NCRP report no 108 conceptual basis for calculations of absorbed dose distributions

In the so-called deterministic approach, the radiation field is characterized by functions describing the phase-space density of particles a t a given point and its distributions with re

NATIONAL COUNCIL ON RADIATION

PROTECTION AND MEASUREMENTS

Issued March 31, 1991

Sexond Reprinting February 1, 1995

National Council on Radiation Protection and Measurements

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This report was prepared by the National Council on Radiation Protection and Mea- surements (NCRP) The Council strives to provide accurate, complete and useful information in i t reports However, neither the NCRP, the members of NCRP, other persons contributing to or assisting in the preparation of this report, nor any person acting on the behalf of any of these parties: (a) makes any warranty or representation, express or implied, with respect to the accuracy, completeness o r usefulness of the information contained in this report, or that the use of any information, method or process disclosed in this report may not infringe on privately owned rights; or (b) assumes any liability with respect to the use of, or for damages resulting from the use

of any information, method or process disclosed in this report, under the Civil Rights

Act of 1964 Section 701 et seq as amended 42 U.S.C Section 2000e et seq (Titk VII1

or any other strrtutory or common law theory gooerning liability

Library of Congress Cataloging-in-Publication Data

Conceptual basis for calculations of absorbed-dose distributions

p cm.-(NCRP report; no 108)

Includes bibliographical references and index

ISBN 0-929600-16-9

1 Radiation dosimetry 2 Ionizing radiation-Measurement

I National Council on Radiation Protection and Measurements

Preface

The idea for this report emerged, in the early 1970's, from the need

of a n NCRP Scientific Committee to characterize the beta-ray depth- dose distribution in connection with immersion doses It was realized, however, that the calculation of such a distribution was only a small part of the very much larger task concerned with the theoretical, mathematical and computational concepts involved in the develop- ment of absorbed-dose distributions in general To address this issue

in an allencompassing manner, the NCRP formed Scientific Commit-

tee 52 on the Conceptual Basis of Calculations of Dose Distribution

In either external or internal irradiation, the absorbed dose is usually non-uniform in any structure and, in particular, in the human body This non-uniformity is to be distinguished from the stochastic variations that exist even in regions where the dose is uniform and that are the subject of microdosimetry and not this report Many illustrations of absorbed dose non-uniformity come to mind: for example, the absorbed-dose distributions from hot parti- cles, from internal emitters, from radiation therapy, from radiation accidents and from environmental radiation There can even be addi- tional non-uniformity with respect to time of the non-uniform distri- bution, for example, in the redistributions of administered radioac- tivity in the body

For all absorbed-dose calculations, there is a source (or sources) of radiation and a receptor (or receptors) of some of the energy of this radiation, with or without intervening material between the source and receptor The calculation of absorbed-dose distributions requires specification of the sources and receptors, characterization of their geometrical relationships and consideration of the physical interac- tions of the radiations involving attenuation, scattering and the production of secondary radiations All these processes are consid- ered in the basic transport equation, the general theorems and prop- erties and the methods of solution of which are described in the transport theory

The report is a systematic presentation, discussion and compila- tion of all the concepts involved It contains some complicated mathe- matics that will be of interest to the mathematically knowledgeable, but that should not discourage those not mathematically inclined

iv / PREFACE

The text of the report contains detailed explanations of all the con- cepts and of the consequences of the equations so that, even omitting the mathematics, a broad and comprehensive understanding can be obtained of what is entailed in the calculation of an absorbed-dose distribution

The cutoff date for the report is about two years ago and, hence, the report is lacking in the most current references However, this field does not evolve at a rapid pace and the current literature is, therefore, not abundant and can be reviewed easily

In accord with the recommendations of NCRP Report No 82, SI Units in Radiation Protection and Measurements, as of January 1990, only SI units are used in the text Readers needing factors for conver- sion of SI to conventional units are encouraged to consult Report No

82

This report was prepared by NCRP Scientific Committee 52 on Conceptual Basis of Calculations of Dose Distributions Serving on the Committee during the preparation of this report were:

Upper Nyack, New York

Oak Ridge National Laboratory

Oak Ridge, Bnnessee

Martin J Berger

5011 Elm Street

Bethesda, Maryland

Lewis V Spencer Post Office Box 87 Hopkinsville, Kentucky

Columbia University New York, New York

NCRP Secretariat: Thomas Fearon (1976-80)

Contents

1 Introduction 1

1.1 The Concept of Absorbed Dose 1 1.2 Dose Measurement and Dose Calculation 3

1.3 Elements of Dose Calculations 4

2 Transport Formalisms 6

2.1 Concepts in Dose Calculations 6 2.2 Transport Equation 8

3 Sources 14

3.1 Specification of Sources 14

3.2 Simplified Representations of Sources 15

4 Receptors 17

5 Cross Sections 20

5.1 Schematization 20

5.2 General Aspects of Required Cross Sections 22 6 Transport Theory-General Theorems and

Properties 26 6.1 Integral Form of the Transport Equation 26

6.2 Iterative Solutions (Orders of Scattering) 27

6.3 Density Scaling Theorem 28

6.4 Fano's Theorem 29

6.5 Energy Conservation 29

6.6 Superposition 30

6.7 Adjoint Transport Equation 31 6.8 Reciprocity 33

6.9 Transport Equations in Commonly Used Coordinate Systems 34

7 Transport Theory-Methods of Solution 36

7.1 Introduction 36 7.2 Radiation Equilibrium and Space-Integrated

Radiation Fields 38 7.3 Continuous Slowing-Down Approximation (CSDA) 40

7.4 Numerical Integration Over Energy 43

7.5 Elementary Problems Involving Particle Direction ' 45

7.5.1 Thin-Foil Charged Particle Problems 45

7.6 Penetration Studies 47

7.6.1 The Moment Method 47

7 .6.2 Discrete-Ordinates Transport Codes 49

CONTENTS 1

7.6.2.1 Neutron-Photon Transport

7.6.2.2 Dosimetry Calculations By the Method of Discrete Ordinates

7.7 Spectral Equilibrium and Related Concepts

7.7.1 Aspects Applicable to All Radiations

7.7.2 Electrons

7.7.3 Photons and Neutrons

7.8 Radiation Quasi-equilibrium

7.8.1 Transient Equilibrium 7.8.2 Non-uniform Sources

7.8.3 Non-uniformity in the Internal Dosimetry of Radionuclides

7.8.4 Non-uniform Media 8 Monte-Carlo Methods

8.1 Principles

8.2 Analog Monte-Carlo and Variance-Reduction

Bchniques 8.3 Transport Codes

8.3.1 Neutron-Photon Transport a t Energies 520 MeV

8.3.2 Electron-Photon Cascades 8.3.3 Nucleon-Meson Transport a t Energies >20 MeV

8.3.4 Dosimetric Calculations

9 Geometric Considerations

9.1 Absorbed Dose in Receptor Regions

9.2 Reciprocity Theorem

9.3 Isotropic Point-Source Kernels 9.4 Point-Pair Distance Distributions and Geometric Reduction Factors

10 Calculation of the Dose Equivalent

List of Symbols

Appendix A Information about Cross Sections for Transport Calculations

A.l Photon Cross Sections

A.l.l Photoelectric Effect

A.1.2 Fluorescence Radiation and Auger Electrons

A 1.3 Incoherent (Compton) Scattering A.1.4 Pair Production

A.1.5 Coherent (Rayleigh) Scattering A.1.6 Photonuclear Effect

A.1.7 Attenuation Coefficient

A.1.8 Energy-Absorption Coefficient A.1.9 Photon Cross-Section Compilations

vii

49

A.2.2 Elastic Scattering of Protons by Atoms 117

A.2.3 Scattering of Electrons by Atomic Electrons 119

~ 4 Nuclear Cross Sections for Charged Particles a t

B.2 Shielding of Manned Space Vehicles Against

Galactic Cosmic-Ray Protons and Alpha Particles 172

Appendix C A Compilation of Geometric Reduction

1 Introduction

1.1 The Concept of Absorbed Dose

The effects of radiation on matter are initiated by processes in which atoms and molecules of the medium are ionized or excited Over a wide range of conditions, i t is a n excellent approximation to assume that the average number of ionizations and excitations is proportional to the amount of energy imparted to the medium by ionizing radiation1 in the volume of interest The absorbed dose, that

is, the average amount of energy imparted to the medium per unit mass, is therefore of central importance for the production of radia- tion effects, and the calculation of absorbed-dose distributions in irradiated media is the focus of interest of the present report It should be pointed out, however, that even though absorbed dose is useful a s a n index relating absorbed energy to radiation effects, it is almost never sufficient; it may have to be supplemented by other information, such a s the distributions of the amounts of energy imparted to small sites, the correlation of the amounts of energy

imparted to adjacent sites, and so on Such quantities are termed stochastic quantities Unless otherwise stated, all quantities consid- ered in this report are non-stochastic A discussion concerning sto- chastic quantities is given in ICRU Report 33 (ICRU, 1980)

The absorbed dose, D, is defined (ICRU, Report 33) as the quotient

of d by dm:

where d2 is the mean energy imparted by ionizing radiation to matter

of mass dm

The energy, E, imparted to the volume containing dm is defined as

'Ionizing radiation consists of directly ionizing and indirectly ionizing radiation Directly ionizing radiations are charged particles (electrons, positrons, protons, alpha particles, heavy ions) with sufficient kinetic energy to ionize or excite atoms or molecules Indirectly ionizing radiations are uncharged particles (photons, neutrons) that set in motion directly ionizing radiation (charged particles) or that can initiate nuclear transformations

2 / 1 INTRODUCTION

where Ei, (E,,,) is the sum of energies of all the charged and uncharged ionizing particles that enter (leave) the volume, excluding rest mass energies, and XQ is the algebraic sum of all changes (decreases: positive sign; increases: negative sign) of rest-mass energy in mass-energy transformations occumng in the volume

A few clarifications of Equation (1.1) are necessary a t this point The energy imparted results from random discrete energy deposition events by individual ionizing particles andlor their secondaries The quantity E is therefore stochastic in nature and governed by a (nor- malized) probability distribution function fv(d where V is the volume containing m The mean value of e,

is the quantity referred to in the definition, Equation (1.1) In addi- tion, this equation implies a limiting process2 such that V + 0 The

absorbed dose, Dfi), is thus defined a t a given position ? in the irradiated object (see footnote 2), and is a non-stochastic quantity Furthermore, in general D c ) changes with and this variation is termed the "dose distribution."

A second important aspect refers to the temporal pattern of dose accumulation Let

be the dose increment at ? during the time interval CtJt+dtl This equation defines the absorbed-dose rate, D@,t) This aspect is impor- tant in understanding the relation between dose and biological effect

A final remark concerns the relation between.dose and its stochas- tic counterpart, the specific energy, z, defined as

The specific energy is always measured in a non-zero volume, V, and its mean value, 2, has the value of the average absorbed dose, Dv, in that volume of

2Because of the discrete manner in which energy is imparted, the limiting process

V- 0 is obviouely an idealization At all times the volume V should contain a large number of atoms and molecules

1.2 DOSE MEASUREMENT AND DOSE CALCULATION / 3

It follows (ICRU, Report 33) that

DP) = lim Z

v + 0 FEV, where the limiting process is such that ? is contained in the volume

v

The stochastic variations of specific energy that occur even in a medium that is uniformly irradiated must not be confused with variations of absorbed dose in a medium that is not uniformly irradi- ated The latter often occurs with internally deposited radionuclides, especially with alpha particles For example, if 2S9pU particulates, having a diameter of the order of a micrometer, are lodged in tissue, substantial variations i n absorbed dose occur over comparable dimensions

The theoretical and experimental study of the distributions in z is the objective of microdosimetry (see ICRU, 1983) and is beyond the scope of the present report

No formal definitions of other radiation quantities are given here because they can be found in many publications and because their meaning is usually obvious from the context The units employed are those of the International System (SI) except that, in accord with common practice, the electron-volt (eV) is frequently used as a unit

of energy

The absorbed dose of ionizing radiation may be determined by two entirely different methods: measurement and calculation

Measurement has the decided advantage in that it requires-at least in principle-only one auxiliary quantity, the calibration factor

of the instrument employed The construction of the instrument from materials of suitable atomic composition can largely obviate even the need to identify the incident radiation The experimental deter- mination is also often simpler and more rapid

Calculations, on the other hand, usually require extensive subsid- iary information, including configuration of sources, disposition and atomic composition of matter intervening between them and the points of interest, and corresponding data for objects that are merely

in the vicinity of source and receptor if they produce significant scattered or secondary radiation Information is also needed on the nature and energy distributions of the primary radiations, on the

4 1 1 INTRODUCTION

cross sections for the production of secondary radiation, and on the magnitude of material constants such as attenuation coefficients and stopping powers These must again be known for each type and as a function of energy for all significant primary and secondary radia- tions Errors or uncertainties in any of the variables in this lengthy catalog can result in incorrect dose assessment and further inaccura- cies may also be due to simplifications that must often be made Even with such simplifications, the calculations tend to be complex and expensive

Theoretical assessments are nevertheless essential in a number of cases Measurements can be made only when dose is actually received Retrospective determinations of doses that have been received, or evaluations of the doses to be received, are generally done by calculations Calculations are also necessary when it is impractical to place a dosimeter at the point of interest as in most instances of radiotherapy or the dosimetry of internal emitters Cal- culations can be performed when the required spatial resolution cannot be attained with a physical device or when radiation intensit- ies are too high or too low to permit measurement with available equipment

Calculations can also play a significant role in the interpretation

of dose measurements For example, the quantity actually measured

is often not the absorbed dose itself, but the ionization in an air-filled cavity chamber Theory, in the form of the Bragg-Gray principle and its elaborations, can then be used to estimate what the absorbed dose would be in the medium, a t the point of measurement, in the absence

of the dosimeter Furthermore, the presence of the dosimeter usually perturbs the radiation flux A theoretical model of radiation trans- port in and near the dosimeter may be needed for the appropriate perturbation corrections

Apart from any immediate utility, theoretical dosimetry often results in a deeper understanding of basic aspects of the propagation and absorption of radiant energy This has led to the development of concepts and theorems that are of value in their own right and whose application may also lead to simplifications of measurements or reductions in their required number

1.3 Elements of Dose Calculations

This report is designed to present an outline of the methodology

of theoretical dosimetry From the definitions above, it is clear that the calculation of dose requires a description of the radiation fields

1.3 ELEMENTS O F DOSE CALCULATIONS / 5

in terms of sources of particles, of the physics of their interactions, and of receptors With this description, one should be able, in princi- ple a t least, to predict the flow of energy in and out of volumes of interest and to calculate the dose

Two general methods, not always equivalent, are used in these calculations In the so-called deterministic approach, the radiation field is characterized by functions describing the phase-space density

of particles a t a given point and its distributions with respect to energy, angle and particle type The term "deterministic" refers to

the fact that these density functions (or their distributions) are sub- ject to mathematical laws termed transport equations, and, there- fore, given the sources, boundary conditions and interaction coeffi- cients, one can calculate their values a t each point and time

In the second approach, the particle trajectories are simulated individually according to the stochastics of the physical interactions The treatment is in terms of probability distributions, for instance, the probability that a given number of particles (in a n energy and solid angle interval) traverse a n element of area The stochastic

approach (e.g., a Monte Carlo method) allows the calculation of any deterministic quantity as the average of the corresponding stochastic one over its probability distribution, although, of course, a large body

of additional information is available (e.g., the moments of these distributions) The simulation of particle trajectories is performed with Monte-Carlo techniques

Because, in the calculation of dose (a non-stochastic quantity), the additional information provided by the stochastic method is not needed, one prefers to use the transport equations For many practi- cal situations, however, the complexity of the source-receptor sys- tems makes a deterministic approach impractical and then the Mon- te-Carlo method remains the only suitable one

The basics ofthe deterministic approach to dose calculation (trans- port formalisms) are presented in Section 2 together with several simple examples Specific elements of radiation transport are covered

in succeeding sections They are: sources (Section 3), receptors (Sec- tion 4), and interaction coefficients (Section 5)

Dose calculations are sometimes based on general theorems (Sec- tion 6), or on the use of transport methods (Section 7), or they may utilize the stochastic approach (Section 8) Section 9 presents some

geometric considerations relevant to this subject

Section 10 deals with calculations of the dose equivalent, which is the absorbed dose weighted by the biological effectiveness of the charged particles producing it These calculations are performed for use in radiation-protection evaluations

2 Transport Formalisms

2.1 Concepts in Dose Calculations

The primary objective of dose calculations is to determine the absorbed dose a t a given location (the point of interest) in a finite mass (the receptor) due to ionizing radiation originating from a region (the source) that is outside the receptor for external irradia- tion and within the receptor for internal irradiation A further objec- tive of dose calculations can be the determination of the dose equiva- lent a t the point of interest From Equation (1.21, it is obvious that the physical information on which a dose calculation is based includes the distribution in number, energy and direction of all particles entering or leaving a differential volume surrounding the point of interest or originating within it (sources) This type of infor- mation, in a non-stochastic formulation, is the object of transport theory In the following, the basic concepts and quantities of trans- port theory are introduced and the relation between these quantities and the calculation of dose is established in a formal manner Consider a medium Define

f i , E, 2, t)d3rdEdu = the expected (mean) number of

radiation particles or photons (hereafter referred

to as "particles") in a volume element d3r about 3, (2.1)

with energy E in dE and moving with directions in

the solid angle du about ii (unit vector) a t time t

If 8 and a are the spherical coordinates of Ti, then du=sin8 d8 da, and n is the particle density distribution with respect to E and ti The particle density, NG, t) is then

Let 3 be the velocity of a particle of energy E Then

where v is the magnitude of ;.

2.1 CONCEFTS IN DOSE CALCULATIONS / 7

A closely associated concept is the particle current-density distribu- tion with respect to E and 2:

7 G, E,Z, t)-dSd.Edu = expected number of particles

that cross an area dS per unit of time with energy

E in dE, moving in directions within du about Z at (2.4) time t (by definition, & is perpendicular to the

surface element dS)

The particle current density is:

A simple relation holds between n and 7:

The quantity in square brackets in Equation (2.6) is the flux-density distribution, 4, in E and 2,

which, when integrated, yields the flux density (also called fluence mte)

It is important to note that, although @and z4re measured in the same units, they have very different meanings: J - & is the net flow rate of particles through dS (i.e., outgoing minus ingoing particles) while specifies the total rate of particles going through dS

The rate of change in the energy absorbed in a volume can be related directly to jG,E,ii,t) If sources are present in the volume, one can define, in analogy with Equations (2.1) and (2.4),

sG, E,ii,t)d3rdEdu = the expected number of particles

produced per unit of time in d3r about 7, etc (2.9) One has then:3

3Q is the change in r e ~ t maas in interactions within the volume

8 / 2 TRANSPORTFORMALISMS

With the use of Green's theorem (conversion of a surface integral to

a volume integral), the first term on the right-hand side of Equation

(2.10) becomes:

(The minus sign above results fkom the convention of dS pointing

outward, i.e., in the direction of particles leaving the volume)

By considering an infinitesimal mass dm of density p,

one arrives at the final expression:

or, by integrating over the time, t, to an equivalent relation for D This is the desired (albeit formal) relation between the dose, D G , ~ )

a t a point 7, over time t, and the field quantities 4 and s (or 3 and

s) The calculation is thus reduced to the question of calculating $J

or y, which constitutes the objective of transport theory 4, s and Q

in Equation (2.13) include all types of particles present in the field The flux density distribution, +(T, E,;,t), satisfies the transport equation Because of its fundamental importance in dose calcula- tions, this equation is derived in full detail in the next section

Consider a volume V with surface area S The net change per unit time in the total number of particles of a given type in V within the phase-space element dEdu is given by4:

'All the quantities used in the discussion of the transport equation refer to a single type of particle as opposed to the more general definitions given in the previous section For simplicity the same notation is used

2.2 THE TRANSPORT EQUATION 1 9

One should note that s includes here not only "true" sources, but also contributions from other types of radiation in the field undergo- ing transformation to the particle type to which Equation (2.14) refers This is discussed later in more detail The term (3mll - repre- sents changes in n (per unit time) due to collisions in the medium Equation (2.14) is simply the balance of the particles entering or leaving the phase-space element dEdu throughout the volume V, per unit time

With:

Equation (2.14) becomes:

Because this equality is valid for any volume V, it follows that

which is the transport equation for nG, E,z,t)

Consider now the collision term Let

p&, E ) = probability per unit pathlength of a particle (2.18)

a t ;' and with energy E to have a n interaction

If the interactions are uncorrelated, then f i is proportional to the density of scattering centers in the medium, n,:

The interaction coefficients and a are also called macroscopic and microscopic cross sections, respectively Also:

Vp&, E ) = number of interactions per unit time expe-

rienced a t 7 by a particle with energy E (2.20) One can defme various distributions oft.+ with respect to its vari- ables For example:

cL,G, E+E$+zr)dE'du' = the probability per unit dis-

tance traveled that a particle with E and < will

produce, as a result of an interaction at 7, a particle (2.21) (including the primary one itself, i.e., a scattering)

with energy E' and direction ii',

10 / 2 TRANSPORT FORMALISMS

and

where pa corresponds to particle absorption The absorption term, however, will be formally omitted here with the convention that such interactions are represented by transitions E + E' = 0 All the quantities defined above, Equations (2.18 to 2-22], correspond to mechanisms contributing to the rate of change in n due to interac- tions Thus, particles are removed from the phase-space element dEdu by collisions ( k ) while other particles scatter in from other elements [p,(7, E '+E,ii1-Z)]:

Introducing Equation (2.23) into the transport equation, Equation (2.171, and changing from n to 4 [Equation (2.711 gives:

This is the classical form of the linear transport equation with the interaction probabilities explicitly shown

It is important to remark also that Equation (2.24) is linear because

it involves C#I only to the first power (for instance, the other well known transport equation, the Boltzmann equation for gases, belongs to a nonlinear transport formalism; see also Subsection 6.1)

Examples

In order to further elucidate the concepts introduced above, several examples of practical application of the transport equation, Equation (2.24), are given here These examples also allow the introduction of

a number of additional concepts widely used in dose calculations The presentation here follows closely that of Rossi and Roesch (1962) and Roesch (1968)

Possibly the most typical dose calculation involves the situation where a constant radiation source is used to irradiate over a given time interval (O,t,) One is interested in the total dose delivered

2.2 THE TRANSPORT EQUATION 1 11

The transport equation, Equation (2.241, can be simplified by observing that

and by defining:

where t is the fluence distribution in E and 2, and

sOG, E,G) dEdu = dEdu 6 sG, E,Z,t) dt, (2.27) where So is the expected number of particles produced per unit vol- ume during [O,t,] in dEdu about E and ii With these simplifications and definitions, Equation (2.24), on integration over time, becomes

For a field with only one type of charged particle and Q = O (no secondary particle transport), one obtains the dose delivered during [O,tol by inserting Equation (2.28) in the time integral of Equation (2.13):

12 / 2 TRANSPORT FORMALISMS

Because F~ is defined per unit length, x , one recognizes in Equation

(2.29) the stopping power of the medium

The dose, then, is

The quantity in square brackets in Equation (2.31) is called the

mass stopping power of the medium for the charged particles under

consideration Equation (2.31) equates the energy lost by the parti-

cles (- dEldx) and the energy absorbed in the volume (- D), an equality that holds only because transport by secondary particles was neglected

Consider now a slightly more complicated situation, namely a field with two kinds of particles, charged and uncharged With subscripts

c and u indicating the particle types, one has now a system of coupled transport equations similar to Equation (2.28):

The dose is then given by

The first integral in Equation (2.33) can be transformed in a man- ner similar to that of Equation (2.29) Assume further that S, = 0

(no charged particle sources) Then:

2.2 THE TRANSPORT EQUATION / 13 Now, let q, (Eu+E:) be the change in rest mass energy following a transition Eu+E: and in which a secondary charged particle of energy E, - E: - q, is produced The energy transfer coefficient of the uncharged particles, pk(Eu), defined as the fraction of E, trans- ferred to kinetic energy of the charged particles per unit distance traversed, is

pk(Eu) = - ' I dE: p, (E,+E:) [E, - E: - Q,(E,+E:)I (2.35)

Eu

The first integral in Equation (2.34), with the correction for rest mass energy changes, q,, represents, then, the sum of the initial kinetic energies of all the charged particles created by the uncharged particles per unit mass:

1

KG) = IdE, E u ' P u ~ , E,) - pk(Eu) (2.36)

P

K is called the kermu of the uncharged particles

Frequently the second integral in Equation (2.34) is negligibly small compared to K (the same holds, then, for dQJdm) The condition when ly,(?, Ec) does not depend on F' (i.e., is constant over the field)

is called charged particle equilibrium Then, because $'PC = 0, one has

a relation useful'in photon or neutron dosimetry

3 Sources

Ionizing radiation is produced by accelerators that impart kinetic energy to charged particles such as electrons, protons or other atomic nuclei It is also emitted in radioactive decay and in both fission and fusion, and it is incident upon the earth from outer space (cosmic rays) The radiation field a t a point consists not only of the particles and photons from these original sources, but of secondary radiations produced in intervening or proximal matter: deceleration of charged particles produces x rays, nuclear reactions produce neutrons and other particles and quanta, and absorption and scattering usually produce secondary radiations as well as changes in the energy and direction of the primary Any one of these original or secondary processes can be designated as the "source" for the purposes of dose calculations, as is most convenient in the circumstances

3.1 Specification of Sources

A calculation of the absorbed dose starts with the solution of the transport equation, Equation (2.28) above The sources can be repre- sented in this equation in two ways: The quantity So, the number of particles or quanta with a given energy emitted per unit volume of the medium, is specified For example, for radionuclides distributed

in a medium, the emission can be obtained from the activity per unit volume of the radioactive material Alternatively, the sources themselves are not specified; instead, the number of particles or quanta per unit area and their energy incident on the surface of a medium are given The specification of such a boundary condition is

a convenient way of dealing with a source whose structure plays no significant role in determining the radiation field in the material Examples are beams of radiation from accelerators and cosmic rays Both methods of specifying sources can be used in a single calcula- tion, but joint use is avoided here Instead the calculation is broken into parts For example, in the calculation of secondary radiation fields, the primary radiation may be given as a boundary condition and the distribution in energy, direction, and position of the radia- tions it produces is then calculated These results are then used as the source for calculation of the remainder of the radiation field

3.2 SLMPLIFIED REPRESENTATIONS OF SOURCES 1 15

Instead of using boundary conditions, one can devise functions for fictitious sources on the surface of the medium that will produce the same radiation field The radiation emerging from these fictitious sources must have the same distribution in energy and direction a t the surface of the medium as the radiations from the actual sour- ces Whenever appropriate, this replacement will be assumed in the rest of this report because i t simplifies discussion of the tran- sport equation and its solution by using only one method of source representation

3.2 Simplified Representations of Sources

The analytical methods for solving the transport equation are difficult to apply to complex configurations of sources The Monte- Carlo method is better in this regard, but increasing complexity may make its cost prohibitive A common practice, therefore, is to replace the complicated source structure with a simpler structure whenever doing so will retain the desired accuracy For example, a real 60Co gamma-ray source is a small pellet of material containing the cobalt inside a sealed capsule to prevent loss of radioactive material This arrangement can often be simulated by a point source because the radiation field of the point source differs significantly from that of the real source only a t small distances Furthermore, the pellet source can usually be assumed to emit photons isotropically Even complicated sources can often be simplified Sources, such as nuclear reactors, massive accelerator targets, etc., have structures

of such complexity that calculating the radiation emerging from them is in itself a major task apart from the computations of the subsequent absorption and transmission of that radiation In such cases, the emerging radiation and the subsequent transport are fre- quently calculated separately For example, people in the vicinity of

a nuclear reactor, but not employed there, are apt to be a t distances large enough that the fission source can be simulated by a point source emitting the spectrum calculated to emerge from the real source On the other hand, workers may be close to the reactor; therefore, for them, its geometrical structure may have to be taken into account

Another way of simplifying complex sources is to consider them divided into very small pieces, each of which can be treated as a point source This method is particularly valuable if the radiation field a t

a given distance from each of the points is the same except for a factor (proportional to how much radiation is emitted a t that point)

16 / 3 SOURCES

independent of that distance Then, the field need be calculated for

a point source only once and the total field obtained by combining this point-source field for all the point sources that make up the complex source For this method to work, the source and surrounding media must be reasonably homogeneous out to distances from the

source a t which no one point source makes a significant contribution

Sources other than the point source can sometimes be treated in the same way If the sources and media are such as to make the radiation field one- dimensional, "plane" sources that emit uniformly over an infinite plane may be useful For example, in calculations of doses from radioactivity, the emission from each element ofthe plane

is isotropic and the plane source is called a "plane-isotropic source" Plane sources are also useful in simulating the boundary conditions for radiation incident from outside the me&um Then the emission from the plane should not be isotropic; rather the emission should

be only in the direction or directions of the incident radiation These are "plane-directional sources"

If a source emits radiation with a spectrum of energies, calcula- tions can be made for sources that emit monoenergetic particles or quanta ("monoenergetic sources") and the solutions combined in proportion to the abundance of each type in the emitted spectrum Further simplification results if the spatial conditions, a s described above, permit straightforward superposition of point or plane sources; then one need only calculate monoenergetic point or plane sources with suitable directions of emission

Superposition is similarly used for fields containing different kinds

of particles Provided that the different kinds of radiation are not coupled (i.e., one does not give rise to the other), calculations are made for quanta and for one kind of particle a t a time and the solutions added Actually, most mixed fields are coupled (e.g., elec- trons and photons produce each other) but often to a small enough degree that separate calculations for each radiation are adequate, with coupling taken care of by iterative corrections In high-energy phenomena such a s cosmic-ray showers, however, the coupling is so large that it must be included ab initio

4 Receptors

The aim of absorbed-dose calculations is usually to determine the dose in tissue But doses in other media also are required sometimes, for example, in the parts of a n instrument for measuring radiation Because of this diversity, the non-specific term receptor is used for the object in which the dose is to be determined

For the same reasons that lead one to simplify the structures of the sources of radiation in transport calculations, one also simplifies the structures of receptors For example, early calculations aimed a t estimating neutron doses for purposes of radiation protection used a receptor consisting of a 30-cm thick slab of tissue-equivalent mate- rial that was infinite in lateral extent (NCRP, 1957) Later calcula- tions used increasingly more realistic receptors The next one used was a tissue-equivalent cylinder 30 cm in diameter and 60 cm high (Auxier et al., 1968) More recently, a n anthropomorphic receptor was used in which body surfaces, organ boundaries, and bone vol- umes were defined by simple mathematical equations (Snyder et al., 1969) Figure 4.1 illustrates this receptor Further refinements of this receptor have been made to simulate the two sexes and to simu- late children (Cristy and Eckerman, 1985)

In complex irradiation conditions, i t may be economical to calcu- late first the radiation field with no receptor present and then the field in the receptor, using the receptor-free field for the boundary conditions on or near the surface of the receptor Clearly, such a two- step calculation is possible only if introduction of the receptor does not materially alter the receptor-free field outside the receptor For example, the transmission through thick shields is ordinarily evalu- ated without considering a receptor; subsequently, one starts with the energies and directions of the radiation emerging from the shield

to calculate the dose distribution in the receptor An extension of this step-wise procedure is followed, for example, in calculating the doses from nuclear weapons in the following sequence: the radiation emerging from the weapon; the radiation from the rising fireball; the transport of these radiations to points on the bare ground; the effects of houses and buildings; and the penetration in a receptor simulating the human body In the first two steps, the sources are fissions, nuclear reactions, and radioactive emissions; in the follow-

18 / 4 RECEPTORS

Fig 4.1 An anthropomorphic model receptor Dimensions and coordinate system

of adult human phantom (After Snyder et a[., 1969.)

ing steps, the input conditions are the fluences calculated in the preceding step

Apart from simplifying the calculation, the step-by-step approach has the advantage that calculation of one receptor-free field suffices for calculations in a number of different receptors or in any one receptor by a number of different methods

The calculation of the dose in the receptor essentially follows the propagation of radiation from the receptor-free field after it crosses the receptor boundary One can do just the reverse: start with radia- tion a t a point in the receptor and follow it backwards in time till

it crosses the receptor boundary Then one must superpose these backward solutions in proportion to the receptor-free fluences a t the boundary These backward calculations are done by what are called

adjoint methods In principle, the radiations could be followed back- ward to the points a t which they emerged from the original source The advantage of the adjoint calculation is that one such calculation suffices for any number of different receptor-free field calculations Receptor-free calculations and measurements in nearly receptor- free conditions are frequently employed in radiation protection The results can, for convenience, be characterized by the kerma in some material at a given point Because of differences in how they attenu- ate radiation, different receptors placed a t that same point will pro- duce different depth-dose distributions

Calculations relating to the measurement of radiation present a somewhat different aspect of the receptor concept: here one is con-

cerned with the receptor as a n instrument rather than as a biological

target Occasionally the receptor-free condition is employed as a n intermediate step Even when the instrument is calibrated to read some quantity that will exist in the biological target, design compro- mises, differences in calibration conditions, and other conditions, almost always exist to make the reading and the quantity differ The

art of making measurements is to be aware of these differences and

to avoid (or minimize) them Calculations of radiation fields are often used to understand and correct for the differences between the instmment-receptor and body-receptor conditions

5 Cross Sections

5.1 Schematization

The penetration, diffusion, and slowing down of ionizing radiation

in an extended medium, and the associated transfer of energy to the medium, can be treated a s resulting from a large number of binary interactions between radiation particles and certain constituents of matter The radiations of interest in radiation dosimetry are photons

(X rays, gamma rays, continuous bremsstrahlung, characteristic x rays), neutrons, and charged particles (electrons, mesons, protons, alpha particles, heavy ions) The interactions of interest are either electromagnetic or nuclear, and take place with various scattering centers such a s atoms, molecules, atomic electrons, nuclei, or individ- ual nucleons in nuclei The choice of the appropriate scattering ten-

ters in a transport calculation depends on the nature of the interac- tions, and also on the approximation in which they are treated In any case, a dosimetry calculation usually involves the solution of a multiple scattering problem, starting with a n assumed knowledge

of the cross sections for single interactions

In order to have a framework for the discussion of cross sections needed for dosimetry calculations, i t is useful to discuss the schemati- zation in which the cross sections are used The schematization, which is assumed in the derivation of the linear transport equation,

Eq (2.241, and which also underlies the formulation of the Monte- Carlo method, is based on the following assumptions and approxima- tions:

A The various radiations are treated a s particles with well-defined trajectories, and the effects of successive collisions with scattering centers are treated in terms of probabilities rather than in terms of probability amplitudes, that is, quantum-mechanical interference effects a r e disregarded This approach precludes the treatment of phenomena of minor importance for dosimetry such as the diffraction

of photons, electrons or neutrons a t low energies, where the particle wavelengths are of the same order- of magnitude a s the spacing between atoms This schematization assumes that the radiation par- ticle wavelengths are small compared to the sizes of collision targets and also small compared to the average distances between collisions

5.1 SCHEMATIZATION 1 21 Finally, i t should be noted that quantum-mechanical interference effects are neglected only in the treatment of multiple collisions, whereas the underlying single-scattering cross sections are derived

on a quantum-mechanical basis

B The particle trajectories in the medium are assumed to be zig-

zag paths consisting of free flights interrupted by sudden localized interactions with scattering centers The particle momentum and energy are assumed to be constant during each free flight, changing suddenly a t the time of the interaction which terminates the free flight

C The scattering centers are assumed to be distributed throughout the medium with a specified average density distribution, but other- wise randomly This assumed randomness precludes the treatment of channeling phenomena, that is, the modification of charged-particle penetration in crystalline materials that occurs when the direction

of the motion of the particle coincides, within certain critical angles, with one of the symmetry axes or planes of the crystalline medium

D The probability that an interaction of type j will occur in a small

segment A t of the flight path is assumed to be equal to yAt where

p, is the interaction coefficient [see Equation (2.18)l It is assumed further that, in the limit of vanishingly small At, the probability of more than one interaction of type j in A t is negligibly small The probability that no interaction of type j will occur in A4 is therefore equal to 1 - p,Al The interaction coefficient can be expressed as the product of two factors [see Equation (2.19)l

where n is the density of scattering centers, and oj is the cross section for the interaction Note that p, is a probability per unit pathlength,

n is the number of scattering centers per unit volume, and the cross section uj has the dimensions of an area The total probability of an interaction per unit pathlength, taking into account all modes of interaction, is given by

= Z p ,

j

(5.2) The probability that a free flight will be terminated when its length

is between C and t + d t is given by

and the mean free path between collisions is

E The irradiation intensity is assumed to be low enough to satisfy

22 1 5 CROSSSECTIONS

two conditions: (i) The interactions between the radiation particles are negligible compared to their interactions with the medium This condition does not hold for certain electron accelerators with intense beam currents of lo4 to lo6 A used in radiation effects studies (ii) The properties of the medium are not changed significantly as the

result of the irradiation, i.e., it is extremely unlikely that the radia-

tion particles have interactions with scattering centers whose prop- erties or distribution have been altered as the results of earlier interactions with other particles

Within the framework of the schematization indicated above, the following information is needed in terms of cross sections, interaction coefficients or related quantities:

A Location-of next collision Suppose the particle has emerged from the previous interaction (or has been injected into the medium

by the radiation source) at the point 7, with energy E and in direction

,

u The next interaction will occur a t the point

-+

where e is the length of the free flight The probability distribution

of the distance t o the next interaction is thus determined by the distribution of lengths of free flights, p(e), as given by Equation (5.31, and thus depends only on the total interaction coefficient C(T

B Type of collision With probability M h , the collision is of type j

C Outcome of collision Detailed differential cross sections are required that provide the following information, in terms of probabil- ities for various contingencies:

(i) What happens to the particle making the collision? Is it scat- tered or absorbed? If scattered, through what angle is it deflected, and how much energy does it lose?

(ii) What happens to the struck collision target? How much recoil energy does it receive, and in what direction is it set in motion? Is the collision elastic, with the target left in its ground state, or is it inelastic, with the target left in an excited state? Is there any long-lasting modification of the target,

e.g., as the result of an ionization or fragmentation? What is the expected chain of events when the target, or some of its fragments, are eventually de-excited?

(iii) What kinds, and how many, of secondary particles are emitted

by the struck target, either promptly or after some delay? What are their energies and directions of motion?

5.2 GENERAL ASPECTS OF REQUIRED CROSS SECTIONS 1 23

In order to specify the various outcomes of radiation interactions, one needs a vast set of electromagnetic and nuclear cross sections Fortunately, the complete array of cross sections may be required only in the most detailed types of calculations Many other transport methods are based on approximations that reduce the cross-section requirements For example, in the treatment of neutron diffusion, the simplifying assumption is often made that elastic scattering

is isotropic in the center-of-mass system In the treatment of the penetration and diffusion of charged particles, the continuous-slow- ing-down approximation is often used, in which the kinetic energy

of the charged particles is assumed to decrease continuously along the particle tracks a t a rate given by the stopping power (mean energy loss per unit pathlength) The applicability of this approxima- tion derives from the circumstance that the transfer of energy from charged particles to the medium takes place via many thousands of transfers, most of which are small (less than - 30 eV), with only rare collisions involving large energy transfers In the continuous- slowing-down approximation, only stopping-power data are needed

as input, but detailed inelastic-scattering cross sections are needed when energy-loss straggling is taken into account

Other reductions of the cross-section requirements can result from the nature of the problem under consideration For example, in a calculation of the intensity and energy spectrum of the gamma rays emitted from a large encapsulated 60Co teletherapy source, one can concentrate almost entirely on the penetration, diffusion and slowing down of the gamma rays in the source and treatment head, and can disregard the penetration of the secondary electrons also produced Similarly, when calculating the spatial distribution of absorbed dose from x rays, it may not be necessary to consider the transport of energy by photoelectrons and Compton-recoil electrons Because the ranges of these secondary electrons are small compared to the mean free paths of the x rays, it is a good approximation to compute the absorbed dose in terms of the integral in Equations (2.36) and (2.37) where their transport is neglected

Neither experimental data nor theoretical results by themselves are, in general, adequate to provide the comprehensive and complete cross-section sets required for transport calculations It is often nec- essary to use a n amalgam of experimental and theoretical cross sections derived from a critical data analysis Such an analysis involves the reconciliation of different experimental and theoretical results, the interpolation and extrapolation of cross sections, and the application of consistency checks

The critical evaluation of cross sections is particularly difficult for interactions in which the energy of the incident particle is of the

24 1 5 CROSSSECTIONS

same order of magnitude as the differences between internal energy levels of the scattering center The cross sections, then, are character- ized by thresholds and resonances, and vary irregularly as functions

of particle energy and atomic number These conditions make the theory, and even interpolation with respect to energy and atomic number, quite difficult Such difficulties are particularly severe for neutron cross sections, and also arise for photoelectric cross sections

at low energies near the K-, L-, , absorption edges

It is often necessary to adapt the format of the cross-section input

to the requirements of the method for a dosimetry calculation For

example, when the moment method is used (Fano et al., 1959), the

angular dependence of the scattering cross section must be available

in the form of expansion coefficients that allow the cross section to

be written a s a sum of Legendre polynomials For neutron transport calculations in the multi-group approximation, the various cross sections must be available in the form of averages over specified energy bins The application of the Monte-Carlo method may require cross sections converted to cumulative probability distributions Thus, there is a need for specialized data libraries adapted to the needs of particular transport computer programs Many such librar- ies have been collected and are available from the Oak Ridge Radia- tion Shielding Information Center (RSIC, 1981)

To give a complete and detailed account of the cross sections needed for dosimetry calculations is beyond the scope of this report Table 5.1 lists a number of textbooks and monographs that contain authori- tative discussions of the various cross sections In Appendix A, the characteristics of some of the more important photon, neutron and charged-particle cross sections are briefly described and illustrated, and references are given to review articles and data compilations of up-to-date cross-section information

5.2 GENERAL ASPECTS OF REQUIRED CROSS SECTIONS 1 25

TABLE 5.1-Textbooks and monographs dealing with cross sections

Bethe (1933) electrons, protons

Bohr (1948) charged particles

Rossi (1952) photons, electrons

Bethe and Ashkin (1953) photons, charged particles

Heitler (1954) photons, electrons

Evans (1955) photons, charged particles, neutrons Price et al (1957) photons, neutrons

Fano et al (1958) photons

Birkhoff (1958) electrons

Weinberg and Wigner (1958) neutrons

Fano (1963) protons, alpha particles, mesons

Northcliffe (1963) heavy ions

Beckurts and Wirtz (1964) neutrons

Mott and Massey (1965) electrons

Auxier et al (1968) neutrons

Bichsel (1968) charged particles

Hubbell and Berger (1968) photons

Goldstein (1971) photons, neutrons

Bichsel (1972) charged particles

Jackson (1975) charged particles

Ziegler (1980) heavy ions

Starace (1982) photons

ICRU (1984) electrons, positrons

6 Transport Theory-

General Theorems

and Properties

6.1 Integral Form of the Transport Equation

A number of useful properties and theorems concerning the solu-

tions of the transport equation can be obtained if the differential representation of Equation (2.24),

is changed to an integral one In Equation (6.1):

Consider a change of variables:

One can rewrite Equation (6.1) in these new variables as

In this representation, the gradient term in Equation (6.1) has been

eliminated, and Equation (6.4) can be solved formally to yield:

P

x exp [ - J dpl&G -plZ Ell

6.2 ITERATIVE SOLUTIONS (ORDERS OF SCATTERING) 1 27 This is the integral form of the transport equation Equation (6.5) is not a n actual solution, because S contains 4 in its definition (Equa- tion (6.2)) which is of course the unknown function The interpreta- tion of Equation (6.5) is straight forward: contributions to the flux

density distribution, 4 , come from particles generated by the general- ized source S a t any position (7 -pi& O s p ~ m and time (t -plv) and are attenuated exponentially according to the coefficient h , inte- grated between their point of origin and ;' When the medium is

homogeneous (i.e., C(T does not depend on ?) the exponential factor

in Equation (6.5) simplifies to exp [ - h(E)pl

6.2 Iterative Solutions (Orders of Scattering)

Consider for simplicity a homogeneous mediuin and define two operators T and K as follows:

where f is an arbitrary function The transport equations, Equations (6.2) and (6.5), can be written as

S = K 4 + s

where K plays the role of a "scattering" operator, which generates a source term from a flux density, and T performs rather the alternate operation ("translation" operator)

Equations (6.7) can be combined to yield

where I is the unit operator and, for reasons apparent below, the notation So = s is introduced

By introducing the operator (I - Tk)-I one can rearrange Equation (6.8') a s

28 1 6 TRANSPORT THEORY-THEOREMS AND PROPERTIES

Kdo, whose output was transported, TS,, to the point ;, and so on, hence the name "orders of scattering." It is clear from Equations (6.10) that the Neumann expansion, Equation (6.91, expresses a n iterative solution to the transport equation

6.3 Density Scaling Theorem

A simple but important theorem holds that, if two configurations differ in physical dimensions by only a scaling factor 5 4 , then one can multiply the densities everywhere in the smaller by the inverse scaling factor to bring about precise agreement between radiation fields in these two configuations a t all corresponding points (Spencer

et al., 1980) This theorem requires, a s a precondition, a similar scaling relationship between the radiation source functions in the two configurations and the density-independent interaction coeffi- cients

This scaling theorem can be understood from Equation (6.8) or (6.8'1, where the translation operator, T, and the scattering operator,

K, operate sequentially; any factor t h a t cancels when both are applied is essentially irrelevant Thus, if distances are decreased by

a factor t< 1, then scattering probabilities are increased by the same factor But the last term in Equation (6.8) remains unchanged only

if the source is also intensified by the factor 5-l

6.4 FANO'STHEOREM 1 29

6.4 Fano's Theorem

This well-known and important theorem (Fano, 1954) states that,

in any medium with only density variations from one point to another, if there is also a radiation source function that differs from one point to another only in proportion to the density ratio, the resulting radiation field is everywhere the same, independent of local densities Two cases with different density functions can then be

equated by equating source strengths per unit density, i.e., using a single characteristic number

To prove this theorem, note that in the combined operator (TK) of Equation (6.81, the integrals involve only the differential dp multi- plied by the local density because both and p, are proportional to

the local density Thus, if distances are expressed by a variable 77

a consequence, must be independent of position

This theorem can be applied in plane symmetry to uniform radia- tion sources confined to a plane There results a plane-scaling theo- rem which applies to configurations with differing plane-density variations (Spencer et al., 1980)

6.5 Energy Conservation

While energy conservation in radiation transport is a basic expec- tation, i t is instructive to formulate a n expression of it One formula- tion of energy conservation is that for particles of a given type, in a given band of kinetic energies, the input of kinetic energy into this band through radiation sources exactly matches the losses through the various interactions by the particles

To obtain a n expression of energy conservation, it is easiest to utilize the differential form, Equation (2.28), for the transport equa- tion There is no interest in time dependence; hence, one may ignore,

or integrate out, this variable Each term is multiplied by the energy

E, and integrated over energy above a n arbitrary cutoff designated

by A With the use of a n expression for the source term that sepa-

30 1 6 TRANSPORT THEORY-THEOREMS AND PROPERTIES

rately identifies scattering interactions, i.e., Equation (6.21, the energy input into the particle field above A is expressed by the term

IAmdE Es

The main complication, then, concerns the scattering interactions that may regenerate a particle still in the beam, albeit of lower energy By changing the order of integrations in the scattering term this possibility can be formulated precisely The resulting expression

is shown in Equation (6.12), with explicit indication of most variables omitted to clarify the meaning

One needs mainly an understanding of the expression in the curly brackets The quantity, p&, clearly refers to the total energy of particles that interact The integral term represents the energy

remaining in the beam, i.e., the interactions that leave particles with

energy above A Because this subtractive term represents the beam energy that remains, the full curly-bracket factor weights in accor- dance with the part of the interaction energy that leaves the beam The irrelevant variables, position and direction, can be removed

by integrations that also remove the gradient term The remaining terms equate total energy removed to total energy injected, i.e.,

energy is consenred

6.6 Superposition

The transport equation is a linear equation, which means that any linear representation of a given source function, s, can be translated into a solution using the same superposition for solutions for the component sources Thus, if

r E,ZA g(z),

then the solution is

where +f is the solution when f is used as the source function in the same configuration Note that the variable z may be discrete, and

6.7 ADJOINT TRANSPORT EQUATION 1 31

the integral, then, is replaced by a sum The individual source terms

f may be products of functions of the different variables in arbitrary combinations The main requirement is that the configuration, the interaction coefficients, and the boundary conditions remain the same for different 4, and for +8

This theorem is essential for procedures based on Fourier trans- form methods, Legendre transform methods, etc It will be utilized

in Section 7 of this report, which deals with methods of solution of the transport equation

One of the more interesting aspects of superposition in the space variables is the full generality of sources uniformly distributed on infinite plane surfaces It can be shown that point-delta functions

6G -7,) can be represented by superpositions of functions

6 6 C;: -;,)I (Morse and Feshbach, 1953; Berger and Spencer, 1959)

corresponding to arbitrary planes passing through 3, This represen- tation constitutes a proof of the completeness of such systems ofplane sources

6.7 Adjoint Transport Equation

The adjoint operator Ht corresponding to a n operator H is defined by:

where f and g are arbitrary functions and the inner product may be

defined a s

where g* is the complex conjugate of g

Consider the transport equation, Equation (2.24), and define a n operator L such that

With this definition Equation (2.24) becomes:

Without significant loss of generality, one can assume th2t the boundary condition on 4 (and therefore on L) is such that +(R.,E,i)

32 1 6 TRANSPORT THEORY-THEOREMS AND PROPERTIES

is zero at a surface& for directions pointing into the medium Then, the adjoint of L can be obtained from Equations (6.15) and (6.17):

In analogy with Equation (6.18), one can write, now, an adjoint transport equation for the adjoint flux density 4t a s

where the adjoint source, st, is arbitrary

The physical significance of the adjoint equation can be illustrated

as follows Consider the time-independent versions of Equations (6.18) and (6.201, as in Equation (2.28)

and let Si be the mass stopping power of a receptor at a point 7'

where p is the local density,

Then Equation (6.22) yields

where the notation shows that 1+9 is a solution of the transport equa- tion corresponding to the point source, Equation (6.23) But $' of Equation (6.25) is the dose at the point ;f' in the receptor due to the point source in Equation (2.31)