OFFICIAL ORGAN OF THE RADIATION RESEARCH SOCIETY

RADIATION RESEARCH

MANAGING EDITOR: TITUS C. EVANS

Volume 47, 1971

Academic Press • New York and London

Copyright © , 1971, by A C A D E M I C P R E S S , INC.

A L L R I G H T S R E S E R V E D

N o p a r i of t h i s v o l u m e m a y be r e p r o d u c e d i n a n y

f o r m b y p h o l o s t a t , m i c r o f i l m , b y r e t r i e v a l s y s t e m t

o r a n y o t h e r m e a n s , w i t h o u l w r i l l e n p e r m i s s i o n

f r o m t h e p u b l i s h e r s .

Made in the United States of America

RADIATION RESEARCH

OFFICIAL ORGAN OF THE RADIATION RESEARCH SOCIETY

BOARD OF EDITORS: Managing Editor: TITUS C. EVANS, College of Medicine, Universitv of Iowa, Iowa City, Iowa 52240

G . E. ADAMS, Alt. Vernon Hospital, England

M . J . BERGER, National Bureau of Standards

J. G . CARLSOX, Universitv of Ten­nessee

11. A . CONARD, Brookhaven National Laboratorv

S. B. CURTIS, Universitv of California W. GORDV, Duke Universitv D. G R A H N , Argonne National Labora­

torv M . L. GRIE.M, Universitv of Chicago II. H . H A Y N E S , York University,

Canada

J. JAGGER, University of Texas R . H . JOHNSEN, Florida State Uni­

versity R . F. K A L L M A X , Stanford University S. KOXDO, Osaka University, Japan T. L. PHILLIPS, University of Cali­

fornia P. RIESZ, National Institutes of Health W. C. SNIPES, Pennsylvania State

University H . D. SUIT, University of Texas J. K . THOMAS, University of Notre

Dame L. J. TOLMACH, Washington University G. M . WOOD WELL, Brookhaven Na­

tional Laboratorv

OFFICERS OFTHE SOCIETY: President, N . F. BARR, Atomic Energy Commis-sion, Washington, D.C.

Vice President (and President Elect), A. D. CONGER, Temple University, Philadelphia, Pennsylvania.

Secretary-Treasurer, FALCONER SMITH, Biology Department, American Uni­versity, Washington, D. C. 20016

Managing Editor, TITUS C. EVANS, University of Iowa, Iowa City, Iowa Executive Secretary. RICHARD J. B U R K , JR. , 4211 39th Street, N . W., Washington,

D . C. 20010 Assistant to M a n a g i n g Editor: DOROTHY D. SCHOTTELIUS, University of Iowa, Iowa

City, Iowa

ANNUAL MEETINGS:

1972: Mav 14-18, Portland, Oregon 1973: April 29-May 3, St. Louis, Missouri 1974: July 13-20, V International Congress Radiation Research, Seattle,

Washington

VOLUME 47, 1971

Copyright © 1971, by Academic Press, Inc., New York, N . Y . 10003, U . S. A.

Second class postage paid at Baltimore, M d . 21202

Councilors Radiation Research Society 1971-1972

PHYSICS

R. J. Shalek, University of Texas M . Ter-Pogossian, Washington University

BIOLOGY

R. F. Kallman, Stanford University S. Wolff, University of California

MEDICINE

V. P. Bond, Brookhaven National Laboratory L. K . Bustad, University of California

CHEMISTRY

Jack Schubert, University of Pittsburgh J. Weiss, Brookhaven National Laboratory

A T - L A R G E

E. L. Alpen, Battelle Memorial Institute G. F. Whitmore, Ontario Cancer Institute

R A D I A T I O N R E S E A R C H 47, Ü i - V (1971)

CONTENTS OF VOLUME 47

N ü M H E R 1, J ü L Y 1971

E . K . H U S S M A N N A N D W . L . M C L A U G H L I N . Dose-Distribution Measurement of High-Inten-sity Pulsed Radiation by Means of Holographie Interferometry 1

A L H K K C H T M . K E L L E R E R A X D H A R A L D H . R O S S I . R B E and the Primary Mechanism of

Radiation Action 15 A . R u s s E L L J O N E S . The Radiolysis of Aliphatic Carboxylic Acids: The Decarboxylation

of Normal Acids at 38° 35 W . E . F A L C O N E R , R. S A L O V E Y , W . A . S U N D E R , A N D L . G . W A L K E R . Effects of Dose, Dose

Rate, and Trace Impurities on n-Hexadecane Radiolysis 41 E . L U K A S O V Ä A N D E . P A L E C E K . Changes in D N A Secondary Structure after X-Irradiation 51 A . C H E C C U C C I , G . C O L O M H E T T I , A . M . C O L Ü M B Ü , P . R. C R I P P A , F . L E N C I , A N D G . M O N -

T A G N O L T . Influence of the Molecular Organization on Radical Yield of Organic Com­pounds 66

P A U L M I L V Y . Dose Dependence of the Transfer of Radiation-Induced Free Radicals from D N A to Cysteamine 83

K r UT G . H O F E R A N D W A L T E R L . H U G H E S . Radiotoxicity of Intranuclear Tr i tum, 1 2 5Iodine

and 1 3 1Iodine 94 8. C H I U A N D A . M . R A U T H . A Comparison of the Sensitivity to Ultraviolet Light of Mouse

L Cells and Mouse Bone Marrow Cells Assayed i n v i t r o 110 J . T H O M A S P A Y N E A N D E D W A R D I. S H A W . A Radiation Effect from Low Doses of Tritiated

Thymidine or X-Rays 123 J O S E P H E . T R A Y N O R A N D H A R O L D W . C A S E Y . Five-Year Follow-up of Primates Exposed to

55 Mev Protons R O N A L D F . H A G E M A N N , C U R T I S P. S I G D E S T A D , A N D S. L E S H E R . Intestinal Crypt Survival 143

and Total and Per Crypt Levels of Proliferative Cellularity Following Irradiation: Fractionated X-ray Exposures 149

R O N A L D F . H A G E M A N N AXD S . L E S H E R . Intestinal Crypt Survival and Total and Per Crypt Levels of Proliferative Cellularity Following Irradiation: Age Response and Animal Lethality 159

S O L M . M I C H A E L S O N A N D B E R N A R D F. S C H R E I N E H , J R . Cardiopulmonary Effects of Upper-

Body X-Irradiation in the Dog 168 R O B E R T S R C G I I , M A R L I S W O H L F R O M M , A N D R E V A R M A , N A N C Y S P E N C E R , A N D W . S T A N ­

FORD. A Reexamination of the Mouse Embryonic Radiation Cataract Studies 182 P. V . V I T T O R I O , J . F . W H I T F I E L D , A N D IL H . R I X O N . The Radioprotective and Therapeutic

Effects of Imidazole and Krythropoietin on the Erythropoiesis and Survival of Irradi-ated Mice 191

W . H . K N O S P E , J . B L O M , II. B . G O L D S T E I N , A N D W . H . C R O S H Y . Delayed Bone Marrow

Aplasia in Rats Protected Against Lethal Irradiation by Allogeneic Marrow Trans­plantation 199

M A R G A R E T W I E N E R . Effects of X-Irradiation on Steroid Synthesis in the Human Placenta 213 K . R. T R A E L N E S , H . E R N S T , K . I. A L T M A N , A N D L . H . H E M P E L M A N N . The Effect of X-Ir ­

radiation on Collagen Metabolism in Polyvinyl-Sponge Granulomas 224

Abstracts of Papers for the Nineteenth Annual Meeting of the Radiation Research Society, Boston, Massachusetts, May 9-13, 1971 233

A N N O U N C E M E N T 357

N U M B E R 2, A U G U S T 1971

A L B R E C H T M . K E L L E R E R . Considerations on the Random Traversal of Convex Bodies and Solutions for General Cylinders 359

A L B R E C H T M . K E L L E R E R . An Assessment of Wall Effects in Microdosimetric Measure-ments 377

M A R I E M . G R E N A X A N D E D M U N D S. C O P E L A N D . Structure-Function Studies of the Amino-

thiol Radioprotectants. Effect of Carbon Chain Length in Mercaptoethylamine Ho-mologs-Mammalian Radioprotection 387

J O H N C H R Y S O C H O O S A N D D A V I D S. S H I H A B J . X-Radiolysis of Aqueous Erythrosin 392

R. K A T Z , B . A C K E R S O N , M . H O M A Y O O N F A R , A N D S. C . S H A R M A . Inactivation of Cells by

Heav}^ Iron Bombardment 402 J A M E S S H A E F F E R A N D T I M O T H Y M E R Z . A Comparison of Unscheduled D N A Synthesis, D 0 ,

Cell Recovery, and Chromosome Number in x-Irradiated Mammalian Cell Lines 426 W. U . S H I P L E Y , M . M . E L K I N D , A N D W. B . P R A T H E R . Potentiation of X - R a y Kill ing by

5-Bromodeoxyuridine in Chinese Hamster Cells: A Reduction in Capacity for Sub-lethal Damage Accumulation 437

H O W A R D B. N E W C O M B E A N D J O H N F . M C G R E G O R . Persistence of Heritable Changes in an

Irradiated Rat Population After Cessation of the Exposures 450 F . F . H A H N , E . G O L D S T E I N , A N D D . L . D U N G W O R T H . Effect of Whole-Body X-Irradiation

on Pulmonary Bactericidal Function 461 T H O M A S C . W E L B O U R N E A N D G . B . S P U R R . Mechanism of Renal Acid Excretion Following

X-Irradiation of the Dog 472 S H I E L D S W A R R E N A N D O L I V E G A T E S . The Induction of Leukemia and Life Shortening in

Mice by Continuous Low-Level External Gamma Radiation 480 Y . N . K A B A K O V . Factors Influencing Clasmatosis i n V i v o : Observation of Periodicity of

Clasmatosis in Lymphocytes and Hematopoietic Recovery After Irradiation 491 K U O - P A O Y A N G A N D O T T O W. N E U H A U S . I n V i t r o Uptake of Model Amino Acids by Rat

Liver Following Whole-Body 7-Irradiation 500 C . T . G A F F E Y . Relation Between R B E and L E T to Inhibit Neural Impulse Conduction 511 J O H N M . Y U H A S . Biological Factors Affecting the Radioprotective Efhciency of S-2-13-

AminopropylaminoJEthylphosphorothioic Acid (WR-2721): LDÖO.7) Doses 520 C H A R L E S O. M C K I N N E Y A N D F R E D E R I C K B . T U R N E R . Genetic Variation in Irradiated and

Nonirradiated Populations of the Lizard, U t a s l a n s b u r i a n a 530 J O H N B . S T O R E R . Chemical Protection of the Mouse against Radiation-Induced Life Short­

ening 537 H . Y O S H I K U R A . Radiation Effects on Macromolecular Synthesis in Contact Inhibition-

Sensitive Cells Synchronized by Medium Change 548 H . I. P O P E S C U A N D D. T. S T A F A N E S C U . Cytogenetic Investigation of lndustrial Workers

Occupationally Exposed to Gamma Ra>̂ s 562 B O O K R E V I E W 571

A N N O U N C E M E N T 572

N U M B E R 3, S E P T E M B E R 1971

A N T O N I J E D U L C I C A N D J A N K O N . H E R A K . Crystal Structural Dependence of Radiation-

Induced Radicals in Thymine: An E S R Study 573 T . W. A R M S T R O N G A N D B. L . B I S H O P . Calculation of the Absorbed Dose and Dose Equiva-

lent Induced by Medium-Energy Neutrons and Protons and Comparison with Ex­periment 581

S T A N L E Y K R O N E N B E R G , R O B E R T L U X , A X D K R I S T I A N N I L S O N . Relative Biological Effec-

tiveness of X-Rays Delivered at Very High Dose Rates to Radish Seeds (Raphanits sativus) 589

P. C . S H R A G G E , II. B . M I C H A E L S , A X D J . W . H U N T . Factors Affecting the Rate of Hydrated Klectron Attack 011 Polynukleotides 598

P. N E T A A N D R O B E R T H . S C H U L E R . Rate Constants for Reaction of Hydrogen Atoms with Compounds of Biochemical Interest 612

M . H . W Y K O F F . Distribution of 1 3 4 Cs in the Conceptus of the Pregnant Rat 628 M . R. R A J U , M . G N A N A P U R A N I , B . S T A C K L E R , B . I . M A R T I N S , U . M A D H V A N A T H , J . H O W A R D ,

J . T . L Y M A N , A N D R . K . M O R T I M E R . Induction of Heteroallelic Reversions and Lethal-ity in Saccharomyces cerevisiae Exposed to Radiations of Various L E T ( 6 0Co 7 Rays, Heavy Ions, and 7r~Mesons) in Air and Nitrogen Atmospheres 635

H O W A R D H . V O G E L , J R . A N D R O B E R T Z A L D I V A R . Cocarcinogenesis: The Interaction of Chem­

ical and Physical Agents 644 B A L A L A K C H A U R A A N D J . B . C L A R K . Recoveiy from X - R a y and Nitrogen Mustard Inacti-

vation of Escherichia coli by 2,4-Dinitrophenol Treatment 660 W. C . D E W E Y , L . E . S T O X E , H . H . M I L L E R , A N D R. E . G I B L A K . Radiosensitization with

5-Bromodeoxyuridine of Chinese Hamster Cells X-Irradiated D ü r i n g Different Phases of the Cell Cycle 672

G A R Y S. S H A B E R . Alteration of Taste Thresholds in the Rat Following Low Dose X-Irradi­ation 689

A D O L F O P O R T E L A , M A R C E L O G A R F U N K E L , J O R G E V A C C A R I , A N A M A R I A D E L B U E , P E T E R

A . S T E W A R T , A N D J U A N C A R L O S P E R E Z . Radiation Effects on Water Permeability and

Distribution in Frog Muscle Cells 704 J . L . C O O L E Y A N D F . L . M I L L E R , J R . Effects of Chronic Irradiation on Laboratory Popu-

lations of the Aquatic Snail Physa heterostropha 716 L A W R E N C E W. D A V I S A N D T H O M A S A . S T R I K E . Effectiveness of Tranquilizing Drugs in

Rats Ivxposed to Mixed Gamma-Neutron Radiations 725 D . K . M Y E R S . The Initial Effect of X-Radiation on Thymidine Incorporation into D N A

in Thymus Cells 731 S. P H Y L L I S S T E A R N E R A X D E M I L Y J . B . C H R I S T I A N . Mechanisms of Acute Injury in the

7-Irradiated Chicken. I n V i v o Studies of Dose Protraction on the Microvasculature 741 B O O K R E V I E W 756

E R R A T U M 758

^AUTHOR I N D E X 759

S U B J E C T I N D E X 761

RADIATION RESEARCH

OFFICIAL ORGAN OF THE RADIATION RESEARCH SOCIETY

BOARD OF EDITORS: Managing Editor: TITUS C. EVANS, College of Medicine, University of Iowa, Iowa City, Iowa 52240

CT. E . ADAMS, Mt . Vernon Hospital, England

M . J . BERGER, National Bureau of Standards

J. G . CARLSON, University of Ten-nessee

R. A. CONARD, Brookhaven National Laboratory

S. B. CURTIS, University of California W. GORDY, Duke University D. G R A H N , Argonne National Labora­

tory M . L. G R I E M , University of Chicago R. H . H A Y N E S , York University,

Canada

J. JAGGER, University of Texas R . H . JOHNSEN, Florida State Uni­

versity R . F. K A L L M A N , Stanford University S. KONDO, Osaka University, Japan T. L . PHILLIPS, University of Cali­

fornia P. RIESZ, National Institutes of Health W. C. SNIPES, Pennsylvania State

University H . D. SUIT, University of Texas J. K . THOMAS, University of Xotre

Dame L. J . TOLMACH, Washington University G . M . WOODWELL, Brookhaven Na­

tional Laboratory

OFFICERS OF THE SOCIETY: President, N . F. BARR, Atomic Energy Commis-sion, Washington, D. C.

Vice President (and President Elect), A. D. CONGER, Temple University, Philadelphia, Pennsylvania

Secretary-Treasurer, FALCONER SMITH, Biology Department, American Univer­sity, Washington, D. C. 20016

Managing Editor, TITUS C. EVANS, University of Iowa, Iowa City, Iowa

Executive Secretary: RICHARD J . B U R K , JR. , 4211 39th Street, N.W., Washington, D. C. 20016

Assistant to M a n a g i n g Editor: DOROTHY D. SCHOTTELIUS, University of Iowa, Iowa City, Iowa

ANNUAL MEETINGS:

1972: May 14-18, Portland, Oregon 1973: April 29-May 3, St. Louis, Missouri 1974: July 13-20, Fifth International Congress of Radiation Research, Seattle,

Washington

VOLUME 47, NUMBER 2, AUGUST 1971

Copyright © 1971, by Academic Press, Inc., New York, N . Y . 10003, U . S. A.

Second class postage paid at Baltimore, M d . 21202

CONTENTS

Albrecht M . Kellerer. C o n s i d e r a t i o n s o n the Random Traversal of Con-vex Bodies and S o l u t i o n s for General Cylinders 359

Albrecht M . Kellerer. A n Assessment of Wall Effects i n Microdosi-m e t r i c Measurements 377

Marie M . Grenan and Edmund S. Copeland. S l r u c t u r e - F u n c t i o n Studies of the A m i n o t h i o l R a d i o p r o t e c t a n t s . Effect of Carbon C h a i n L e n g t h i n Mercaptoethylamine H o m o l o g s - M a m m a l i a n R a d i o -p r o t e c t i o n 387

John Chrysochoos and David S. Shihabi. X - R a d i o l y s i s of Aqueous E r y t h r o s i n 392

R. Katz, B. Ackerson, M . Homayoonfar, and S. C. Sharma. I n a c t i v a -t i o n of Cells by Heavy I r o n B o m b a r d m e n t 402

James Shaeffer and Timothy Merz. A C o m p a r i s o n of Unscheduled D N A Synthesis, D 0 , Cell Recovery, and Chromosome Number i n x - I r r a d i a t e d M a m m a l i a n Cell L i n e s 426

W. U . Shipley, M . M . Elkind, and W. B. Prather. P o t e n t i a t i o n of X-Ray K i l l i n g by 5 - B r o m o d e o x y u r i d i n e i n Chinese H a m s t e r Cells: A R e d u c t i o n i n Capacity for S u b l e t h a l Damage A c c u m u l a t i o n 437

Howard B. Newcombe and John F. McGregor. Persistence of H e r i t -able Changes i n a n I r r a d i a t e d Rat P o p u l a t i o n After Cessation of the Exposures 450

F. F. Hahn, E. Goldstein, and D. L. Dungworth. Effect of Whole-Body X - I r r a d i a t i o n on P u l m o n a r y B a c t e r i c i d a l F u n c t i o n 461

Thomas C. Welbourne and G. B. Spurr. M e c h a n i s m of Renal A c i d E x c r e t i o n Following X - I r r a d i a t i o n of the Dog 472

Shields Warren and Olive Gates. The I n d u c t i o n of L e u k e m i a and Life S h o r t e n i n g i n M i c e by C o n t i n u o u s Low-Level E x t e r n a l G a m m a R a d i a t i o n 480

Y. N . Kabakov. F a c t o r s I n f l u e n c i n g C l a s m a t o s i s v a N ' w o : Observations of P e r i o d i c i t y of Clasmatosis i n Lymphocytes and Hematopoietic Recovery After I r r a d i a t i o n 491

Kuo-Pao Yang and Otto W. Neuhaus. In Vitro Uptake of Model A m i n o A c i d s by Rat Liver Following Whole-Body y - I r r a d i a t i o n 500

C. T. Gaffey. R e l a t i o n Between RBE and L E T to I n h i b i t N e u r a l I m ­pulse C o n d u c t i o n 511

John M . Yuhas. Biological Factors Affecting the Radioprotective Effi-ciency of S - 2 - [ S - A m i n o p r o p y l a m i n o \ E t h y l p h o s p h o r o t h i o i c A c i d ( W R - m i ) : LDß0(7) Doses 526

Charles O. McKinney and Frederick B. Turner. Genetic V a r i a t i o n i n I r r a d i a t e d and N o n i r r a d i a t e d P o p u l a t i o n s of the L i z a r d , Uta stansburiana 530

John B. Storer. Chemical P r o t e c t i o n of the Mouse a g a i n s t R a d i a t i o n -I n d u c e d Life S h o r t e n i n g 537

H . Yoshikura. R a d i a t i o n Effects on Macromolecular Synthesis i n Con-tact I n h i b i t i o n - S e n s i t i v e Cells Synchronized by M e d i u m Change 548

H . I. Popescu and D. T. Stefanescu. Cytogenetic I n v e s t i g a t i o n of I n ­dustriell Workers Occupationally Exposed to G a m m a Rays 562

Book Review 571 Announcement 572

Published monthly at M t . Royal & Guilford Aves., Baltimore, M d . 21202, by Academic Press, Inc. 111 Fifth Avenue, New York, N . Y . 10003.

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RADIATION R E S E A R C H 47, 359-376 (1971)

Considerations on the Random Traversal of Convex Bodies and Solutions for General Cylinders1

A L B R E C H T M . K E L L E R E R Radiological Research Laboratory, Department of Radiology, College of Physicians d* Surgeons,

C o l u m b i a University, New York, New Y o r k

K E L L E R E R , ALBRECHT, M . Considerations on the Random Traversal of Convex Bodies and Solutions for General Cylinders. R a d i a l . Res. 47, 359-376 (1971).

The basic concepts applicable to the random traversal of convex bodies are discussed, and a set of theorems relevant to the problem is presented. Formulae are derived for the chord length distribution in cylinders of arbitrary convex cross section in a uniform isotropic field of straight random tracks.

1. I N T R O D U C T I O N

Chord length distributions resulting from the random intersection of convex bodies by straight lines have been discussed in such different fields as acoustics ( 1 - 3 ) , reactor design ( 4 - 6 ) , ecology (7), and microscopy ( 8 , 9 ) . In radiation physics chord length distributions are needed for the evaluation of pulse height spectra obtained with proportional counters ( 1 0 , 1 1 ) ; the distributions are also relevant to various other problems of microdosimetry ( 1 2 , 1 3 ) , general dosimetry ( 1 4 ) , and radiation shielding.

For spheroids chord length distributions have been given in analytical ( 1 5 ) and in numerical (16) form. For circular cylinders the problem is more complicated and several different ways have been chosen to derive chord length distributions. Schwed and Ray (17) have attempted analytical Solutions. Wilson and Emery ( 1 8 ) have applied mixed analytical and numerical methods. Birkhoff et a l . ( 1 9 ) have used Monte Carlo techniques; they have also given a useful survey of the topic and have pointed out that the results from the various methods are not in füll agreement. Little work has been done for cylinders of noncircular cross section. But Coleman (20) has obtained the Solution for a cube.

1 Based on work performed under Contract AT-(30-1)-2470 for the U.S. Atomic Energy Commission.

359 Copyright © 1971 by Academic Press, Inc.

3(>0 K K L L K K K R

In this paper a formula for chord length distributions in cylinders of arbitrary cross section will be given. The idea of the Solution is that the problem is reduced to one in two dimensions. The chord length distribution in the cylinder is expressed in terms of the chord length distribution for the cross section of the cylinder.

Sections 2 and 3 contain considerations of general interest to problems involving the random traversal of convex bodies. Some of the theorems listed in section 3 are used in the Solution for cylinders.

2. C O N C K P T S

An excellent outline of the problems arising in the random traversal of convex bodies is given by Kendall and Moran in "Geometrical Probability" ( 2 1 ) . Recently Kingman ( 3 , 2 2 ) and Coleman (20) have derived various results on random chords. Some of these results will be used in the following considerations. The original works should be used for reference, but the basic concepts will in brief and simple form be stated in the present paper.

Coleman defines five different kinds of randomness of the secants of a convex body. Three of these are relevant in the present context:

M e a n free p a t h randomness (^.-randomness). A chord of a convex body K is cle-fined by a point in Euclidian space and a direction. The point and the direction are from independent uniform distributions. This randomness results if the convex body is exposed to a uniform, isotropic field of straight infinite tracks; it is the main object of the present paper.

Surface radiator randomness (S-randomness). A chord of a convex body K is defined by a point on its surface and a direction. The point and the direction are from independent uniform distributions. S-randomness results if the surface is a uniform, isotropic radiation source.

I n t e r i o r radiator randomness (/-randomness). A chord is defined by a point in the interior of K and a direction. The point and the direction are from independent uniform distributions.

The distributions of chord length are different for the various types of ran­domness. This difference and the use of the word randomness in an unspecified sense are the basis of some of the so-called paradoxes of probabilit.y ( 2 1 ) . In the follow­ing, indices will be used to identify the different kinds of randomness. The ex-pectation values will be labeled in the same way. For example/„(Z) is the probability density of chord length l under ^-randomness, while ZM and l * are the mean and the second moment of the distribution.

Sum distributions will be symbolized by the capital letter corresponding to the small letter used for the differential distribution. For convenience and because this simplifies some of the formulae the sum distributions are summed from the tail of the distribution. Thus l \ ( l ) is the probability that the chord length exceeds

Ii A N D O M T R A V 1 0 1 1 S A L 3(51

/ under /z-randomness:2

l ' \ ( D = l U x ) d x . (1)

Surface radiator randomness has been discussed in the context of radiation physics; specifically it has been shown that for a sphere of diameter d one obtains the constant density ( 2 3 ) :

/*(/) = d~\ 0 < / ^ d, (2)

while for /z-randomness one has

1 , ( 1 ) = 2 l / d \ 0 < l ^ d. (3)

AS-randomness will not be further discussed in this paper. It appears that there is no simple relation between the distributions f s ( l ) and/„(/) . It is an opcn question whether there exists a unique relation between f s ( l ) and/M(Z), and to what degree these distributions determine the shape of the body K .

Kingman ( 3 ) has shown that /-randomness is closely related to ^-randomness:

/ / ( / ) cc I U I ) . (4)

From this relation one finds Ihat the mean chord length // under /-randomness always exceeds the mean chord length l ß under /z-randomness,

l = J l / i ß = ( V ß + 1) (5 )

where Vß is the fractional variance of the d i s t r i b u t i o n :

V . - S r - ^ - i . ( 6 )

For a sphere FM is equal to 0.12") as can readily be derived from relation (3). One may assume that all nonspherical convex bodies have a larger fractional variance of their chord length distribution. It seems, however, that a proof of this simple assertion lias yet to be found.

One must note that interior radiator randomness is not what one obtains if the interior of a convex body K is a uniform source of straight particle tracks. The definition of /-randomness refers to füll straight lines while an actual radiation source produces tracks which, even if they are assumed to be infinite, represent only half lines. If one generalizes the concept of a chord so far as to include the

2 The upper limit of Integration is given as * in all cases where the Integration extends to the maximal chord length. This is correct because the densities and the sum di.slribulions are zero for / > / m n x . One must, however, be careful to observe this condition in actual cases where the distributions may be given analytically.

362 K E L L E R E l l

segments of these half lines which are contained in the body K one may introduce the corresponding concept which shall be named i n t e r n a l source randomness and labeled by the small letter i.

For z-randomness one obtains the density,

MD - F , ( l ) / l ß , (7)

where F ^ l ) is the sum d i s t r i b u t i o n of l under /x-randomness according to the formula:

F*V) = [ M x ) dx. (8)

Proof Relation (4) has already been derived by Kingman. Here its validity will be

demonstrated again in a consideration which can then be varied to yield relation (7).

M D is the chord length distribution if K is exposed to an isotropic field. Consider an infinitesimal sphere of cross section A a centered around a position distributed uniformly over the interior of K . The probability that a chord of length l intersects the infinitesimal sphere is l Aa/V, where V is the volume of the body K . If one samples the chords which intersect the infinitesimal sphere in all its possible posi-tions one fulfills the conditions for /-randomness (see above), and one therefore obtains,

M i ) <* y W ) « l U D , O )

or with proper normalization,

M l ) = l M l ) / \ * M * ) dx = l M l ) / l > (10)

In case of an internal source one counts the half tracks originating from the point of emissiori separately (z-randomness). In this case the probability that the chord of length l intersects the infinitesimal sphere is still l Aa/V, but the point of inter-section on the chord must be considered. This point of intersection is distributed uniformly over the length of the chord. The chords of length l therefore give rise to an equidistribution of half segments s:

M s ) = 0 < s ^ l (11)

Integration over all chords occurring in /-randomness yields Eq. (7):

M s ) = £° ljMD d l / l = F ß ( s ) / l ß . (12)

From the preceding considerations it is apparent that the mean chord length for

R A N D O M T R A V E R S A L 363

an internal source is half the mean chord length under /-randomness and is there­fore related to the first and second moment of the chord length distribution under M-randomness in the following way [see Eq. (o)]:

l = 1/2 = /7/2/M = 7, . (13)

The relation can also be derived by partial Integration from Eq. (12), and it can be generalized to

j N + l

l " = v ^ -1 (14)

In section 4 results are given for chord length distributions in uniform, isotropic fields (/x-randomness). Relation (7) means that the results can also be applied to the case of interior sources.

One should note that all considerations in this section, with exception of relations (2) ancl (3) which refer to a sphere, equally apply to three-dimensional and two-dimensional Euclidian space.

3. A U X I L I A R Y R E L A T I O N S

This section contains theorems relevant to the problem of chord length distri­butions. Some of the relations will be applied in section 4. Theorems 3.1-3.7 have been derived earlier ( 3 , 10, 2 0 ) . For convenience proofs and some applications are included here.

3.1 DISTRIBUTION OF THE A N G L E OF INCIDENCE TO A SURFACE E L E M E N T

Theorem If a surface element is exposed to an isotropic Held (/x-randomness) and if 6

is the angle between the incident line and the normal of the surface element at the point of incidence, then 6 is distributed according to the density/M(0):

f ß ( e ) de = 2 sin e cos e de, o ^ e < ~ , (15)

while the sum distribution, i.e. the probability that the angle of incidence exceeds 0, is

? \ ( e ) = cos20. (16)

Proof The distribution of 6 on a surface element in an isotropic uniform field does not

depend on the orientation of this surface element. One may therefore consider the surface of a sphere. For a sphere one can, however, assume a unidirectional field instead of an isotropic field. This is schematically represented in Fig. 1. The total

K K L L R R E R

F I G . 1. Determination of the distribution of angles of incidence to a surface element.

cross section of the spliere of radius r is equal to vir. The disc whose protection in Fig. 1 is the line A B represents the cross section for angles of incidence equal to or less than 0. Its area is r V sin20, and the fraction of tracks incident on the sphere with angles exceeding 0 is therefore given by Eq. (16). By differentiation one obtains relation (15).

As an example of the application of relation (16) one can derive the chord length distribution in an infinite slab of thickness h . This distribution will be used in section 4. The chord length as a function of the angle of incidence is

l = h/cosd. (17)

Inserting this in (16) one has the sum distribution

F \ ( l ) = h2/t, l ^ h , (18)

and therefore the density

f ß ( l ) = 2h*/f. (19)

3.2 DISTRIBUTION OF T H E A N G L E OF INCIDENCE TO A L I N E E L E M E N T

Theorem In 2-space the angle 0 of incidence to a line element in an isotropic field has the

sum distribution

F ß ( 0 ) = 1 - sin 0, 0 ^ 6 < TT/2, (20)

and therefore the density3

fß(0) = cos0. (21)

The angle of incidence 0 is the angle between the random track and the normal to the line element.

The proof can be omitted. It is analogous to the proof in 3.1. 3 A frequeut source of confusion in cjnsiderations on chord length distributions is failure

to distinguish between /u-randomness and <S-randomness. For a surface source in three di-mensions the angle 6 of entrance into the body K is distributed according to the density: /.s(0) = sin 0; while the density in the two-dimensional case is constant: f a ( 6 ) = 2/TT.

R A N D O M T R A V E R S A L 36f)

One can use Eq. (20) to derive the chord length distribution in a circle, which will be used in section 4. For a circle of diameter d one has:

l = d cos 0. (22)

By Substitution in (20) and transition to the complementary probability in order to account for the inverse Variation of 0 with / one obtains the sum distribution

F ß ( l ) = V i - W l ^ d, (23)

and the density

^ l ) = d v h i > - (24)

Another example is an infinite strip of height h . In this case one has

sin0 = V i - h*/P, (25)

and therefore

F ß ( l ) = 1 - y/Y^lfß, l ^ A, (26)

and

^ = r v f ^ - (27)

3.3 TOTAL T R A C K L E N G T H IN A V O L U M E

Theorem In a uniform field of fluence 4> the expected4 total track length per unit volume

element is equal to ^.

Proof The fluence is defined as the expectation value of the number of traversals of a

sphere of unit cross section. The total track length per unit volume is a function of * only. In order to obtain the numerical relation one may consider a cylinder of unit cross section and height h and a uniform unidirectional field of fluence * parallel to the axis of the cylinder. The expected number of traversals is 3>, the expected total track length is $ h , and the expected total track length per unit volume is therefore 3>.

4 Here and in the following subsections the term expected is used as an abbreviation for expectation value of. The term m e a n is not used in order to avoid confusiou with the mean of the chord length distributions.

366 K E L L E R R R

3.4 N U M B E R OF TRAVERSALS THROUGH A SURFACE E L E M E N T

Theorem In an isotropic uniform field of fluence $ the expected number of travers-

als through a unit surface element is $/2. The expected number of chords through a convex body of surface S is therefore aS'$/4.

Proof As discussed in 3.1 one can consider a sphere exposed to a unidirectional field of

fluence <f>. The expected number of chords is ?*Vf>. Each chord intersects the surface of the sphere twice. The expected number of intersections per unit surface element is therefore equal to $/2; the expected number of chords is equal to $£/4.

3.5 TOTAL T R A C K LENGTH AND N U M B E R OF TRAVERSALS IN Two DIMENSIONS

Theorem In two-dimensional space fluence 3> is defined as the expected number of traversals

of a circle of unit diameter. The expected total track length per area element in an isotropic uniform field is <!>, the expected number of intersections with a unit line element is 2 $ / i r .

The proof is analogous to the proof of 3.3 and 3.4.

3.6 M E A N CHORD LENGTH IN T H R E E DIMENSIONS

From 3.3 and 3.4 one obtains a relation which is sometimes called the Cauchy theorem. The relation, implicity deduced by Cauchy (24) in 1850, seems to have first been stated by Czuber ( 2 5 ) .

Theorem

If a convex body has the volume V and the surface S, then the mean chord length l ß is equal to

Iß = 47/S. (28)

Proof The theorem follows from the fact (see 3.3 and 3.4) that the mean chord length

is equal to the total track length $ V in the body divided by the number S<£/4 of chords through the body.

3.7 M E A N CHORD LENGTH IN Two DIMENSIONS

Theorem For a two-dimensional convex figure of area A and circumference C one has

l ß = t A / C . (29)

R A N D O M T R A V E R S A L 367

Proof This theorem which has also been implicitly deduced by Cauchy follows from the

fact (see 3.5) that the mean chord length is equal to the total track length A $ in the figure divided by the number <£>C/TT of chords through the figure.

From the proofs of 3.6 and 3.7 one concludes that, with the Interpretation of a tracksegmentasasimply connected piece of the track within the region, the Cauchy theorem is valid for a uniform isotropic field even if the random tracks are not straight (13). The generalization applies equally to two and three dimensions. One can furthermore show that the tracks may be branched; but they must not be closed curves. The present discussion will, however, be confined to straight random tracks. The generalization of the Cauchy theorem to the case of finite straight tracks is given in the following subsection.

A similar simple relation exists for the fourth moment of the chord length dis­tribution ( 2 1 , 2 6 ) . But the case is more complicated for the second and third mo­ment, and such characteristics of the distribution as fractional variance or skewness must therefore be evaluated numerically in each particular case.

3.8 M E A N SEGMENT LENGTH IN THE C A S E OF FINITE TRACKS

The following theorem deals with the mean segment length which results in a convex body when exposed to random tracks of finite length. It can be considered as the generalized form of the Cauchy theorem. Al l considerations in this and the following section refer to /x-nindomness. The index p is therefore omitted. Through-out this paper the letter F will be used for distributions referring to infinite random tracks, while P will be used for distributions which result if the random tracks are of finite length.

Assume that the finite random tracks have an orientation. A random track is then defined by a starting point, a direction, and a length. The point and the direc­tion are from independent uniform distributions, the length is distributed according to r ( u ) . This is the generalization of ^-randomness to finite tracks. r ( u ) is called the ränge distribution.

If a convex body of mean chord length l = 4F/& is exposed to a uniform, iso­tropic field of random tracks of mean ränge ü, then the resulting mean segment length s in the body is given by the relation

The total track length in the body is V$ (see 3.3), and the number of end points plus starting points of tracks in the body is therefore 2V$/ü> the number of traversal

Theorem

(30)

Proof

368 K E L L E R E U

points on the surface of the body is S&/2 (see 3.4). Euch track segment has two t e r m i n a l points which are either starting points or end points of tracks, or traversal points. The mean segment length s is therefore equal to the total track length divided by half the number of terminal points:

F* = (ü~l + S/47)" 1 . (31) &V/Ü + S&/4:

One can readily show that Eq. (30) holds also in the two-dimensional case.

3.9 CHORD LENGTH DISTRIBUTIONS IN THE C A S E OF FINITE RANDOM T R A C K S

Theorem Assume that the convex body K has the sum distribution F ( l ) of chord length

under /i-randomness and that it is exposed to an isotropic uniform field of straight random tracks whose length u has the sum distribution R ( u ) . Then the resulting sum distribution of segment length in K is

P ( s ) = k (^F(s) ^ R ( x ) dx + R(s) ^ F ( x ) d x j . (32)

The normalization constant has the value k = (ü + / ) _ I , where / is the mean chord length 4 F / S , ^ n d ^ i ' s the mean ränge of the tracks.

In the special case where all random tracks are of a fixed length u, the formula reduces to

p M = ( u - s ) F i s ) + 1^ F i x ) dx^j , 8 £u. (33)

Note that the sum distributions are summed from the right, i.e., equal to the probability that the random variable exceeds a given value.

Proof First an auxiliary relation has to be given: If two intervals of length u and l on a

straight line overlap randomly, then the length s of the overlap has the sum dis­tribution :

H ( s ) = i u + l - 2 s ) / i u + Z), s ^ u, l (34)

The relation follows from the geometrical consideration indicated in Fig. 2. The solid line represents the interval of length w, the dotted line those positions of the left border of the otlier interval which correspond to overlaps. The positions which

s S

A B C D

F I G . 2. D e t e r m i n a t i o n o f t h e o v e r l a p p r o b a b i l i t y o f t w o i n t e r v a l s .

R A N D O M T R A V E R S A L 369

belong to an overlap greater than s are inside B C . The ratio of the interval length B C to A D is equal to H ( s ) .

Assume a sphere S which is centered around the body K and has a diameter large as compared to the dimensions of K and the maximum length of the random tracks. Consider a track of length u inside S. The probability that the straight-line exten-sion of the track intersects K is independent of u. If an intersection occurs the prob­ability for a chord length l in K is proportional to f ( l ) . Given the values u and Z, the probability that the track itself intercepts K is proportional to u + Z.

The resulting sum distribution of segment length is therefore «CO * 0 0

P i s ) = k l / f ( l ) r ( u ) ( u + l ) H ( s ) d u d l J s J s

/»CO /.CO

= k / i u + l - 2 s ) f ( l ) r ( u ) d u d l (35)

= k ( F ( S ) t t r ( u ) du - s R ( s ) ] + R ( s ) //(/) dl - sF(s)]

By another partial Integration one obtains Eq. (32). The value of k results from the condition P(0) = 1. This ends the proof.

By differentiation of Eq. (32) one obtains the probability density

p(s) = k ( j ' i s ) [ s R i x ) dx + v i s ) j s F i x ) dx + 2 F i s ) R i s ) y (36)

and in the special case of a fixed track length u,

p i s ) = k ( f i s ) i u - s) + öis - u ) ^ F i x ) dx +2Fis)y s ^ u. (37)

One can show that the three terms in Eqs. (36) and (37) correspond to those segments which Caswell (12) categorizes as crossers, insiders, and as Starters and Stoppers.

In Eq. (32) the first term represents the contribution of the interior source (z-randomness) and the second term the random tracks originated outside K . If Eq. (32) is split up into the contribution of crossers, insiders, and Starters and stoppers it takes the form

P i s ) = fc (F(S) ^ R ( x ) dx - js F ( x ) R i x ) dx

/co /»co

F ( x ) dx - / F ( x ) R ( x ) dx (38)

+ 2 F i x ) R i x ) dx

370 K E L L E R E R

or in the case of a fixed track length u,

P i s ) = k ( i u - s ) F ( s ) - ^ F ( x ) dx

+ f F ( x ) dx (39)

+ 2 ^ F i x ) d x j , s g u.

These formulae are important for microdosimetric calculations where one deals with finite charged particle tracks incident on a sensitive volume of cylindrical or spherical shape. In the case of a sphere which has been treated by Caswell one has to insert the functions

/ ( * ) = 2s/d2, F i s ) = 1 - s2/d\ f F i x ) dx = j d - s + ^ - 2 , (40)

8 < d. For cylinders the corresponding functions will be derived in the next section. Theorems (32) and (33) have been derived here for 3-space. Going through the

arguments in the proof one finds that they equally hold in 2-space. This will be used in section 4.

4. S O L U T I O N F O R G E N E R A L C Y L I N D E R S

This section deals with the derivation of chord length distributions for right cylinders in an isotropic, uniform field (/z-randomness). The principle underlying the Solution is to reduce the problem to one in two dimensions. Al l distributions and expectation values are understood to refer to ju-randomness, if not otherwise stated. The index p is therefore omitted.

4.1 DERIVATION OF THE CHORD LENGTH DISTRIBUTIONS

Assume that the cylinder has height h and an arbitraiy convex cross section, and consider the orthogonal projection of the random tracks and of the cross section of the cylinder onto a plane parallel to the faces of the cylinder. Figure 3 represents this projection. The heavy line segments are the projections of those segments of the random tracks which lie inside the infinite slab formed by the two planes through the faces of the cjdinder. The length u of the projection is related to the length v of the actual segment in the slab by

v2 = h 2 + u2. (41)

H A N D O M T R A V E R S A L 371

F I G . 3. Projection of a cylinder exposed to random tracks.

With this relation and Eq. (18) one obtains the sum distribution of the projection length:

G ( U ) = h*/i? = 1/(1 + u / h 2 ) . (42)

The projections are uniform and isotropic in the plane. The straight lines through these projections form the chord length distribution in 2-space, f ( t ) , in intercepting the cross section of the cylinder. The distribution f ( t ) is known for a circle and for a rectangle [see Eqs. (24) and (61)]; for more complicated convex figures it can be derived numerically. The mean chord length in the cross section is t.

When the intervals u and t overlap, then the segment s of overlap is the projection of the actual chord in the cylinder. The problem thus reduces to a determination of the distribution of these segments. This problem has been solved in 3.9. For a fixed projection length u> one obtains the following sum distribution of segments s in the projection of the cylinder in accordance with Eq. (33):

C ( s ) = y-J— ( ( u - s ) F ( s ) + f F ( x ) d x } , s£u. (43) t + u \ Js /

The true chord length l in the cylinder is proportional to s:

l = 8 V i + W/u*. (44)

With this relation and with the abbreviation

F * ( s ) = f F ( x ) dx - s F ( s ) , (45)

one obtains the chord length distribution for a given angle of incidence of random tracks relative to the axis of the cylinder:

c^ = i h (>* ( v r r m ) + F * ^ r r m ^ • 1 = ( 4 6 )

Except for a stretching by the factor y / l + h 2 / u 2 the distribution is equal to the distribution (43) which results when the cross section of the cylinder is exposed to random tracks of length u.

372 K E L L E R K R

Integration over the distribution of u yields the füll Solution. As in Eq. (35) and for the same reasons one has to include the weight factor ( l + u ) in the integral

cco - - K f ( U F ( ^ 7 = ^ ) + p * ( v r n ^ ) ) ( 4 7 ' where the lower hmit a of Integration is 0 for l ^ h and ^ / l 2 — h 2 for l > h . The chord length distribution in the cylinder is designated by the letter C and not F because F is used for the 2-space chord length distribution of the cross section of the cylinder. With the Substitution

1 hx

and with G ( u ) from Eq. (42),

G ( u ) = 1 - x2, (49)

one obtains the general formula for the chord length distribution in right cylinders:

C ( l ) = 2fc f * ( ^ = = F ( l x ) + x F * ( l x ) ^ d x ,

fO f o r i * A ( 5 0 )

a = [ V i - h 2 / l 2 for l > h .

4.2 RESULTS

In this section the result of 4.1 will be restated, and the formulae for the chord length density will be added. The auxiliary function/(0 will be given for the special cases of a circle and a Square.

The chord length distribution in a right cylinder exposed to a uniform isotropic field of straight infinite tracks (/i-randomness) can be expressed in terms of the chord length distribution of the two-dimensional figure which is the cross section of the cylinder.

Let / (0 and tbe the differential distribution and the mean value of chord length t for the cross section. The sum distribution of t is

F(t) = J"f(x)dx, (51)

and F * ( t ) is an auxiliary function:

F * ( t ) = J F(x)äx-lF(l) = J x f ( x ) d x - 2 t F ( t ) . (52)

Then the sum distribution of chord length in the cylinder, i.e., the probability that

l i A N D O M TLIAV15IISAL 373

(55)

a chord exceeds length /, is given by the formula:

C ( l ) = 2Ä: jT1 ( ^ = = F ( l x ) + x F ^ l x ^ d x . 0)3)

// is the height of the cylinder. The lower limit a of the Integration is 0 for £ ^ h and V i — h 2 / l 2 for / > Ii. The normalization constant is

k = r+ü = r ( A / C + A/2) ' ( ° 4 )

as follows from 3.7 and from Eq. (42). A is the area and C the perimeter of the cross section of the cylinder. By differentiation of Eq. (53) one obtains

ein - - * ( ' ( - 7 f = 3 / < ' * ) + *''/(<*) - 2»*F ( i .o )&

The probability density of chord length is therefore equal to

HO = 2k ( Y ( - ^ = - x ' l ) f ( l x ) + 2 x i F ( l x ) ) d x ' W 1 " *" ' (56)

+ ^ f F U ) d*.

where the second integral applies only to values l ^ h . Equations (53) and (56) are the formulae for chord length distributions of

cylinders of arbitrary convex cross section. In the general case the chord length distribution f ( t ) for the cross section has to be evaluated numericalry. For circular and square cross sections one can, however, give/(£), F ( t ) , and F * ( t ) in analytical form.

For a circle of diameter d one obtains aecording to Eqs. (23), (24) and (52),

= d V T ^ l ' F ( i ) = v / l " ? / c P ( 5 7 )

f F ( x ) dx = \ arecos L - L V i - t2/d2 (58)

t St F * ( t ) = - arecos - - - V i - P/d* . (59)

All functions are zero for t > d. Coleman (20) has derived the chord length distribution f ( t ) for a rectangie One.

374 K E L L E H E R

may note that one can obtain this function from Eq. (47 ) . This equation is valid in a space of arbitrary dimensions, and a rectangle can be considered as a cylinder in two dimensions. In the special case of a Square of side length c one inserts the dis­tribution G ( u ) for an infinite strip of height c [see Eq. (26)] and a delta function 8(t — c) for f ( t ) . Equation (47) with the Substitution (48) then takes the form:

for t ^ c a = 0

F ( t ) = k f (-/== + c - 2 t x ) dx b = \ - ^

L V V i - x2 ) a = V i - eye i o r c < f ^ ^ b = c/t

1 ~ t/2c for t ^ c

t/2c - V i - c 2 / f for c < t £ \/2c

By differentiation one obtains the chord length density for the Square:

' l / 2 c i o v t ^ c

(60)

tfyß

? - ( 6 1 ) - l/2c for c < t £ \/2c

Evaluation of the integrals in (53) and (56) for the special cases of circular or Square cross section, analysis of the moments of the distribution, and discussion of the limiting cases of very long and very flat cylinders will be the object of a forth-coming paper. In general it is practical to evaluate the integrals numerically. Nu­merical results for circular cylinders have already been given (27).

In work with cylindrical proportional counters the case of a unidirectional radia­tion field is of some interest. One must then use Eq. (46) instead of Eq. (53) . The value of u is equal to h tgß, where 6 is the angle between the field and the axis of the cylinder. In the general case of an anisotropic field one has to use Eq. (47) and the distribution G ( u ) which belongs to the angular distribution of the field.

If one deals with finite tracks of a ränge distribution R ( u ) or of the fixed ränge u, one can first derive the chord length distribution C ( l ) for infinite tracks and then apply the theorem (32) or (33) . The distribution C takes the place of F in these equations.

Finally one should notice that except for a normalization constant Eq. (53) gives the probability density of segments created in a cylinder by an internal source. This follows from the considerations given in section 2.

A C K N O W L E D G M E N T S

This study has profited greatly from the interest and advice of Dr . William Gross and D r . Harald H . Rossi. I am also grateful to my brother, D r . Hansgeorg Kellerer, for various suggestions.

R E C E I V E I K January 18, 1971

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