ICRU REPORT 16

L i n e a r E n e r g y

T r a n s f e r

Issued JUNE 15, 1970

INTERNATIONAL COMMISSION ON RADIATION UNITS AND MEASUREMENTS

4201 CONNECTICUT AVENUE, N.W. WASHINGTON, D.C. 20008

USA

UBR069010932145

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Copyright © International Commission on Radiation Units and Measurements 1970

Library of Congress Catalog Card Number 72-113962

Copies of this report can be purchased for U.S. $3.00 each from ICRU Publications P.O. Box 4869 Washington, D.C. 20008 Ü.S.A.

(For detailed information on the availability of this and other ICRU Reports see page 48)

P r e f a c e

Scope of ICRU Activities

The International Commission on Radiation Units and Measurements ( I C R U ) , since its inception in 1925, has had as its principal objective the development of internationally acceptable recommendations regarding:

(1) Quantities and units of radiation and radioactivity,

(2) Procedures suitable for the measurement and application of these quantities in clinical radiology and radiobiology,

(3) Physical data needed in the application of these procedures, the use of which tends to assure uniformity in reporting.

The Commission also considers and makes recommendations in the field of radiation protection. I n this connection, its work is carried out in close cooperation with the International Commission on Radiological Protection ( I C R P ) .

Policy

The I C R U endeavors to collect and evaluate the latest data and information pertinent to the problems of radiation measurement and dosimetry and to recommend the most acceptable values for current use.

The Commission's recommendations are kept under continual review in order to keep abreast of the rapidly expanding uses of radiation.

The I C R U feels it is the responsibility of national organizations to introduce their own detailed technical procedures for the development and maintenance of standards. However, it urges that all countries adhere as closely as possible to the internationally recommended basic concepts of radiation quantities and units.

The Commission feels its responsibility lies in developing a system of quantities and units having the widest possible range of applicability. Situations may arise from time to time when an expedient solution of a current problem may seem advisable. Generally speaking, however, the Commission feels that action based on expediency is inadvisable from a long-term view

point; it endeavors to base its decisions on the long-range advantages to be expected.

The I C R U invites and welcomes constructive comments and suggestions regarding its recommendations and reports. These may be transmitted to the Chairman.

Current Progra m

I n 1962 the Commission laid the basis for the development of the I C R U program over the next several years. A t that time it defined three broad areas of concern to the Commission:

I . The Measurement of Radioactivity I I . The Measurement of Radiation

I I I . Problems of Joint Interest to the I C R U and the International Commission on Radiological Protection (ICRP)

The Commission divided these three areas into nine subareas with which it expected to be primarily concerned during the next decade. The division of work agreed upon is as follows:

I . Radioactivity A. Fundamental Physical Parameters and Measure

ment Techniques B. Medical and Biological Applications

I I . Radiation A. Fundamental Physical Parameters Β. X Rays, Gamma Rays and Electrons C. Heavy Particles D. Medical and Biological Applications (Therapy) E . Medical and Biological Applications (Diagnosis) F . Neutron Fluence and Kerma

I I I . Problems of Joint Interest to the I C R U and the I C R P A. Radiation Protection Instrumentation and its Ap

plication

The Commission established a separate planning board to guide I C R U activities in each of the subareas. The planning boards, after examining the needs of their respective technical areas with some care, recommended, and the Commission subsequently approved, the constitution of task groups to initiate the preparation of reports. The substructure which resulted from these actions is given below.

I V · · · Preface

Planning Board I.A. Radioactivity—Fundamental Physical Parameters and Measurement Techniques

Task Group 1. Measurement of Low-Level Radioactivity

Task Group 2. Specification of Accuracy in Certificates of Activity of Sources for Calibration Purposes

Task Group 3. Specification of High Activity Gamma-Ray Sources (Joint with P.B. II .B)

Planning Board I .B . Radioactivity—Medical and Biological Applications

Task Group 1. In Vivo Measurements of Radioactivity

Task Group 2. Scanning Task Group 3. Tracer Kinetics Task Group 4. Methods of Assessment of Dose in

Tracer Investigations Planning Board H.A. Radiation—Fundamental Physical

Parameters Planning Board I I . B . Radiation—X Rays, Gamma Rays

and Electrons Task Group 1. Radiation Dosimetry; X Rays from 5

to 150 kV Task Group 2. Radiation Dosimetry; X and Gamma

Rays from 0.6 to 100 MV Task Group 3. Electron Beam Dosimetry

Planning Board L L C . Radiation—Heavy Particles Task Group 1. Dose as a Function of L E T Task Group 2. High Energy and Space Radiation

Dosimetry Planning Board I I . D . Radiation—Medical and Biological

Applications (Therapy) Task Group 1. Measurement of Absorbed Dose at a

Point in a Standard Phantom (Absorbed Dose Detenriination)

Task Group 2. Methods of Arriving at the Absorbed Dose at any Point in a Patient (In Vivo Dosimetry)

Task Group 3. Methods of Compensating for Body Shape and Inhomogeneity and of Beam Modification for Special Purposes (Beam Modification)

Task Group 4. Statement of the Dose Achieved (Dosage Specification)

Planning Board U . E . Radiation—Medical and Biological Applications (Diagnosis)

Task Group 1. Photographic Materials and Screens Task Group 2. Image Intensifier Radiography Task Group 3. T V Systems

Planning Board I I . F . Radiation—Neutron Fluence and Kerma

Task Group 1. Neutron Fluence, Energy Fluence, Neutron Spectra and Kerma

Planning Board I I I . A . Radiation Protection Instrumentation and its Application

Task Group 1. Radiation Protection Instrumentation Handbook—Part I

Task Group 2. Neutron Instrumentation and its Application to Radiation Protection

Because the Commission's basic recommendations on radiation quantities and units relate to the work of all of the planning boards, the Commission decided to establish a separate committee with membership drawn largely from the Commission itself to initiate the revision of I C R U Report 10a, Radiation Quantities and Units. Thus, the Committee on Fundamental Quantities and Units was added to the above substructure.

I n 1962 the Commission decided to abandon its past practice of holding a meeting together with all of its sub-units every three years. Instead, it was decided that the Commission would receive reports from the subgroups at the time of their completion rather than at fixed deadlines. Meetings of the Commission and of the subgroups are held as needed.

The adoption of the above substructure and mode of operation was intended to alleviate some of the problems associated with the expanded program required in recent years. I n the past, the Commission's attempt to administer and review the work of each of the working groups imposed a very considerable burden on the Commission itself. The need to concern itself with each detail, which was inherent in such a scheme of operation, when coupled with the procedure of completing all reports at one time, subjected the Commission members to an intolerable work load if rigorous standards were to be maintained. The above substructure and mode of operation have now produced results in the form of reports drafted by the task groups and reviewed by the planning boards. Present evidence in dicates that the substructure and mode of operation, has to a substantial extent succeeded in alleviating the problems previously experienced. Recently, however, the Commission has begun the examination of further modification of the substructure.

ICRU Reports

I n 1962 the I C R U , in recognition of the fact that its triennial reports were becoming too extensive and in some cases too specialized to justify single-volume publication,, initiated the publication of a series of reports, each dealing with a limited range of topics. This series was initiated with the publication of six reports:

I C R U Report 10a, Radiation Quantities and Units I C R U Report 10b, Physical Aspects of Irradiation I C R U Report 10c, Radioactivity I C R U Report lOd, Clinical Dosimetry I C R U Report lOe, Radiobiological Dosimetry I C R U Report lOf, Methods of Evaluating Radiological Equip

ment and Materials

Preface · · · ν

These reports were published, as had been many of the previous reports of the Commission, by the United States Government Printing Office as Handbooks of the National Bureau of Standards.

I n 1967 the Commission determined that in the future the recommendations formulated by the I C R U would be published by the Commission itself. This is the sixth report to be published under this new policy. With the exception of I C R U Report 10a, which was superseded by I C R U Report 11, the other reports of the " 1 0 " series have continuing validity and, since none of the reports now in preparation are designed to specifically supersede them, will remain available until the material is essentially obsolete. Al l future reports of the Commission, however, will be published under the I C R U ' s own auspices. Information about the availability of I C R U Reports is given on page 48.

ICRU Relationships With Other Organizations

One of the features of I C R U activity during the last few years has been the development of relationships with other organizations interested in the problems of radiation quantities, units, and measurements. I n addition to its close relationship with the International Commission on Radiological Protection and its financial relationships with the International Society of Radi ology, the World Health Organization, and the International Atomic Energy Agency, the I C R U has also developed relationships of varying intensity with several other organizations. Since 1955, the I C R U has had an official relationship with the World Health Organization (WHO) whereby the I C R U is looked to for primary guidance in matters of radiation units and measurements, and in turn, the W H O assists in the worldwide dissemination of the Commission's recommendations. I n 1960 the I C R U entered into consultative status with the International Atomic Energy Agency. The Commission has a formal relationship with the United Nations Scientific Committee on the Effects of Atomic Radiation ( U N S C E A R ) , whereby I C R U observers are invited to attend U N S C E A R meetings. The Commission and the International Organization for Standardization (ISO) informally exchange notifications of meetings and the I C R U is formally designated for liaison with two of the ISO Technical Committees. The I C R U also corresponds and exchanges final reports with the following organizations:

Bureau International des Poids et Mesures Council for International Organizations of Medical Sciences Food and Agriculture Organization International Council of Scientific Unions International Electrotechnical Commission

International Labor Organization International Union of Pure and Applied Physics United Nations Educational, Scientific and Cultural Organization

Relations with these other international bodies do not affect the basic affiliation of the I C R U with the International Society of Radiology. The Commission has found its relationship with all of these organizations fruitful and of substantial benefit to the I C R U program.

Operating Funds

Throughout most of its existence, the I C R U has operated essentially on a voluntary basis, with the travel and operating costs being borne by the parent organizations of the participants. (Only token assistance was originally available from the International Society of Radiology.) Recognizing the impracticability of continuing this mode of operation on an indefinite basis, operating funds were sought from various sources in addition to those supplied by the International Society of Radiology.

Prior to 1959, the principal financial assistance to the I C R U had been provided by the Rockefeller Foundation which supplied some $11,000 to make possible various meetings. I n 1959 the International Society of Radiology increased its contribution to the Commission providing $3,000 for the period 1959-1962. For the periods 1962-1965 and 1965-1969 the Society's contributions were $5,000 and $7,500 respectively. I n 1960 the Rockefeller Foundation supplied an additional sum of some $4,000 making possible a meeting of the Quantity and Units Committee in 1960.

I n 1960 and 1961 the World Health Organization made available the sum of $3,000 each year. This was increased to $4,000 per year in 1962 and $6,000 per year in 1969. I t is expected that this sum will be allocated annually, at least for the next several years.

I n connection with the Commission's Joint Studies with the I C R P , the United Nations allocated the sum of $10,000 for the joint use of the two Commissions.

The most substantial contribution to the work of the I C R U has come from the Ford Foundation. I n December 1960, the Ford Foundation made available to the Commission the sum of $37,000 per year for a period of five years. This grant was to provide for such items as travel expenses to meetings, for secretarial services and other operating expenses. I n 1965 the Foundation agreed to a time extension of this grant making available for the period 1966-1970 the unused portion of the original grant. To a large extent, it is because of this grant that the Commission has been able to move forward actively with its program.

I n 1963 the International Atomic Energy Agency al-

v i · · · Preface

located the sum of $6,000 per year for use b y the I C R U . T h i s was increased to $9,000 per year i n 1967. I t is expected t h a t th is sum w i l l be allocated annual ly at least for the next several years.

F r o m 1934 through 1964 va luable indirect c o n t r i but ions were made b y the U.S . N a t i o n a l Bureau of Standards where the Secretariat resided. T h e Bureau prov ided substantial secretarial services, pub l i ca t i on services and t rave l costs i n t h e amount of several thousands of dollars.

T h e Commission wishes to express i ts deep appreciat i o n to a l l of these and other organizations t h a t have contr ibuted so i m p o r t a n t l y t o i t s w o r k .

C o m p o s i t i o n o f t h e I C R U

I t is of interest to note t h a t t h e membership of the Commission and i ts subgroups totals 140 persons d r a w n f r o m 16 countries. T h i s gives some ind i ca t i on of the extent to wh i ch the I C R U has achieved i n t e r n a t i o n a l breadth of membership w i t h i n i ts basic selection requirement of h igh technical competence of i n d i v i d u a l par t i c ipants .

T h e membership of the Commission d u r i n g the preparat ion of th is report was as fo l lows:

L . S. TAYLOR, Chairman M . TUBIANA, Vice Chairman H . O . WYCKOFF, Secretary A . ALLISY J . W . BOAG ( 1 9 6 5 - 1 9 6 6 ) R . H . CHAMBERLAIN F . P . COWAN F . E L L I S (1965) J . F . FOWLER H . FRANZ (1965) F . GAUWERKY J . R , GREENING Η . E . JOHNS ( 1 9 6 5 - 1 9 6 6 ) K . LIDFIN R . H . MORGAN V . A . PETROV ( 1 9 6 5 ) Η . H . ROSSI A . TSUYA

T h e current membership of the Commiss ion is as f o l l ows :

H . O . W Y C K O F F , Chairman A . A L L I S Y , Vice Chairman K . L I D E N , Secretary F . P . C O W A N F . G A U W E R K Y J . R . G R E E N I N G A . M . K E L L E R E R R . H . M O R G A N Η. H . Ross i W . K . SINCLAIR F . W . SPIERS A . TSUYA A . W A M B E R S I E

C o m p o s i t i o n o f I C R U S u b g r o u p s R e s p o n s i b l e for t h e I n i t i a l D r a f t i n g o f t h i s R e p o r t

Serving on t h e T a s k G r o u p on Dose as a F u n c t i o n of L E T d u r i n g the preparat i on of th i s repor t were:

W . K . SINCLAIR, Chairman P. R . J . B U R C H A . COLE D . V . CORMACK W . GROSS A . M . K E L L E R E R

Serving on the P l a n n i n g B o a r d o n R a d i a t i o n — H e a v y Particles d u r i n g t h a t t i m e were:

W . K . SINCLAIR, Chairman G. J . N E A R Y W . C. ROESCH

Η . H . Ross i served as Commiss ion Sponsor for the P lann ing B o a r d .

T h e Commission wishes to express i t s appreciation t o the ind iv idua ls i n v o l v e d i n the preparat ion of th i s report for the t i m e and effort t h e y devoted to this task .

HAROLD O. WYCKOFF Chairman, I C R U

Washington , D . C. January 15, 1970

Contents

Preface i i i 1. I n t r o d u c t i o n 1

1.1 R a d i a t i o n - I n d u c e d Changes and R a d i a t i o n Q u a l i t y 1 1.2 Specif ication of R a d i a t i o n Q u a l i t y : H i s t o r i c a l 1 1.3 C u r r e n t D e f i n i t i o n of L inear Energy Transfer ( L E T ) 2 1.4 L E T D i s t r i b u t i o n s and Averages 2 1.5 Other Methods of Specifying R a d i a t i o n Q u a l i t y 3 1.6 Scope of th i s Repor t 3

2. I n t e r a c t i o n o f R a d i a t i o n w i t h M a t t e r 3 2.1 General 3 2.2 Absorbed Dose, Part ic le Fluence and L E T 4 2.3 D e l t a Rays 4

3· D e f i n i t i o n a n d C o n c e p t s o f L E T 6 3.1 C u r r e n t D e f i n i t i o n of L E T 6 3.2 1962 D e f i n i t i o n of L E T and F u r t h e r Considerations 6 3.3 Concepts 6 3.4 L E T D i s t r i b u t i o n s and T h e i r Averages 8 3.5 Recommendations 8

4. C a l c u l a t i o n s of D i s t r i b u t i o n s of A b s o r b e d Dose i n L E T 8 4.1 I n t r o d u c t i o n 8 4.2 Calculat ions of Part ic le Fluence Spectrum Using the C o n t i n

uous Slowing D o w n A p p r o x i m a t i o n (csda) 9 4.3 Calculat ions of Absorbed Dose D i s t r i b u t i o n Based on a T w o -

Group M o d e l 10 4.4 L E T D i s t r i b u t i o n s i n Water 11 4.5 V a r i a t i o n of L E T w i t h D e p t h i n a M e d i u m 14 4.6 Average L E T 14

5. A p p l i c a t i o n s of L E T C a l c u l a t i o n s 15 5.1 General 15 5.2 General Appl icat ions , R B E vs. L E T 15 5.3 L i m i t a t i o n s of R B E vs. L E T Plots 15

6. L E T i n R a d i a t i o n P r o t e c t i o n 16 6.1 Q u a l i t y Factor 16 6.2 Measurement of Absorbed Dose D i s t r i b u t i o n i n L E T for Pro

tec t i on Purposes 17 7. L i m i t a t i o n s o f t h e L E T C o n c e p t 17 8. O t h e r M e t h o d s o f S p e c i f y i n g Q u a l i t y 19 9. C o n c l u s i o n s 19 A p p e n d i x A l . F o r m u l a e for S t o p p i n g P o w e r or L E T 21

A l . l General 21 A1.2 H e a v y Particles 22 A1.3 Electrons 22 A1.4 L o w Energy H e a v y Part ic le Stopping Power and

Ranges 22 A p p e n d i x A 2 . D e f i n i t i o n a n d M e a s u r e m e n t o f R a n g e s 24

A 2 . 1 Range and Stopping Power 24 vii

VÜi · · · Conients

A2.2 Semi -Empir i ca l Range-Energy Relations 25 A2.3 Exper imenta l Determinat i on of Ranges 25

A p p e n d i x A 3 . M e a s u r e m e n t of dE/dl 26 A 3 . 1 General 26 A3 .2 Energy D i s t r i b u t i o n M e t h o d — V e r y T h i n A b

sorber, ΔΙ ^ IQ 26 A3.3 M e a n Energy Loss—Very T h i n Absorber, AI ^ k 26 A3 .4 Energy A b s o r p t i o n — T h i n Absorber R» Μ > k 26 A3 .5 Slope of Energy versus Range Curve 26 A3 .6 T h i n Detectors 26

A p p e n d i x A 4 . T h e o r e t i c a l a n d E x p e r i m e n t a l V a l u e s for R a n g e , d £ / d U n d L E T 27 A 4 . 1 General 27 A4 .2 Electrons 28 A4.3 H e a v y Particles 31

A p p e n d i x A 5 . M e a s u r e m e n t o f L E T D i s t r i b u t i o n s 34 A p p e n d i x A 6 . D i s t r i b u t i o n of I o n s i n C l u s t e r s 35 A p p e n d i x A 7 . M e a n E x c i t a t i o n E n e r g y 36

A7 .1 General 36 A7.2 E m p i r i c a l Relat ions 36 A7.3 Recommended Values 36 A7.4 M i x t u r e s and Compounds 36

A p p e n d i x A 8 . A p p l i c a t i o n of L E T i n Radiobiology a n d C h e m i c a l D o s i m e t r y 37 A 8 . 1 D i r e c t and I n d i r e c t Ac t i on 37 A8 .2 Exponent ia l Absorbed Dose-Survival Curve 37 A8.3 Simple Target Theory 37 A8 .4 Target Theory and Inac t i va t i on b y M u l t i p l e I o n i

zations 38 A8.5 Target Theory , Complex Target 38 A8 .6 Ind i re c t A c t i o n and Appl i cat ion to Chemical D o

simeters 39 A p p e n d i x A 9 . L i s t o f S y m b o l s 40

A9 .1 General 40 A9.2 Absorbed Dose 40 A9.3 Energy 40 A9.4 Fluence 41 A9.5 I o n i z a t i o n 41 A9.6 L inear Energy Transfer 41 A9.7 Range 42 A9.8 Mic rodos imetry 42

R e f e r e n c e s 43 I C R U Reports 48 I n d e x 50

Linear Energy Transfer

1. I n t r o d u c t i o n

1.1 R a d i a t i o n - I n d u c e d C h a n g e s a n d R a d i a t i o n Q u a l i t y

I o n i z i n g r a d i a t i o n can induce many physical , chemic a l and biological changes. T h e k i n d and the extent of •change o f ten depend on the physical conditions of i r r a d i a t i o n . Foremost among such conditions is the •energy dissipated per u n i t mass (absorbed dose) i n the regions of interest . However , the q u a l i t y of the radia t i o n and the temporal d i s t r i b u t i o n of the transferred energy sometimes exert a profound influence. A l t h o u g h factors such as absorbed dose rate and absorbed dose f r a c t i o n a t i o n can be most i m p o r t a n t , especially i n biological systems, these temporal aspects w i l l not concern us here.

The subject of th is report is radiat ion q u a l i t y . The t e r m quality1 i n this report refers to those features of the spatial d i s t r i b u t i o n of energy transfers—along and w i t h i n the tracks of par t i c l es—that influence the effectiveness of an i r r a d i a t i o n i n producing change, w h e n other physical factors such as t o t a l energy dissipated, absorbed dose, absorbed dose rate and absorbed dose f rac t ionat ion are kept constant. I n this report part i cu lar emphasis is given to the description of q u a l i t y i n terms of linear energy transfer ( L E T ) . M a n y rad ia t ion- induced phenomena (such as in t ra - t rack ion comb i n a t i o n , l i g h t emission f r o m organic and inorganic sc int i l lators , chemical y ie ld , gene m u t a t i o n and cell k i l l i n g ) depend on the spatial d i s t r ibut i on of discrete energy transfers f r o m the ioniz ing partic le to the i r radiated medium. I n some systems a large number of energy transfers per u n i t l ength of t rack of a partic le favors a h igh yield of one product b u t a low y i e l d of another.

N o single in terpretat ion of the influence of L E T on radiat ion- induced change has yet been given t h a t is

1 A l t h o u g h s t r i c t l y the t e r m q u a l i t y refers to the r a d i a t i o n o n l y , independently of the m e d i u m i r r a d i a t e d , the d i s t r i b u t i o n of events produced i n a m e d i u m can also be used to describe r a d i a t i o n q u a l i t y (as w e l l as the i r r a d i a t i o n circumstances) . A precedent exists i n the use of H a l f Value Layer i n a given m a t e r i a l for q u a l i t y specification ( I C R U , 1962b).

v a l i d for a l l circumstances. I n cer ta in S3Tstems m u l t i p l e energy transfers w i t h i n a g iven smal l target region or regions m a y be needed t o effect change. I n others, t h e immediate physical or chemical products of an i r r a d i a t i o n may in terac t w i t h one another , along or close to the t r a c k of the i on iz ing part i c le . W e m a y be concerned w i t h the u l t i m a t e y i e l d of products t h a t either escape f r o m , or are produced b y , the i n t r a - t r a c k interact ions . W e m a y also be interested i n t h e physical , chemical , or biological effects produced b y such i n t r a - t r a c k products . A theory of r a d i a t i o n ac t i on i n any g iven system must be able t o expla in the different effects produced by radiat ions of d i f ferent q u a l i t y .

I n some circumstances, w h e n the effect of a g iven absorbed dose of one t y p e of r a d i a t i o n is k n o w n , we m a y wish to predic t the effect of a s imi lar absorbed dose of a different t y p e of r a d i a t i o n . T o do th i s , the q u a l i t y aspects of b o t h i r r a d i a t i o n s m u s t f i r s t be described i n q u a n t i t a t i v e terms. A complete descr ipt ion w o u l d l i s t the spat ia l and t e m p o r a l coordinates of every act ive product i n the system, t h r o u g h o u t the i r r a d i a t i o n , and throughout the subsequent per iod d u r i n g w h i c h change can be effected. However , t h e stochastic features of t h e in terac t i on of r a d i a t i o n w i t h m a t t e r alone p r o h i b i t any such exhaustive and u n i q u e descr ipt ion . T h e prac t i ca l problem, therefore, is t o find a convenient, b u t i n ev i tab ly incomplete , character izat ion of r a d i a t i o n q u a l i t y t h a t w i l l enable predict ions t o be made w i t h sufficient accuracy for the purpose i n question. A re la t i v e l y crude account of r a d i a t i o n q u a l i t y may be adequate for some purposes, for example i n rad ia t i on p r o tect ion , where o f ten even absorbed dose need n o t be accurately assessed. I n other appl icat ions, such as chemical dos imet iy , a r e l a t i v e l y detai led descr ipt ion becomes ob l igatory .

1.2 S p e c i f i c a t i o n o f R a d i a t i o n Q u a l i t y : H i s t o r i c a l

Lea (1946) ca lculated a n d t a b u l a t e d the p r i m a r y ionizat ion densities, s topp ing powers and the spectra of

1

2 · · · 7. Introduction

secondary or delta t racks produced i n water b y electrons (o f energy 100 eV t o 384 k e V ) , protons ( 1 M e V t o 10 M e V ) , and alpha particles ( 1 M e V t o 10 M e V ) .

G r a y (1947) in t roduced t h e parameter mean linear ion density, wh i ch , for x , 7, and β rays could be defined as fo l lows:

M e a n linear i on dens i ty

EQ

~ REQ-W

where E 0 = average i n i t i a l k i n e t i c energjr of p r i m a r y

electrons R EQ = range of electrons of energy EQ

W = average energy expended per i o n pair formed i n a gas.

Cormack and Johns (1952) calculated complete d i s t r ibut i ons of electron fluence as a f u n c t i o n of l inear i o n density , and used a more r igorous averaging p r o cedure for mean l inear i o n densi ty , i n w h i c h the mean va lue is obtained b y d i v i d i n g the t o t a l number of i o n pairs per c m 3 b y the t o t a l l e n g t h of t h e electron tracks per c m 3 . The mean values obta ined using th is averaging procedure were about 3 0 % lower for l ow L E T rad ia t ions t h a n those calculated b y G r a y (1947) .

I o n i z a t i o n is d i f f i cu l t or impossible t o measure or even to define i n l iqu ids a n d solids, and other types of energy transfer, n o t a b l y exc i ta t ion , can also lead t o radiat ion- induced change. Z i r k l e et a l . (1952) i n t r o duced the concept of linear energy transfer ( L E T ) , f o r m e r l y called l inear energy absorpt ion b y Z i rk l e (1940) . T h i s refers t o t h e l inear density of a l l forms of energy transfer i n c l u d i n g exc i ta t i on and ionizat ion.

B u r c h considered t h e problems raised b y delta t r a c k f o r m a t i o n , and b y t h e v a r i a t i o n i n the l inear density of energy transfers along the t r a c k of the decelerating i on iz ing part ic le . H e de termined the d i s t r i b u t i o n of absorbed energy i n L E T (see Sec. 1.4) b y calculat ing the f rac t i on of t o t a l energy deposited w i t h i n each L E T i n t e r v a l and defined an energy-weighted mean L E T for th i s d i s t r i b u t i o n ( B u r c h and B i r d , 1956; B u r c h , 1957a, b ) . [Energy transfers t o electrons i n excess of 100 eV were regarded b y B u r c h ( a n d earlier b y Lea, 1947) as cons t i tu t ing separate, independent (or de l ta) tracks. ]

1.3 C u r r e n t D e f i n i t i o n o f L i n e a r E n e r g y T r a n s f e r ( L E T )

L E T has been defined i n terms of local energy t rans fers. U n f o r t u n a t e l y local has h a d var ious connotations and recent I C R U def ini t ions (no te modi f i cat ion f r o m

1962 to 1968) have sought to avoid confusion. T h e m a t t e r is discussed i n more detai l i n Section 3.4.

T h e fo l lowing is the most recent def init ion g iven b y I C R U (1968):

The linear energy transfer or restricted linear collision stopping power (LA) of charged particles i n a med ium is the quot ient of dE b y dl, where dl is the distance traversed b y t h e particle and dE is the mean energy-loss due to collisions w i t h energy transfers less t h a n some specified value Δ.

- G& N O T E : A l t h o u g h the def ini t ion specifies an energy cut-off and no t a range cut-off, the energy losses are sometimes called "energy locally i m p a r t e d " .

B y th is def init ion, L100, for example, designates the L E T when Δ = 100 eV. The symbol L „ is used when a l l possible energy transfers are included, i n accordance w i t h previous usage ( R B E Committee , 1963). However , the subscript co should not be taken to mean t h a t in f in i t e ly large transfers of energy are possible. The m a x i m u m energy transfer ( Q m a x ) is governed by the type and velocity of the inc ident part ic le and w i l l be discussed i n Appendix 1.

I n this report, the symbol L w i l l designate the L E T w i t h o u t reference to any part icular value of Δ.

1.4 L E T D i s t r i b u t i o n s a n d Ave rages

The " R e p o r t of the R B E C o m m i t t e e " (1963) discusses two types of L E T d is t r ibut ion . I n one type , i.e. t ( L ) , t(L)d.L represents the fract ion of t o t a l t r a c k length, T , hav ing values of L E T between L and L + d L . 2 Thus , i f T ( L ) is defined as the t r a c k l ength associated w i t h L E T up to L , d iv ided by t o t a l t r a c k length, Ί\ t ( L ) = d T ( L ) / d L . I n the other, i.e. d(L), d(L)dL represents the fract ion of the absorbed dose, D , delivered between L and L + d L . T h u s i f D(L) is t h a t p a r t of the absorbed dose w i t h L E T up to L , d iv ided by the t o t a l absorbed dose, D , d ( L ) = d D ( L ) / d L . Thus t ( L ) and d(L) are the d istr ibut ions of t rack length and absorbed close i n L E T respectively. 3 Associated w i t h the first d i s t r i b u t i o n is a t rack average L T and w i t h the second, an

2 The f rac t i on of the t r a c k length w i t h L E T between L and L + dL is equal to the p r o b a b i l i t y t h a t the part ic le is f o u n d w i t h L E T between L and L + d L . Th i s alternate i n t e r p r e t a t i o n m a y be preferable i n instances where the concept of a t rack is considered to be i l l -def ined.

3 S t r i c t l y , absorbed dose d i s t r i b u t i o n s of L E T should be described as absorbed energy d i s t r ibut i ons of L E T as these

2. Interaction of Radiation with Matter · · · 3

absorbed dose average, L D . Unless there is only a single v a l u e of L , these two averages have d is t inct ly di f ferent values for the same circumstances because in the one equal weight is assigned to each u n i t of t r a c k l e n g t h , whi le i n the other equal weight is assigned t o each u n i t of energy deposited along the track. T h e mean l inear i o n density of Gray (1947) and of Cormack a n d Johns (1952) corresponded to a t rack average L E T , whi le Burch 's calculations determined an absorbed dose average L E T . The differences in values of L D for dif ferent types of rad iat ion are usually n o t as large as for L T . T h e relationship between these d i s t r i b u t i o n s , t ( L ) and d(L), their averages and the effect of an energy cut-off Δ w i l l be discussed i n Section 3.-1.

L inear energ}* transfer d istr ibut ions have been calculated b} r various authors for many radiations i n common use (Boag , 1954; B u r c h , 1957a, b ; H o w a r d -Flanders , 1958; Danzker , Kessaris, and L a u g h l i n , 1959; H a y n e s and D o l p h i n , 1959; Cormack, 1956; Bruce , Pearson and Freedhoff, 1963; Snyder, 1964; Lawson and W a t t , 1967; Bewley, 1968a and 1968b). Examples of such d is tr ibut ions are shown i n Section 4.

1.5 O t h e r M e t h o d s of Spec i fy ing R a d i a t i o n Q u a l i t y

A l t h o u g h L E T distr ibut ions can be calculated for m a n y i on iz ing particles, i t is di f f icult to measure L E T d i s t r ibut i ons . Furthermore , the concept of L E T has l i m i t a t i o n s ; these are discussed i n Section 7. Rossi (1959, 1964, 1966, 1967) introduced the concepts, local energy density ( Z ) , incremental local energy density ( Δ Ζ ) and i n d i v i d u a l event size ( F ) which overcome some of these difficulties and also make i t

possible to relate the energy deposition to the size of any s tructure w h i c h m a y be thought relevant.

T h e local energy density, Z , has been defined as the energy dissipated i n a smal l sphere d iv ided b y i t s mass (Rossi , 1966, 1967) . T h e probab i l i t y d i s t r i b u t i o n of t h e local energy densi ty , ' P ( Z ) , can be determined for di f ferent radiat ions and the f o r m of the d i s t r i b u t i o n depends on the absorbed dose, Z>, to the i r r a d i a t e d m e d i u m and on the d iameter of the test sphere. T h e mean value of Ζ is equal t o D .

T h e energy dissipated b y i n d i v i d u a l events d i v i d e d by t h e mass of the content of the test sphere is called t h e incremental local energy density ΔΖ. I t can be shown (Rossi, 1966, 1967) t h a t Δ Ζ = Κ Y/d2 where Y, t h e individual event size, is defined as the energy expended i n the event d iv ided b y d, the diameter of t h e test sphere. Κ is a constant = β /π i f the u n i t s used are coherent. H o w e v e r the uni ts used for Y and ΔΖ are often n o t coherent and then the value of Κ depends upon t h e m . These concepts are discussed f u r t h e r i n Section 8.

1.6 S c o p e o f t h i s R e p o r t

T h e progress t h a t has been made i n specifying r a d i a t i o n q u a l i t y i n t e rms of L E T is described here, and some of the o u t s t a n d i n g problems remaining are discussed. Descr ipt ions are given of the physical and theoret i ca l premises on w h i c h calculations of L E T and L E T d i s t r ibut i ons , are based. Examples of L E T d i s t r ibut i ons are g iven . Examples of applications of average L E T and L E T d i s t r ibut i ons to pract i ca l a n d theoret i ca l problems are described and the l i m i t a t i o n s of such procedures are discussed. A l t e r n a t i v e methods of specifying r a d i a t i o n q u a l i t y are brief ly considered.

2. Interact ion of R a d i a t i o n w i t h M a t t e r

2.1 G e n e r a l

Energetic charged particles lose energy i n traversing a m e d i u m m a i n l y b y processes of electronic excitation i n w h i c h an o r b i t a l electron is raised to a higher energy level , and b y ionization i n whi ch an orb i ta l electron is ejected. Energy losses by other processes are less i m p o r t a n t , except for radiat ion losses f r om very

d i s t r i b u t i o n s are not necessarily related to mass. I n accordance •with past pract i ce , however, the t e rm absorbed dose d i s t r i b u t i o n i n L E T w i l l be used throughout .

energetic l i g h t part ic les , for nuclear interact ions b y v e r y energetic heavy partic les and , at very low speeds, for losses b y elastic collisions.

Energet ic photons such as χ rays and gamma rays lose energy m a i n l y b y three mechanisms: (a ) the p h o t o electric effect, i n w h i c h t h e t o t a l energy of the p h o t o n is expended i n the e ject ion of an o r b i t a l e lectron; ( b ) Comp t o n scatter ing, i n w h i c h a p a r t of the p h o t o n ejaergy is transferred t o an o r b i t a l electron; and ( c ) pa i r product ion , i n w h i c h the pho ton energy is conv e r t e d to the mass and k inet i c energy of an electron-

4 · · · 2. Interaction of Radiation with Matter

pos i t ron pa ir . T h u s the b u l k of the inc ident photon energy is expended i n the l i b e r a t i o n of energetic electrons ( a n d sometimes pos i trons) w h i c h t h e n lose energy t h r o u g h mechanisms of atomic exc i tat ion and i on izat ion .

Energet ic neutrons lose energy m a i n l y b y elastic coll ision processes w h i c h i m p a r t energy to the atomic nucle i of the m e d i u m . I n hydrogenous mater ia l the b u l k of the energy of the fast n e u t r o n is g iven to hydrogen t o produce p r o t o n recoils. Protons t h e n lose energy b y exc i ta t i on and i o n i z a t i o n processes. A t energies below a few k e V and above tens of M e V , neutrons i n t e r a c t w i t h m a t t e r p r i n c i p a l l y b y inelastic nuclear reactions. These processes m a y give rise t o b o t h heavy partic les and γ r a d i a t i o n .

A l l d i r e c t l y i on iz ing rad ia t i ons transfer most of the i r energy t o m a t t e r b y col l is ion processes i n v o l v i n g ion izat ion and e x c i t a t i o n ; these p r i m a r y events occur r a n d o m l y along t h e t racks of charged particles. T h e ra t i o of exc i ta t i on t o i on i za t i on energy losses and the re lat ive frequency of i on i za t i on clusters of dif ferent size are considered t o be near ly independent of the nature and energy of the p r i m a r y part ic le . Hence differences i n biological effectiveness of different ioniz ing radiat ions should be due m a i n t y t o differences i n the spatial distribution of the p r i m a r y events and not to differences i n the nature of the events themselves. T h e factors re levant to q u a l i t y are therefore the spacing of the p r i m a r y collisions and t h e frequency of the more energetic de l ta rays along the t r a c k of the d irect ly ioniz ing part i c le . These are discussed i n more detai l i n Section 2.3.

Because energy losses are r a n d o m i n nature , the physical quant i t i es p e r t a i n i n g t o a specific i r r a d i a t i o n should be described e i ther i n terms of mean values, or better , the i r p r o b a b i l i t y d i s t r ibut i ons . Procedures for a r r i v i n g at such descriptions are discussed i n Section 4.

2.2 A b s o r b e d D o s e , P a r t i c l e F l u e n c e , a n d L E T

T h e absorbed dose, D, is defined ( I C R U , 1968) as follows:

The absorbed dose (D) is the q u o t i e n t of AED

b y Am, where AED is the energy i m p a r t e d b y i on iz ing r a d i a t i o n t o t h e m a t t e r i n a vo lume element and Am is the mass of the mat te r i n t h a t v o l u m e element.

D = ^ 2.2.(1) Am

The particle fluence, Φ, i n t h e region of interest is defined as t h e q u o t i e n t of t h e number of particles

ΔΑΤ w h i c h enter a sphere, by i ts cross-sectional area Δα.

Φ = AN/ Aa 2.2.(2)

T h e fluence spectrum i n energy Φ{ΕΫ is def ined by

φ(Ε) = An{E)/Aa 2.2.(3)

where An(E)dE is the number of incident partic les w i t h energies between Ε and Ε + dE en ter ing Δα. One m a y fur ther specify an angular d i s t r i b u t i o n of fluence of a rad ia t i on f ield.

D i r e c t l y ioniz ing particles of k inet ic energy Ε w i l l transfer energy local ly to the med ium according to L = AE/Al where AE is the average energy t r a n s ferred when the part ic le moves through the distance ΔΖ.

L , the L inear Energy Transfer ( L E T ) , depends on the ve loc i ty , charge and mass of the part i c le . T h e part ic le fluence spectrum i n L E T is given b y

<t>(L) = An{L)/Aa 2.2.(4)

where An(L)dL is the number of particles w i t h L E T between L and L + dL which enter Δα. N o t e t h a t b y normal i z ing the spectrum <f>(L) one obtains the t r a c k l eng th d i s t r i b u t i o n i n L E T .

φ(Ζ,) 0 ( L ) An(L) (TS5 9 0 / r x M L ) d L = " φ " " = -AN- = t ( L ) 2 " ( o )

F r o m 2.2.(3) and 2.2.(4) and the fact t h a t An(L)dL = An(E)dE, the re lat ion between φ ( L ) and φ(Ε) is

* ( L ) = φ(Ε) (g) 1 2.2.(6)

2.3 D e l t a R a y s

T h e types of interact ion (excitat ion- ionizat ion) w h i c h occur along the tracks of i n d i v i d u a l ionizing particles are i l l u s t r a t e d i n Figure 1. T w o m a i n types may be dist inguished, (a ) a localized excitation or i on izat ion i n the t r a c k of the ionizing particle, ( b ) a larger energy transfer leading t o the ejection of an atomic electron of sufficient energy to produce further ioniz ing events. I n the l a t t e r case the energy transferred m a y be so

4 T h r o u g h o u t th i s repor t the di f ferent ial d i s t r i b u t i o n of one q u a n t i t y (A) w i t h respect to another (B) w i l l be w r i t t e n i n the f o r m A (B) r a t h e r t h a n i n the f o r m Ab to avoid complicat ions i n the use of a d d i t i o n a l suffixes. For example, the d i s t r i b u t i o n of fluence w i t h respect to L E T when an energy l i m i t Δ is i m posed, is w r i t t e n Φδ(ΙΪ) i n this report instead of φι^ wh i ch m i g h t have been chosen.

6 See f ootnote 2, p. 2.

2.3 Delta Rays · · · 5

l ow t h a t o n l y an i o n cluster of 2,3,4, etc. i on pairs is formed or i t m a y be large enough to produce a separate t r a c k k n o w n as a del ta ray . The d is t inc t ion between clusters a n d de l ta rays, a l though largely a r b i t r a r y , has been used t o construct models of t rack s t ruc ture (e.g., M o z u m d e r and Magee, 1966).

Tracks of heavy charged particles are essentially s tra ight a n d except at higher energies they are densely ioniz ing, i .e. the mean spacing between successive p r i m a r y collisions is very small . Single ions and i o n clusters a long the t r a c k constitute the t r a c k " core" . A l t h o u g h the m a x i m u m delta-ray energy is on ly a smal l f r a c t i o n of the energy of the p r i m a r y part i c le , the more energetic delta rays generated by heavy charged partic les m a y be clearly separated f r o m the t r a c k core because the i r range greatly exceeds the mean spacing of t h e p r i m a r y collisions i n the t rack core.

The s i t u a t i o n is different w i t h fast electrons. Ener getic de l ta rays can be formed w i t h a range comparable to the range of the p r i m a r y partic le . The m a x i m u m energy t h a t can be i m p a r t e d to a delta ray is hal f t h a t of the p r i m a r y electron. On the other h a n d the distances between successive pr imary collisions are often larger t h a n the range of the m a j o r i t y of the del ta rays. Therefore, the not ion of a t rack core d i s t inc t f r o m the de l ta ray has l i t t l e meaning for fast electrons. For slower electrons, inc lud ing delta rays, the s i tua t i on is also complicated by the fact t h a t the tracks are devious (see Figure 1 ) .

The p r o b a b i l i t y per scattering center per u n i t area t h a t a charged part ic le of energy Ε w i l l undergo an interact ion i n v o l v i n g a given energy transfer, Q, is expressed by the collision cross section, a ( Q ) . Classical collision theory indicates t h a t the probab i l i t y of an

Incident Particle

X

Single Ionizations (or Excitations)

• J>- Clusters

F i g . 1 . D i a g r a m m a t i c representation of the t r a c k of an ioniz ing part i c le i n m a t t e r .

1 τ ι 1 1 Γ

Electron Energy Loss/eV F i g . 2 . T h e single co l l i s i on energy loss d i s t r i b u t i o n f o r 20

k e V electrons passing t h r o u g h a layer of F o r m v a r 13 n m t h i c k . The percentage of the i n e l a s t i c a l l y s ca t tered electrons per 10-eV energy loss i n t e r v a l is p l o t t e d against t h e energy loss ( R a u t h and Simpson, 1964). [ B y courtesy of the authors and R a d i a t i o n Research ( c o p y r i g h t he ld b y Academic Press).]

energy transfer Q is p r o p o r t i o n a l t o 1/Q2 and the recoil angle of the de l ta ray is

[Ε 4ra 0 Mo J

where Μ0 is the mass of the inc ident part i c le and m0

is the mass of the s t r u c k part i c l e . A s the value of the cosine of the recoi l angle ranges f r o m 0 t o 1, the recoil angle itself w i l l range f r o m 90° t o 0°. T h e m a x i m u m delta ray energy is [ 4 m 0 i l f 0 / ( m 0 + Mo)2] E, except for collisions i n v o l v i n g ident i ca l part ic les , e.g. negative electrons, for w h i c h i t is E/2.

These simple re lat ionships h o l d on ly w h e n the t w o particles are considered u n b o u n d a n d w*hen no q u a n t u m mechanical , r e l a t i v i s t i c , or m u l t i - b o d y kinet ics are invo lved .

Lea (1946) calculated t h e de l ta - ray energ\r d i s t r i b u t i o n of di f ferent i on i z ing part ic les on t h e basis of the 1/Q2 dependence. Theore t i ca l considerations and exper imenta l evidence suppor t t h i s re lat ionship for Q > 200 eV, i n l ow atomic n u m b e r media. However theoret ical considerations i m p l y t h a t t h e relationship cannot h o l d for smaller values of Q a n d t h a t the d i s t r i b u t i o n is considerably steeper a t energies below a few hundred eV. T h u s the shape of the spectrum at l ow energies depends on t h e k i n e t i c energy of the ioniz ing part i c le . U s i n g Bethe 's (1933) theory , Walske (1952, 1956) has der ived theoret i ca l re lat ions for the collision spectra produced b y energetic particles inter act ing w i t h K - and L-she l l electrons; numer i ca l evaluat ions have been per formed b y Bichsel (1968 ) . Choi and Merzbacher (1969) have t r ea ted t h e p r o b l e m numer i cal ly for protons.

6 · 3. Definition and Concepts of LET

Exper imenta l data on the shape of the energy-loss spectrum are scarce. C loud chamber studies have been made by Alper (1932) . Exper iments have been conducted by R u t h e m a n n (1948) and by R a u t h and Simpson (1964) on the d i s t r i b u t i o n of energy losses f r o m low energy electrons i n t h i n plastic foils. R u t h e -m a n n used 5 keV electrons and col lodion foils and he f ound t h a t collision losses of about 25 eV had the highest frequency. A similar peak was ev ident i n early seconda r y electron emission w o r k (e.g., Rudberg 1930, 1931) and has been observed i n m a n y other materials ( for a review see M a r t o n et a l , 1955). A n example f r o m t h e w o r k of R a u t h and Simpson, w h o used 20 keV elec

trons and F o r m v a r foils, is shown i n F igure 2. T h e d i s t r i b u t i o n m a x i m u m occurred at 22 cV and an average energy loss of about 60 eV was determined .

T h e simple classical relationships for angular dist r i b u t i o n of delta rays are certainty n o t v a l i d for moderate or small delta ray energies. I n classical theory each recoil angle or related scattering angle has a unique value of AE associated w i t h i t . However , R a u t h (1962) observed t h a t a broad d i s t r i b u t i o n of energy loss values was associated w i t h a l l scatter ing angles between 0 and 50°. Detai ls are avai lable i n R a u t h (1962) and R a u t h and Simpson (1964) .

3· Def init ion a n d Concepts of L E T

3.1 C u r r e n t D e f i n i t i o n of L E T

T h e current def in i t ion of L E T b y I C R U ( I C R U , 1968) is quoted on page 2, Section 1.3 of th is report . T h i s def init ion was modif ied f r o m the previous def ini t i o n ( I C R U , 1962a) as a result of a recommendation developed by the Task Group responsible for th is report . The background of the current de f in i t ion is relevant to an understanding of the concept of L E T .

3.2 1962 D e f i n i t i o n of L E T a n d F u r t h e r C o n s i d e r a t i o n s

Repor t 10a of I C R U ( I C R U , 1962a) defined L E T as fo l lows:

T h e linear energ^y transfer (L) of charged particles i n a m e d i u m is the quot i ent of dEL

b y dl where dEL is the average energy locally i m p a r t e d to the m e d i u m b y a charged par t ic le of specified energ}^ i n travers ing a distance dl.

T dEL

T h e t e r m " l oca l ly i m p a r t e d " m a y refer either to a m a x i m u m distance f r o m t h e (part i c le ) t rack or to a m a x i m u m value of discrete energy loss by the part ic le beyond w h i c h losses are no longer considered as local. I n either case, the l i m i t s chosen should be specified.

T h e report of the R B E C o m m i t t e e of the I n t e r n a t i ona l Commissions on Radio logical Protect ion and

on Radiological U n i t s and Measurements (1963) p r o v i d e d add i t i ona l in format ion . " . . . Var ious cut-off levels of energy have been selected to separate delta ray tracks f r o m clusters, and i t is l i k e l y t h a t different cut-off levels are appropriate for dif ferent reactions. I t is, therefore, suggested t h a t the cut-off level be indicated b y a subscript, e.g. L E T 1 0 o w o u l d be an L E T obtained when tracks due to secondary particles w i t h energy of 100 eV or more are counted as separate tracks. The simplest parameter to use is the LEToo , defined as the energy loss per u n i t distance of the charged particles or iginal ly set i n m o t i o n b y electromagnetic radiat ion or neutrons, or of the charged particles w h i c h originate i n rad iat ion sources (a-rays, ß-rays, e tc . ) . LEToo is the same as " s t opp ing power" . . .

A discussion of the basic concepts and the l imi ta t i ons of these earlier definitions follows.

3.3 C o n c e p t s

Consider the possible types of energy loss b y charged particles of specified energy, E, which are incident n o r m a l l y on an absorber of thickness AL I t is assumed t h a t Al is sufficiently t h i n so t h a t mul t ip l e scattering events can be neglected. A particle loses energy AE at a discrete site, is deflected at an angle 0, and passes out of the absorber w i t h energy E'. AS i l lustrated i n F igure 3, the several types of energy loss may be characterized by the fo l lowing:

Ο represents a partic le traversal w i t h no energy inter-change

U is the energy transferred to a localized in te r act ion site

3.3 Concepts 7

q is the energy transferred to a short range seco n d a r y part ic le for which q < Δ where

Δ is a selected energy cut-off level Q' is the energy transferred to a long range second

a r y part i c le for wh i ch Q* > Δ 7 is the energy transferred to photons ( u p to a

m a x i m u m equal to E ) r is a geometric cut-off distance f r o m the part ic le

t r a c k θ is the scattering angle of the incident part ic le

T h e interac t ions q, Q'', and 7 are further d iv ided i n t o three c o m p a r t m e n t s : compartment 1 represents energy spent 6 w i t h i n b o t h Al and a cylinder of radius r sur round i n g t h e part ic le t rack , compartment 3 represents energy spent outside Al b u t w i t h i n the cyl inder surr o u n d i n g the t rack , and compartment 2 represents energy spent outside the cylinder.

stopping power dl is replaced by pdl, wh i ch is mass per u n i t area.

W e now t u r n t o the problem of defining L E T w i t h either energy or distance cut-off l i m i t s imposed. L E T A (or L A ) is defined as t h a t par t of the t o t a l s topping power, dE/dl, wh i ch is associated w i t h exc i tat ion- ionizat ion energy transfers up to a cut-off value Δ . F r o m the classification of energy losses th i s can be seen t o be

LA = —r or Al ( - ) \dl JA

3.3.(2)

where AEA represents the sum of the energies expended i n categories i n 0 , U, and q only d iv ided b y the t o t a l number of inc ident particles. LA is the same as t h e " res tr i c ted s topping power" and can be readi ly calculated for a wide range of energies.

ABSORBER

F i g . 3. D i a g r a m of the passage of partic les of energy Ε t h r o u g h a thickness Al of m a t e r i a l i l l u s t r a t i n g the several types of energy loss t h a t may occur.

D i s t r i b u t i o n s i n the various energy compartments are generated as the result of many inc ident charged particles of energy Ε traversing the absorber. A E represents the sum of the various energies expended i n categories 0 , U, qi, q2, Qi, Q2, Qz\ 7 1 , 7 2 , and 7 3 , d iv ided by the t o t a l number of incident particles. T h e l inear stopping power of the absorber is then defined as

3.3.(1)

Hence dE/dl is an average rate of energy' loss and the distance travel led is dl. I n the case of the mass

6 N o t e t h a t contr ibut ions f r o m 7 rays are not inc luded i n the de f in i t i on of s topping power as explained i n A p p e n d i x A l , page 21 .

7 The convention ( I C R U , 1968) of t r ea t ing energy losses as pos i t ive is adopted here.

L E T r (or L R ) is defined as t h a t p a r t of the t o t a l energy loss dE/dl w h i c h is deposited w i t h i n a cy l inder of radius r and l e n g t h Al, centered along the part i c l e t rack .

F r o m figure 3, th i s is seen to be

L T = 3.3.(3) Al

where AEr represents an average for the contr ibut ions i n 0 , U, qi, Qi, and 7 1 . However , on the average, compensation w i l l occur and the energy por t i ons , Qz and 73 w h i c h are expended outside Al b u t w i t h i n the cyl inder w i l l be compensated b y s imi lar t racks or ig inat ing outside Al; hence Qz and 73 should also be included i n AEr.

I t is evident f r o m the previous discussion t h a t LA is easy to evaluate analyt i ca l ly b u t di f f i cult to measure

8 · · · 4. Calculation of Distribution of Absorbed Dose in LET

d i r e c t l y ; whereas L r is di f f icult to evaluate analyt i ca l ly , b u t can be measured i n principle b y replacing the cyl inder w i t h a suitable measuring device such as a (mi c ro ) dosimeter.

3.4 L E T D i s t r i b u t i o n s a n d T h e i r Averages

A s a fast charged partic le loses energy i n i t s passage t h r o u g h an absorber, the value of L w i l l also change. T h e resultant d i s t r i b u t i o n i n L can be expressed i n t w o ways, the t r a c k length d i s t r i b u t i o n , t ( L ) and the absorbed dose d i s t r i b u t i o n , d ( L ) , (see Section 1.4). T h e two d i s t r i b u t i o n functions are re lated as follows

d(L) = k ^ l 3.4.(1) L T

and are each normalized to u n i t y . T h e integra l forms of these d istr ibut ions are denoted b y

T ( L ) = f t(L)dL and D ( L ) = Γ d(L)dL

respectively. The mean L E T associated w i t h the t rack d i s t r i b u t i o n is the t r a c k average L E T , L T , where

Λ 00

LT = / t ( L ) T A L 3.4.(2)

T h e mean L E T associated w i t h the absorbed dose d i s t r i b u t i o n is the absorbed dose average L E T , L D , where

Λ CO

L D = / d(L)LdL 3.4.(3)

As discussed i n Section 3.3, LA represents the L E T calculated when energy transfers above an energy Δ ( i n eV or k e V ) are considered to generate separate tracks . T h e t r a c k length and absorbed dose d i s t r i b u t ions corresponding to values of L E T restricted i n th is

way are t a ( L ) and e fo (L) , and i n the i n t e g r a l f o r m , TA(L) and DA(L). LA,D and LA,τ represent t h e absorbed dose and t r a c k averages of these d i s t r ibut i ons . T h e d i s t r ibut ions and averages when Δ is equal to the m a x i m u m delta r a y energy are designated tw(L) and dUL) and L ^ , T and L ^ , D . F u r t h e r discussion of these modif ied d i s t r ibut i ons of L E T and their averages is presented i n Section 4.

3.5 R e c o m m e n d a t i o n s

T h e use of the energy cut-off f o r m of L E T w h i c h can be evaluated i n a s t ra ight - f o rward manner using restr icted s topping power formulae, is recommended when a restr icted f o r m of L E T is desired. T h u s the l inear energy transfer, LA , is defined as t h a t p a r t of the t o t a l l inear energy loss of a charged part ic le wh i ch is due to energy transfers up to a specified energy cut-off value, Δ . T h i s de f in i t ion corresponds t o t h a t for restr icted s topping power.

Loo signifies the value of l inear energy transfer whi ch includes a l l energy losses up to the m a x i m u m allowed and is therefore numerica l ly equal to the t o t a l mass stopping power. T h e subscript oo is used for convenience and to conform w i t h recent usage b u t i t should no t be taken to mean t h a t an a r b i t r a r i l y h i g h energy transfer could occur.

L r is an interest ing physical q u a n t i t y of po tent ia l significance. However the question of whether L r or LA is of more significance or usefulness i n the evaluat i o n of rad ia t i on effects w i l l n o t be discussed fur ther here.

T h e most recent I C R U def in i t ion of L E T , ( I C R U , 1968), see page 2, is i n accord w i t h this discussion.

A special prob lem arises i n the definit ion of the L E T of l ow energy electrons where the t o t a l p a t h or penet r a t i o n l ength is comparable t o the cut-off distance, r. T h i s p o i n t is discussed i n Section A4.2.2.

4· C a l c u l a t i o n of D i s t r i b u t i o n of Absorbed Dose 8 i n L E T

4.1 I n t r o d u c t i o n

E q u a t i o n 2.2.(3) defines the part i c le fluence spect r u m i n part ic le energy. However , w h a t is usually of more interest i n rad ia t i on biology is the d i s t r i b u t i o n of the fluence, no t i n kinet ic energy b u t i n L E T .

8 See footnote 3, page 2.

Because L E T is a unique f u n c t i o n of kinetic energy for a given t y p e of part i c le , the fluence d i s t r ibut i on i n k inet i c energy m a y be converted direct ly to a d i s t r i b u t i o n i n L E T . T h e absorbed dose delivered by par ticles w i t h a g iven k inet i c energy (or L E T ) may be f ound b y m u l t i p l y i n g the part ic le fluence by the corresponding L E T .

A fluence spectrum m a y be calculated b} r means of

4,2 Calculations o f Particle Fluence Spectrum · · · 9

I0 ;

I0 4

21 ο

ΙΟ3

; - ι ι ι ι ι ι ι ι J ι ι ι ι ι ι 11

\ 50 kV Χ Rays

\ \ / \

I I I I 1 I I I 1 I I I I I I L

\ Ιο*

I I

I I

I I

I ι

200 kV Χ Ray

: ί

ι ι ι ι ι ι ι ι 1 ι ι ι I ι ι ι ι

J \ ι

I ι I I I I I Μ

° v co i

ι ι ι ι ι I I ι

0.1 I 10

ELECTRON KINETIC ENERGY;

100 1000 ε

keV

F i g . 4. E l e c t r o n fluence d i s t r i b u t i o n s i n k i n e t i c energy f or water f o r var ious r a d i a t i o n s . The electron fluence d i s t r i b u t i o n f or 50 k V χ rays is based on the x - r a y spec t rum g iven b y B u r k e and P e t t i t (1960). T h e e lectron fluence d i s t r i b u t i o n for 200 k V x rays was deduced f r o m d a t a of B u r c h (1957b). The fluence d i s t r i b u t i o n s inc lude the de l ta rays or knock -on electrons.

the continuous-slowing-down approximation (csda) i n w h i c h charged particles are assumed to lose energy cont inuously along the ir paths according to the l inear energy transfer, L . Such a model ignores the discrete nature of energy decrements removed f r o m the part ic le t r a c k by delta rays. As indicated i n the previous sections a better approx imat ion to the real physical s i tuat i on is obtained b y a p p l y i n g a " t w o - g r o u p m o d e l " i n w h i c h the collisions are d iv ided i n t o t w o groups b y an energy cut-off, Δ . T h i s model uti l izes the t o t a l part i c le fluence inc luding t h a t of the delta rays whereas, i n calculations based on the continuous-slowing-down approx imat ion , the fluence of p r i m a r y particles only is needed. Furthermore , i n ca lculat ing the d i s t r i b u t i o n of absorbed dose f r o m the fluence b y the two-group model , one uses the appropriate restr icted stopping power, (dE/dl)A rather t h a n the t o t a l s topping power, dE/dl

4.2 C a l c u l a t i o n s of P a r t i c l e F l u e n c e S p e c t r u m U s i n g t h e C o n t i n u o u s - S l o w i n g - D o w n

A p p r o x i m a t i o n (csda)

Suppose t h a t charged particles are emi t t ed u n i f o r m l y i n a homogeneous m e d i u m such t h a t the number of particles w i t h i n i t i a l k inet i c energies E0 to E0+dEo per u n i t volume is n(E0)dEQ. Us ing the continuous-slowing-down approx imat ion , the charged partic le fluence spectrum φ(Ε) (number of particles per u n i t

area per u n i t i n t e r v a l of E ) at a po in t of the med ium, under conditions of rad ia t i on e q u i l i b r i u m , 9 is

φ(Ε) = L ~ l f ?i(E0)dEo 4.2.(1)

T h e fluence d i s t r i b u t i o n as a funct ion of ma} ' then be found f r o m :

* » ( L ) = φ(Ε) 4.2.(2)

T h e d is tr ibut ions of absorbed dose as a func t i on of part ic le k inet ic energy, d^(E), and of L E T , dcc(L), are given b y

d»(E) = Leo Φ(Ε)/ΡΌ 4.2.(3)

and

dco(L) = Leo 4>*(L)/PD 4.2.(4)

where the absorbed dose D = ( 1 / p ) / L x φ(Ε)άΕ. Jo

Calculations of fluence and absorbed dose d i s t r i b u t i o n based on the csda have been made b y a number of authors ( for example Cormack and Johns, 1952; Boag, 1954; Johns et a l . , 1954; Danzker et a l , 1959; B i a v a t i et a l . , 1963; Snyder, 1964; Bewley, 1968a).

9 The p o i n t is assumed to be surrounded on a l l sides b y a thickness of m e d i u m greater t h a n the m a x i m u m range of the charged part ic les .

1 Ο · · · 4. Calculation of Distribution of Absorbed Dose in LET

or

ο

< ζ: ο ΙΟ < or

30 50

LINEAR ENERGY TRANSFER; • keV^nf

F i g . 5. D i s t r i b u t i o n of absorbed dose i n L E T for water for the electrons set i n m o t i o n by 6 0 Co gamma rays . The L E T values for various electron energies used i n F igs . 5 to 10 were t a k e n f r o m Table A l .

4.3 C a l c u l a t i o n s of A b s o r b e d D o s e D i s t r i b u t i o n s B a s e d o n a T w o - G r o u p M o d e l

T h e l imi ta t i ons of the csda model i n b o t h c a v i t y -ionizat ion theory and i n ca lculat ing the absorbed dose d i s t r ibut i on i n L E T have been po inted out b y a number of workers ( B u r c h , 1955, 1957a, b, c; Spencer and A t t i x , 1955; N C R P , 1961). T h e two-group model described above has been used i n t w o wa tys to calculate absorbed dose d istr ibut ions da(E) or da(L). I n the first of these, developed by B u r c h (1955, 1957b), the particles are fol lowed f r o m the highest energy down t o the cut-off energy, Δ , i n a step-wise manner, the d i s t r i b u t i o n of absorbed dose among a number of k inet i c energy intervals being calculated f r o m stopping power theory. The second method (Spencer and A t t i x , 1955) proceeds v i a the calculation of the t o t a l part ic le fluence, φ{Ε), ( inc lud ing delta rays ) f r o m whi ch the absorbed dose d i s t r i b u t i o n da(E) may be found b y m u l t i p l y i n g φ(Ε) b y LA .

4.3.1 C a l c u l a t i o n s U s i n g t h e S t e p - W i s e M e t h o d

As i n Section 4.2, i t is assumed t h a t charged particles are generated u n i f o r m l y i n the v o l u m e of m e d i u m surrounding the p o i n t or small region of the m e d i u m a n d t h a t the number per u n i t v o l u m e and u n i t k inet ic

energy is 7i(E0). I n slowing down, these part ic les w i l l transfer most of the ir kinetic energy to the m e d i u m i n the f o r m of excitat ion and ionization b u t some energy w i l l appear as brernsstrahlung w h i c h w i l l generally escape f r o m the region of interest. P a r t of the energy is dissipated along the tracks of the p r i m a r y particles and par t is carried away by the de l ta rays.

S t a r t i n g w i t h the spectrum of i n i t i a l electrons i n the m e d i u m , the electrons w i t h i n the highest i n i t i a l k inet ic energy range are considered f irst . The energy these electrons dissipate ( i n collisions i n v o l v i n g energy transfers below the chosen cut-off) as wel l as a l l the secondary electrons and brernsstrahlung they generate i n passing to a lower energy range is then calculated. T h e calculation is continued by fo l lowing the h i s tory of the electrons i n the next lower energy range and so on, u n t i l the t o t a l energy dissipated is accounted for. T h e particles i n any energy range consist, i n general, of (1 ) electrons whose i n i t i a l energy is w i t h i n the range (2 ) electrons which enter f r om a higher energy range as the result of dissipative or discrete energy losses and (3 ) secondaries, tertiaries, etc. generated b y the p r i m a r y electrons. The technique has been extended to various types of ionizing particles ( B u r c h and B i r d , 1956; B u r c h , 1957a, b ; Brustad , 1962; H a y lies and D o l p h i n , 1959).

4.3.2 C a l c u l a t i o n s U s i n g the E l e c t r o n F l u e n c e D i s t r i b u t i o n as a F u n c t i o n of K i n e t i c E n e r g y

Spencer and Fano (1954) developed the theory of the s lowing-down of fast electrons and performed sample calculations of the fluence spectrum of electrons, i n c luding delta rays, for a few situations. The method was used b y McGinnies (1959) to prepare an extensive set of tables of the electron fluence spectrum per i n i t i a l part ic le , y(Eo, E ) , at energy Ε for electrons w i t h various i n i t i a l energies EQ. Here y(E0, E ) is the elect r o n fluence spectrum which results i f one particle of i n i t i a l energy Ε ο is generated per u n i t volume. M c G i n nies' tables are for values of Ε ο f rom 6 keV to 6 M e V , for values of Ε f r om 0.45 keV to 5 M e V and for several materials inc luding air, water, muscle and bone.

T h e fluence spectrum under equi l ibr ium conditions for a d i s t r ibu t i on of i n i t i a l energies n(E0) can be f ound b v integrat ion :

φ(Ε) = / y(E0, E ) n(E0)dE0

J Ε 4.3.(1)

T h i s fluence spectrum pertains to a l l values of the cut-off energy, Δ , b u t values of φ(Ε) are required only for Ε > A.

T h e absorbed dose delivered by particles w i t h energy Ε is t h e n found b y m u l t i p l y i n g by the appropriate L E T . T h u s :

4.4 LET Distributions in Water · 11

ddE) = LA4>(E)/PD 4.3.(2)

The absorbed dose delivered b y particles w i t h a par t i cu lar l inear transfer, L , is then found f r o m

M L ) = ( w ) 1 M E ) 4 · 3 · ( 3 )

Calcu lat ions of part ic le fluence spectra for water and of the corresponding absorbed dose spectra have been made b y Danzker et a l . (1959) for Δ = 5.1 keV, by C o r m a c k (1956, 1966) for Δ equal t o 5, 0.5 and 0.1 keV a n d by Bruce et a l . (1963) for Δ equal to 0.5 and 0.1 k e V .

4.4 L E T D i s t r i b u t i o n s i n W a t e r

T h e m e t h o d of der iv ing the d is tr ibut ions of absorbed dose i n L E T f r o m the fluence spectra has been used i n o b t a i n i n g the d is t r ibut ions given i n Figures 4 to 10. A l l of these apply to water as the med ium.

F i g u r e 4 shows the partic le fluence d i s t r ibut ions as a f u n c t i o n of electron kinet ic energy for 50 k V χ rays, 200 k V χ rays, and 6 0 Co y rays. These and s imi lar

data f o r m the s t a r t i n g po in t for the calculat ion of d i s t r ibut ions of absorbed dose i n L E T , the next step being the ca lculat ion of the fluence as a func t i on of L E T f r o m th is d i s t r i b u t i o n and of the data of Table A l (Append ix 4 ) .

F igure 5 shows the d i s t r i b u t i o n of absorbed dose i n L E T for 6 0 Co τ r a d i a t i o n . T h e effect upon the d i s t r i b u t i o n of various choices of cut-off energy for the de l ta rays invo lved i n LA is also shown.

I n Figure 6 the same data for 6 0 Co are p l o t t e d using the cumulat ive d i s t r i b u t i o n , i.e. the cumulat ive fract i o n a l absorbed dose DA(L) rather t h a n d A { L ) is p l o t t ed against L E T . T h e advantages of th is m e t h o d are t h a t i t d i rec t ly indicates the f ract ional absorbed dose between any l i m i t s of L E T and t h a t the curves are i n v a r i a n t w i t h coordinate transformations . T h e method is used i n Figures 7-10, for a single value of the cut-off energy, Δ = 100 eV i n Figures 7-9, and w i t h out cut-off, Doo(L) , i n F igure 10.

A l l L values and d is t r ibut ions presented i n th i s report are based on energy cut-off e\^aluations or estimates. As discussed i n Section A4.2.2, the determ i n a t i o n of proper L values t o be used for l ow energ

< ο

0.1 0.3 0.5

LINEAR ENERGY TRANSFER;

10

LA

1 1 1 1 1 1 I 1 I 1 1 I 1 H l

> ^ Δ=Ε/2

- / ^ 4 = 10,000 eV

ι I ' M 1 1 '

/ A-\C\CS a\J

— /

~r ZI-IVJU cV

~ In/ 6 ° C o

- I

Δ--Ε/Ζ (CSDA)

1 ! 1 1 1 1 . t 1 1 , 1 , , , , 30 50 100

keV./im '

F i g . 6. C u m u l a t i v e absorbed dose d i s t r i b u t i o n i n L E T f o r water for 6 0 C o . T h e o r d i n a t e values give the f r a c t i o n of the absorbed dose del ivered by electron tracks w i t h L E T ' s less t h a n the value on the abscissa and are i n t e g r a l representat ions of the curves i n F i g . 5. The curve for Δ = 100 eV is based on e lectron fluence d a t a ca lculated us ing M c G i n n i e s ' (1959) tables and is i n agreement w i t h data f r om B u r c h (1957b).

ÜJ CO Ο Q Q ÜJ OD er ο CO ω < < Ο F-o < er

Lü >

3

Ο

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0 2

0.1

ι ι 1 J 1 1 ι 1 1 ] ι ι ι ι ι I 1 j . ι ι . ι ι ι

2 MeV electrons (complete

\ 22 MV · X R a y s - S ^

track) s / I / ~

/

~ ^ 200 kV X Rc

/ / / η/ / 50 kV X

Tritium / /

Rays

ι ι , , , ,

Beta Rays / /

5.3 MeV Alpha Rays

• 1 , 1 , , . . 1

0.1 0.3 05 30 50 100

L I N E A R ENERGY T R A N S F E R ; ^ ' keV.^m 1

F i g . 7. C u m u l a t i v e absorbed dose d i s t r i b u t i o n s i n L E T for w a t e r for var ious r a d i a t i o n s f o r an energy cut -o f f of 100 eV T h e curves for 22 -MV x rays , 2 M e V electrons , 6 0 C o , 200 k V χ rays , t r i t i u m beta rays and 5.3 M e V a lpha rays were t a k e n f r o m B u r c h (1957b) and ad jus ted t o correspond to the L E T values g iven i n Tab le A l . T h e curve for 50-kV χ rays was ca lcul a t e d using the part i c l e fluence f r o m the p h o t o n spectrum of B u r k e and P e t t i t t (1960).

ω <

< ζ: ο

000

LINEAR ENERGY TRANSFER; 100 keV./i.m~'

F i g . 8. C u m u l a t i v e absorbed dose d i s t r i b u t i o n s i n L E T for w ater for heavy i o n beams w i t h energies of 10 M e V per nucleon. The de l ta - ray s p e c t r u m has the same shape and contr ibutes the same f r a c t i o n of the dose for a l l beams.

12

4.4 LET Distributions in Water · · 13

LINEAR ENERGY TRANSFER; J ^ — . keV.^irrf1

F i g . 9. C u m u l a t i v e absorbed dose d i s t r i b u t i o n s i n L E T for water for heavy i o n beams w i t h energies of 100 M e V per n u c leo l i .

LINEAR ENERGY TRANSFER; · - — : keV.̂ m"1

F i g . 10. C u m u l a t i v e absorbed dose d i s t r i b u t i o n s i n L E T for water for the charged par t i c l es set i n m o t i o n b y fas t neutrons a n d b y 6 0 C o gamma rays . T h e L E T includes a l l coll isions and therefore i t is assumed t h a t a l l the energy is deposited a long the p r i m a r y t r a c k s . T h e curves for 0 .1 , 0.34, 0.65, and 1.8 M e V neutrons were ob ta ined b y i n t e g r a t i o n of the da ta of B i a v a t i et a l . (1963) and those for 3 and 14.6 M e V neutrons f r o m Bewley (1968a).

1 4 · · 4. Calculation of Distribution of Absorbed Dose in LET

electrons is somewhat ambiguous. T h i s may be a par t i cu lar problem when L d i s t r i b u t i o n s are given for mixtures of l i g h t and heavy particles.

4.5 V a r i a t i o n of L E T w i t h D e p t h i n a M e d i u m

T h e data given i n Figures 6 and 7 are based upon p r i m a r y x- or 7 -ray spectra w i t h no allowance for scattered radiat ion . Scattered r a d i a t i o n should be taken i n t o account when situations a t depths i n an extended m e d i u m are considered.

L E T distr ibut ions at var ious depths i n water p h a n toms i rradiated w i t h 280 k V χ ravs, 1 3 7 Cs and 6 0 Co have been calculated b y Bruce et a l . (1963) w i t h cut-off energies of b o t h 100 eV and 500 eV. The v a r i a t i o n of the L E T spectrum w i t h d e p t h due to increasing scatter is very small and i t was concluded t h a t the change is unl ike ly to be b io logical ly significant.

T h e problem of v a r i a t i o n of absorbed dose average L E T w i t h h igh energy electrons has been discussed b y Ha3-nes and D o l p h i n (1959) . Calculat ions of the energy d i s t r i b u t i o n of electrons as a f u n c t i o n of depth for p r i m a r y electron beams of 10 M e V ( H a r d e r , 1964) show t h a t the spectrum does n o t r e m a i n constant.

For neutrons, p a r t i c u l a r l y at l ow energies, subs tant ia l changes i n L E T d i s t r i b u t i o n w i t h i n large organisms are to be expected.

4.6 Average L E T

T h e absorbed dose average LA.D and the t rack average LA,τ have been previously defined (Section 3.4). Values of LA,D and LA,τ have been calculated for a range of radiations w i t h Δ = 0.1 k e V and these are

T A B L E 1—Track-average and absorbed close-average values of LET in water irradiated with various radiations

Radiation Cut-off Energy

Δ eV keV//um

/-Δ.Ζ) keV/μηι

6 0 C o gamma rays U n r e s t r i c t e d 0 .23 9 0.31 10,000 0 .23 2 0.48

1,000 0 .23 0 2.8 100 0 .22 9 6.9

2 2 - M V χ rays 100 0.19 6.0 2 -MeV electrons

(whole t r a c k ) 100 0.20 6 .1 200-kV χ rays 100 1.7 9.4 3 H beta rays 100 4.7 11.5 50 k V χ rays 100 6.3 13.1 5.3 M e V alpha rays

(whole t r a c k ) 100 43 63

N o t e : The t r a c k ends (E < A) were assigned an L E T of A/R(A) using values f r o m T a b l e A l .

given i n Tab le 1. I n a d d i t i o n values of LA,D a n d LA,τ are given for e o Co 7 rays for Δ = 1 keV and 10 keV. I t is clear f r o m these data and, for example, f r o m Figure 5, t h a t LA,D and LA,τ ( w i t h Δ = 0.1 k e V ) are more widely separated for l ow L E T radiat ions t h a n for heavy part ic le h i g h L E T radiat ions .

These and add i t i ona l features m a y be derived d i rec t ly f r o m the definit ions of these averages, as fol lows. T h e mean L&,D, of the d i s t r i b u t i o n da(L) of absorbed dose i n L E T is equal t o the second moment of the d i s t r i b u t i o n t ^ ( L ) of t r a c k - l e n g t h i n L E T d i v i d e d b y the mean LA,τ, of th is d i s t r i b u t i o n . The variance, σ 2, of a d i s t r i b u t i o n is equal t o the second m o m e n t minus the square of the mean, thus one obtains the fo l l owing re la t ion for the variance of the d i s t r i b u t i o n t A ( L ) [see 3.4.(1) and 3 .4 . (2 ) ] :

σ 2 = LA,T(LA,D — LA,T) 4 .6. (1)

F r o m this i t fol lows t h a t LA,D is always greater t h a n LA,τ except i n the case where one has to deal w i t h only one value of L ( i .e . i n the t r a c k segment t y p e of exp e r i m e n t ) .

A useful dimensionless q u a n t i t y wh i ch d i rec t ly indicates the spread of L E T values is the re lat ive variance of L E T - v a l u e s .

4~ = - 1 4.6.(2) L&TT LA,τ

This re lat ion also aids i n understanding the dependence of LA,τ and LA.D on r a d i a t i o n q u a l i t y and on t h e cut-οίΐ value Δ . For h i g h Δ , and especially for L ^ , the difference between LA,τ and LA,D indicates the spread of L E T values i n the field of p r i m a r y charged particles. The effect of a smal l cut-off Δ is one of separ a t i n g the contr ibut ions f r o m the t r a c k core and f r o m delta rays. T h i s is most conspicuous for l o w - L E T radiations where the c o n t r i b u t i o n of the delta-rays lies i n a much higher L E T - r a n g e t h a n the contr ibut ion of the t r a c k core. Therefore, for l ow L E T - r a d i a t i o n s the absorbed dose average of L E T increases steeply w i t h a decrease of Δ ; as can be seen f r o m the data of Table 1, and, for example, f r o m Figure 5 where the values of LA,υ and LA,τ are wide ly separated. For small Δ and low L E T rad ia t i on , LA,D is (as pointed out b y B u r c h (1957a) ) almost independent of the spectrum of i n i t i a l part ic les ; i t depends m a i n l y on the delta-ray contr ibut ions . LA,τ , however, depends on the spectrum of i n i t i a l particles and therefore varies widely for different rad ia t i on qualit ies . F o r heavy charged p a r t i cles the effect of broadening the L E T spectrum b y in t roduc ing a cut-off is m u c h less pronounced. I n this case the c o n t r i b u t i o n of the delta-rays is near the L E T - r a n g e of the p r i m a r y particles.

5.3 Limitations of RBE vs LET Plots · · · 1 5

5. A p p l i c a t i o n s of L E T C a l c u l a t i o n s

5.1 G e n e r a l

Calcu la t i ons of average L E T values, and of L E T d i s t r i b u t i o n s , have t w o m a i n applications. I n f u n d a m e n t a l studies, observed relations between r a d i a t i o n effect a n d L E T for di f ferent types of rad ia t i on may be used e i ther t o suggest, or t o test, models of the mechanism of r a d i a t i o n act ion . W h e n , for several types of r a d i a t i o n , the effectiveness as wel l as the L E T d i s t r i b u t i o n or average is k n o w n , th i s i n f o r m a t i o n can somet imes be used b y i n t e r p o l a t i o n or l i m i t e d extrapo lat ion , to p red i c t the effectiveness of other s imi lar ly specified rad ia t i ons . F o r example, the effectiveness of 1 3 7 Cs γ rays is expected t o be in termediate between t h a t of 6 0 Co 7 rays and 200 k V p χ rays. I n a second appl i cat ion L E T values are used i n the est imat ion of dose equivalent i n r a d i a t i o n pro tec t i on (see Section 6 ) .

5.2 G e n e r a l A p p l i c a t i o n s , R B E vs L E T

M a n y authors have publ ished graphs of the effectiveness of di f ferent radiat ions against some f o r m of average L E T . Representat ive examples f r o m rad ia t i o n b io logy are g iven b y Howard -F landers (1958) , S inc la ir and K o h n (1964) , Tobias and M a n n e y (1964) and N a k a i and M o r t i m e r (1967) a l l of w h i c h show s imi lar features.

A n example, t a k e n f r o m N a k a i and M o r t i m e r (1967) is shown i n F igure 11 . W i t h i n a c t i v a t i o n as the end -po int , the re lat ion between Re la t ive Biological Effectiveness ( R B E ) a n d L E T ( L „ ) differs f r o m one biological system t o another . F o r the i n a c t i v a t i o n of certain bacteria, viruses, enzyme molecules, and for some chemical y ie lds—such as the ox idat ion of ferrous su l fa te—the curve of R B E (or y ie ld ) vs L E T has no m a x i m u m : beyond about 10 keV/μιη i t declines w i t h i n creasing values of L E T (cf . curve for T - l phage i n Figure 11) . However , the R B E for the i n a c t i v a t i o n of t rans f o rming D N A shows a peak at 30 keV/μηι ( F i g u r e 11) . I n more complex systems, inc lud ing m a m m a l i a n cells, the curve of R B E vs L E T generally has a m a x i m u m at L E T values i n the v i c i n i t y of 100 keV/μηι. T h e actual value of the L E T at wh i ch the m a x i m u m occurs appears to differ somewhat depending upon the biological m a t e r i a l s tud ied and the condit ions of i r r a d i a t i o n . I n F i g u r e 11 L x was employed because the experiments were done b y t h e t r a c k segment method .

5.3 L i m i t a t i o n s o f R B E vs L E T P l o t s

( 1 ) B r o a d generalizations of response vs L E T i n biological m a t e r i a l are useful for establishing the genera f o r m of response a n d are suitable for l i m i t e d i n t e r po la t ion and for gross comparisons between studie

10

RBE ι

0.1

ι ι ι ι ι ι ι 11 ι ι ι ι ι ι 11 - INACTIVATION EFFICIENCY

AS A FUNCTION OF LET

1 1 1 1 1 I Μ Human kidney

^^jf T-l (initial s cells, Ξ ope) -

- FOR DIFFERENT BIOLC

Transforming DN/ (Hutchinson)

)GICAL SYSTEMS

" ^ ^ v ^ Haploid yec >w (Manney,

>v Ny and

odd)

ist X T Irustad Tobias)

Bach (Brus

-

7-] ρ (ßrus

! I ! ! ! ! ! !

hage in broth tad) -

10 100 1,000

LINEAR ENERGY TRANSFER; keV.^rr.

F i g . 11 . R B E vs . L E T (i .e. L J ( N a k a i and M o r t i m e r , 1967) for i n a c t i v a t i o n of h u m a n k i d n e y cells ( T o d d , 1965); hap lo id yeast (Manney , B r u s t a d and Tob ias , 1963); bacter ia ( B r u s t a d , 1961); T - l phage (Brustad , 1961); a n d t r a n s f o r m i n g D N A ( H u t c h i n s o n , 1965). [By courtesy of the authors , M e d i c a l College of V i r g i n i a Q u a r t e r l y , W i l l i a m s and W i l k i n s Co. , and R a d i a t i o n Research. (Copy r i g h t on the paper b y N a k a i and M o r t i m e r held by Academic Press. C o p y r i g h t on the paper b y T o d d held by the M e d i c a l College of V i r g i n i a Q u a r t e r l y . ) !

16 · · · 6. LET in Radiation Protection

i n v o l v i n g different bio logical endpoints , e.g. i n a c t i v a t i o n vs m u t a t i o n frequency ( N a k a i and M o r t i m e r , 1967) .

( 2 ) One serious d i f f i c u l t y i n such plots is the choice of L E T . Since the i r r a d i a t i o n circumstances are such t h a t on ly an L E T d i s t r i b u t i o n , and n o t a single value (see Section 4 ) describes t h e m proper ly , the choice of a single average va lue is a r b i t r a r y . I n m a n y b io log i cal circumstances, e.g. i n a r e la t i ve ly large organism such as a dog, even t h e d i s t r i b u t i o n of L E T m a y change substant ia l ly t h r o u g h o u t the an ima l .

( 3 ) T o a f i rst a p p r o x i m a t i o n radiat ions of equal Loo are expected t o y i e l d s imi lar bio logical results. However , differences have been c i ted (Bewley, 1968a). These differences m a y be due t o di f ferent associated de l ta r a y spectra. N o t w o radiat ions h a v i n g the same LA d i s t r ibut ions appear t o have been tested i n th i s respect.

( 4 ) T h e value of a s impl i f ied p l o t such as shown i n F igure 11 is l i m i t e d because average values of L E T are necessarily invo lved . T h i s is p a r t i c u l a r l y t r u e when

the k i n d of average chosen is inappropr iate to the L E T dependence of the system.

( 5 ) T h e da ta i n F igure 11 apply to the p a r t i c u l a r experiments c i ted and great ly di f ferent values of R B E are f o u n d i n a number of test systems for other condi t ions ( N e a r y et a l . , 1963; Davies and B a t e m a n , 1963; D u m a n o v i c and Ehrenberg , 1965; S m i t h and Rossi, 1966). Each biological system must be analyzed separately a n d no generally accepted models of the mechanism of r a d i a t i o n act ion are y e t available.

( 6 ) I n a p l o t of R B E vs L E T such as F igure 11 , no feature of the biological m a t e r i a l w h i c h relates the deposit ion of energy t o the size, shape or nature of par t i cu lar structures is t a k e n i n t o account. These aspects can a t best be approx imated b y the choice of a suitable value of A i n LA . A l t e r n a t i v e l y , methods of q u a l i t y specification w h i c h do take these features i n t o account could be employed (see Section 8 ) .

Append ix 8 shows some of the ways i n whi ch L E T can be applied to exper imental s i tuat ions i n rad iat ion biology.

6. L E T i n R a d i a t i o n P r o t e c t i o n

6.1 Q u a l i t y F a c t o r

T h e I n t e r n a t i o n a l Commission on Radiological Pro te c t i on ( I C R P ) and the I n t e r n a t i o n a l Commission on R a d i a t i o n U n i t s and Measurements ( I C R U ) ( I C R U , 1962a) have recommended t h a t the evaluat ion of r a d i a t i o n hazards be based upon the dose equivalent ( D E ) w h i c h is the produc t of the absorbed dose (D) a n d a number of m o d i f y i n g factors one of w h i c h is t h e q u a l i t y factor ( Q F ) appropr iate for the rad ia t i on . T h e Q F is an estimate of the effectiveness of a g iven r a d i a t i o n , as evaluated f r o m i ts L E T , for the requirements of rad ia t i on protec t ion . I t is apparent t h a t , because of the l i m i t e d accurac}^ required i n r a d i a t i o n pro tec t i on and the u n c e r t a i n t y i n the selection of a suitable value of Δ, t h e use of L x on ly is acceptable.

Values of Q F recommended b y the I C R P (1966) as a func t i on of L w are shown i n Tab le 2.

T A B L E 2—Values of QF as a function of LM

Ζ » in water Q F keV/μηι Q F

3.5 or less 1 7.0 2

23 5 53 10

175 20

For a mixed rad ia t i on f ield the dose equivalent is to be obtained f r o m an average Q F by an integral over the L E T spectrum of the f o r m

Q F = [ Q F ( L ) doo(L)dL 6.1.(1)

where the q u a l i t y factor is assumed to be a continuous f u n c t i o n of L « , , doo(L) is the d i s t r i b u t i o n of absorbed dose as a f u n c t i o n of L w and the l i m i t s of the integral span the range of L E T values i n the radiat ion field.

Knowledge of d ^ { L ) is seldom available i n sufficient deta i l to enable an i n t e g r a t i o n of th i s type to be performed. T h e most common procedure is t o separate the absorbed dose i n t o l ow and h i g h L E T components and t h e n assign a q u a l i t y factor to the component of h i g h L E T a l lowing for a suitable safety marg in . Another possible method is t o assume t h a t Q F is l inearly related to L E T . T h u s the re lat ion ( R B E Committee , 1 9 6 3 )

Q F = 0.8 + 0.16· Loo-Mrn/keV 6.1.(2)

represents an approx imat i on w h i c h , i n the range up to 100 keV/μηι, is usual ly sufficiently conservative. T h i s includes the most i m p o r t a n t range i n many r a d i a t i o n protect ion situations i n v o l v i n g heavy particles ( p a r t i c u l a r l y neutrons ) . E q u a t i o n 6.1.(1) may then be w r i t t e n as

7. Limitations of the LET Concept · · · 1Ζ

Q F = 0.8 + 0.16 j u m / k e V ( Ld*>(L)dL Jo

= 0.8 + 0.16-Loo,D-Mm/keV 6.1.(3)

where L X , D is the absorbed dose average L E T . H o w e v e r i n most s i tuat ions even simpler approx i

m a t i o n s are employed t h a n the appl i cat ion of values g iven i n T a b l e 2. Examples include separate assessment of absorbed doses due to neutrons and gamma rays w i t h a n assumed Q F of 10 for the former ; or a simple measurement of t o t a l absorbed dose and an assumed Q F of 10 regardless of L E T d is t r ibut ions . These approaches are progressively more simple b u t also increasingly more conservative.

6.2 M e a s u r e m e n t o f A b s o r b e d Dose D i s t r i b u t i o n i n L E T for P r o t e c t i o n * P u r p o s e s

Ä m e t h o d of o b t a i n i n g the absorbed dose d i s t r i b u t i o n i n L E T us ing a spherical propor t i ona l counter developed b y Rossi and Rosenzweig (Rossi and Rosen-zweig, 1955a, 1955b) is discussed i n Appendix 5 of this report . T h e app l i ca t i on of th is method to the evaluat i o n of r a d i a t i o n hazards a t h i g h energy accelerators has been described b y Rossi et a l . (1962) . T h i s method can supply the d*>(L) d i s t r i b u t i o n for r a d i a t i o n of u n k n o w n composit ion. T h e several assumptions re q u i r e d t o derive t x ( L ) or d X ( L ) f r o m the pulse he ight spec t rum obta ined w i t h the spherical counter are described i n Append ix 5. T h e i r effect on an actual survey has been est imated b y Rossi et a l . (1962) and f ound

ί t o be smal l . Sol id state detectors have also been used I t o derive the d M ( L ) d i s t r i b u t i o n of a rad ia t i on of an

u n k n o w n composit ion ( R a j u et a l . , 1967). T h e measurements necessary t o per form a d x ( L )

analysis are f a i r l y compl icated and require m u c h more

effort t h a n a simple d e r i v a t i o n of absorbed dose f r o m an i on chamber measurement. However , the ir usefulness i n accurate eva luat ion of the hazards of a r a d i a t i o n field i n d o u b t f u l cases is considerable (Cowan, 1966).

E x p l i c i t eva luat ion of the f u n c t i o n d U ( L ) is n o t always required since methods have been developed t o estimate the dose equivalent d i rec t ly , u t i l i z i n g i n s t r u ments w i t h special response characteristics. One system util izes the Rossi -type L E T spectrometer, discussed above, i n c o m b i n a t i o n w i t h a suitable non- l inear amplif ier to o b t a i n a read-out d i rec t ly i n dose equivalent or effective q u a l i t y factor for any type of r a d i a t i o n ( B a u m , 1.967 and B a u m et a l . , 1968).

A n o t h e r system w h i c h ut i l izes the dependence of columnar recombinat ion on the L E T of the part i c le was developed b y t w o groups (Zielczynski , 1968; S u l l i v a n and B a a r l i , 1963). A n ionizat ion chamber is employed w i t h a l ow col lect ing voltage so t h a t on ly a f rac t i on of the i o n pairs f o r m e d are collected. B y comparison of the i on i za t i on i n th i s chamber w i t h t h a t i n a f u l l y saturated chamber, the average q u a l i t y factor and the dose-equivalent can be estimated (Lebedev, 1966). S t i l l another m e t h o d of determining the dose-equivalent i n m i x e d n e u t r o n and gamma ray fields also uses two i on i za t i on chambers (Goodman et a l . , 1964). T h e basic pr inc ip le here is to use an i on iza t i on chamber whose response to neutrons is related t o the q u a l i t y factor appropr iate t o these neutrons. T h i s is accomplished b y the use of Te f l on i n the walls and b y provis ion of a gas m i x t u r e i n the chamber selected for i t s response to neutrons . I n a d d i t i o n , a tissue equiva lent i on izat ion chamber is used simultaneously. B y elect ron i ca l l y adding appropr ia te fractions of the current f r o m these t w o chambers, a d i rec t determinat ion of the absorbed dose a n d dose-equivalent is possible. T o date on ly p re l iminary exper imenta l development of t h i s method has been accomplished.

7. L i m i t a t i o n s of the L E T C o n c e p t

T h e purpose of the foregoing i n f o r m a t i o n on L E T has been to provide the exper imental r a d i a t i o n research worker w i t h definite i n f o r m a t i o n on q u a l i t y specificat i o n for given i r r a d i a t i o n circumstances. D a t a per t inent t o most of the radiat ions i n common experimental use are included.

Whenever feasible, f u l l L E T d is tr ibut ions should be quoted ; i n general these are obtained b y calculat ion only . As discussed later i n th is section, measurements are beset b y severe l i m i t a t i o n s . As has been emphasized throughout th is report , an average value of L E T is rare ly satisfactory. I f , however, only an average is

g iven, i ts na ture should be clearly ident i f ied . LA,τ and LA,D differ m a r k e d l y for low L E T rad ia t i ons p a r t i c u l a r l y w i t h a l o w energy cut-off (e.g. see F igure 5 for 6 0 C o ) , b u t the difference is s t i l l m u c h too large to be neglected even w i t h r e l a t i v e l y h igh L E T rad iat ions (e.g. see Table 1 for a lpha rays ) or for h i g h values of Δ . Since the variance of t h e d i s t r i b u t i o n of t r a c k l e n g t h i n L E T is determined b y b o t h LA,τ and LA,D (see Sect i o n 4.6) i t is useful t o give b o t h values i n order t o indicate the spread of L E T values i n a p a r t i c u l a r s i tuat ion .

For purposes of approx imate q u a l i t y specification,

1 8 · · · 7. Limitations of the LET Concept

e.g. i n the assignment of q u a l i t y factors for rad ia t i on protect ion , the absorbed dose average, L D , is appropr ia te , since to a f i rs t a p p r o x i m a t i o n t h e q u a l i t y factor is a linear func t i on of L E T . I n general, however, the v a r i a t i o n of response w i t h L E T means t h a t no single average or effective value is universa l ly appropriate . T h u s the f u l l L E T d i s t r i b u t i o n is re levant unless experiments are done i n w h i c h single values of L E T can be realized or at least approx imated . I n " t r a c k segm e n t ' ' experiments a t h i n b io logical specimen is exposed to monoenergetic charged particles. Prov ided the specimen is t h i n enough, the L E T stays effectively constant throughout the targe t zone. However , i f de l ta rays are considered t o be separate f r o m the m a i n t r a c k , a d i s t r i b u t i o n , ra ther t h a n a single value of L E T , is invo lved .

Despite reservations expressed i n t h e I n t r o d u c t i o n , the remainder of t h i s report m i g h t be taken to i m p l y t h a t the L E T concept permi ts a precise description of the d i s t r ibu t i on of absorbed r a d i a t i o n energy. H o w ever, attempts made t o employ i t for th is purpose encounter serious problems. I f charged particles expended their energy continuously and uniformly along tracks hav ing negligible diameter and curvature, L E T w o u l d be ideally suited to define r a d i a t i o n q u a l i t y . I n r e a l i t y , these conditions are never m e t and rarely approached w i t h i n l i m i t s t h a t are of pract ica l interest. Indeed, i n the most commonly encountered situation—• t h a t of electrons secondary t o χ rays h a v i n g energies of the order of 100 keV—one can often n o t even discern a t rack when the p a t t e r n of energy deposition is rendered visible i n a cloud chamber. T h e adopt ion of a cut-off can at least give a l i m i t e d meaning to the L E T concept i n these cases.

T h e average energy lost b y a part i c l e of L E T , L , i n going a distance Al is LAI; b u t because of stat ist ical fluctuations, particles generally lose more or less t h a n t h i s average. M o s t target theory analyses have allowed for the random occurrence of p r i m a r y collisions along a t rack . Usual ly , a fixed va lue for the magnitude of t h e energy transferred i n a col l is ion is assumed and the mean number of collisions (def ined i n th i s conventional sense) per u n i t l ength of t r a c k is determined b y the r a t i o of the L E T to the energy per col l ision. T h i s mean number defines the Poisson d i s t r i b u t i o n of the prob a b i l i t y t h a t any specified n u m b e r of collisions w i l l occur i n a certain segment of a t r a c k . C lear ly , the absolute values of the various probabi l i t ies w i l l depend on the estimate of the energy per col l is ion and on the energy l i m i t , Δ, above w h i c h a col l ision is regarded as g iv ing rise to a separate de l ta t r a c k . I n spite of these elements of arbitrariness, calculations on such a basis may some

times be useful, p a r t i c u l a r l y i n considering t h e f o r m of the dependence of a biological effect on L E T . T h e u n d e r l y i n g assumption of th is k i n d of analysis is t h a t the biological effect is determined b y the n u m b e r of collisions i n a target rather t h a n b}^ the t o t a l energy transferred.

T h e stat is t i ca l spread of energy loss can be neglected only i f the number of p r i m a r y collisions w i t h i n the t r a c k segment is large, corresponding t o an energy loss of the order of one k e V or more, since t h e mean energy transferred per p r i m a r y collision is a r o u n d 60 eV (F igure 2 ) . F o r l ow L E T radiat ions t h i s impl ies a res tr i c t ion t o structures w i t h dimensions of a t least a few microns. O n the other h a n d , for densely i on iz ing radiat ions , the use of the L E T concept is l i m i t e d t o smal l structures since a t r a c k segment can be characterized b y a single L E T value only i f the energy lost is smal l as compared t o the t o t a l energy of the ion iz ing part ic le . F o r low energy electrons, the radius of curvature or the l ength of the t rack m a y be comparable to the sample size; t h e n the average energy deposited b y a part i c le of L E T , L , i n a layer of t h i c k ness Al m a y be greater t h a n LAI. Also, the difference between energy lost b y the charged part ic le and energy absorbed b y the sample is a factor w h i c h can be accounted for only very roughly b y an energy or range cut-off . For sparsely ioniz ing rad ia t i on , an L E T value is representative only over distances of at least a few microns, hence the cut-off values have to be correspondi n g l y large.

T h e pr inc ipa l result arising f r o m these diff iculties is t h a t L E T d i s t r ibut ions cannot be observed a l though sometimes t h e y can. be deduced f r o m measurements. T h i s is due to the fact t h a t real d i s t r ibut ions of energy deposition are due to various stochastic factors of w h i c h t h e L E T d i s t r i b u t i o n is on ly one.

A complete set of L E T spectra for a l l dif ferent c u t offs contains m u c h of the essential i n f o r m a t i o n on fluence d i s t r ibut ions . However , the re lat ion is very complex and i t w o u l d appear t h a t no a t t e m p t has been made to calculate one set of d is tr ibut ions ent ire ly f r o m t h e other. T w o m a i n difficulties are t h a t L E T d i s t r i but ions do n o t conta in i n f o r m a t i o n on the rate of change of L E T along the t r a c k and t h a t there is no unique re lat ion between fluence and L E T distr ibut ions w h e n several types of particles are present. For th is reason and also because one fluence spectrum contains more i n f o r m a t i o n t h a n a whole set of LET-spec t ra , the d i s t r i b u t i o n of fluence i n energ}^ is more fundamenta l t h a n the L E T d i s t r i b u t i o n as a basis for theoretical analysis of i r r a d i a t i o n circumstances.

9. Conclusions · · · 19

8. O t h e r Methods of Specifying Q u a l i t y

I n order to prov ide a more d irect specification of energy d i s t r i b u t i o n s Rossi (1959) has defined the quan t i t y , F (sometimes called the event size), w h i c h is the energy deposited i n a single event i n a smal l sphere d i v i d e d b y i t s d iameter . T h e dimensions of Y ( energy / l e n g t h ) are the same as for L E T . P ( F ) , the p robab i l i t y d i s t r i b u t i o n of a l l the events i n F , is used to characterize t h e r a d i a t i o n q u a l i t y for a g iven size of sphere. I f charged part ic les expended the i r energy continuously and u n i f o r m l y along tracks h a v i n g negligible diameter and c u r v a t u r e , P ( F ) w o u l d be independent of sphere size. R a d i a t i o n of a single L E T , L , w o u l d y ie ld a t r i a n g u l a r spec trum, P ( F ) = 2 Y / L 2 ( 0 < F < L ) , w i t h P ( F ) d r o p p i n g discontinuously t o zero at Y = L . R a d i a t i o n fields of mixed L E T w o u l d then y ie ld P ( F ) spectra w h i c h w o u l d be sums of such triangles. As discussed above, there are considerable deviations f r o m th is ideal behavior and the extent t o w h i c h P ( F ) does v a r y w i t h the size of the sphere is a measure of the i n a b i l i t y of L E T proper ly to characterize local energy deposition.

The essential feature of F is t h a t i t represents the actual energy deposited b y an absorption event. N o add i t i ona l corrections, as for energy loss fluctuations, de l ta-ray escape, or t rack l ength v a r i a t i o n , have to be applied to the d i s t r i b u t i o n P( Y). Fur thermore , the F d is tr ibut ions m a y be measurable even when i t is d i f f i cu l t t o characterize the r a d i a t i o n field b y other means.

One factor, the fluctuation i n the number of events at a certain dose, is n o t expl i c i t ly inc luded i n either the LET-concept or the F-concept. T o overcome this deficiency, Rossi, B i a v a t i and Gross (1961) introduced the not ion of local energy density w h i c h they designate

as Z. T h e local energy density is defined as the energy absorbed i n a sphere d i v i d e d b y the mass of the sphere. For reasonably large spheres a n d / o r h igh doses, Ζ and the absorbed dose are essentially equal. W h e n smaller spheres a n d / o r lower doses are considered i n the l i g h t of a l l the factors described above, there is a wide range i n the possible values of energy absorption for a single dose. T h u s , a d i s t r i b u t i o n T?(Z) i n Ζ is obtained, P ( Z ) d Z being the p r o b a b i l i t y of obta in ing values of local energy dens i ty i n t h e range Ζ to Ζ + dZ. P ( Z ) is a f u n c t i o n n o t on ly of t h e t y p e of rad ia t i on and sphere size, b u t also of t h e absorbed dose. For large spheres a n d / o r h i g h absorbed dose, P ( Z ) is a sharp Gaussian d i s t r i b u t i o n centered at Ζ values equal t o the absorbed dose. A t the other extreme, smal l spheres a n d / o r l ow absorbed doses, P ( Z ) is ident ical i n shape to the F d i s t r i b u t i o n since the probab i l i t y for more t h a n one event i n a g iven sphere is negl igibly small . There is, however, a h i g h p r o b a b i l i t y for Ζ = 0. T h e P ( Z ) d i s t r ibut ions can be measured (Gross, B i a v a t i and Rossi, 1962; B i a v a t i , Gross and Rossi, 1962), or comp u t e d f r o m the F d i s t r i b u t i o n ( B i a v a t i and B i a v a t i , 1964).

These quant i t i es prov ide i n f o r m a t i o n not easily available f r o m a knowledge of absorbed dose and L E T alone, b u t i t remains t o be seen whether they w i l l be f ound generally useful i n the specification of r a d i a t i o n q u a l i t y . T h e i r greatest va lue m a y be i n the fact t h a t they make possible predict ions concerning the size of c r i t i ca l structures w h e n exper imental biological d a t a for dif ferent rad ia t i ons are taken i n t o account and thereby help i n the i d e n t i f i c a t i o n of mechanisms of radiobiological ac t ion .

9. Conclus ions

( 1 ) Because the in terac t i on of charged particles w i t h mat ter is a stochastic process, the spatial and temporal d i s t r i b u t i o n of the products of an i r r a d i a t i o n can be described on ly i n stat is t i ca l terms. P r o b a b i l i t y distr ibutions can be computed or measured, i n greater or lesser detai l , and these can be useful for theoretical and pract ical purposes.

(2) A m o n g approximate descriptions of rad ia t i on qua l i ty , L E T m a y serve useful functions. Di f ferent radiations h a v i n g the same d W ( L ) ( i .e. absorbed dose d is t r ibut ion i n L E T ) w i l l no t necessarily have the same biological or chemical effects, b u t the possibi l ity

exists t h a t th i s is the case for some value of Δ i n d & ( L ) Y

w i t h Δ depending on circumstances. W h e n there is no basis for choosing a p a r t i c u l a r cut-off energy, the L W

d i s t r i b u t i o n can be a h e l p f u l approx imat ion . (3 ) Average values of L E T dis tr ibut ions are l i k e l y

to be useful on ly i n v e r y r o u g h approximations. T h e pr inc ipa l indices are the t r a c k average, LA,τ, and the absorbed dose average, LA,D , b u t neither should be c i ted w i t h o u t c learly i d e n t i f y i n g i ts character. C o m plete specification is p a r t i c u l a r l y i m p o r t a n t w i t h l ow L E T radiat ions and smal l Δ for which LA,T and LA,D differ very m a r k e d l y . I t should be noted t h a t the range

20 · · 9. Conclusions

of values of LA,D among various radiations is usual ly m u c h narrower t h a n t h a t of LA,τ .

( 4 ) I n calculations, the source of L E T values should be clearly identi f ied.

( δ ) I n rad iat ion protect ion, L«, serves as a suitable

specification of rad iat ion q u a l i t y . The q u a l i t y factor ( Q F ) is related to L M > D .

I n f o r m a t i o n on calculated L E T distr ibut ions and averages applicable to part i cular i r r a d i a t i o n c i r c u m stances w i l l be found i n Section 4.

A P P E N D I X 1

A I . F o r m u l a e for Stopping Power or L E T

A 1.1 G e n e r a l

T h e coll ision cross section, σ, represents the co l l i sion p r o b a b i l i t y of an incident partic le per scattering center a n d u n i t area. Di f ferent ia l forms are used t o express p a r t i a l probabil it ies such as a(Q)dQ w h i c h represents the probab i l i ty (per scattering center per u n i t area) for collisions which involve energy transfers between Q and Q + dQ. M u l t i p l y i n g a(Q)dQ by Q gives, for a number of particle traversals, the average energy loss per scattering center per u n i t area due t o interact ions i n v o l v i n g the energy transfer Q. T h e i n t e gra l

Λ 00

ΔΕ' = Qa(Q)dQ A l . ( l ) Jo

t h e n represents the average energy loss 1 0 per scattering center per u n i t area due to interactions i n v o l v i n g a l l possible energy transfers. The integrand w i l l have non-zero values only over a suitable range of p e r m i t t e d values of Q; elsewhere a(Q) is zero.

I n the subsequent formulae the fo l lowing symbols w i l l be used. F u n d a m e n t a l Constants:

ΝΛ = Avogadro 's constant e = electronic charge c = veloc i ty of l i ght

I n c i d e n t Fast Charged Part ic le : Mo = rest mass Mi = relat ive atomic mass ( 1 2 C = 12) ζ = charge, i n multiples of the electronic charge Ε = kinet ic energy ν = velocity β = v/c

Absorb ing M e d i u m : ρ = density Ζ = atomic number Ma = relative atomic mass ( 1 2 C = 12) MA = molar mass i n g / m o l and is numerical ly the

same as M a

Recoil Part ic le of Absorbing M e d i u m : mo = rest mass of recoil electron μ = m0c2 = rest mass energy for recoil electron vio — rest mass of nuclear recoil 1 0 The convention ( I C R U , 1968) of t r e a t i n g energy losses as

posit ive is adopted here.

I n general, a(Q) w i l l depend on factors other t h a n Q such as Ζ, 0, etc. A l t h o u g h these dependencies are n o t indicated they are assumed to be included i n the subsequent formulae.

T h e stopping power of a m e d i u m is defined as the average energy loss per u n i t p a t h l ength of a charged part ic le . I f we consider a l l o r b i t a l electrons as p o t e n t i a l scattering centers, t h e n for a u n i t area there are pNAZ/MA electrons i n a u n i t l ength , and expression A . 1.(1) can be modified t o

or

l ^ = ^ f QVQMQ A l . ( 3 ) ρ at ΜA Jo

I t is convenient to express a(Q) as

A l . ( 4 )

where s is expressed as a re lat ive dif ferential cross sect i o n per energy i n t e r v a l . I f k i is subst i tuted for the numerica l and physical constants the f o l l owing is obtained

(l/p)(dE/dl) is expressed i n M e V - c m ^ g " 1 , when k i has the value 0.1536 and the terms inside the in tegra l are expressed i n appropriate M e V uni ts . I n q u a n t u m mechanical formulae the cross-section s is made u p of the t rans i t i on probabi l i t ies , sn, appropriate t o the exc i tat ion energy levels, En . T h e equat ion then becomes

K t H j & f ? * - * αι·(6)

I n th is f o rmula the summat ion over En may be accomplished by u t i l i z i n g the re lat ion

l n ( / ) = Σ / a l n ^ n A I . ( 7 ) η

where I and En are i n the same uni ts , usually eV. / is called the mean exc i tat ion energy for the p a r t i c u l a r absorber and the fn are termed oscillator strengths and represent the f rac t ion of electron orbitals avai lable

21

22 · · Ah Formulae for Stopping Power or LET

to be excited t o the level w i t h va lue En . E m p i r i c a l l y determined values for / are c ommonly used i n these equations because t h e values for the i n d i v i d u a l fn

are k n o w n o n l y i n cer ta in cases (see Walske, 1952, 1956). / is discussed f u r t h e r i n A p p e n d i x 7.

A1.2 H e a v y P a r t i c l e s

T h e principles o u t l i n e d above were used b y Bethe (1933) t o derive t h e f o l l o w i n g expression for the coll ision s topping power of heavy part ic les :

1 (dE\ =

Ρ \dl)

I n

Zz2

M& β*

(2μ/3 2 ) 2

J 2 ( l _ 0 2 ) 2 2β2 A l . ( S )

T h e shell correct ion 2{C/Z) al lows for b o t h the inaccuracy of t h e B o r n a p p r o x i m a t i o n ( B o r n , 1926) used i n the d e r i v a t i o n ( w h i c h assumes t h a t the inc ident part i c le wave f u n c t i o n is u n d i s t u r b e d b y the scattering field) and the n o n - p a r t i c i p a t i o n of exc i tat ion- ion izat ion levels when these levels exceed the m a x i m u m possible energy transfer,

+ (2M0c/E) (Mo + m 0 ) 2 cV2m 0 £ ' , t ]

Ε A l . ( 9 )

due to the restr ic t ions of m o m e n t u m and energy conservation.

T h e density correct ion δ accounts for the fact t h a t energy losses are reduced because a po lar i zat i on i n duced i n the absorber tends to shield d i s tant atoms f r o m the electric field of the inc ident part i c le . For the energies considered i n t h i s r epor t , th i s effect is ap preciable on ly for condensed m a t e r i a l ( i .e. l iquids or sol ids) .

I t is o f ten of interest t o determine t h e " l o c a l " rate of energy loss due t o energy interchanges up to a specified value Δ. T h i s res tr i c ted mass stopping power is determined b y i n t e g r a t i n g equat i on A l . ( 5 ) only up t o the value Δ. I n th i s case the Bethe equat ion for heavy partic les takes t h e f o r m

1 / c L B \ _ , Zz2 Γ 2μβ2ί Ρ \ dZ Λ " M^ß2 L Ρ Ϊ Ϊ ^ ß2)

(1 - ß2)A ß2

C -] A l . ( l O ) 2μ Ζ

where Δ < Q m a x ( a n d Q r a a x = 2μβ2/(1 - β2)).

A 1.3 E l e c t r o n s

Stopping power formulae for negative electrons m u s t account for the fact t h a t the inc ident and recoil particles

arc bo th electrons and cannot be dist inguished. T h i s requires sett ing Q m a x = E/2 rather t h a n Q m a x = Ε and proper summation over a l l the events.

The electron stopping power as f o r m u l a t e d b y Rohr l i ch and Carlson (1954) is

1 /dA'\T = 1 . Ζ Ρ \dl / t o t a l ν ΐ ϋ ί , · 0 2

I n ßß-E

+ 1

2Γ-(1 - β2)

A I . ( 1 1 )

and

l(dE\ , Ζ j 2(Ε + 2μ)(Ε - Α)(Λ)

- ι - β2 + Ε

Ε - Λ + .ν

+ ( f - ) ' »0 Ε

δ Α Ι . ( 1 2 )

where Δ ^ Ε/2. I n equation A I . ( 1 1 ) , (l/p)(d7^ 7 /d0rad represents the

energy given to photon rad iat ion (Bremsstrahlung) due to the deceleration of the charged particle i n the atomic or nuclear fields. Rad iat i on losses represent an appreciable contr ibut ion to the t o t a l stopping power only for energetic l i g h t particles. These losses are no t included i n restricted stopping power formulae because most of the energy w i l l , i n general, be dissipated at an appreciable distance f rom the particle track.

A1.4 L o w E n e r g y Heavy P a r t i c l e S topping P o w e r a n d R a n g e s

I n the range below 1 M e V / a t o m i c mass un i t (amu) for protons and at s t i l l higher speeds for heavier n u clei, electrons tend to be captured b y the moving particle so t h a t the net charge is reduced and the stopping power correspondingly altered (Bohr and L i n d h a r d , 1954; L i n d h a r d and Scharff, 1961; Northcl i f fe , 1963; Mozumder et a l . , 1968). As β decreases to below 2Z/ 137, the net charge tends to 0 and the excitation-ionizat ion collision losses consequently decline to zero. L i n d h a r d and Scharff (1960) have derived an expression for electronic collision losses i n the range β < Ζ2'3/137

AI.4 Low Energy Heavy Particle Stopping Power and Ranges · · · 23

\ CD = k* [m i ? 4 w J - w A L ( 1 3 )

where Κ is the value of the expression i n the brackets and k 2 = 1.584 Χ 10 6 when (1/p) (dE/dl) is i n M e V -

_ 1 . A n a l ternat ive normalized f o r m is cm"

P \dlj k 3 -j ν *M

z2

A l . ( 1 4 )

where (1/p) (de/dl) = (1/p) ( d # / d Z ) - l / z 2 and eM = Ε/Mi (see Appendix 4 ) . k 3 = 7.355 Χ 10 4 when Ε is i n M e V and (1/p) (dE/dl) is i n M e V - c m ^ g - 1 .

Collisions of the incident particle w i t h the atom as a whole ( t e rmed "nuc lear " collisions) become a relat i v e l y i m p o r t a n t mode of energy loss as electronic exc i tat ion losses decline. For β > 10~ 3, nuclear energy losses have been described by Bohr (1948) i n the f o l l owing f o r m

l ( m ) = k i ( w ] t l 0 S w [ k i K t u ] A L ( 1 5 )

where

1 zZ (Mi + ΜΛ) (ζ21* + Z 2 / 3 ) 1 / 2

and for (1/p)(dE/dl) i n M e V - c m 2 - g ~ \ k 4 = 1.S2 Χ 10" 1

and k 5 = 6.49 Χ 10 4. F o r heavy particles w i t h β < 10~ 3 L i n d h a r d and Scharff (1960) have der ived another re lat ion for (l/p)(dE/dl) due t o nuclear collisions w h i c h reaches a m a x i m u m value g iven b y

ZMi (^2/3 + 2 2 / 8 ) 1 / 2 . ^ . 2 ^ . + M&)

A l . ( 1 6 )

where k 6 = 2,15 Χ 10 3 for (1 /p ) (de/dl) i n M e V - c m 2 -g - 1 . T h e m a x i m u m va lue occurs at the normal ized part i c le energy

u Z(Mj + MM" + Z™)w

tM = k l Mjfi A L ( 1 7 )

where k 7 = 9 Χ 1 0 " 6 for eM i n M e V / a m u . Below th is value of part i c l e energy, t h e energy losses

decline t o zero a t zero eM . F o r higher values of eM, the value of (1 /p) (dE/dl) declines to an approx imate correspondence w i t h t h e B o h r t h e o r y .

A P P E N D I X 2

Α2· Def ini t ion a n d M e a s u r e m e n t of Ranges

A2.1 R a n g e a n d S t o p p i n g P o w e r

T h e stopping power can be u t i l i z e d t o determine the range of a part ic le . I f dE/dl represents t h e t o t a l average ra te of energy loss per u n i t p a t h l e n g t h ( w i t h p a t h l e n g t h measured along the t r a c k ) , t h e n dl/dE represents t h e average p a t h l e n g t h per u n i t of energy loss. T h e integra l

represents the p a t h l e n g t h traversed b y the part ic le for the t o t a l loss of the i n i t i a l energy, E0, and is termed t h e continuous slowing down approximation range or csda range ( 7 ? C S d a has also been t e rmed variously , reciprocal s topping power range, average t ra jec tory l ength , average t r u e range ) . RCSd& is the p a t h length a part i c le wou ld traverse w h e n s lowing d o w n to a stop, i f i t s rate of energy loss along the t r a c k were equal t o the mean rate of energy loss defined by the stopping power. Because of f luc tuat ions i n the energy lost, a d i s t r i b u t i o n of p a t h lengths w i l l be generated for partic les of ident i ca l i n i t i a l energy so t h a t a range " s t r a g g l i n g " is observed. T h e mean of the p a t h l ength d i s t r i b u t i o n is somewhat greater t h a n the csda range ( K o c h and M o t z , 1959; Lewis , 1952; Fano, 1953) a l t h o u g h appreciably so on ly for l i g h t particles.

τ — [ — ι — ι — ζ — ι — ι — ι — ι — ι — Γ

F i g . A l . The d i s t r i b u t i o n of p a t h lengths OS) and of range (R) for 19.6 keV electrons i n oxygen a t 0°C and 1 atmosphere pressure : R = 0.32 cm = m e a n p ro j e c t ed range ; Ro — 0.52 cm = ex t rapo la ted pro jec ted range ; S = 0.64. cm = mean p a t h l e n g t h ; and $o = 0.82 cm = e x t r a p o l a t e d p a t h l e n g t h .

[ F r o m T H E A T O M I C N U C L E U S b y R . D . Evans . Copy r i g h t © 1955 b y M c G r a w - H i l l , I n c . Used w i t h permission of M c G r a w - H i l l Book Co.]

T h e actual p a t h length d i s t r ibut i on can best be observed using detectors which visualize three d i m e n sional p a t h configurations such as Wi lson cloud c h a m bers, photographic emulsion, bubble chambers, or spark chambers.

T h e ra ther smal l values of the range straggl ing variance and the small average scattering angles associated w i t h heavy particles permit measurements of ' i pro jec ted ranges" to be accurately correlated w i t h the theoret ical p a t h length range, j R c s c l a . Pro jected ranges refer t o actual penetration depths i n absorbers, i.e., the pro jec t ion of the p a t h of the particle i n the d i rec t i on of incidence to the material . The mean projected range, R, represents the thickness of absorber t h a t stops 5 0 % of the incident particles. 1 1 R w i l l be shorter t h a n i ? C S d a b u t can be accurately related t o i t i f corrections are made for the energy straggl ing (Seltzer and Berger, 1964). For protons of 10 M e V to 1 M e V , R is less than 7?csi'.a by about 0 .5% to 0 . 8 % i n a l u m i n u m , 1 % to 2 % i n copper, 2 % to 4 % in t i n , and 3.5 % to 7 % i n lead (Berger and Seltzer, 1964a).

H i g h e r energy protons or heavier particles of comparable velocities w i l l generally show an even closer corre lat ion between R and i ? c s < i a · However, the corre lat ion is m u c h poorer for l ight particles, because b o t h large fract ional energy losses and large angular deviations occur. Figure A l shows a plot of measurements of W i l l i a m s (1931) of the track length and projected range distr ibutions shown i n integral f o r m (i .e. , the curves represent the fractional number of paths w h i c h exceed the given distance) for 19.6 keV electrons as measured i n a Wilson, cloud chamber.

T h e figure i l lustrates four of the six range values n o r m a l l y referred to i n the l i terature :

R, the mean projected range corresponds to the absorber thickness t h a t transmits 50 % of the perpendicularly incident particles. 1 1

S, the mean p a t h length, corresponds to the Ä C S ( j a range to a reasonable approximation.

Ro, the extrapolated projected range, represents an absorber thickness determined by ext rapo la t i on of the slope of the number versus range curve, at R, to the range axis. (7?0 is sometimes defined as an extrapolation of the

1 1 F o l l o w i n g general usage, the t e r m wean, is used rather t h a n the correct ad ject ive which is median.

24

Λ2.3 Experimental Determination of Ranges · · · 25

l inear ly declining port ion of the curve to the range axis) .

So , T h e extrapolated p a t h length , represents the distance determined by extrapolat ion of the slope of the S curve, at S, to the range axis.

ß m a x , the (absolute) projected range represents an absorber thickness just sufficient to attenuate the incident partic le flux to an undetectable level . Because the meaning of ' 'undetectable l e v e l " depends on the measuring system, R m a x usual ly cannot be precisely specified.

Rq , a specified projected range represents an absorber thickness t h a t attenuates an i n c i dent beam to a specified level , q ; such as 3 7 % , 1 0 % , 1 % , etc. The a t tenuat i on m a y refer to partic le number, beam energy, absorbed dose deposited by the beam i n a subsequent absorber, etc.

A compl icat ion arises i n a t tempt ing to assign a range to ß-rays or positrons f rom radioactive isotopes because a spectrum of energies and therefore of ranges is present. Several techniques have been devised to assign range values i n th i s s i tuat ion ( K a t z and Penfold, 1952). Essential ly a l l of these methods amount to a scaling of the u n k n o w n absorption curve by comparison w i t h a standard reference curve for the absorption of a spectrum of k n o w n m a x i m u m energy.

A2.2 S e m i - E m p i r i c a l R a n g e - E n e r g y R e l a t i o n s

Several semi-empirical range-energy (R vs E ) re lat ions have been proposed for b o t h electrons and heavy particles. These generally take on the f o r m

R = AEn — Β A 2 . ( 2 )

The values g iven for the constants A , Β and η depend on the part ic le , the s topp ing m e d i u m and the energy range considered.

A2.3 E x p e r i m e n t a l D e t e r m i n a t i o n of R a n g e s

General ly a co l l imated beam is attenuated w i t h a series of p lanar absorbers and a part ic le counter ut i l i zed to measure the n u m b e r of particles t r a n s m i t t e d . D i r e c t visual izations of part i c l e tracks w i t h a c loud chamber, photographic emulsion, or bubble chamber are a l ternat ive b u t m u c h more demanding techniques.

Gas absorbers m a y be used i n a number of wa3^s. A part ic le count m a y be made as a funct ion of the surface density of i n t e r v e n i n g gas w h i c h can be v a r i e d either b y changing the absorpt ion distance or the gas pressure. T h e i on i za t i on produced i n the gas m a y also be used to determine the range, either by collecting the ion izat ion produced between t w o planar electrodes of f ixed separation placed at increasing absorption distance, or by col lect ing the t o t a l ionizat ion produced between a fixed and a movable electrode. B o t h methods represent an energy detect ion method rather t h a n a part ic le count m e t h o d and consequently w i l l y i e l d absorption curves of shape dif ferent f r om t h a t for a part ic le number absorpt ion curve. T h i s is true for any method t h a t ut i l izes t r a n s m i t t e d beam energy (as a funct ion of absorber thickness) as the parameter measured. T h e reason is t h a t an energy sensitive detector takes account of b o t h the loss of particles f r o m the beam and t h e loss of energy of the t r a n s m i t t e d particles. Accurate values of Ro cannot be determined f r o m such p lo ts ; however, f r o m these plots either an Ä m a x ( i .e. , " c o m p l e t e " energy absorption range) or an Ä q range (where q m a y indicate , for example, 1 % energy transmission) can be defined.

A P P E N D I X 3

A3. M e a s u r e m e n t of dE/dl

A3.1 G e n e r a l

One can measure dE/dl i n various ways. I n a funda m e n t a l approach a part ic le detector is placed i n such a way as t o intercept a l l of the p r i m a r y particles scattered b y a t h i n absorber. A l t e r n a t i v e l y a series of measurements can be made w i t h a smal l detector placed at angles to the part ic le beam, 0, f r o m 0 t o π (one usually m a y assume cy l indr i ca l s y m m e t r y ) . T h e detector should possess h i g h energy resolution and measure only p r i m a r y particles and exclude or d i s c r imi nate against secondary particles or photons. I n most experimental studies, particles have been collected for θ < π /2 . T h i s usually can be just i f ied because cross sections for single scattering beyond τ/2 are smal l . Var ious experimental approaches are out l ined below. I n these descriptions Al is the absorber thickness and Z0 is a thickness such t h a t m u l t i p l e scatter ing occurs w i t h small p robab i l i t y .

A3.2 E n e r g y D i s t r i b u t i o n M e t h o d — V e r y T h i n A b s o r b e r , Al ~ U

A direct method is to ut i l i ze a part i c l e detector of good energy d iscr iminat ion to determine the d i s t r i b u t i o n of energy losses, AE, for particles passing t h r o u g h an absorber t h a t is so t h i n t h a t m u l t i p l e interactions rare ly occur.

A3.3 M e a n E n e r g y L o s s — V e r y T h i n A b s o r b e r , AI ~ l0

A less direct method is to measure t h e integrated t r a n s m i t t e d energy w h i c h is propor t i ona l to the average inc ident energy, E; so t h a t Ν AE = NE — NE', where Ν is the t o t a l number of particles collected and Er

is the average t r a n s m i t t e d energy.

A3.4 E n e r g y A b s o r p t i o n — T h i n A b s o r b e r , Ä » Al > l0

I n th i s method a p l o t of E* vs absorber thickness, I, is made and the slope a t the or ig in determined. The slope, (AE /Δΐ)ι+ο, gives the correct stopping power values w i t h i n the l i m i t s of accuracy of the extrapo lat ion and the restr icted geometry.

A3.5 S lope o f E n e r g y V e r s u s R a n g e C u r v e

Consider the case where inc ident particles of energy, E, are j u s t stopped b y an absorber. I f the part ic le energy is increased s l ight ly b y AE*, an absorber of thickness AR mus t be added to j u s t stop the beam. T h u s the energy AE* can be said t o be absorbed i n AR. T h e slope of a p l o t of Ε vs R, i.e. dE/dR, represents the value (AE*/ AR)AE*^0. T h i s is not a measure of s topping power i n a rigorous sense, but i n practice, dE/dR values generally agree we l l w i t h the dE/dl values determined ana ly t i ca l ly or experimental ly (Cole, 1969).

A3.6 T h i n D e t e c t o r s

A detector m a y be used as a t h i n (R » ΔΖ > k) or very t h i n (Δϊ ~ k) absorber to measure the energy expended b y i n d i v i d u a l particles or by a k n o w n number of particles. Such detectors would include: t h i n p lanar ionizat ion chambers or proport ional counters, t h i n photographic emulsions (which could be used d o w n to thicknesses of a f rac t i on of a mic ron ) , t h i n s c in t i l l a t i on crystals, semi-conductor detectors w i t h t h i n sensitive regions, possibly t h i n chemical detectors or even t h i n biological systems.

26

A P P E N D I X 4

A4. T h e o r e t i c a l a n d E x p e r i m e n t a l V a l u e s for R a n g e , dE/dl a n d L E T

A4.1 G e n e r a l

Rev iews are avai lable w h i c h summarize da ta on range a n d mass s topp ing powers for various charged

particles i n various materials ( K a t z and Penfold, 1952; Sternheimer, 1959; A l l i s on , 1964; Barkas and Berger, 1964; Berger and Seltzer, 1964b; Jann i , 1966; Green and Peterson, 1968) . I n th is section selected i n f o r m a -

T A B L E AI—Vahles of range and mass stopping power for electrons in water** b

£ /MeV £ P / (g/cm2) [see note c]

Mass Stopping Power; - ( ^ J " ) / MeV.cmVg

1 ρ \dl / c o l . ρ \dl / rad Ρ \dl / tot- ρ \dl / l o o eV Ρ \dl ) ιοοο eV ρ \άΙ / ιο,οοο eV

1 χ i o - 5 4 Χ ΙΟ"» ~ 3 - -30 2 8 2 X 10~8 133 133 133 133 133 3 1 55 X 10~7 139 139 139 139 139 5 2 85 232 232 232 232 232 1 X 10~4 4 48 303 303 303 303 303 2 8 00 220 220 220 220 220 3 1 23 X 10" 6 215 215 211 215 215 5 2 20 195 195 183 195 195 1 X 10" 3 5 34 130 130 112 130 130 2 1 64 X 10 - 6 77.5 77. 5 60 77. 5 77.5 3 3 22 57.8 57 8 42.2 56.6 57.8 δ 7 70 39.2 39 2 27 1 36.9 39.2

1 X 10~2 2.50 X 10~4 23.2 23 2 15.1 20.2 23.2 2 8 33 13.5 13.5 8 5 11 1 13.5 3 1.71 X 10~3 9.88 9.88 6 12 7.9 9.7 5 4 22 6.75 6.75 4 12 5.26 6.35 1 x l o - i 1.40 X I O " 2 4.20 4 20 2.52 3 15 3.78

2 4 40 2 .84 4 0.006 2 85 1 67 2 08 2.44 3 8 26 2 .39 4 0.007s 2 40 1 39 1 72 2 .01 5 1 74 X 10 - 1 2.06 0.01 2 07 1 17 1 44 1.69 1 X 10° 4 30 1.87 e O .OI7 1 89 1 05 1 28 1.48 2 9 61 I.864 0 .03 2 1 89 1 02 1 23 1.41 3 1 49 X 10° 1.88 0.04 8 1 93 1 01 1 22 1.41 5 2 50 1.93 0.08 2 01 1 00 1 23 1.41 1 X lOi 4 .88 2.00 0.18 2 18 1 00 1 24 1.42 2 9 18 2.06 0.41 2 47 1 02 1 25 1.42 3 1 .30 Χ 101 2.10 0.64 2 74 1 03 1 25 1.42 5 1 .97 2.14 1.13 3 27 1 04 1 26 1.43 1 X 10 2 3 .25 2.20 2.40 4 61 1 06 1 27 1.44 2 4 .96 2.26 5.01 7 27 1 07 1 28 1.46 3 6 .13 2.30 7.65 9 95 1 08 1 29 1.47 5 7 .74 2.34 12.9 6 15 30 1 09 1 30 1.49 1 X 103 1 .01 X 10 2 2.40 26.3 28 7 1 12 1 31 1.50 2 2.50 49.0 51 5 1 125 1 32 1.51 3 2.55 72.0 74 5 1 13 1 34 1.53 5 2.61 117.4 121 1 14 1 34 1.54 1 X 104 2.70 230 233 1 .17 1 36 1.57

a T h i s tab le appears i n three sections. T h e t o p sect ion is based on exper imenta l d e t e r m i n a t i o n s of άΕ/dR o n l y (Append ix 3.5) ; t h e midd le section is based b o t h on exper imenta l d a t a and t h e o r e t i c a l ca lculat ions and t h e b o t t o m sect ion on ca lculat ions o n l y . D a t a for electron energies below 10" 4 M e V are to be considered t e n t a t i v e at present.

b M u l t i p l y values of mass s t o p p i n g power i n M e V - c m 2 / g b y 0.1 i n order t o o b t a i n values of L E T i n keV/μπι i n u n i t d e n s i t y m a t e r i a l , i .e . , a mass s t o p p i n g power of 10 MeV« c m 2 / g corresponds t o an L E T of 1 keV/μπι i n u n i t dens i ty m a t e r i a l .

c R represents an a p p r o x i m a t i o n t o RCSd& and is ob ta ined f r o m var ious theore t i ca l a n d exper imenta l de terminat i ons as descr ibed i n Append ix 2.

27

28 · · · A4. Theoretical and Experimental Values for Range, dE/dl and LET

0 [' 1 t • • • m l ι ι ι 111 i l l ι ι ι ι ι n i l 1—ι ι ι ι m l 1—ι ι ι ι m l 1—ι ι 11 i n I 1—ι ι ι m i l '—' ι n m i 1 — ' ' 1 ' " "

icr5 icr4 icr3 ιο - 2 icr1 io° ιο' ιο 2 ιο3 io4

ELECTRON ENERGY; - 7 -MeV

F i g . A 2 . Mass s t o p p i n g power f o r w a t e r f o r electrons versus e lectron energy. Below 10 k e V t h e o r e t i c a l formulae are i n appl i cab le (see t e x t ) . L i m i t e d e x p e r i m e n t a l d a t a on ly are ava i lab le ; consequently t h i s region is s h o w n dashed. I n the re g i o n between 10 keV and 100 k e V b o t h measured and calculated values are used. R e s t r i c t e d s t o p p i n g power f or Δ values of 100, 1000, and 10,000 eV are s h o w n .

t i o n of interest p r i m a r i l y t o the r a d i a t i o n biologist is presented, w i t h emphasis placed on the s topping power and energy transfer of electrons a n d protons i n water. Other generalized and representative i n f o r m a t i o n provides some unders tand ing of energy loss by other charged particles i n var ious mater ia ls .

A4.2 E l e c t r o n s

A4.2 .1 F a s t E l e c t r o n s (E > 10 k e V )

T h e calculations f or energy loss and range for electrons based on. the B e t h e (1933) and R o h r l i c h and Carlson (1954) f o r m u l a t i o n s t h a t have been made by L e a (1946) , H a l p e r n a n d H a l l ( 1948 ) , Ne lms (1956) , a n d Berger and Seltzer (1964b ) , have been incorporated i n Tab le A l and Figures A 2 and A 3 . T h e density correc t i on g iven b y Sternheimer (1952, 1953, 1956, 1966) has been inc luded a n d the exc i tat ion - ion izat ion (i .e. , col l is ion) losses are i n d i c a t e d separately f r o m rad iat ive a n d / o r t o t a l energy losses. Var i ous experimental determinat ions of e lectron mass s topp ing power and ranges agree w i t h the theoret i ca l values to w i t h i n about 5 % (see K a t z a n d Penfo ld , 1952). A d d i t i o n a l calculations given for t h e col l is ion losses using m a x i m u m energy transfer l i m i t s of 10 2 , 10 3 a n d 10 4 eV are based o n the R o h r l i c h a n d Car lson res tr i c ted mass stopping power f o r m u l a t i o n g iven i n A p p e n d i x 1.

A4.2 .2 L o w E n e r g y E l e c t r o n s (E < 10 k e V )

Theoret i ca l e lectron mass s topp ing power formulae are no t applicable below 10 keV, hence, experimental ly

determined ranges i n air and i n col lodion foils were ut i l i zed i n the energy range between 20 eV and 10 keV. Figure A 4 summarizes data wh i ch include c loud chamber measurements by W i l l i a m s (1931) and A l p e r (1932) , air absorption chamber measurements by Cole (1969), collodion fo i l absorpt ion data by Lane and Zaffarano (1954) and Cole (1969), and prote in layer i n a c t i v a t i o n data b y Dav is (1954, 1955). Values calculated b y Lea (1946) and Berger a n d Seltzer (1964b) are also shown.

T h e def in i t ion of the range measured depends on the method ut i l i zed (see Section A 2 ) . However , where overlap of methods was available i t was found t h a t Ro (extrapolated projected range) corresponded, w i t h i n 20 %, to the ranges based on the 5 % part ic le transmission, 1 % energy transmission and RCSd& · A l l these ranges are ut i l i zed i n the p l o t of F igure A 4 .

Accord ing t o the Bethe formulae the mass stopping power of collodion should be about 5 % greater t h a n t h a t of air , b u t inasmuch as the uncertainties i n the data are about ± 1 0 % , no correct ion has been applied. D a t a w h i c h are included f r o m measurements i n other materials (pro te in , H 2 , A r , H 2 0 ) have been converted t o air absorption using appropriate re lat ive stopping powers derived f r o m the Bethe dependence at and above 10 k e V energy. Conversion factors are indicated i n the capt ion to Figure A 4 . A l t h o u g h the use of such re lat ive stopping powers i n the l ow energy region may not be warranted , reasonable agreement is found among the various data except for those f r o m pro te in inact iva t i o n studies. T h e cloud chamber va lue of A lper (1932) at 200 eV is an extrapolat ion f r o m measurements

A4.2 Electrons · · · 29

Lü Ο < 0C

I 0 b

ΙΟ 4

ΙΟ 3

102

10 1

ιοο

ιο-'

ΙΟ" 2

-3 10

10 -4

10"

Ι Ο " 6

10-7

ι ι I ι i > ι I I I . . , . . I I J ι I ι ι I ι I

-

- -

ELECTROfv

/ / / / / t /

PROTONS

-

-} /

/

/ /

/ S / S

/ /

/ / -

- // -

I I I I 1 I ! I I t I 1 Ι I I , ι I . , I I I , Γ 10-5 10-4 10-3 10-2 10-1 100 ΙΟ' 102 ΙΟ 3 10« 105

E L E C T R O N ENERGY Ε MeV

F i g . A 3 . P l o t s of range (2?C3da) versus energy for electrons and pro tons absorbed i n w a t e r . T h e e lec tron p l o t refers t o values f r o m e x p e r i m e n t a l measurements below 10 k e V energy, f r o m b o t h e x p e r i m e n t a l measurement a n d theore t i ca l f o rmulae between 10 k e V and 100 k e V energies (dashed l ine ) and f r o m theore t i ca l f o rmulae above 100 k e V energy . (Ranges of elect rons beyond ^ 1 0 M e V are n o t t r u e ß C 3da ranges, because (1/P) (d2£/dZ) r ad t h e n dominates the t o t a l s t o p p i n g power w h e r e as the jKcsda refers to co l l i s ion s t o p p i n g power only . ) The p r o t o n p l o t refers to values f r o m t h e o r e t i c a l f o rmulae below 1 k e V energy, f r o m b o t h e x p e r i m e n t a l measurement and theoret i ca l f ormulae between 1 k e V and 1 M e V energies (dashed l ine ) a n d f r o m theore t i ca l f o rmulae above 1 M e V energy.

above 1,400 eV and therefore, is no t expected to be accurate.

The surpr is ingly good correspondence of Lea's calculated values w i t h the measured values can be considered somewhat accidental since the numer i ca l results of the Bethe f o r m u l a t i o n i n th i s energy range depend very c r i t i c a l ^ on the selection of the value of / ; hence, Lea's selection for I can be considered to be fortuitous .

T h e smoothed p l o t of energy versus range i n air a n d col lodion was converted t o a corresponding p l o t for water b y d i v i d i n g b y the mass s topping power of w a t e r re lat ive t o a ir w h i c h is equal t o 1.165. T h e la t t e r p l o t , inc luded i n F igure A 3 , coincides w i t h calculated range values i n the region of over lap f r o m 10 to 100 k e V . However , below 10 k e V t h e accuracy is uncer ta in because the use of the r e la t i ve s topping power is n o t f u l l y just i f i ed i n t h i s range. T h e stopping power r a t i o

30 · · A4, Theoretical and Experimental Values for Range, όΕ/dl and LET

lO1'—I 1 ' I 1 I I I 11 __1 1 1 I I 1 I l l I 1 I I I I I I 1 I I 1 1 1 ,1 111 1 1 I 1 1111, I0" 7 I0" 6 I0" 5 I0" 4 I0" 3 I0" 2

R A N G E : -̂JT g . cm"^

F i g . A 4 . Energy versus range for low-energy electrons for a i r and co l lod ion . (Various range def in i t ions , see t e x t , were used i n t h i s p l o t . ) + , C o l l o d i o n — Lane and Zaffarano (1954); X , P r o t e i n i n a c t i v a t i o n ( X 1.15)—Davis (1954, 1955); V , A i r : c loud chamber—Alper (1932); • , Co l l od i on—Cole (1969); O , A r g o n : c loud c h a m b e r — W i l l i a m s (1931); Ht, Oxygen: cloud chamber — W i l l i a m s (1931); 0, H y d r o g e n : c loud chamber ( X 2 .62 )—Wil l iams (1931); O , A i r : ion chamber—Cole (1969); Δ , W a t e r : ca l cu la ted ( X 1.165)—Lea (1946); A i r : calculated—Berger and Seltzer (1964b).

is expected to increase as the electron energy decreases (Cole , 1969).

D i f f e rent ia t i on of the Ε vs R re la t ion yields dE/dR for water and (1/p) (dE/dR) is p l o t t e d i n F igure A 2 . T h i s p l o t also coincides w i t h calculated values of (1 /p) (dE/dl) i n t h e region of overlap. As stated i n A p p e n d i x 3, (1/p) (dE/dR) can be t a k e n as a reasonable approx imat ion for (1/p) (dE/dl).

F o r an electron of energy E, the restr icted mass s topp ing power for a cut-off of Δ = E/2 is ident ica l ly equal t o the t o t a l mass stopping power, hence the calculated restr icted mass stopping power plots of F i g u r e A 2 were a l l extrapolated to meet the t o t a l mass s topp ing power p l o t at values of Ε = 2 Δ .

Because of the uncertainties i n the d a t a and calculat ions a t low electron energies, the graphs and plots should be considered accurate to no bet ter t h a n about ± 1 0 % f r o m 10 keV to 1 k e V ; and ± 2 0 % f r o m 1 k e V t o 100 eV. Other recent semi-empirical formulae ( M o -zumder and Magee, 1966; Green and Peterson, 1968) for the stopping of l ow energy ( < 1 k e V ) electrons i n w a t e r y i e l d values w h i c h are i n the one case larger, and i n t h e other smaller, t h a n those given i n Tab le A l and Figures A 2 and A 3 .

A number of studies have been made of energy loss a n d range of re lat ive ly l ow energ}^ electrons i n various

metal l ic foils (see Cosslet and Thomas, 1964a, 1964b, 1965). I n general, the measured electron ranges i n a l u m i n u m correlate well (considering differences i n Z , A , and range definitions) w i t h the ranges shown i n F igure A 4 .

Deta i l ed computations for positrons are no t presented. However , calculated restricted and t o t a l mass stopping powers are generally w i t h i n about 3 % of those for electrons of the same energy, except at lower energies where, for 10 keV for example, the rat io

dE/dl+

dE/dl-

becomes 1.10 i n water . As defined here, (l/p)(dE/dl), refers t o the average

rate of energy deposited, for a d i s t r ibut ion of energy exchanges f r o m 0 to Δ , along a very short or inf initesimal p a t h l ength . T h e p a t h of a l ow energy electron is very tor tuous and convoluted, thus a considerably larger value for (1/p) (dE/dl) m a y be determined i f the rate of energy deposition includes energy expended w i t h i n a smal l b u t finite distance (1 -100 n m ) surrounding the i n i t i a l short electron t rack . T h i s la t ter evaluation act u a l l y represents a distance cut-off L value (i.e. L R ) rather t h a n an energy cut-off L value (i.e. L A ) (see Section 3.3). For electron energies below 1 keV, such

A4.3 Heavy Particles · · · 31

Lr values m a y be some four times larger t h a n the est i mates presented i n th is report (Cole , 1969). These po ints are re levant to the i n t e r p r e t a t i o n of r a d i a t i o n effects due t o l ow energy electrons ( o r electron " t r a c k -ends" ) for , i f rad ia t i on effectiveness depends on the t o t a l energy deposited w i t h i n a region of dimensions 1 to 100 n m , the relat ive effect for such electrons m a y be comparable to t h a t for energetic heavy particles w h i c h have L values approaching 1000 M e V c m 2 g " 1 .

A4 .3 H e a v y P a r t i c l e s

A 4 . 3 . 1 M o d e r a t e a n d H i g h E n e r g y H e a v y P a r t i c l e s

H e a v y particles are particles w i t h rest mass large compared w i t h t h a t of an electron. Examples are mesons, kaons, protons, hyperons, α-particles, and accelerated nuclei . A l t h o u g h the da ta presented i n Tab le A 2 and Figures A 5 and A3 refer to p r o t o n absorp

t i o n i n water , data for other heavy ions can be der ived f r o m t h e m , as w i l l be discussed later .

T h e values for the t o t a l mass stopping powers and ranges for energies between 1 and 500 M e V are t a k e n f r o m the calculations of Barkas and Berger (1964) . T h e density effect correction is included, b u t h igh energy nuclear collision losses are no t included. I n add i t i on , separate calculations using the Bethe f o rmula for restr icted mass stopping power were made for t h e energy transfer cut-off values Δ = 100, 1000, and 10,000 eV. T h e restr icted mass stopping powers become ident ica l ly equal to the t o t a l mass stopping power a t a p r o t o n energy Ε — 458 Δ , because the m a x i m u m a l lowed energy transfer for (non-re lat iv i s t i c ) protons s t r i k i n g electrons is g iven b y

Amax ( — Qmax) — Ε 4m 0 i¥ 0 Ε ( m 0 + Mo)2 458

I f we define

1 d_€ = Ι Λ Ε 1 ρ dl ρ dl z2

A 4 . 3 . ( l )

A4.3.(2)

32 · · · Theoretical and Experimental Values for Range, dE/dl and LET

T A B L E A2—Values of range and mass stopping power for protons in water*' b- c

Mass Stopping Power; - p ( f ) / M e V - C m V g

Ε/MeV Ä p / ( g / c m » ) Ε/MeV Ä p / ( g / c m » )

P \di /nucl. - (—λ 1 (dE\ ρ \ dl /1000 cV P \di /nucl. Ρ \dl /coi. P \di / tot. ρ \dl / 100 eV ρ \ dl /1000 cV p \dl J 10,000 eV

ι χ 10- 5 9 Χ ΙΟ"» 96 18 114 114 114 114 2 1.68 Χ ΙΟ" 7 100 26 126 126 126 126 5 3.85 99 41 140 140 140 140 1 x 10- 4 7.2 90 58 148 148 148 148 2 1.37 X 10~ 6 78 82 160 160 160 160 5 3.14 58 130 188 188 188 188 1 x 10- 3 5.58 43 180 223 223 223 223

2 9.43 29 260 289 289 289 289 δ 1.80 Χ ΙΟ" 5 14.7 410 425 425 425 425 1 Χ ΙΟ" 2 2.79 7.5 580 587 587 587 587 2 4.22 —4.5 730 734 734 734 734 5 7.3 910 910 910 910 910 1 Χ ΙΟ" 1 1.25 Χ ΙΟ" 4 915 915 715 915 915 2 2.42 750 750 488 750 750 5 8.0 437 437 254 433 437 1 X 10° 2.25 Χ ΙΟ" 3 268 268 147 240 268 2 7.13 170 170 91.8 141 170 4 2.30 X 10" 2 100 100 54 78 100 6 4.71 71.4 71.4 38.5 54.2 69.2

1 x 101 1.18 X 10- 1 46.8 46.8 25.3 34.2 43.0 3 8.64 19.2 19.2 10.3 13.4 16.5 5 2.18 X 10° 12.7 12.7 6.88 8.76 10.8 1 X 10 2 7.57 7.42 7.42 4.00 5.04 6.08 1.5 1.55 X 10 1 5.54 5.54 3.00 3.77 4.55 3 5.06 3.58 3.58 1.93 2.46 2.90 5 1.15 X 10 2 2.79 2.79 1.50 1.90 2.25 1 χ 103 3.21 2.24 2.24 1.23 1.52 1.81 3 1.29 X 103 2.03 2.03 1.13 1.39 1.63 5 2.26 2.07 2.07 1.18 1.42 1.67 1 X 104 4.62 2.17 2.17 1.29 1.54 1.78 3 1.335 X 104 2.37 2.37 1.41 1.68 1.94 5 2.162 2.46 2.46 1.46 1.74 2.02 1 Χ 10 δ 4.142 2.58 2.58 1.54 1.83 2.12

a T h i s tab le appears i n three sections. The t o p sect ion and the b o t t o m sect ion depend u p o n theoret i ca l calculations on ly , the m i d d l e sect ion is based b o t h on exper imental d a t a and theore t i ca l ca l cu lat ions .

b M u l t i p l y values of mass s t opp ing power i n M e V - c m 2 / g b y 0.1 i n order t o o b t a i n values of L E T i n keV/jum i n u n i t density m a t e r i a l , i .e . , a mass s t opp ing power of 10 M e V - c m 2 / g corresponds t o an L E T of 1 keV/μπι i n u n i t densi ty m a t e r i a l .

c Above c e r t a i n l i m i t s (see T a b l e A3) t h i s tab le m a y also be used for part i c les other t h a n protons i f the f i rst co lumn is t r e a t e d as a l i s t i n g of values of €M ( M e V / a m u ) and the second co lumn is considered t o be a l i s t i n g of values of normal ized range (z2/Md-Rpf

and the s t o p p i n g powers considered t o be a l i s t i n g of the normal ized f o r m , (1/p) (dE/dl) · (1 / z 2 ) .

and

eM = jjf A4.3. (3)

where Ε is normal ly i n M e V and w r i t e the Bethe mass stopping power f o r m u l a i n the f o r m

i ^ = z 2 / ( / 3 , . . . ) A4.3. (4) p at

w rhere β is a funct ion of Ε/Mi only, then a l l heav} ' particles of a given eM ( M e V / a m u ) value have the same normalized mass stopping power ( l / p ) ( d e / d £ ) prov ided charge exchange effects ( t o be discussed la ter ) or

rad ia t ive losses are negligible. ( A n addit ional dis crepancy occurs at extreme velocities due to differences i n the m a x i m u m energy transfer for different particles.)

G i v e n these same restrictions, i t can be shown t h a t a normalized range, (z2/Mi) R, is also approximately the same for a l l heavy particles of equal e M . Thus i f the energy, range, and stopping power values given i n Table A 2 and Figures A 5 and A 3 for protons (2 = 1, Mi = 1) are considered as eM , (ζ/Μ·/) R and (1/p) (de/dZ) values, these data can be applied to other heavy particles over a suitable range indicated i n Table A 3 . "

A t energies below the range of appl icabi l i ty , the

A4.3 Heavy Particles · · · 33

T A B L E A 3 —Range of applicability of data in Table A2 for heavy particles other than protons

Incident Particle Energy Range Applicable* for

MeV/amu

M u o n s Pions K a o n s Pro tons H y p e r o n s Deuterons T r i t o n s α-Particles,

> 1 < eM < 10 δ

N u c l e i \vith ζ < 15 10 < 6M < 10 5

a I n t h i s range charge exchange and nuclear and r a d i a t i o n losses are negl ig ib le .

(1 /p) (cU/dZ) plots for var ious ions diverge as is i l l u s t r a t e d i n F igure A 6 , w h i c h is t a k e n f r o m a review of N o r t h c l i f f e (1964) .

A4.3 .2 L o w E n e r g y H e a v y P a r t i c l e s

T h e mass stopping power and range i n f o r m a t i o n presented i n Table A 2 and i n Figures A 5 and A 3 for p r o t o n energies between 0.003 and 1 M e V is based on s imi lar smoothed data taken f r o m Northcl i f fe ' s review paper (Nor thc l i f f e , 1964) and data of A l l i son and W a r s h a w (1953) , V a n Wi jngaarden and D u c k w o r t h (1962) and Glass and Samsky (1967) . For values of e,i/ below about 1 M e V / a m u for protons and alpha par

ticles, and 10 M e V / a m u for heavier particles, electrons are captured by t h e m o v i n g part ic le and the net charge is reduced. T h e mass stopping powers below 0.02 M e V g iven i n Table A 2 and p l o t t e d i n Figure A 5 are p a r t l y based on the L i n d h a r d a n d Scharff (1960) equat ion (see Append ix 1) for electronic collision losses w h i c h takes i n t o account charge reduct ion. T h e dashed l ine m a r k e d "e lec tronic " i n Figure A 5 indicates the L i n d h a r d -Scharff re lat ion. Below about 0.002 M e V the data are based on the L i n d h a r d and Scharff and B o h r formulae (Append ix 1) for "nuc l ea r " interact ions. T h e solid curve represents the sum of b o t h electronic and nuclear collision losses.

T h e ranges for protons given i n Table A 2 and Figure A 3 for energies below 1 M e V were based on an approx i mate numerical in tegrat i on of the reciprocal of the stopping power p lo t ted i n F igure A 5 . A l t h o u g h there are various data on mass stopping powers for heavy particles of energies above 4 keV (Al l i son , 1964; Tep lova et a l . , 1962; Phi l l ips , 1953) there are few data on range determinations i n the lower energy region (Tep lova et a l . , 1962) and there is a part i cu lar scarcity of data for water (either as gas, l i q u i d , or so l id ) . Hence the lower energy range da ta must be considered t enta t ive . Fur thermore , a rather large discrepancy arises when these data are compared w i t h range measurements for l ow energy protons (2-75 k e V ) i n pro te in (Person et a l . , 1963). Some of the measurements (Glass and Samsky, 1967) indicate t h a t the m a x i m u m L E T of the p r o t o n m a y be i n excess of 1000 MeV-cmVg.

τ 1 1 1 1 1

N O R M A L I Z E D E N E R G Y ; — — ψ — ' MeV/amu

F i g . A6. Smoothed data stopping-power curves for various ions f or a l u m i n u m . (Dashed l ines, theoret i ca l result of L i n d h a r d and Scharff, 1960; solid l ines, exper imenta l data , N o r t h c l i f f e , 1964.) [ B y courtesy of the authors , Phys ica l Review, A n n u a l Reviews, I n c . and the N a t i o n a l Academy of Sciences—National Research Counci l . ]

A P P E N D I X 5

A5. M e a s u r e m e n t of L E T D i s t r i b u t i o n s

T h e L E T , as opposed to L E T distr ibut ions , can be measured by any of the methods out l ined i n Append ix 3. However, the measurement of L E T distributions entails two kinds of difficulties. F i r s t , the thickness of the detector m u s t be v e r y small , and second, as discussed i n Section 7, the q u a n t i t y measured is energy deposition (or i t is proport iona l to energy deposition) ra ther t h a n L E T .

A method w h i c h was designed specifically to evaluate L E T d is tr ibut ions was developed b y Rossi and Rosenzweig (1955a, 1955b). A spherical propor t i ona l counter (or more s t r i c t l y a spherical ionizat ion chamber t h a t is gas-coupled to a cy l indr i ca l proport ional count e r ) is employed. T h i s counter has a wa l l of tissue-equivalent conducting plastic and is f i l led w i t h tissue-equivalent count ing gas i n order to obta in the d i s t r i b u t i o n appropriate for tissue. The reduct ion of the pulse height data obtained w i t h th is counter t o L E T spectra depends upon the p a t h length d i s t r i b u t i o n of t h e ioniz ing particles i n traversing the sphere. T h e p r o b a b i l i t y of a traversal w i t h a p a t h l ength i n the i n t e r v a l I to I + dl when the sphere is u n i f o r m l y i r r a d ia ted is d irect ly proport ional to I. Possible p a t h lengths l ie between zero and twice the sphere radius. I f the conditions set out below are met , the L E T d i s t r i b u t i o n m a y be calculated f r o m the f o rmula

t ( L ) = k [η(Λ) - h-^P] A 5 . ( l )

Here t(L) is the t rack d i s t r ibuted i n L E T , k is a cons tant , and n(h) the number of pulses of height (or energy) h per u n i t pulse height i n t e r v a l . L is related t o h b y L = h/2r where r is the radius of the equivalent tissue sphere and is g iven b y the product of counter radius and the density of the counter gas relat ive t o tissue.

T h i s conversion of the experimental data to L E T d is t r ibut ions is based on several assumptions w h i c h are m e t to a v a r y i n g degree of exactness i n actual practice. T h e most l i m i t i n g assumption is t h a t of s t ra ight l ine traversals of the sphere b y the charged particles. T h i s requirement makes the conversion of the data f r o m x- or gamma-ray fields di f f i cult as electron scatter usually cannot be ignored. I n the case of heavier d i rec t ly i on iz ing particles, the assumption is usually just i f ied .

A n o t h e r assumption i m p l i c i t i n the use of th is counter f o r obta in ing L E T spectra is also connected w i t h

part ic le traversal . I t is assumed t h a t particles suffer negligible change i n L E T i n traversing the counter and i n par t i cu lar no partic le either begins or ends i t s t r a c k i n the counter gas. H o w we l l this assumption can be met i n practice for any given radiat ion depends on r. As r is increased, changes i n L E T become i m p o r t a n t and i f i t is sufficiently large, a significant number of d i rec t ly i on iz ing particles w i l l be created i n the gas and an equal number of the particles formed i n the w a l l w i l l be stopped. T h e occurrence of such particles interferes w i t h the proper analysis of the experimental data (Caswel l , 1966).

Decrease of the gas pressure to reduce the magni tude of th is d i s t o r t i on causes an addi t ional problem. W h e n the energy loss of a particle i n traversing the sphere is smal l , the stat ist ical fluctuations i n energy loss are large. These fluctuations again prevent proper analysis i n terms of L E T which is a concept dealing w i t h average energy loss. Thus , there is no pressure range where the shape of the pulse height spectrum is independent of pressure. W h i l e such energy f luctuations lessen the accuracy of the measurement technique, they i l lustrate l i m i t a t i o n s of the L E T concept as a measure of radia t i o n q u a l i t y as discussed i n Section 7.

I n general, the combined effect of these sources of error is d i f f i cu l t to assess. Where scattering is significant, the spectrum derived w i l l be biased towards the h i g h L E T region whereas the influence of incomplete t r a versal is i n the opposite direction. Statist ical f luctuat ions for h i g h energy losses i n addit ion to being small , are also symmetr ica l , thus s imulat ing poor counter resolution. F o r small energy losses, however, the dist r i b u t i o n is asymmetrical and thus distorts the calculated d i s t r i b u t i o n i n a less definite fashion. However, i n a l l cases, the device broadens the L E T d is tr ibut ion .

Rossi and Rosenzweig (1955b) have used this i n s t r u m e n t to determine the L E T d is t r ibut ion of absorbed dose i n tissue when the counter was irradiated b y monoenergetic neutrons i n the energy range of 0.2 to 8 M e V . I n comparisons of these experimental results w i t h theoret ical calculations the l imitat ions of the la t t e r must be kept i n m i n d . Stopping power values chosen for these calculations were not of high accuracy. Nevertheless, the discrepancies between theory and experiment were found to be minor and should be u n i m p o r t a n t i n most applications related to theoretical radiobiolog} ' .

34

A P P E N D I X 6

A6. D i s t r i b u t i o n of Ions i n Clusters

A l t h o u g h the cross sections for energy transfers above several h u n d r e d eV can be calculated accurately, theory becomes unreliable when the energy transfers are smal l and the number of i on pairs per cluster is i n the region of 1 to 5.

Useful i n f o r m a t i o n for gases has been obtained i n c loud chamber studies, b u t , unfor tunate ly , m a r k e d discrepancies exist between the two pr inc ipa l sets of available data , those of Wi lson (1923) and the higher resolut ion data of Beekman (1949). These results are l isted i n Table A 4 .

Accord ing to Ore and Larsen (1964) m u c h of the discrepancy arises f r om the poor resolution of Wilson 's observations w h i c h gives rise to cluster overlap. Ore and Larsen conclude t h a t Beekman's da ta are subs tant ia l l y reliable a l though these were also uncorrected for cluster overlap.

I t must be realized t h a t no precise de f in i t ion of a cluster can be provided. Furthermore , i t should be

emphasized t h a t the data i n Table A 4 provide i n f o r m a t i o n on the ions produced i n the gaseous state. F o r the condensed state (e.g. i n biological mater ia l ) d i s t r i b u t ions are no t necessaril\ r s imilar .

T A B L E A4—Relative frequencies of clusters

Beekman (1949) Number of Ion Pairs

per Cluster Wilson (1923)

"0-Rays" a

Number of Ion Pairs per Cluster

Wilson (1923) "0-Rays" a

154 keV electrons

322 keV electrons

% % % 1 43 60 62 2 22 23 20 3 12 8 9 4 10 4 4

> 4 13 — — δ — 2 2

> 5 — 3 3

a T h e "jo-rays' ' employed by Wi l son i n th i s exper iment were electrons generated b y χ rays . The e lectron energies appear to have been i n the range 9-36 keV (Wi lson , 1923).

35

A P P E N D I X 7

A7. M e a n E x c i t a t i o n E n e r g y

A7.1 G e n e r a l

T h e formulae for s topping power conta in the p a r a m eter, 7, the mean exc i ta t i on energy (See Appendix 1 and equations A l . ( 7 ) t o A l . ( 1 2 ) ) .

7 is defined, (see equat ion A l . ( 7 ) ) as:

I n / = Σ / η 1 η # η A 7 . ( l ) η

where I and En are i n the same u n i t s , n o r m a l l y eV, and where fn is the opt i ca l dipole osci l lator s trength for the exc i tat ion of the a t o m f r o m i t s g r o u n d state, to the excited level En . B l o c h (1933) has shown t h a t I should be approximately p r o p o r t i o n a l t o the atomic number of the medium, and t h a t the p r o p o r t i o n a l i t y constant should approximate the R y d b e r g energy 13.5 eV. Thus ,

/ ~ 13 .5Z-eV A 7 . ( 2 )

Because of the inexact n a t u r e of th i s relationship, and the di f f i cul ty of d e r i v i n g more accurate theoretical values, estimates of I have been obta ined f r o m measurements of the stopping-power or range of heavy particles. I n such der ivat ions , i t is necessary to evaluate the shell corrections i n t h e Bethe (1933) equation (see equat ion A l . ( 8 ) ) .

A7.2 E m p i r i c a l R e l a t i o n s

Several empir ica l formulae have been proposed wh i ch relate I more exact ly t o the atomic number . A m o n g these is one given b y Jensen (1937)

I = k ' Z ( 1 + w ) A7 . (3)

a n d another by D a l t o n and T u r n e r (1968)

I = (11.2 + 11 .7Z)eV Ζ < 13

/ = (52.8 + 8 .71Z)eV Ζ > 13 A 7 . ( 4 )

A l t h o u g h such relat ions give reasonable agreement w i t h most experimental da ta , discrepancies can be large for certa in elements.

A7.3 R e c o m m e n d e d V a l u e s

T h e Subcommittee on Penet ra t i on of Charged Par ticles of the C o m m i t t e e on Nuc lear Science, N a t i o n a l

Academy of Sciences—National Research Counc i l has publ ished ( T u r n e r , 1964) a l ist of suggested I values. These represent a consensus of opinion and are suffi c ient ly accurate for most stopping power calculations. I t must be remembered t h a t the stopping power depends upon the l o g a r i t h m of I and is therefore a s lowly v a r y i n g f u n c t i o n of I . Other compilations of 7 values have been made by the N C R P (1961) and by D a l t o n and T u r n e r (1967) .

A7.4 M i x t u r e s a n d C o m p o u n d s

O n the basis of the Bragg a d d i t i v i t y rule , the mean exc i tat ion- ionizat ion po tent ia l can be calculated approx imate ly for a compound or mix ture , prov ided the I values of the const i tuent atoms are k n o w n . T h u s :

, T ZNiZi In Ii I n 7 = — r /

where AR\ is the re lat ive number of atoms of atomic number Z j i n the medium.

Values of 7 for a fe\v substances of interest to rad ia t i o n biologists and rad ia t i on chemists have been calculated b y th is rule . Tab le A 5 lists these values along w i t h the assumed elemental 7 values and compositions.

T A B L E A5—Mean excitation energies for selected substances

Substance / / e V a

W a t e r 65.1 M e t h a n e 44 .1 L i t h i u m fluoride 87.2 A i r b 80.8 M u s c l e b 65.9 B o n e b 85.1

a E l e m e n t a l I values (eV) are as fo l lows : Hydrogen—18.7, L i t h i u m — 3 8 . 0 , Carbon—78.0, Nitrogen—85.0 , Oxygen—89.0, Fluorine—115.0 , Sodium—140, Magnesium—152, Phosphorus— 183, Sul fur—192, Argon—210, Potassium—218 and C a l c i u m — 287.

b Compos i t i on f r o m I C R U Repor t 10b ( I C R U , 1962b). B y we ight these are as f o l l ows :

A i r : N i t r o g e n — 7 5 . 5 % , Oxygen—23.2% and Argon—1.3%. M u s c l e : Hydrogen—10 .2%, Carbon—12.3%, N i t r o g e n —

3.5%, Oxygen—72.9%, Sod ium—.08%, Magnes ium— .02%, Phosphorus—0.2%, S u l f u r — 0 . 5 % , Potassium—0.3%, and C a l c ium—.007%.

Bone : H y d r o g e n — 6 . 4 % , Carbon—27.8%, N i t r o g e n — 2 . 7 % , Oxygen—41.0%, Magnes ium—0.2%, Phosphorus—7.0%, S u l f u r — 0 . 2 % and Calc ium—14.7%.

36

A P P E N D I X 8

A8. A p p l i c a t i o n of L E T i n Radiobiology a n d C h e m i c a l D o s i m e t r y

A8.1 D i r e c t a n d I n d i r e c t A c t i o n

M a n y radiobiological theories have been based on the assumpt ion t h a t certain radiat ion- induced biological changes such as gene mutat ions , c h r o m a t i d and chromosomal aberrations, and cell death, depend upon app r o p r i a t e energy transfers t o one or more specific subcel lular targets. T h e effectiveness of a g iven rad ia t i on per par t i c l e , or per u n i t absorbed dose, w i l l therefore depend upon the p r o b a b i l i t y t h a t the appropriate energy transfers w i l l be made to the specific target or targets . T h i s mechanism is called direct action and target theory seeks to explain i t . However , short - l ived chemical species generated b y the rad ia t i on may migrate to the target region to produce chemical and biological change. I n th i s s i tua t i on , biological or chemical change w o u l d exh ib i t the kinetics of direct act ion a l though s t r i c t l y , the mechanism of change is ind irec t (Lea , 1946). I n some systems, and notably i n chemical dosimeters, the mechanism of rad ia t i on act ion is ent i re ly , or almost ent i re ly , t h a t of indirect action. The ox idat ion of ferrous or ferric ions i n t h e Fricke dosimeter depends upon the i n i t i a l f o rmat i on of oxidizing species such as O H , H O 2 , and H2O2 w i t h i n , or close to the t rack , and the i r subsequent diffusion, a n d chemical transformations , before a ferrous i on is encountered. T h e dependence of the direct and ind i rec t actions of r a d i a t i o n on L E T is generally complex, and i t differs f r o m one system to another.

A8.2 E x p o n e n t i a l A b s o r b e d D o s e - S u r v i v a l C u r v e s

I n some radiobiological studies, the re lat ion between the i n a c t i v a t i o n , or the s u r v i v i n g f rac t i on ( S D ) of, for example, certain enzymes, viruses, spores and cells, and the absorbed dose ( D ) , is a negative exponential of the f o r m :

SD = e~D/Do A 8 . ( l )

where Do is called the mean i n a c t i v a t i o n absorbed dose. T h i s re lat ion , when taken i n con junct ion w i t h various other considerations, suggests t h a t i n a c t i v a t i o n depends upon the random deposition of a discrete amount of rad iat ion energy (called a hit) i n a defined region (called a target) of the biological e n t i t y (Lea , 1946). The efficiency w i t h which a given q u a l i t y of rad ia t i on produces inac t ivat i on depends upon the effective shape

and size of the target , a n d the a m o u n t and f o r m of energy t h a t must be deposited w i t h i n the target zone. [Note , however, t h a t exponent ia l s u r v i v a l curves are not necessarily the result of single h i t k inet ics on ly ( H a l l , 1953, Z immer , 1961).]

A8.3 S i m p l e T a r g e t T h e o r y

Targe t theory was i n t r o d u c e d b y Dessauer (1922) and Crowther (1926) and developed m a i n l y b y Lea (1946) . Lea showed t h a t w h e n biological effects are a t t r i b u t a b l e to a single i on iz ing event occurr ing w i t h i n the target s tructure ( t h e size of w h i c h is smal l compared w i t h the spacing between i on i za t i ons ) , the vo lume of the target can be deduced f r o m the fac t t h a t the mean i n a c t i v a t i o n dose corresponds to one event per targe t vo lume. Thus , i f the s u r v i v a l k inet i cs of equat ion A 8 . ( l ) apply , the v o l u m e of the target , V = l/D0 , where D0 is measured i n ionizat ions ( or clusters of i on izat ion) r a n d o m l y d i s t r i b u t e d per u n i t vo lume.

I f more t h a n one i on i za t i on is produced i n the targe t vo lume when only one is sufficient for i n a c t i v a t i o n , the extra ionizations are " w a s t e d " and the r a d i a t i o n is used ineffectively. Because ionizat ions are more closely spaced along the t r a c k of h i g h L E T particles, t h i s tendency t o w a r d ineffectiveness becomes more m a r k e d w i t h h i g h L E T r a d i a t i o n . T h u s s imple target theory-predicts a decline i n efficiency w i t h increasing L E T (cf . the curve for T - l phage i n F igure 11) , a s i tua t i on for w h i c h a t rack average L^)T is applicable .

T h i s simple approach m u s t be modi f ied w i t h larger targets and Lea devised a m e t h o d , the associated v o l ume method , to account for the fac t t h a t clusters are, i n general, not r a n d o m l y d i s t r i b u t e d . L e a was able t o show t h a t for the i n a c t i v a t i o n of smal l viruses, the inferred target sizes correspond to the size of the viruses themselves (measured by o ther methods ) and t h a t the response declined w i t h increasing L E T (de l ta rays inc luded) as predicted.

Simple target t h e o r y i n special circumstances was f u r t h e r supported b y the w o r k of P o l l a r d (1953 ) , P o l l a r d et al . (1955) and D o l p h i n and H u t c h i n s o n (1960) . These workers es t imated molecular weights f r o m the target sizes of i r r a d i a t e d d r y enzymes such as t r y p s i n and they f o u n d t h e m t o agree closely w i t h those determined b y p l^s i cochemica l methods.

37

38 · · · A8. Application of LET in Radiobiology and Chemical Dosimetry

A8.4 T a r g e t T h e o r y a n d I n a c t i v a t i o n by M u l t i p l e I o n i z a t i o n s

I n many biological systems, the effectiveness of dif ferent radiations differs f r o m t h a t described i n the preceding paragraph and, as ind i ca ted earlier i n F igure 11 , R B E rises w i t h increasing L E T to a peak and then declines. Fur thermore , the f o r m of the surv iva l curves is often more complicated: c o m m o n l y an i n i t i a l shoulder is fol lowed b y an approx imate ly exponential p o r t i o n . Simple target theory is inappl icable i n such cases b u t target concepts m a y s t i l l be re levant . F o r example, i f biological change requires m u l t i p l e ionizat ions w i t h i n a target , the size of w h i c h is larger t h a n the t y p i c a l spacing between successive p r i m a r y ionizations along the tracks of, say, α-particles, t h e influence of r a d i a t i o n q u a l i t y can be accounted for, a t least i n broad terms.

T h e surv iva l of i r rad ia ted h a p l o i d yeast, bacter ial spores, and cells cu l tured f r o m m a n y higher organisms show this f o r m of dependence on r a d i a t i o n q u a l i t y . T h e peak response for these systems is usual ly f ound at an L X , D of about 100 keV/μηι, fo l lowed b y a decline i n effectiveness at higher L E T ( H o w a r d - F l a n d e r s , 1958; Barendsen, 1964a, b , 1967), see F igure 11. For chrom a t i d aberrations i n Tradescantia, R a n d o l p h (1964) demonstrated a marked peak i n sens i t i v i ty at an Lioo,r> of 65 keV/μΐη. Llsing accelerated heavy ions to i r rad iate m a m m a l i a n cells, T o d d (1967) f o u n d a peak for l e tha l damage ( fa i lure to f o r m colonies) t o h u m a n k idney cells at L„ ~ 220 keV/μηι; Skarsgard et a l . (1967) compared chromosome damage and colony f o r m i n g a b i l i t y i n Chinese hamster cells and f ound b o t h end po ints to have a peak response at an of ^ 1 5 0 k e V / μηι.

Howard-Flanders (1958) analysed the mechanism of rad ia t i on action b y the track-segment method , i n an a t t e m p t to account for th is quality-dependence of radiosens i t iv i ty . H e postulated t w o basic types of r a d i a t i o n i n j u r y , one of which , R 7 , is oxygen-dependent, whi le the other, R " , is independent of oxygen tension. H e proposed t h a t R ' is the result of 1 or more, b u t fewer t h a n η ions, formed w i t h i n a distance t along the t r a c k of the ioniz ing part ic le , whi le R " is the result of η or more ions w i t h i n th is distance t. T h e n , using calculated L E T spectra, he adjusted values of η and t to give the best f i t to the experimental da ta for the dependence of the 37 % surv iva l absorbed dose on r a d i a t i o n q u a l i t y .

H e concluded t h a t the da ta for p l a n t viruses, and bacteriophage T l , are consistent w i t h the v iew t h a t i n a c t i v a t i o n is the result of on ly one or t w o ionizations, w h i c h act independently i f spaced about 0.5-1.0 n m apart for phage and 2.5-5.0 n m a p a r t for tobacco mosaic v i r u s i n vacuum. For bacter ia l spores i r rad ia ted i n v a c u u m , i n a c t i v a t i o n requires 8 or more ions formed w i t h i n a distance of 3/p n m (p is the effective density

of the target region relative to w a t e r ) . I n Viciafaba l l ' results f r o m 20-30 ions and R " f rom more t h a n 30 ions w i t h i n 50 n m .

B r u s t a d (1962) applied Howard-Flanders 's (1958) track-segment theory to the analysis of data obta ined w i t h heavy i o n beams. Some of the data are shown i n F igure 11. L T sing further refinements i n the analysis he derived results s imilar to those of Howard-Flanders b u t w i t h somewhat different numerical values. For t h e cases i n which the R B E first increases w i t h increasing L E T , and then decreases (Figure 11), η is always greater t h a n u n i t y . F o r the i n h i b i t i o n of the co lony- forming a b i l i t y of haplo id Saccharomyces cerevisiae, B r u s t a d est imated t h a t η = 10 and t = 6.9/p n m , and for the i n d u c t i o n of dominant lethals i n haploid Saccharomyces cerevisiae he f o u n d η = 10 and t = 4/p n m .

Barendsen (1964b, 1967) extended the track-segment f o r m of analysis t o the inact ivat ion of m a m m a l i a n cells. H e concluded t h a t between 10 and 15 ionizations are required w i t h i n a distance of between 6.6 and 9.9 n m , t o produce i m p a i r m e n t of the prol i ferative capacity of m a m m a l i a n ( h u m a n kidney) cells.

N e a r y (1965) has proposed t h a t observable biological effects i n a cell usually require a single energy-loss event i n a macromolecule, and a chromosome aberrat ion (and possibly ce l l -k i l l ing ) is said to result f r om an exchange between two such damaged macromolecules. H e analysed the LET-dependence of aberrations produced by one-track and two- t rack mechanisms and has obtained satisfactory agreement between theory and observation ( N e a r y and Savage, 1966; Neary , Preston and Savage, 1967).'

A8.5 T a r g e t T h e o r y , Complex T a r g e t

T h e track-segment f o rm of analysis assumes t h a t when more t h a n one ionization is needed to produce an R " f o r m of i n j u r y , the required mul t ip l e ionizations (n or more) m a y be formed anywhere w i t h i n a distance t. However , i n complex biological targets, i t is more l i k e l y t h a t specific mult ip le regions w i t h i n the sensitive region have t o be affected concurrently ( B u r c h , 1967).

Tobias and Manne} ' (1964) considered the inact iva t i o n cross-section ( σ ) for bacterial spores, haploid yeast, and h u m a n k idney cells, as a funct ion of the L E T for heav}^ i on irradiat ions . Because the L E T distr ibutions for heavy i o n irradiat ions are generally quite narrow (see Figures 8 and 9 ) , curves relat ing biological effect t o average L E T are suitable for theoretical analysis. M o s t of the curves show a linear relation between σ and Ζ » , r at low L E T , followed b y an intermediate rising section between 10 and 100 keV/μηι, and finally a plateau at about 200 keV/μηι. For the inact ivat ion of h u m a n k idney cells irradiated i n nitrogen, Tobias and

A8.6 Indirect Action and Application to Chemical Dosimeters · · · 39

Μ am icy est imated t h a t σ increased approximate ly w i t h ( L ^ . T ) 2 in the intermediate region. They proposed t h a t this corresponds to irreversible l e t h a l i t y and indicates in terac t i on w i t h i n the nucleus of t w o parts of the same ioniz ing track . For lysozyme, however, the dependence of σ on ϊ ο ο , τ is sub-linear between about 20 and 500 keV/μηι, and even at the lat ter L E T , no p lateau is reached.

B u r c h (1967) found t h a t at h igh L E T (above L m ~ 40 keV/μηα or L w ~ 60 keV/μηι) the experimental results for the heavy ion i r rad iat ion of m a m m a l i a n cells (Deer ing and Rice, 1962; Barendsen, 1964b) are described b y the equation:

σ = σι(1 - e " k L 3 ) A 8 . ( 2 )

and the re lat ion holds for a cut-off value of 200 eV and for . Here σ is the inac t i va t i on cross-section, σ χ is the l i m i t i n g i n a c t i v a t i o n cross-section found i n the p lateau at very h i g h L E T , and k is a constant. I n other words, at h i g h L E T and before saturation effects intervene, σ is proport ional to (L 2oo) 3 or ( L x ) 3 .

Whether or not analyses of the type described above w i l l u l t i m a t e l y become established remains for f u t u r e experiment to decide. They are presented here t o i n d i cate the manner i n which the d is tr ibut ion of energy transfer as described by L E T can be used to in terpre t the mechanism of radiat ion action.

A8.6 I n d i r e c t A c t i o n a n d A p p l i c a t i o n to C h e m i c a l Dos imeters

Analysis of the qua l i t y dependence of indirect act ion presents some formidable and at present insoluble

problems. Outs tand ing among these is the d i f f i cu l ty of assessing the behavior of low energy electrons i n aqueous media.

G r a y (1955) t r i e d t o account for the molecular y i e l d i n aqueous media b y assuming t h a t electrons whose energy lies between 60 and 500 eV y ie ld only molecular hydrogen and hydrogen peroxide, and t h a t electrons w i t h energy less t h a n 60 eV or greater t h a n 500 eV y i e l d only hydrogen atoms and h y d r o x y l radicals.

B u r c h (1959) t r i e d t o explain the effect of r a d i a t i o n q u a l i t y on y ie ld i n t h e ferrous and eerie sulfate dos imeters on the basis of a model w h i c h , a l though m u c h more complicated t h a n Gray ' s (1955) model, s t i l l f e l l far short , as he po inted out , of the complexi ty of a c tua l events.

T h e experimental results for chemical yields w i t h 6 0 Co y rays and p l u t o n i u m a particles were used to derive the u n k n o w n parameters i n the Jaffe co lumnar re -combinat ion f ormulae (Jaffe, 1913). Chemical yields for other qualit ies of rad ia t i on were t h e n predic ted . Despite the approx imate nature of several of the assumptions i n the model , i t gives good agreement w i t h experimental results (Col l inson et a l . , 1960; Fregene, 1967). Fregene (1967) finds t h a t when G F e + + + ( t h e y ie ld of ferric ions per 100 eV) is p l o t t ed against the mean i n i t i a l energ3 r of electrons f r o m x, y and β rays , a l l the data lie on the same smooth curve. I t is interest ing t h a t (?Fe+++ is s t i l l r is ing at the lowest mean L E T invest igated ( for 15 M e V χ rays ) . For pract i ca l p u r poses, GFe+++ values can be interpolated f r o m such a graph, for most sources of x, y, and β rays, prov ided t h e i r dose-weighted mean L E T can be calculated.

A P P E N D I X 9

A9. L i s t of Symbols 1

A9.1 G e n e r a l

a Aa

ß e h d

I PI

k

dl

Al

m Am Mo Mi

m0

Ma = MA = ΝΛ = AN =

An{E) =

An(L) =

n(h) =

Ρ = r = σ =

a(Q)dQ =

cross sectional area. element of cross sectional area. v/c where c — ve loc i ty of l i g h t . electronic charge. pulse height. diameter of sphere. distance. surface density measured i n mass/ u n i t area. distance such t h a t mul t ip l e scatteri n g occurs w i t h smal l p robab i l i t y , distance travel led b y part i c le i n losing energy dE. small element of distance i n w h i c h part ic le loses energy AE. mass (genera l ly ) . small element of mass. rest mass of inc ident part ic le . re lat ive atomic mass (of inc ident par t i c l e ) . rest mass of s truck partic le . ( ra 0 = mass of recoil electron) (vio = mass of recoil nucleus) . moc2 = rest mass energy of recoil electron. re lat ive atomic mass (of absorber) . molar mass (of absorber) . Avogadro 's number. number of particles entering Aa. number of particles w i t h energy Ε per u n i t energy i n t e r v a l . number of particles of L E T L per u n i t L E T i n t e r v a l . number of pulses of height h. density. geometric cut-off distance, collision cross-section, p r o b a b i l i t y for collisions i n v o l v i n g energ}^ transfers between Q and Q + dQ.

4 4

21 21 34

3 21

21

26

4 4 4

21

5 21

21 21 21 21

4

4 34 21

7 5

21

s = cross section = a(Q) / 27Γβ9^0.

/ moc2ß2

21

σ = i n a c t i v a t i o n cross section. 38 θ = angle of deflection of inc ident par

t ic le . 7 t = distance along a track 38

Τ = t o t a l t r a c k l ength . 2 ν = ve loc i ty of the part ic le . 21

V = vo lume of target . 37 ζ = nuclear charge number of the par

t ic le . 21 Ζ = atomic number of absorber. 21

1 2 T h i s l i s t of s} 'mbols is inc luded to f a c i l i t a t e the reading of t h i s repor t and n o t to propose these symbols for general use.

1 3 F i r s t use or d e f i n i t i o n .

A9.2 A b s o r b e d Dose

D = absorbed dose = AED/Am. 4 d ( L ) = d i s t r i b u t i o n of absorbed dose w i t h

respect t o L . 2,8 D ( L ) = cumulat ive d i s t r i b u t i o n of absorbed

dose w i t h respect to L . 2,8 dta(L) = d i s t r i b u t i o n of absorbed dose w i t h

respect to L w i t h cut-off energy A . 8 DA(L) = cumulat ive d i s t r i b u t i o n of absorbed

dose w i t h respect to L w i t h cut-off energy A . 8

d * { L ) = d i s t r i b u t i o n of absorbed dose w i t h respect t o L (no cut-off Δ). 8

Doo(L) = cumulat ive d i s t r i b u t i o n of absorbed dose w i t h respect to L (no cut-off A ) . 8

d(E) = d i s t r i b u t i o n of absorbed dose w i t h respect to part ic le energy (=Σφ(Ε)). 9

UA{E) = d i s t r i b u t i o n of absorbed dose w i t h respect to part ic le energy w i t h cut -off Α(=ΣΑΦ{Ε)). * 11

d«>{E) = d i s t r i b u t i o n of absorbed dose w i t h respect t o part ic le energy w i t h no cut-off (=Σΰ0φ(Ε)). 9

D E = dose equivalent . 16 Do = mean i n a c t i v a t i o n absorbed dose. 37

= reciprocal of the f inal slope of the log surv iva l vs. absorbed dose curve.

Q F = q u a l i t y factor. 16

A9.3 E n e r g y

dE = mean energy loss due t o collisions w i t h energy transfers less t h a n some

40

A9.6 Linear Energy Transfer · · · 41

specified value Δ . ( I C R U , 1968). = AEA also ( w h i c h is the average of a

d i s t r ibut i on , i n O, U and q on ly , Figure 3 ) .

dEL = mean energy loss due t o collisions i n traversing dl ( I C R U , 1962a).

Ε = incident part i c le k inet i c energy. (Ε' = average inc ident energy.)

Eo = i n i t i a l part ic le k inet i c energy. (E0 = average i n i t i a l k inet i c energy) .

Ε' = t ransmi t ted part i c le k inet i c energy. (E — average t r a n s m i t t e d energy.)

AE = energy transferred b y part ic le of energy Ε i n m o v i n g t h r o u g h a distance AL

AE = average of d i s t r i b u t i o n of AE's for particles of inc ident energy E.

AEf = average energy loss per scattering center per u n i t area.

AE* = energy lost b y a part ic le of energy Ε i n t rave l l ing a thickness AR.

AED = energy absorbed f r o m a part ic le of energy Ε i n a smal l element of mass Am.

AEr = energy absorbed w i t h i n a cyl inder of radius r about the interac t i on site (includes d is t r ibut ions i n 0 , U} q, Qf

and 7 ' , Figure 3 ) . €JI/ = normalized energy

= E/Mi A = cut-off energy transfer, i.e. energy

transfers less t h a n Δ are considered p a r t of the m a i n t rack , those greater t h a n A const i tute a separate delta track.

Q = energy transferred b y a part ic le of energy Ε d u r i n g a single interact i o n .

Qmax = m a x i m u m possible value of Q. Q' = an energy transfer greater t h a n

A(Qi\ Qz', QZ ···). q = an energy transfer less t h a n A(qi,

φ · · · ) ·

Φ(Ε) = fluence spectrum of particles of energy E( = An(E)/Aa per u n i t energy i n t e r v a l ) .

An(E) — d i s t r ibut i on i n energy of particles entering Δα.

n(Eo)dEo = number of particles w i t h i n i t i a l kinetic energies between Eo and E0 + dEo, per u n i t volume.

U = energy transferred to a local site. 7 = energy transferred t o photons ( 7 1 ,

Y2 , 73 · · · ) ·

A9.4 F l u e n c e

4 9 2

21

26

23

2,7

ο 2

9 6

Page 1 3

Φ = part ic le fluence = AN Aa

_ number of inc ident particles inc ident area

Φ(Ε) = charged part ic le fluence d i s t r i b u t i o n w i t h respect to energy Ε = An(E)/ Aa.

φ ο ο ( ί / ) = charged part ic le d i s t r i b u t i o n w i t h respect t o Loo = An(L)/Aa = Φ(Ε)/ ( d L / d L ) .

y(E0, E ) — electron fluence spectrum w h i c h results i f one partic le of i n i t i a l energy E0 is generated per u n i t vo lume.

A9.5 I o n i z a t i o n

E N = exc i tat ion or ion izat ion levels i n absorber atoms.

fn = oscillator strengths. I = mean exc i tat ion- ionizat ion energy

for absorber atoms. W = average energy expended per i o n

pair i n a gas. δ = density correction due to polariza

t i o n .

A9.6 L i n e a r E n e r g y T r a n s f e r

L = Linear energy transfer (or stopping power ) , of a part ic le of energy Ε i n m o v i n g dl or Al = dE/dl = Ä E / A I [= dEL/dl, 1962 I C R U def ini t ion] . L inear energy transfer w i t h cut-off

\AIJA \άΙ/ΑΛ

l inear energy transfer w i t h Δ set a t 100 eV. linear energy transfer w i t h no cut-off Δ , i.e. a l l transfers of energy up t o a m a x i m u m possible considered p a r t of the m a i n t r a c k = dE/dl.

Φ(Ε) — part ic le fluence d i s t r i b u t i o n i n L E T = An(L)/Aa where An(L) is the d i s t r i b u t i o n i n L E T of particles entering Aa. A E R / Δ Ι ~ where A E R is the energy deposited w i t h i n a cyl inder of radius r f r o m the interac t i on site and

LA =

L100

Lea

energy Δ

L r =

41

10

21 21

21

22

2,7

i n cludes 0 , U, q, Q', and y.

42 · · . A9. List of Symbols

Page 1 3

L = dE/dR, an a p p r o x i m a t i o n for dE/dl w h e n R is v e r y smal l . 26

cU/d / = normal ized s topp ing power [ = ( l / * 2 ) . ( d E / d Z ) ] . 23

L E T - A v e r a g e s

L T

LA,τ

/.oo t r a c k average L E T = / t(L)LdL. 2,8

J/»CO

' U(L)LdL ο

= t r a c k average L E T w i t h cut-off Δ . 8

L o o , r = / t o o ( L ) L d L

= t r a c k average L E T w i t h no cut-off . 8 L D = absorbed dose average L E T 3

L A ,

ί d(L)LdL

J/»oo dA(L)Ldl

ο absorbed dose average L E T w i t h 8 cut-of f Δ .

L o o . D = / doo(L)LdL

absorbed dose average L E T w i t h no cut-off .

L E T - D i s t r i b u t i o n s

d(L)

D ( L )

ML)

DA(L)

d o o ( L )

D c o ( L )

d i s t r i b u t i o n of absorbed dose w i t h respect to L E T . 2 c u m u l a t i v e d i s t r i b u t i o n of absorbed dose w i t h respect t o L E T . 2 d i s t r i b u t i o n of absorbed dose w i t h respect t o L E T w i t h cut-off energy Δ = ( d / V d ^ r 1 * S c u m u l a t i v e d i s t r i b u t i o n of absorbed dose w i t h respect t o L E T w i t h cut-off energy Δ . 8,11 d i s t r i b u t i o n of absorbed dose w i t h respect t o L E T (no cut-off Δ ) . 8 c u m u l a t i v e d i s t r i b u t i o n of absorbed dose w i t h respect t o L E T (no cut-of f Α ) · variance of t & ( L ) = LA,T(LA,D — LA,T)>

11

11

t ( L ) = d i s t r i b u t i o n of track length w i t h respect to L E T .

T ( L ) = cumulat ive d i s t r ibut ion of t rack length w i t h respect to L E T .

IA(L) = d i s t r i b u t i o n of track lengths w i t h respect to L E T w i t h energy cut-off Δ .

T A ( L ) — cumulat ive d i s t r ibut ion of t rack lengths w i t h respect to L E T w i t h energy cut-off Δ .

t a o ( L ) = d i s t r i b u t i o n of t rack lengths w i t h respect t o L E T (no cut -o f f ) .

A ]

nj R --

R --

Rcsoa

R E 0 —

Rmux

7?o =

s = SQ =

Y =

1 3 F i r s t use or d e f i n i t i o n .

i n d i v i d u a l event size. = the amount of energy deposited i n a

sphere d iv ided by i ts diameter d. P ( F ) = probab i l i t y d i s t r ibut i on of a l l events

i n Y. local energy density, incremental local energy density, p robab i l i t y d i s t r i b u t i o n of local energy density.

Ζ AZ

P ( Z )

8

24

24

A9.7 R a n g e

constants of particle range-energy equation 25

range 25 mean projected range = absorber thickness t r a n s m i t t i n g 5 0 % of the perpendicularly incident particles, the "cont inuous slowing down app r o x i m a t i o n " range = t o t a l t rack length traversed by a particle , range of particles of average incident energy E0. absolute projected range (reduces transmission to an undetectable level ) . extrapolated projected range, specified projected range, e.g. R1% . mean p a t h length , corresponds to Äcsda approximately . extrapolated p a t h length.

A9.8 Microclosimetr v

25 24 25

24 25

19 3 3

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1 Discussion on International Units and Standards for X-ray Work, B r i t . J . Rad io l . 23, 64 (1927)

2 International X-ray Unit of Intensity, B r i t . J . R a d i o l . (new series) 1, 363 (1928)

3 Report of Committee on Standardization of X-ray Measurements, Radio logy 22, 289 (1934)

4 Recommendations of the International Committee for Radiological Units, Radio logy 23, 580 (1934)

5 Recommendations of the International Committee for Radiological Units, Radio logy 29, 634 (1937)

6 Report of International Commission on Radiological Protection and International Commission on Radiological Units, N a t i o n a l B u r e a u of Standards H a n d b o o k 47 (U.S. Government P r i n t i n g Office, W a s h i n g t o n , D . C . , 1951)

7 Recommendations of the International Commission for Radiological Units, Radio logy 62, 106 (1954)

8 Report of the International Commission on Radiological Units and Measurements (ICRU) 1956, N a t i o n a l B u r e a u of Standards Handbook 62 (U.S . Government P r i n t i n g Office, Wash ington , D .C . , 1957)

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o ther languages.

Index

Absorbed dose, 1, 2, 4, 9, 16, 40 Average of d i s t r i b u t i o n i n L E T , 8 ,14 ,17 ,18 ,19 , 42 Calculat ions of d i s t r i b u t i o n s i n L E T for w a t e r , 8-14 D i s t r i b u t i o n i n L E T , 2, 8-14, 16,17, 18 19, 20, 40, 42 M e a n i n a c t i v a t i o n (Z>0), 37, 38, 40

Absorber thickness, 6, 7, 18, 24, 25, 26 A t t e n u a t i o n , 25

Bio log ica l effect, 1, 15, 16, 18, 19, 37, 38, 39 B o r n approx imat ion , 22 Bremsst rah lung , 10, 22

C a v i t y i on iza t i on , theory , 10 Charge exchange, 32, 33 Chemical dos imetry , 1, 37, 39 Chemical y i e l d , 1, 39 Chromosome aberrat ions , 37, 38 C l o u d chamber, 18, 24, 25, 35, 37 C luster , of ions, 4, 5, 6, 35, 37

frequency, 35 overlap, 35

Co l l i s ion , cross-section, 5, 21, 35, 40 p r o b a b i l i t y , 21, 40 theory , classical, 5

Co lumnar recombinat ion , 17 Compensation (of energy loss), 7 C o m p t o n scat ter ing , 3 Concepts of L E T , 6, 17, 18, 34, 41 Cont inuous s lowing down a p p r o x i m a t i o n (csda), 9, 10

calculat ions, 9, 10 csda model , 10 csda range ( R c s d a ) , 24, 28, 29, 42

Cross section, d i f f e rent ia l , 21 , 40 C u m u l a t i v e d i s t r i b u t i o n s (of absorbed dose i n L E T ) , 11, 12,

13, 40 Cut-o f f , energy, 2, 3, 6, 8, 9, 10, 11, 14, 18, 30, 31 , 32, 33, 39, 41

distance or range, 2, 6, 7, 18, 30, 40

D e l t a t racks (or rays ) , 2, 4, 5, 6, 9 ,10 ,14 , 18,19, 37 d i s t r i b u t i o n , 5

D e n s i t y correct ion (for p o l a r i z a t i o n ) , 22, 28, 3 1 , 41 Detectors , 18, 24, 25, 26, 28, 34, 35 D i r e c t act ion , 37 * Distance (or range) cut-off , 2, 6, 7, 18, 30, 40 Dose equivalent , 15, 16, 17, 40

E l e c t r o n fluence spec t rum, 2, 10, 41 Electrons , low energ} r , 8, 11-14, 30 Energy cut-off (Δ) , 2, 3, 6, 8, 9 ,10 ,11 ,14 ,18 , 30, 31, 32, 33, 41 Energy dens i ty , 3

incrementa l loca l , 3, 19, 42 local , 3, 19, 42

Energy loss, 2, 3, 6, 7, 18, 19, 21, 24, 27, 28, 29, 30, 31, 41 spectrum, for electrons, 5, 6, 41

E n e r g y per co l l i s i on (e lectrons) , 6, 18, 41 E n e r g y t r a n s f e r (Q), 5, 7, 22, 23, 41 E q u i l i b r i u m , r a d i a t i o n , 9 E v e n t size, i n d i v i d u a l ( F ) , 3, 19, 42 E x c i t a t i o n energy levels (En)t 21 , 22, 36, 41 E x c i t a t i o n energy (mean) / , 2 1 , 36, 41

elements and compounds, 36 r e l a t i o n t o Z, 36

E x c i t a t i o n vs . i o n i z a t i o n , 2, 3, 4, 7, 10, 22

Fluence d i s t r i b u t i o n i n energy, 8, 10, 18, 41 Ferro us- ferr ic (Fricke) dosimeter, 1, 15, 37, 39

H e a v y charged part ic les or ions, 5, 14 effects, 38, 39 h i g h energy, 31 low energ} ' , 33

H i t , s ing le , 37 m u l t i p l e , 38

I o n c luster , 4, 5, 6, 35, 37 I o n i z a t i o n chamber, 17, 25 I n a c t i v a t i o n , of bacter ia and yeast , 15, 38, 39

enzymes, 15, 37, 39 m a m m a l i a n cells, 15, 38, 39 t r a n s f o r m i n g D N A , 15 v iruses , 15, 37

I n a c t i v a t i o n cross sect ion, 38, 39, 40 I n a c t i v a t i o n , mean dose, 37, 38, 40 I n d i r e c t a c t i o n , 37 I n t e r a c t i o n of r a d i a t i o n w i t h m a t t e r , 1, 3, 4, 21, 22, 23

L indhard -Schar f f r e l a t i o n , 33 L inear energ}^ transfer , L E T

appl i cat ions , 15 concepts, 6, 17, 18, 34, 41 d e f i n i t i o n , 2, 6, 41 l i m i t a t i o n s , 6, 17, 34 measurement , 26

L inear energy transfer d i s t r i b u t i o n s , 2, 3, 8-14, 18, 20, 42 absorbed dose average, 3, 8, 9 ,14 ,16 ,17 ,18 ,19 , 20, 42 at a d e p t h , 14 averages, 2, 3, 8, 17, 18, 19, 20, 42 i n water , 11-13, 42 measurement , 34 t r a c k average, 3, 8, 14, 18, 19, 42

L inear s t o p p i n g power, 2, 7, 8, 22 Loca l energy dens i ty (Z), 3, 19, 42

i n c r e m e n t a l (ΔΖ ) , 3, 19, 42 Loca l energy transfer , 2, 6, 8, 22

Mass s t o p p i n g power, 22-24, 27-33 M e a n l inear i o n dens i ty , 2, 3 Mic rodos imeter , 8 M i c r o d o s i m e t r y , 3, 19, 42

Index

M i x e d r a d i a t i o n field, 16, 19 Models of t r a c k s t r u c t u r e , δ Monoenerget ic charged par t i c l e s , 18

N e u t r o n s , 4, 13, 14, 16, 17, 34 N u c l e a r in terac t i ons , 3, 23, 3 1 , 33

Osc i l la tor s t r e n g t h ( / n ) , 21 , 36, 41

Pa i r p r o d u c t i o n , 3 P a t h l e n g t h , 24, 34, 42

d i s t r i b u t i o n , 34 mean , 24

P a r t i c l e fluence, 4, 8, 9, 10, 11, 41 s p e c t r u m , 4, 8, 11, 41

Photoe lectr ic effect, 3 P o l a r i z a t i o n effect, 22, 41 P o s i t r o n , s t opp ing power, 30 P r o b a b i l i t y d i s t r i b u t i o n 3, 19, 42 P r o p o r t i o n a l counter , 17, 34 P r o p o r t i o n a l i o n i z a t i o n chamber, 34 Pulse he ight spec t rum, 17, 34, 40

Q u a l i t y factor , 16, 17, 18, 20, 40

R a d i a t i o n b io logy , 1 , 15, 16, 34, 37, 38, 39 R a d i a t i o n effectiveness, 31 R a d i a t i o n hazards, 16, 17 R a d i a t i o n p r o t e c t i o n , 1, 15, 16, 17, 20 R a d i a t i o n q u a l i t y , 1 , 3, 14, 18, 19, 20, 34, 38

(methods other t h a n L E T ) , 3, 19, 42 R a d i a t i o n , u n k n o w n compos i t i on , 17 Rad ia t ive loss, 3, 22, 32 Range. 2, 5, 24

absolute pro jected range, 25, 42 data , electrons, 28-32 data , protons , 29, 32 def init ions, 24-25 extrapolated p a t h l e n g t h , 25, 42 Ρ extrapolated pro jec ted range, 24, 28, 42 mean p a t h l eng th , 24, 42 mean pro jected range, 24, 42 measurement, 24-25 normalized range, 32

R e a d u , 24, 29, 42

specified pro jected range, 25, 42 Range—Energy re lat ions , 25, 29, 30, 42 Recoi l angle, 5, 6 Recoi l part i c les , 5, 22 Re la t ive Bio log ica l Effectiveness ( R B E ) , 4, 15, 38

R B E vs . L E T , 15, 16, 38 Restr i c ted s topp ing power, 7, 8, 9, 31 , 41

Second moment (of d i s t r i b u t i o n ) , 14 Secondary electrons, 6, 10, 18 Shell correct ions, 22, 36 Spat ia l d i s t r i b u t i o n , 1, 4 Step-wise method (of ca l cu la t ing d i s t r i b u t i o n s ) , 10 Stochastic features, 1, 18, 19, 34 Stopp ing power, 1, 21 , 24, 41

electrons, 22 heavy part ic les , 22 measurement, 26 res t r i c t ed , 7, 8, 9, 31 , 41 theoret i ca l and exper imental values, electrons, 27

p r o t o n , etc. , 31, 32 Stragg l ing (range), 24

(energy), 24 S u r v i v a l curves, exponent ia l , 37

m u l t i - h i t or target , 38

T a r g e t region, 1, 18, 37, 38 Target theory , 18, 37

m u l t i p l e ionizat ions , 38 s imple , 37

Tissue equivalent gas, 34 counter , 34 i o n chamber, 17 p las t i c , 34

T r a c k average L E T , 2, 8, 14, 18, 19, 37, 42 T r a c k " c o r e " , 5, 14 T r a c k l e n g t h , 2, 8, 18, 19, 24, 40 T r a c k segment method , 14, 15, 18, 38 T r a n s i t i o n probab i l i t i e s , 21 T r a n s m i t t e d beam energy, 25

Variance (σ 2 ) of d i s t r i b u t i o n , 14

Y concept, 3, 19, 42 d i s t r i b u t i o n s 3, 19, 42