in the above section to the Boltzmann transport equation, one must formulate this equation so that it provides the probability density functions necessary for the execution of a stochastic experiment. This is done by converting the integro-differential Boltzmann particle transport equation, Eq. (4.5), into the so-called integral emergent t), the particle density equation [55]. This equation is written in terms of χ( r , E, , density of particles leaving a source or emerging from a collision at phase-space at time t with energy E. Let us introduce the integral coordinates r in direction  operators:  ∞ ≡ T ( r  → r , E, ) dR t ( r , E) exp [−β( r , R, )] (4.144)  → ) ≡ C( r , E → E,  



0 ∞

E  =E

dE





 d

 → ) s ( r , E  → E,  (4.145) t ( r , E  )

  . The operator T ( = R t ( r − R  )dR is a trans with β( r , R, ) r  → r , E, ) 0 port integral operator (kernel) that transports a particle from position r  to position while maintaining the energy E, i.e. with no interaction. r , along direction  causes a particle at r to The collision integral operator, C( r , E  → E,   → ), collide, changing its energy and direction as it scatters. Bookkeeping of particles in the phase-space results in the particle transport equation: t) = S( r , E, , t) + C( r , E  → E,   → ) χ( r , E, , r  , E  ,   , t) ×T ( r  → r , E  , )χ(

(4.146)

t) is the density of particles generated by an external source. where S( r , E, , A solution of Eq. (4.146), like any other integral equation, can be approximately expressed by the sum: t) = χ( r , E, ,

∞  n=0

t) χn ( r , E, ,

(4.147)

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4.4 Stochastic Methods

with t) = S( r , E, , t) χ0 ( r , E, ,

(4.148)

Then using, Eq. (4.146), one has: t) = C( r , E  → E,   → )T ( r  → r , E  , )χ n−1 ( r  , E  ,   , t) χn ( r , E, , (4.149) Recall that T and C are integral operators, defined respectively by Eqs (4.144) and (4.145). Then, by analogy with Eq. (4.111), Eq. (4.149) represents a cumulative density function (cdf), which though complicated can be employed in a Monte Carlo experiment. A particle transport stochastic trial is called a random walk, or a history. Each t), to determine random walk starts by sampling a source particle from S( r , E, , 0 , t0 ). Obviously, S( r , E, , t) has to be given the particle’s coordinates ( r0 , E0 ,  in the form of a pdf that describes the source’s geometry, energy spectrum, and change in intensity with time (if relevant). A flight distance, R, is then sampled where R is the distance the source using the transport kernel, T ( r  → r , E, ), before encountering a collision particle travels with energy E in the direction  at r1 = r0 + R 0 , where r1 is now the site of the first collision. The particle will arrive at this site at time t1 = t0 + R/v0 , where v0 is the particle’s speed corresponding to energy E0 . Time in Monte Carlo experiments is often referred to as the particle’s “age’’. The probability of scattering at the new site, as the collision kernel of r1 ,E0 ) s ( Eq. (4.145) indicates, is  t ( r1 ,E0 ) , where  is the macroscopic cross section of the material at position r1 , and the subscripts s and t designate, respectively, the scattering and total cross sections. In order to avoid “wasting’’ a particle in random walks, particle transport codes perform the so-called “non-analog’’ sampling, a process in which particle absorption is forbidden. This is a form of importance sampling, or biasing, aimed at avoiding early termination of the random walk, and allows the random walk to continue until the tracked particle escapes the system’s geometry or is terminated by other means (e.g. a low-energy cut-off). The bias introduced by this process is compensated for by assigning to each sampled source particle a weight W0 (typically unity, or the source intensity), and multiplying the r1 ,E0 ) s ( weight at the collision point by the non-absorption probability, 1 −  t ( r1 ,E0 ) . After some collisions, when the particle weight becomes low, Russian roulette sampling can be performed, or the random walk can be terminated altogether when the weight becomes too insignificant. At the first collision at r1 , where the weight is adjusted to W1 , post-collision 1 , are sampled from the distribution: particle energy, E1 , and direction,  

d 4π

0 →  1) s ( r1 , E0 → E1 ,  s ( r1 , E0 )

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Chapter 4 Transport

Then, a new collision site is sampled from the transport kernel, and so on. The sampling process is repeated until the particle is terminated by escaping the system (to an artificially created external void), or reaching a pre-assigned cut-off limit for age, energy, or weight.

Estimators The random walk process follows the transport of a particle, and can be repeated N times by sampling new source particles. It does not, however, estimate an answer to a physical quantity of interest, unless a “scoring’’ of this quantity is performed during the random walk, and a “tally’’ is kept of these scores. The quantity of interest is usually the particle fluence, or some related value. The scoring process is determined by a variety of estimators which evaluate the fluence, or fluence-like quantities, at a point or a region. Statistical estimates, including the average and the variance of the average, are calculated at the end of the random walk process. The body crossing estimator evaluates the flux crossing a surface, by accumulating the weight of particles crossing the surfaces divided by the absolute value of the cosine of the angle between the normal to the surface and direction of the incident particle. Provisions must be made to avoid angles with small cosine values. The track length estimator evaluates the fluence by summing the track length of particles crossing a given zone, divided by the volume of the zone. This is usually suitable for evaluating the fluence in void or air regions, and regions containing a low-density material. The collision density estimator adds up the weight of particles colliding within a zone, divided by the total cross section of the material and the volume of the zone. This estimator provides adequate estimates for the fluence in regions of high-density material, where a large number of collisions are anticipated. In all the above estimators, the particle must visit the region or surface of interest. In situations where the probability of the particles reaching the region of interest is low, an expected-value estimator can be used. The next-event estimator is such an estimator. This estimator is particularly useful for point detectors, where there is only one possible position for the“next collision’’.This estimator calculates at every collision the probability of the next event taking place at the detector exp(−λ) site, and scores Wp(μ) , where W is the particle’s weight, p(μ) is the value of 2πR 2 the probability density function at μ, the cosine of the angle between the particle trajectory and the direction to the detector, λ is the total number of mean-freepaths encountered over the trajectory from the collision point to the detector, and R is the distance between the collision point and the detector. This is done by tracing a “pseudo-particle’’ from the collision site to the detector (scoring point), without altering the original random walk path. The same process is also performed for source particles to provide the uncollided component. However, collisions very close to the detector site can lead to unrealistically high scores, as R −→ 0. This estimator should not, therefore, be used in regions with high collision density, and scores close to the detector site should be excluded when this estimator is utilized.

4.4 Stochastic Methods

299

Setup As indicated above, a random walk is controlled by the transport and collision kernels. The distance of flight is sampled from a distribution described by the transport kernel, while the status of the particle after collision is governed by a distribution defined by the collision kernel. Both kernels are defined by the material’s cross sections, presented in the form of probability tables. Throughout the transport process, a particle encounters different geometric configurations and materials. Some biasing techniques may also be applied to discard particles that are unlikely to significantly contribute to the quantity of interest, and to promote particles of importance. Eventually, of course, estimates of the quantity of interest must be obtained, otherwise the entire exercise is fruitless. One has, however, to start with a source. Below we summarize the essential steps that must be specified before executing a Monte Carlo experiment.

Source The position, geometry, directional distribution, and energy distribution of the particle’s source must be specified. In transient analysis, a function describing the change of the source’s intensity with time must also be given. Fission sources and secondary particle sources (e.g. a γ-ray following a neutron interaction, an electron following a photon absorption, or bremsstrahlung photons) are determined by the cross sections of the material and need not be specified as input parameters. The fission distribution with energy, χ0 (E), needs, however, to be specified, in order to determine the energy of the emerging neutrons. A source particle is also assigned a weight, W0 , typically unity. When fission occurs, or when a secondary particle is produced, the attributes of the new particle (weight, position, energy, direction, and age) are stored in a bank for later processing, once the tracking of the initial source particle is completed.

Geometry The Monte Carlo method can handle complex geometries. The geometry must, however, be specified in such a way that enables tracking of the particle throughout the system.This tracking process determines not only the particle’s spatial position but also the type of material encountered, hence cross sections, at the encountered site. The geometry can be specified via analytical geometry procedures, which define the surfaces of different geometric objects. Surfaces are then combined (with logical operators) to define volume cells. This is the method used in the MCNP code [53]. Alternatively, geometry may be specified via a set of elementary bodies, combined together using logical operators to form a zone of a particular material. This is the so-called combinatorial geometry method utilized in the MORSE code [56].

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Chapter 4 Transport

Material cross sections The cross sections for the different materials encountered must be supplied as a function of energy. The Legendre expansion coefficients for each material are also needed, in order to account for anisotropic scattering. The cross sections are processed prior to the execution of the random walks, to provide the probability tables. The tabulated values are then used to determine: the distance to be traveled by a particle until it encounters a collision (an interaction), the outcome of the collision, and the outgoing energy and angle of a scattering event; in addition to the number of neutrons per fission for fissile materials.

Importance sampling Some initial trials should be performed without importance sampling. If the variance of a quantity of interest is high, then these initial trials can provide insight into the problem and assist in assigning the spatial regions and energy regions within which importance sampling can be effective in reducing the variance. Importance sampling can be performed at the source, during the random walk and at scoring. The expected-value estimates of the next-event estimator can be viewed as a form of importance sampling. Source biasing should allow the production of more source particles, with suitably reduced weights, in the more important ranges of each variable: position, energy, and direction. For example, source particles directed to a region of interest should be sampled more often than those directed away from it, provided of course that the weight of the source particle is accordingly adjusted. Variance reduction can also be achieved by the cut-off parameters, discussed below, where insignificant particles are not allowed to continue to score endlessly to the quantity of interest. However, the main methods of importance sampling in particle transport codes are splitting, Russian roulette, and exponential transformation. Splitting should be applied in regions or energies that are expected to significantly contribute to the quantity of interest, but is unlikely to be reached; the opposite is true for Russian roulette. It is important, however, to control the amount of splitting to avoid the unnecessary creation of too many particles. Russian roulette takes a particle of weight W and turns it into a particle of weight W W W  > W with probability W  and kills it with probability 1 − W  . In general, Russian roulette increases the history variance but decreases the time per history, while splitting achieves the opposite effect. As shown in Section 4.1.7, the adjoint flux presents the importance of a particular point in the transport space to the response of a detector. Therefore, the adjoint flux is the best possible importance function for use in Monte Carlo sampling, as it can be shown that it leads to a zero variance [52]. However, knowing the adjoint flux entails solving the Boltzmann adjoint transport equation, Eq. (4.35), which is equal in effort to solving the direct transport problem to begin with. However, a crude solution of the adjoint transport equation can provide an

4.4 Stochastic Methods

301

approximation of the importance function, which in turn can reduce the variance of the original problem significantly. The exponential transformation is a process that stretches or shrinks a particle’s path-length between collisions. This is done by artificially reducing the macroscopic cross section in the preferred direction and increasing it in the opposite direction. A fictitious cross section, ∗ , is related to the actual cross section, , by ∗ = t (1 − pμ), where μ is the cosine of the angle between the preferred direction and the particle’s direction and p is a biasing parameter, |p| < 1; a cona ∗ a stant or equal to  t . For p = t and μ = 1,  = s and the particle path is sampled from the distance to the next scattering, rather than from the mean-freepath ( 1t ) for all interactions. The weight is, consequently, adjusted by a factor of exp(−a d), where d is the distance of travel. Therefore, the exponential transformation works best in highly absorbing media and very poorly in highly scattering media. Exponential transformation is useful in deep penetration problems.

Tallys Tallying is the process of scoring the parameters of interest to provide the required answer for the problem at hand. One or more of the estimators discussed in Section 4.4.7 can be used to calculate various quantities of interest. For example, the particle current (directional flux) over a surface can be evaluated using the body-crossing estimator. This estimator, by including all directions, can be used to estimate the particle fluence (or flux when the source weight (strength) is given in terms of particles per unit time). The particle flux can also be evaluated within a volume using the track-length or the collision density estimators; obviously the latter estimator is not applicable in void and is not reliable in low-density regions where very few collisions take place. The next-event estimator can be used for estimating the flux at point in a voided zone where no collision can occur near the detector site.When using this estimator in a low-density region, an exclusion zone should be assigned around the point detector to avoid singular estimates. Scoring the particle flux multiplied by the material’s total cross section in the region where the flux is evaluated provides an estimate of the interaction rate. For each of these tallies, the user can designate the particle energy, direction or range(s) within which the final answer is desired. The scored quantity for the flux is basically the particle weight.This weight multiplied by the particle energy provides an estimate of energy deposition. One can also supply a detector-response function, by which the particle flux is multiplied to simulate the response of a physical detector.

Termination Since Monte Carlo particle transport codes usually employ non-analog sampling, the user must specify some criteria to terminate a random walk. Termination of a random walk can be affected by one of the following criteria: an upper bound for particle age, an energy threshold, a cut-off weight, and by defining a full

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absorption (also called external void) region outside the domain of interest. In addition, the user must also specify a criterion for terminating the entire Monte Carlo experiment. This is typically done by specifying the number of random walks (histories) to be performed. However, it is often desirable to also assign a maximum computer execution time; just in case a random walk is trapped endlessly within a particular zone in the problem geometry. When performing a criticality calculation, the number of cycles (generations) should also be specified.

4.4.8 Example In order to illustrate the above points, let us consider the relatively simple problem of evaluating the fluence through a shielding slab with a width along the xdirection, and a neutron source and a detector, both on the x-axis but at opposite sides of the slab. Source parameters Let us assume a monoenergetic isotropic point source at 0 of the incident source (x0 , y0 , z0 ), with energy E0 . Then, only the direction,  particle needs to be sampled, in a steady-state problem. The angular probability density function for an isotropic source is:

f ()d =

d sin ϑdϑdϕ d cos ϑ dϕ = = = [ f ( cos ϑ)d cos ϑ][ f (ϕ)dϕ] 4π 4π 2 2π

where ϑ and ϕ are, respectively, the polar and azimuthal angles: 0 = cos ϑ sin ϕ xˆ + sin ϑ sin ϕ yˆ + cos ϕ zˆ  where xˆ is a unit vector in the direction x, and yˆ and zˆ are defined similarly. Since we are not interested in neutrons directed away from the shielding slab, only angles with 0 ≤ cos ϑ < 1 need to be sampled. To compensate for source particles emerging with negative values of cos ϑ, a weight of half (half the source strength) is assigned to each sampled source particle. Then one in effect is concerned with half the distribution domain for ϑ. As such, ρ21 can be equated to the cumulative probability for cos ϑ, where ρ1 is a random number sampled from a uniform distribution in the interval [0,1]:  cos ϑ ρ1 cos ϑ = f ( cos ϑ)d cos ϑ = 2 2 0 The inversion of the above leads to an equation for selecting ϑ: ϑ = cos−1 ρ1 The value ρ1 = 0 should be rejected, since it will lead to ϑ = π2 , and the source particle will never reach the slab. Source particles with ϑ in the neighborhood of π2 , can reach the slab, though will not significantly contribute to the detector,

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4.4 Stochastic Methods

since they result in interactions far away from both the source and the detector. Therefore, one may confine the value of cos ϑ to between some minimum value and 1, so that an “important’’ cone is defined. Then the weight assigned to each particle sampled within this cone should be adjusted to account for this form of importance sampling. The angle ϕ is sampled from:  ϕ  ϕ 1 ϕ   ρ2 = dϕ = f (ϕ )dϕ = 2π 0 0 2π 0 for the direction of where ρ2 is another random number. Now the vector,  the sampled source particle is fully defined.

Distance of travel Next, one needs to determine the distance the neutron will travel until it collides. The probability of a neutron experiencing its first interaction between the distances r and r + dr is equal to t e −t r , where t is the total cross section of the material encountered. The distance, r, the neutron will travel until it collides, is sampled using: r =−

1 ln ρ3 t

where ρ3 is a another random number; see Eq. (4.155) in the Problem’s section. Then the position of next collision (x1 , y1 , z1 ) is given by: x1 = x0 + r cos ϑ sin ϕ y2 = y0 + r sin ϑ sin ϕ z1 = z0 + r cos ϕ

Type of interaction The type of interaction which takes place at the new particle position is sampled from probabilities determined by the macroscopic cross sections for scattering, s , and absorption, a , relative to the total cross section, t . The absorption cross section includes fission, radiative capture, and other production reactions. These absorptive reactions can in turn be sampled from probability tables constructed from the reaction cross sections normalized to a . The interaction tables of probabilities are converted into cumulative probability tables, to facilitate the sampling process. The average number of neutrons produced per fission, ν¯ , is available in cross section libraries; at MT = 451 in the ENDF format. Generated fission particles are then stored in a bank for future processing. If the interaction is determined to be an absorption process, the random walk of the particle may be dismissed. This is called analog Monte Carlo and is not

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usually used as it results in early termination of the random walk. Alternatively, one should use the non-analog process, the particle weight is reduced  in which  t − a by the non-absorption probability and a particle scattering is sampled. t This process allows the particle to fully complete its path within the system, until it escapes the system’s geometry or is terminated by a weight cut-off, or an energy cut-off, or some other pre-specified process. In a material containing more than one element, e.g. in mixtures and compounds, the element with which scattering takes place must be sampled. This is done using a discrete pdf defined by the macroscopic cross section of each element normalized with respect to the total scattering cross section of the material at the incident particle energy.

Energy of outgoing particle In a non-fissile material, the only interaction possible, in a non-analog Monte Carlo simulation, is particle scattering. One needs then to determine the energy and angle of the particle emerging from the collision. Let us assume an elastic isotropic neutron scattering process.Then, the energy of the outgoing particle can lie anywhere from the energy of the incident particle, E0 , to the minimum possible 2  energy αE, where α = A−1 A+1 , with A being the mass number of the element considered. The probability of the particle reaching an energy E is given by: f (E)dE =

dE E0 (1 − α)

Equating the cumulative probability to some random number ρ4 , one obtains: E1 = ρ4 E0 (1 − α) + αE0 The outgoing energy, E1 , is sampled from the above equation. Since isotropic scattering is assumed, the outgoing direction can be sampled using a procedure similar to that used for the source, except that the entire range of the polar angle, ϑ, is sampled. Once the direction and energy of the scattered particle are determined, the distance of flight until the next collision is evaluated, and so on. However, this can direct the tracked particle back toward the source’s position, away from the detector. Important sampling can be employed to avoid unnecessary tracking of particles not moving toward the detector. Moreover, in a deep penetration problem, i.e. a thick shield, the probability of the particle crossing the shielding slab is very low, and very few particles will reach the detector. Importance sampling becomes then useful in promoting more particle transport toward the detector.

Importance sampling Splitting could be used to increase the number of particles traveling away from the source, while Russian roulette should be applied to kill most of the particles

4.4 Stochastic Methods

305

traveling back toward the source. Alternatively, exponential transformation can be employed to stretch the particle’s path-length between collisions, and consequently enable more particles to cross the slab. The slab can be divided, for either of these two importance sampling schemes, into regions of equal thickness, and the importance sampling process can be applied so that the number of particles sampled in each region remains roughly the same. Of course, when applying biasing techniques, the particle weight is to be adjusted, such that the resulting estimates are unbiased.

Scoring A simple scoring process suitable for this example is the boundary crossing estimator at the boundary far away from the source.

4.4.9 Computer codes A number of Monte Carlo computer codes are readily available and can be acquired through the Radiation Safety Information Computational Center, Oak Ridge, TN (http://www-rsicc.ornl.gov), or the OECD Nuclear Energy Agency, France (http://www.nea.fr). However, the most widely used code for particle transport analysis is perhaps the MCNP code [53], for neutrons, photons, and electrons, and its extension the MCNPX code [57] which is applicable to other particle types as well. The COG code [58] can “simulate complex radiation sources, model 3D system geometries with ‘real world’ complexity, specify detailed elemental distributions, and predict the response of almost any type of detector’’10 . The MCBEND [59] code is commercially available, and is designed for “the every day (or occasional) user’’. The EGS4 code [60] and the TIGER series of codes [61] are also used in simulating the transport of photons and electrons. The Geant4 toolkit [62] includes also the simulation of high-energy particles. Special-purpose Monte Carlo codes, as those described in [63] and [64], have been developed for specific tasks.Those interested in writing their own Monte Carlo code for photon transport will find the analytical expressions for the cross sections given in [65] quite useful. The adjoint flux (see Section 4.1.7) can be calculated using the Monte Carlo method, but only when multigroup cross sections are used. Continuous or pointwise cross sections are not amenable for use in adjoint calculations, since they cannot be transposed to provide the upward change in energy required when solving for the adjoint flux. Since knowing the adjoint flux can be helpful, as indicated in Section 4.4.7, in selecting efficient parameters for importance sampling, a crude multigroup Monte Carlo, or for that matter discrete ordinates, 10 J.

M. Hall, J. F. Morgan, and K. E. Sale. Numerical modeling of nonintrusive inspection systems. Substance Detection Systems, Vol. 2092. SPIE – International Society for Optical Engineering, Bellingham,WA, 1994, pp. 342–352.

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calculation can be performed to provide the adjoint flux. The importance sampling results derived from these calculations can subsequently be utilized in a continuous-energy more detailed Monte Carlo calculation.

4.5 Transport of Charged Particles 4.5.1 Special features The transport of charged particles challenges the straight-line assumption of the Boltzmann transport equation, Eq. (4.1), since the path of charged particles is curved by the Coulomb field. The same field also produces very large cross sections, compared to neutron and photons, since charged particles interact continuously with matter. In other words, Coulomb interactions have no meanfree-path. Nevertheless, the total path length of charged particles is quiet short, and the interactions are typically low momentum-transfer events. It is, therefore, often convenient to describe the transport of charged particles in terms of a straight-line (crow-flight) equivalent distance, called the range, and express their energy loss in terms of the stopping power, defined below in Section 4.5.2. Charged–particle interactions also liberate atomic electrons. Heavy charged particles and energetic electrons release atomic electrons by ionization. The liberated electrons are often referred to as “delta’’ rays. However, in the case of electrons, the main mechanism of secondary-electron generation is associated with bremsstrahlung (radiative energy losses). The bremsstrahlung photons produce electrons, as they interact with matter. Secondary electrons themselves may lose energy by the bremsstrahlung process, producing more photons, and so on. Eventually, a cascade of electrons is formed. However, this electron cascading process inevitably tails off, as the bremsstrahlung photons are subjected to photoelectric absorption and electrons are dispersed by Coulomb scattering. These cascade terminating processes, along with the dispersion associated with the electrons produced by Compton scattering, tend to produce a lateral spread in the spatial distribution of electrons, hence the use of the term “shower’’ to describe the electron cascade. At the tail end of the cascade, electron-energy losses are dominated by collisions, leading the energy of the electrons to dissipate into excitation and ionization of the atoms of the medium. The complexity of this electron shower process makes the Monte Carlo method the most viable tool for its simulation. As mentioned in Section 4.4.9, the EGS4 code [60] and the TIGER series of codes [61], as well as the MCNP [53] and MCNPX [57] codes, can be used for the simulation of charged particles and the associated electron shower. The attributes of the electrons of this shower process are stored in a bank for further processing, after the tracking of the initial electron is completed. A similar banking process is performed for the photons produced by bremsstrahlung.

4.5 Transport of Charged Particles

307

4.5.2 Stopping power and range The continuous interaction process of charge particles can be related to the cross section of a particular interaction via the energy transfer cross section, i.e. the difdσ , where Q is the energy transferred ferential cross section with respect to energy, dQ by the charged particle to matter. If the amount of energy transfer is low, chargedparticle collisions are considered to be “soft’’ and the energy loss can be treated as a continuous process. Then the energy loss per unit distance is expressed as:   QH dE  dσ dT (4.150) =N Q  ds soft dQ 0 where QH is the maximum energy transfer allowed for the collision to be soft, or the minimum energy for the collision to be a hard one, N is the number of atoms per unit volume, with σ being the cross section per atom. The value of QH is arbitrary, but must be larger than the binding energy of the electron in the material considered. For energy loss greater than QH , i.e. for hard collisions, the kinematics of particle-on-target interactions must be considered. Nevertheless, in a medium which contains N atoms per unit volume, an “expected value’’ of the energy loss per unit length can be determined by:   Qmax dE  dσ =N Q dQ (4.151)  ds hard dQ QH where Qmax is the maximum allowed energy loss per collision for the interaction considered.The negative of total energy loss per unit distance, after the summation of Eqs (4.150) and (4.151), defines the so-called stopping power:   dE dE  dE  S=− =− − (4.152) ds ds soft ds hard The maximum distance traveled by a charged particle with an initial energy, E0 , before losing its all kinetic energy, is called the range, R, given by:  E0  R(E0 )  0  E0 ds 1 dE dE = (4.153) ds = dE = R(E0 ) = −dE dE S 0 0 0 E0 ds The stopping power and range of positively charged particles (ions) in solids, liquids, and gases can be calculated using the SRIM computer code [66].

4.5.3 Transport The continuous loss of energy of charged particles and the fact that charged particles do not usually move in straight lines due to the Coulomb effect, introduce some difficulties in the simulation of charged particles. Using the range as the

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“crow’’ distance, i.e. a straight-line distance of travel, though satisfactory in some applications, does not faithfully represent the transport of charged particles. Moreover, in small-angle Coulomb scattering the energy loss caused by a single target atom is quite small, and a particle can encounter deflection by many targets before it stops. It is then quite time consuming to simulate the scattering of a charged particle by every target atom it encounters. Instead, Monte Carlo codes resort to the lumping of many deflections into one equivalent step change. A particle’s trajectory then consists of a series of small straight-line steps. The step size can be determined according to a fixed fractional loss of energy. However, this method results in very small step sizes at particle energy below about 1 MeV. Then, the step size can be chosen to be a certain fraction of the particle’s range. Within each step, it is expected that the charged particle has been subjected to many collisions, but suffered a small loss of energy, so that the division of the original trajectory into a number of steps is a satisfactory assumption. This approximation is known as the condensed history method. Within each step, the change in direction is sampled from one of the angular distributions provided by one of the so-called multiple scattering theories, reviewed in [67].

4.6 Problems Section 4.1 4.1 What are the dimensions of n,  s and Q in Eq. (4.1)? Check the dimensions of each term in the equation to ensure consistency. 4.2 Find the proportionality constant in the relationship between the interaction rate per unit volume and nv. crossing an area dA at angle ϑ, such that · nˆ = cos ϑ, 4.3 Consider a current J () where nˆ is a unit vector inward normal to dA. Show that the component of the current density normal to dA is given by Jnormal = cos θ φ(). 4.4 Determine the flux distribution as a function of distance, given an infinite isotropic line source in air.

Section 4.2 4.5 Prove Eq. (4.47) for i = j. ∂ , since Consider only the part of ∇ which is concerned with the angle, ∂μ the transformation is with respect to the angle only.    = 4π . 4.6 Prove that d 3 and perform Hint: Introduce a unit vector nˆ making an angle cos−1 μ with  n ·   · nˆ . the integral on dˆ

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4.6 Problems

2 4.7 Show that, for neutron scattering, μ0 = 3A , where μ0 is the average of the scattering angle, and A is the mass number of the scattering nuclei. State the assumptions used. 4.8 Show that J = −D∇φ with the diffusion coefficient given by:

1 −1 1 D = Ttransport = 3 3(total − elastic removal μ0 )

(4.154)

This is the well-known Fick’s law of diffusion. With this definition of D, show that the diffusion equation is not applicable in void.

Section 4.3 4.9 From Table 4.1, construct a symmetric S4 quadrature sets for a twodimensional problem. 4.10 Construct from first principles an acceptable quadrature set with one dividing point on each directional axis (μ1 = η1 − ξ1 ) symmetrically created on a unit sphere. What Sn level is this set? N and  · ∇ N in a corrdinate system of your choice 4.11 Express both ∇ ·  and show that they are equal. 4.12 Show that 0h,g corresponds to isotropic scattering, while 1h,g describes a cosine distribution.

Section 4.4 4.13 Prove that σ 2 (x) = E(x2 ) − E 2 (x), where E(x) is the mean value (expected value) of x and σ 2 (x) is its variance. 4.14 Show that the following procedure: 1 ln ρ (4.155)  represents a sampling of the distance r from the pdf:  exp (−r), where ρ is a random number and  is the total cross section of the medium. 4.15 Devise a method for determining the outgoing energy for neutron isotropic scattering with an element with mass number A. 4.16 Devise a method for determining the outgoing energy for neutron scattering in a chemical compound such as water. 4.17 In the free gas model with no absorption, thermal neutrons have a Maxwellian energy distribution. r =−

1. Show that the probability density function for this distribution can be written as: 2 √ p(R) = √ R exp (−R) (4.156) π where R is the neutron kinetic energy in units of kT .

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2. Calculate the most probable energy, assuming R varies from zero to infinity. 3. Calculate the average energy. 4. Calculate the energy corresponding to the most probable velocity, and compare its value to the most probable energy.

Section 4.5 4.18 The following is an approximation for the range of α particles: Rα (mm) ≈

E 1.5 (MeV) √ A ρ(kg/m3 )

(4.157)

where E is the particle energy, ρ is the material density, and A is its mass number. The SRIM program (http://srim.org/) gives more accurate estimates of the stopping power and range. Compare the values obtained from the above relationship and from SRIM for air and aluminum at 10 keV, 100 keV, 1 MeV, 4 MeV, and 10 MeV. 4.19 1. The range, Rβ , can be approximated by: Rβ (mm) ≈ 4 × 103

1.4 (MeV) Emax ρ(kg/m3 )

(4.158)

where ρ is the material density. Calculate the range of 2 MeV β particles in air and aluminum. 2. The attenuation law of radiation, Eq. (4.22), is only applicable to neutral radiation. Comment on the validity of this statement. 3. If the statement above is valid, explain why the intensity of β particles can be expressed by the exponential relationship: I = I0 exp[−μx] for x ≤ R

(4.159)

where I refers to intensity, x to distance and μ is the attenuation coefficient, which can be approximated by: μ(mm−1 ) = 2.2 × 10−3

ρ(kg/m3 ) 4

(4.160)

3 Emax (MeV)

4. Calculate the attenuation coefficient (μ) of β particles in aluminum and air, and compare the value of μ1 to R. Can μ1 for β particles be considered equal to its mean-free-path?

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50. RSICC, ‘DOORS3.2: One, two- and three-dimensional discrete ordinates neutron/ photontransport code system,’ Radiation Safety Information Computational Center, Oak Ridge National Laboratory, Oak Ridge, TN, Technical Report, (RSICC Code Package CCC-650). 51. T. R. Hill, ‘ONETRAN: A one-dimensional multigroup discrete ordinates finite element transport code system,’ Los Alamos National Laboratory, Los Alamos, Technical Report LA-5990-MS, June 1975), Radiation Safety Information Computational Center, Oak Ridge, TN, RSIC Code Package CCC-266. 52. J. Spanier and E. M. Gelbard, Monte Carlo Principles and Neutron Transport Problems. Reading: Addison-Wesley, 1969. 53. RSICC, ‘MCNP – Monte Carlo N-particle transport code system,’ Radiation Safety Information Computational Center, Oak Ridge National Laboratory, Oak Ride, TN, Technical Report, Computer Code Number: C00701. 54. H. Kahn,Applications of Monte Carlo. Santa Monica: Rand Corp., 1956. 55. E. Straker, P. Stevens, D. Irving, and V. R. Cain,‘The MORSE code - A multigroup neutron and gamma-ray Monte Carlo transport code,’ Oak Ridge National Laboratory, Oak Ridge, Technical Report ORNL-4585, September 1970. 56. M. B. Emmett, ‘MORSE-CG: A general purpose Monte Carlo multigroup neutron and gamma-ray transport code system with array geometry capability, version 2.’ Oak Ridge National Laboratory, Oak Ridge, Technical Report, November 1997, RSIC Code Package CCC-474. 57. J. S. Hendricks et al., ‘MCNPX: a general-purpose Monte Carlo radiation transport code,’ Los Alamos National Laboratory, Los Alamos, Code Version Release Announcements LA-UR-05-0891 (http://mcnpx.lanl.gov). February 2005. 58. T.Wilcox and E. Lent,‘COG - A particle transport code designed to solve the Boltzmann equation for deep-penetration (shielding) problems,’ Lawrence Livermore National Laboratory, Livermore, Techical Report (http://www-phys.llnl.gov/N_Div/COG/, accessed October 2006) 1989. 59. The ANSWERS Software Service, ‘MCBEND user guide for version 10a,’ Serco Assurance, Cheshire, Technical Report ANSWERS/MCBEND/REPORT/004 (http://www. sercoassurance.com/answers/resource/areas/shield/mcbend.htm). Febuary 2005. 60. W. R. Nelson, H. Hirayama, and D. W. Rogers, ‘The EGS code-system,’ Stanford Linear Accelerator Center, Stanford,Technical Report No. SLAC-265, 1985. 61. J. A. Halbleib andW. H.Vandevender,‘TIGER,A one-dimensional multilayer electron/photon Monte Carlo transport code.’ Nuclear Science and Engineering,Vol. 57, pp. 94–94, 1975. 62. S. Agostinelli, J. Allison, K. Amako, et al.,‘Geant4 - A simulation toolkit.’ Nuclear Instruments and Methods in Physics Research,Vol. A506, pp. 250–303. (http://geant4.web.cern.ch/geant4/) 2003. 63. T. H. Prettyman, R. Gardner, and K. Verghese, ‘The specific purpose Monte Carlo code McENL for simulating the response of epithermal neutron lifetime well logging tools.’ IEEE Transactions on Nuclear Science,Vol. 40, No. 4 Pt 1, pp. 933–938, 1993. 64. E. M. A. Hussein, ‘Approximate estimators for fluence at a point in center-of-mass Monte Carlo neutron transport.’ Nuclear Science and Engineering,Vol. 109, pp. 416–422, 1991. 65. J. Baro, M. Roteta, J. M. Fernandez-Varea, and F. Salvat, ‘Analytical cross sections for Monte Carlo simulation of photon transport.’ Radiation Physics and Chemistry, Vol. 44, No. 5, pp. 531–552, 1994. 66. F. J. Ziegler, ‘The stopping and range of ions in matter (SRIM-2000),’ IBM-Research, Yorktown, NY (http://www.srim.org/, accessed October 2006). October 1999. 67. W. T. Scott,‘The theory of small-angle multiple scattering of fast charged particles.’ Reviews of Modern Physics,Vol. 35, pp. 231–313, 1963.

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CONSTANTS AND UNITS

Exact up-to-date values can be found at http://physics.nist.gov/cuu/Constants/index.html For conversion between various units see http://physics.nist.gov/cuu/Reference/unitconversions.html

Basic constants Symbol

Name

Value

c

Speed of light in free space

2.998 × 108 m/s

e

Elementary charge

1.602 × 10−19 C

h

Planck constant

4.136 × 10−15 eV s 6.626 × 10−34 J s

k

Boltzmann constant

8.617 × 10−5 eV/K 1.381 × 10−23 J/K

me

Electron mass

9.109 × 10−31 kg = 5.486 × 10−4 u ≡ 0.511 MeV

mp

Proton mass

1.673 × 10−27 kg = 1.007 u

mn

Neutron mass

1.675 × 10−27 kg = 1.009 u

mα

Alpha particle mass

6.645 × 10−27 kg = 4.002 u

u

Atomic mass unit (amu)

1.661 × 10−27 kg =

mass of 12 C 12

≡ 931.494 MeV ε0

Permittivity of free space

8.854 × 10−12 C2 /( Jm)

μ0

Permeability of free space

4π × 10−7 N/A2 315

316

Constants and Units

Derived constants Symbol

Name



Rationalized (reduced) Planck constant (also known as Dirac constant)

NA

Avogadro number

α

Fine structure constant

Expression

Value

h 2π

6.583 × 10−16 eV s 1.055 × 10−34 J s

10−3 ×

1 u

e2 4πε0 c μ0 e 2 c = 2h

6.022 × 10−23 /mol 7.297 × 10−3 =

1 137.036

μB

Bohr magneton

e 2me

5.788 × 10−5 eV/T 9.274 × 10−24 J/T

μN

Nuclear magneton

e 2mp

3.152 × 10−8 eV/T 5.051 × 10−27 J/T

c2

(Speed of light in free space)2

1 ε 0 μ0

8.988 × 1016 m2 /s2

317

Constants and Units

SI units and prefixes For information on the SI (Systéme International/International System) of units see the Bureau International des Poids et Mesures (BIPM) at http://www.bipm.fr/enus/3_SI/ The US National Institute of Standard and Technology can also be consulted at http://physics.nist.gov/cuu/Units/index.html

Units Quantity

Symbol

Label

Length Mass Time Electric current Temperature

m kg s A K

meter kilogram second ampere kelvin

Frequency Force Pressure Work & energy Radiation energy

Hz N Pa J eV

Power Electric potential Magnetic flux Magnetic flux density

W V Wb T

hertz ( 1s ) newton (kg m/s2 ) pascal (N/m2 ) joule (Nm) electron-volt (=1.602 × 10−19 J) (=eC × 1V) watt (N m/s = J/s) volt (W/A) weber (V s) tesla (Wb/m2 )

Prefixes Symbol

Name

Multiplies by

Symbol

Name

Multiplies by

a

atto

10−18

E

exa

1018

f

femto

10−15

P

peta

1015

p

pico

10−12

T

tera

1012

n

nano

10−9

G

giga

109

μ

micro

10−6

M

mega

106

m

milli

10−3

k

kilo

103

318

Constants and Units

Natural units Units based on setting c = 1 and  = 1. All physical quantities are then expressed in terms of (energy)d , where d is called the canonical dimension, with energy typically in GeV. Quantity Length (r) Mass (m) Time (t) Velocity (v) Momentum (p) Angular momentum (l) Force (F) Energy (E) Charge (q)

Electric current (i = qt )

Natural form r c mc 2 t  v c pc l F c E √ 4πα √  t

d

Natural unit

Conversion to SI units

−1

GeV−1

1 GeV−1 = 1.973 × 10−16 m

GeV1

1 GeV = 1.783 × 10−27 kg

GeV−1

1 GeV−1 = 6.582 × 10−25 s

0

GeV0

1 = 2.998 × 108 m/s (= c)

1

GeV1

1 GeV = 5.334 × 10−19 kg m/s

0

GeV0

1 = 1.055 × 10−34 J s (= )

2

GeV2

1 GeV2 = 8.119 × 105 N

1

GeV1

1 GeV = 1.602 × 10−10 J

0

GeV0

1 = 5.291 × 10−19 C

1 −1

(e = 0.303 = 1.602 × 10−19 C) 4πα

1

GeV1

1 GeV1 = 8.038 × 105 A

USEFUL WEB SITES

The following are some sites complied at the time of writing this book.

Physical constants and definitions • • • • •

http://physics.nist.gov/cuu/Constants/index.html http://physics.nist.gov/cuu/Reference/unitconversions.html http://physics.nist.gov/cuu/Units/index.html http://www.bipm.fr/enus/3_SI/ http://www.interactions.org/cms/?pid=1002289

Properties of elements • • •

http://www.webelements.com/webelements/elements/text/periodic-table http://www.csrri.iit.edu/periodic-table.html http://www.chemicalelements.com/

Atomic and nuclear data • • • • • • • • • • •

National Nuclear Data Center: http //www.nndc.bnl.gov/ IAEA Nuclear Data Centre: http //www-nds.iaea.org/ Table of Nuclides of Korea Atomic Energy Research Institute http://atom.kaeri.re.kr/endfplot.shtml WWW Chart of the Nuclides 2004 http://wwwndc.tokai-sc.jaea.go.jp/CN04/index.html Isotope Explorer http://ie.lbl.gov/ensdf/ Thermal Neutron Capture γs (CapGam) http://www.nndc.bnl.gov/capgam/ Scattering Lengths and Bound Scattering Cross Sections forThermal Neutrons: http://www.ncnr.nist.gov/resources/n-lengths/ Photon Cross Sections Database http://physics.nist.gov/PhysRef Data/Xcom/Text/XCOM.html X-ray Data Booklet http://xdb.lbl.gov/ X-Ray Mass Attenuation Coefficients http://physics.nist.gov/PhysRef Data/XrayMassCoef/tab4.html RTAB: the Rayleigh Scattering Database http://www-phys.llnl.gov/Research/scattering/elastic.html 319

320 • • •

•

Useful Web Sites

X-ray Form Factor,Attenuation and Scattering Tables http://physics.nist.gov/PhysRefData/FFast/Text2000/contents2000.html Stopping Power and Range of Charged Particles http://srim.org/ Electron Scattering in Solids: Elastic Scattering Differential Cross Sections and Inelastic Properties http://www.ioffe.rssi.ru/ES/ Nuclear Data Processing http://www-nds.iaea.org/ndspub/endf/prepro/ http://t2.lanl.gov/codes/codes.html

Computer codes repositories • •

Radiation Safety Information Computational Center, Oak Ridge,TN http://www-rsicc.ornl.gov OECD Nuclear Energy Agency (NEA), France http://www.nea.fr/html/databank/welcome.html

GLOSSARY

Alpha particle (α): A positively charged nuclear particle consisting of two protons and two neutrons (identical to the nucleus of helium), with a mass mα = 6.645 × 10−27 kg = 4.001 u. Annihilation radiation: Photons emitted when a positron and an electron are combined. Antineutrino: The antimatter of the neutrino. It has the same mass as the neutrino but has an opposite spin. 1 Atomic mass unit (u): 1 u = 12 th the mass of a 12 C atom = 1.660565 ×−27 kg ≡ 931.493 MeV in rest mass. Atomic number (Z): The number of protons in a nucleus, which is also equal to the number of electrons in an atom. Baryons: A class of fundamental particles consisting of three quarks (e.g. protons and neutrons). Beta particle (β): An electron (β− ), mass me = 9.109 kg = 5.486 × 10−4 u, emitted from a radioactive nucleus during radioactive decay. The term beta particle is also used to describe a positron with the designation β+ . Bosons: A class of fundamental particles responsible for transmitting very short range (10−18 m) forces between particles. A boson particle (e.g. a photon, pion, or alpha particle) has a zero or an integral spin quantum number. Any number of identical bosons can occupy the same quantum state. Bremsstrahlung: Electromagnetic radiation emitted during the deceleration of electrons in the electric field of the atom. Charged particle: An elementary particle carrying an electric charge. Cosmic rays: Highly energetic nuclei found in space and penetrate the atmosphere, colliding with other particles and disintegrating into a shower of smaller particles such as pions, muons, etc. Delta rays: Electrons ejected by ionizing particles as they pass through matter. Electron (e− ): An elementary particle with a rest mass me = 9.109 × 10−31 kg = 5.486 × 10−4 u, carrying an electric charge of −1.602 × 10−19 C. Electron volt (eV): A unit of energy equivalent to the kinetic energy acquired by an electron when subjected to an electrical potential of 1V; 1 eV = 1.602 × 10−19 J. Fermions: A class of fundamental particles (e.g. an electron, proton, neutron, or a neutrino) with a half-integral spin quantum number. In a set of identical fermions, no more than one particle may occupy a particular quantum state. Gamma rays (γ): Electromagnetic radiation emitted as a result of the deexcitation of a nucleus. Hadrons: A class of fundamental particles which interact by the strong nuclear force (mesons and baryons). Ion: An atom that carries a positive or negative electric charge due to the loss or gain of one electron or more. Isomers: Nuclides of the same atomic number and mass number that can be at different excitation energy states. Leptons: A class of fundamental particles (e.g. electrons, muons, and neutrinos) with a half-integral spin quantum number, but experience no strong nuclear forces, i.e. they participate in the weak nuclear interactions. Mass number (A): The number of nucleons (neutrons and protons) in a nucleus. Meson: A fundamental particle that participates in the strong nuclear force.

321

322

Glossary

Neutral particle: A particle carrying no electric charge (uncharged). Neutrino: A neutral elementary particle with a zero mass that accompanies the emission of a beta particle. The antineutrino is associated with β− decay, and the neutrino with β+ emissions. Neutron (n): A neutral particle with a rest mass mn = 1.6749543 × 10−27 kg = 1.009 u. Nucleon: A constituent particle of the nucleus, a neutron or a proton. Photon (γ): A particle with zero mass and zero electric charge. A quantum of electromagnetic radiation of frequency ν, energy hν, and a zero rest mass, where h is Planck’s constant. Pion: A meson with a rest-mass energy of 138 MeV. Positron (β+ ): An elementary particle with a rest mass me = 9.109 × 10−31 kg = 5.486 × 10−4 u, carrying an electric charge of +1.602 × 10−19 C (equal in value to that of the electron). Proton (p): An elementary particle with a rest mass mp = 1.673 × 10−27 kg = 1.007 u, carrying a charge of +1.602 × 10−19 C (equal in value to that of the electron). Quark: Any of a number of fundamental particles from which other elementary particles are formed. Along with leptons, quarks are the building block of matter from which mesons (two quarks) and baryons (three quarks) are made. An up (U) quark has a charge of 32 e and a down (D) quark has a − 13 e, where e is the electronic charge. Synchrotron radiation: Radiation emitted by high-energy relativistic charged particles when accelerated by a magnetic field. X-rays: Electromagnetic radiation emitted as a result of atomic transitions of bound electrons in an atom.

INDEX

Abrasion–ablation model, 123 Absorption, 44, 51 Albedo, 282 Alpha decay, 29 hindrance factor, 31 Annihilation positron, 44 Anomalous scattering, 196 Antineutrino, 3 Atomic form factor, 203 Attenuation coefficient, 59 Attenuation law, 59, 254 Auger electrons, 34, 35, 38, 41 Baryon number, 81 Beta decay, 32 double, 36 transitions, 35 Bhabha scattering, 43, 113, 226 Binding energy, 22 Bohr magneton, 238 Boltzmann transport equation, 249 adjoint, 257 integral, 296 Born approximation, 211 Bose–Einstein statistics, 18 Boson, 17, 18 Bragg diffraction, 145 Breit–Wigner formulae, 163, 236 Bremsstrahlung, 46, 136, 220 Buildup factor, 256 Center of mass, 68 relativistic, 78 Central limit theorem, 288 Cerenkov radiation, 45 Charged particle production, 174 range, 307

stopping power, 307 Classical collision theory, 216 Classical electron radius, 140 Collision diameter, 128 hard, 44 nonelastic, 116 radiative, 137 soft, 44 Compound nucleus, 47 Compton scattering, 42, 112, 197 incoherent scattering function, 200 Doppler effect, 200 double, 202 Conservation laws, 80 Coulomb scattering, 43, 215 elastic, 123 inelastic, 132 multiple, 308 Coupling constants, 12 Cross section absorption, 158, 164 angular, 154 barn, 58, 154 Bhabha scattering, 226 Breit–Wigner formula, 163 bremsstrahlung, 220 charged-particle production, 174 Compton scattering, 197 bound electrons, 200 double differential, 200 Coulomb scattering, 215 Delbruck scattering, 213 differential, 154, 160, 162, 193 diffraction, 183, 204 energy grouping, 233 evaporation, 166 fission, 174 library 323

324

Cross section (Continued) BROND, 229 CENDL, 229 ENDF, 229 JEFF, 229 JENDL, 229 XCOM, 229 macroscopic, 58, 155, 237 compound, 237 mixture, 238 microscopic, 58, 155 Moller scattering, 224 Mott scattering, 219, 225 neutron competitive reaction, 172 elastic, 169, 175 inelastic, 172 inelastic gamma, 178 production, 177 thermal, 179, 235 optical model, 156 pair production, 208 photoelectric effect, 205 photon XCOM library, 229 positron annihilation, 227 processing NJOY code, 232 production, 179 radiative capture, 173 Rayleigh scattering, 203 RTAB library, 229 reaction, 164 resonance Breit–Wigner formula, 163 Doppler broadening, 236 resolved, 163 scattering, 164 unresolved, 165 Rutherford scattering, 217 shape elastic scattering, 161 thermal neutron bound atoms, 181 elastic coherent, 183 elastic incoherent, 182

Index

free atoms, 180 inelastic incoherent, 180 S(α, β) treatment, 181 Thomson scattering, 195 Triplet Production, 213 Dalitz plot, 103 de Broglie wavelength, 10 Decay activity, 27 delayed-beta, 39 equilibrium, 28 law, 27 multibody, 103 neutron emission, 39 proton emission, 39 spontaneous fission, 38 statistics, 28 three-body, 101 two-body, 99 Delbruck scattering, 53, 143, 213 Delta rays, 44, 306 Diffraction, 43, 204 thermal neutron, 183 Diffusion equation, 257 Fermi age, 265 Fick’s Law, 256 multi-group, 265 theory, 256, 264 Dirac electron theory, 184 Discrete ordinates, 267 adjoint, 283 computer codes, 283 Sn Pn approximation, 279 Divergence law, 254 Doppler effect, 200, 236 Elastic scattering, 109 Electron atomic binding energy, 5 capture (ε), 34 cascade, 306

325

Index

radius, 195 shower, 306 Energy kinetic, 77 rest mass, 77 total, 77 Excitation energy, 89 Exponential attenuation, 255 Fermi–Dirac statistics, 18 Fermi energy/level, 23 Fermion, 17, 18 Feynman diagrams, 189 Fine structure constant, 12, 14, 131, 193 Fission, 38, 51, 122, 174 energy spectrum, 39 Fluence, 57 Fluorescent radiation, 41 Flux, 56 density, 56 Fragmentation, 122 Fundamental equation de Broglie wavelength, 10 Boltzmann, 249 photon energy, 5 rest-mass energy, 3 Schrödinger, 16 speed of light, 7 Gamma decay, 36 transitions, 36 production, 178 ray, 7 Giant resonances, 166 Half-life, 27 partial, 27 Impact parameter, 73 Importance sampling adjoint, 300 Inclusive collision, 98

Inelastic scattering, 114 Inertial frame of reference, 74 Infrared divergence problem, 202, 223 Internal Conversion (IC), 37 Invariants, 93 Inverse square law, 254 Isobars, 32 Isomeric Transition (IT), 36 Isomers, 36 Isospin, 82 Isotopes, 21 Kinematics Newtonian, 104 non-relativistic, 103 relativistic, 83 reverse, 123 Larmor formula, 242 Laue diffraction, 145 Lepton number, 81 Lethargy, 236 Lorentz transformation momentum and energy, 79 Magneton Bohr, 16 nuclear, 16 Mandelstam variables, 95 Mass attenuation coefficient, 59 Mass defect, 21 Maxwell–Boltzmann distribution, 180, 236 Mean free path, 60 Moller scattering, 44, 113, 224 Monte Carlo estimator expected value, 296 transport, 284 charged particles, 307 computer codes, 305 non-analog, 297 Mott scattering, 43, 49, 219, 225 Multiple scattering, 308

326

Natural units, 191 Neutrino, 3 Neutron, 4 elastic scattering, 109, 169, 175 inelastic scattering, 172 production, 177 temperature, 180 thermal, 179 coherent elastic scattering, 183 incoherent elastic scattering, 182 incoherent inelastic scattering, 180 Maxwell–Boltzmann, 180 scattering length, 180 Nonelastic collision, 116 Nuclear decay, 47 excitation, 47 excited states, 25 fission, 26 interaction elastic, 47 nonelastic, 47 Nucleus binding energy, 22 collective model, 25 Fermi gas model, 26 ground state, 25 liquid drop model, 25 magic numbers, 23 quantum numbers, 23 radius, 26 shell model, 23 Pair production, 45, 53, 118, 208 Parity, 19 Particle density, 55 Pauli exclusion principal, 18 Photodistintegration, 167 Photoelectric effect, 41, 117, 205 absorption edge, 207 Photoneutrons, 121 Photons, 5

Index

Photonuclear reactions, 166 Planck constant, 5 Point kernel, 255 Poisson distribution, 28 Polarization, 194 Positron annihilation, 34, 116, 227 decay, 33 Positronium, 228 Potential field, 12 Potential scattering, 159 Poynting vector, 135 Production reactions, 174, 177–179 Q-value, 30, 89 Quantum numbers, 17 asymmetry, 26, 36 Radiative capture, 51, 101, 119, 173 Radiative collisions, 137 Random numbers, 292 sampling exponential transformation, 301 importance, 294 Russian roulette, 295 splitting, 295 standard, 293 stratified, 296 systematic, 295 zero variance, 300 variables, 285 walk, 297 Range, 307 Rayleigh scattering, 42, 140, 203 Relativity, theory of, 74 Resonance giant, 166 inverted, 171 window, 171 Reverse kinematics, 123 Rutherford scattering, 49, 129, 131, 217

327

Index

S-matrix, 190 Scattering Bhabha, 43, 113 coherent, 42 Compton, 42, 112 Coulomb, 43 elastic, 123 inelastic, 132 Delbruck, 53, 143 elastic, 42, 48, 109 hard ball, 48 potential, 48 smooth region, 49 incoherent, 42 inelastic, 44, 50, 114 inverse Compton, 42 Moller, 43, 113 Mott, 43, 49 potential, 159 Rayleigh, 42, 140 Rutherford, 49, 129, 131 Thomson, 43, 49, 139 Scattering length, 167 Schrödinger equation, 16, 183 Separation energy, 90 Singlet state, 167 Soft collision, 132 Spallation, 51, 52, 122 Spherical harmonics, 260 computer codes, 266 Statistical weight, 167 factor, 164, 169 Statistics Bose–Einstein, 18 Fermi–Dirac, 18 Poisson, 28 Stopping power, 307 Stripping, 166 Strong nuclear force, 13

Tchebycheff theorem, 288 Thermal neutrons, 179 Thomson scattering, 139, 195 with electron, 43 with nucleus, 49 Threshold energy back, 90 forward, 90 Transition radiation, 46, 143 Transport equation, 249 adjoint, 257 diffusion approximation, 264 discrete ordinates, 266 modal solution, 259 Monte Carlo solution, 284 nodal solution, 266 Sn approximation, 269 point kernel, 255 P1 approximation, 262 Pn approximation, 262 spherical harmonics, 260 stochastic solution, 284 Triplet production, 42, 118, 213 Triplet state, 167 Tunnel effect, 29 Uncertainty principle, 10 Virtual particle, 189 photon, 139 state, 187 Watt distribution, 39 X-ray, 7 characteristic, 34, 35, 38, 41 Zeeman effect, 17, 238