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OPTIMISATION AND SYMMETRY

IN EXPERIMENTAL RADIATION PHYSICS

Report

EUR 10968 EN

Blow-up from microf r.he original Commission of the European Communities

nuclear science and technology

OPTIMISATION AND SYMMETRY

IN EXPERIMENTAL RADIATION PHYSICS

A. GHOSE

Visiting scientist

Commission of the European Communities

Joint Research Centre

l-lspra (21020)

PAr\ ' \ "ioth.

Directorate-General Science, Research and Development

Joint Research Centre

1^__

1988 CLEUR 10968 EN Published by the

COMMISSION OF THE EUROPEAN COMMUNITIES

Directorate-General

Telecommunications, Information Industries and Innovation

Bâtiment Jean Monnet

LUXEMBOURG

LEGAL NOTICE

Neither the Commission of the European Communities nor any person acting on behalf

of then is responsible for the use which might be made of the following

information

ECSC—EEC—EAEC Brussels - Luxembourg. 1988 PREFACE

The present monograph is concerned with the optimisation of geometric factors in radiation

physics experiments. The discussions are essentially confined to those systems in which

optimisation is equivalent to symmetrical configurations of the measurement systems. In spite

of this limitation, a surprisingly diverse variety of investigations come under the scope of

the monograph. They include, among others, measurements of interaction cross section of

diverse types, determination of polarisations, development of detectors with almost ideal

characteristics, production of radiations with continuously variable energies and development

of high efficiency spectrometers etc.

The monograph is intended for use by experimental physicists investigating primary interac

tions of radiations with matter and associated technologies. We have illustrated the various

optimisation procedures by considering the cases of the so-called "14 MeV" on d-t neutrons

and gamma rays with energies less than 3 MeV. In recent years a number of small laborato

ries have acquired d-t neutron generators thanks, among others, to the projects supported by

the International Atomic Energy Agency and it is hoped that the present monograph would

be of use to the research workers carrying out experimental measurements using these

generators. Developments in fusion technology are critically dependent on the availability

accurate cross sections of nuclei for fast neutrons of energies at least as high as d-t neutrons.

In this monograph we have discussed various techniques which can be used to improve the

accuracy of such measurements and have also presented a method for generating almost

monoenergetic neutrons in the 8 MeV to 13 MeV energy range which can be used to

measure cross ections in this sparingly investigated region.

The monograph is dedicated to the memory of my first research student and later colleague

Dr. Amalendu Nath whose untimely death cut short a brilliant and promising career in

radiation physics.

The author takes this opportunity to thank his students, colleagues and counterparts in Bose

Institute, Calcutta, India, Brookhaven National Laboratory, Upton, U.S.A., Arts and Science

University, Rangoon, Burma, Universiti Sains Malaysia, Penang, Malaysia, the University of

Surrey, Guildford, U.K., the State University of Mongolia, Ulan Bator, Mongolia and the

European Community Joint Research Centre, Ispra, Italy. He is thankful to Dr. R. Ramanna,

former Chairman of the Atomic Energy Commission of the Government of India, Dr. P.K.

Iyengar, Director, Bhabac Research Centre, India and Dr. A.K. Ganguly, former

Director, Chemical Group, Bhaba Atomic Research Centre, India for their advice, support

and encouragement. He is also thankful to the International Atomic Energy Agency, Vienna

and the U.S. National Bureau of Standards, U.S.A. for sponsoring research projects, the

results from some of which have been reported here.

The author is particularly thankful to Mr. John Hubbell of the U.S. National Bureau of

Standards, Prof. Richard Pratt of the University of Pittsburg, U.S.A., Prof. Daphne Jackson

and Dr. Walter Gilboy of the University of Surrey, U.K., Dr. Ante Ljubicic of the Ruder

Boscovitch Institute, Zagreb, Yugoslavia, Dr. David Bradley of the Royal Marsden Hospital,

London and Dr. Malcolm Cooper of the Warwick University, Coventry, U.K., Dr. A. Hanson

and Dr. K. Jones of Brookhaven National Laboratory, U.S.A., Prof. Suprakash Roy of Bose

Institute Calcutta, Prof. Madhya C.S. Chong of Universiti Sains Malaysia, Penang and Dr. T.

Vilaithong of the University of Chiang Mai, Thailand for supplying the author with

preprints, reprints and reports.

Ill The author expresses his gratitude to Prof. Dr. G.R. Bishop, Director General, European

Community Joint Research Centre, Ispra for granting him the facilities of Ispra as a visiting

scientist, which enabled him to write this monograph.

The monograph would not have seen the light of the day but for the continuous help, advice

and friendship of Dr. Walter Kley, Advisor to the Director General, JRC, Ispra from its

inception to conclusion.

Finally the author is taking this opportunity to express his gratitute to his teacher late Dr.

D.M. Bose, former Director, Bose Institue, for having introduced him to the joys and

challenges of radiation physics research.

A.M. GHOSE

IV CONTENTS

Page

Chapter I Introduction

Chapter II Spherical Symmetry: Sphere Transmission and Equivalent

Techniques in Their Ideal Forms 7

Chapter III The Reciprocity Theorem 19

Chapter IV Sphere Transmission in Practice: Multiple Interactions

in the Target 29

Chapter V Practical Sphere Transmission: Source, Detector and

61

Miscellaneous Factors

113

Chapter VI Axially Symmetric Systems

127

Chapter VII Circular Arc Method and Its Modifications

Chapter VIII Continuously Variable Energy Secondary Radiations Sources 137

Chapter IX Notes on Cylindrical Symmetry 147

Chapter X Further Applications of Symmetry 151

References 157 Chapter I

Introduction

1.1 The Optimisation Problem in Radiation Physics

In the design of experiments in physical sciences an important role is played by those factors

which are concerned with the enhancement of the ratio R of the signal S to the noise Ν

pertaining to the measurements

RS/N . (1.1)

Theoptimumvalueof R is a direct measure of the accuracy which canbeattainedby

followingtheparticular method of measurement used in the experiment.Itisnowonder,

therefore,thatinevery properly designed experiment considerable careisexercisedto

optimisethevalueof this ratio. In this monograph we shall discuss someofthegeneral

designconsiderationswhich are useful in solving this optimisation probleminexperimental

radiationphysics.

The types of experiments with which we shall be mainly concerned here are those aimed at

investigating the interactions of radiations with matter. The essential elements of the

experimental arrangement for such a measurement are a source of radiation, a sample or

target (scatterer or absorber) and a detector. The factors relevant to the optimisation process

can, therefore, be conveniently divided into three groups viz. the source factors the ith

member of which will be denoted by s¡t the target factors the jth member of which will be

denoted by t. and the detector factor the kth member of which will be denoted by dk. If we

now assume that the factors are so chosen that they can be varied independently and are at

the same time complete in the sense that the ratio R can be completely described in terms of

them, the optimisation process would involve the solution of the three sets of partial

differential equations

aR/asjO aR/a^0 3R/adk 0 , (1.2)

where the indices, i.j.k vary over theirrespectiverangesof values. A generalformulation

and solution of the optimisation problemis,therefore,a prohibitingly complextask.

Moreover, the solutions will, in general, refertoidealsystems and it would benecessaryto

extend the analysis further before they becomeusefulinthe practical design of experiments.

We shall, therefore, give up the direct mathematicalapproach to solve the optimisation

problem and search for alternative methods which are, though not strictly general, neverthe

less applicable to a large variety of physical systems. The method which we shall study

extensively in this monograph is based on spatial symmetry.

1.2 Systems with Spatial Symmetries [il

In many investigations in radiation physics, the physical entities under study possess inherent

spatial symmetries which follow, for example, from their dependence on the angular

coordinates appropriate to the measurements. In such cases the optimisation process can be

conveniently developed by utilising these symmetries. In general, the optimisation process

would confer ideal attributes to the source, target and detector factors. For instance, an

optimised system might consist of a point source, an infinitesimially thin target and a

perfectly black detector. In almost all cases the ideal factors would differ from their counterparts in the laboratory in a significant manner. It would, therefore, be necessary to

examine the effects of deviations from ¡deal values on the accuracies of the measurements

performed under realistic conditions. We note the interesting fact that due to the very

existence of spatial symmetries small deviations <1 along the symmetry related directions q1

are such that the resultant correction factors contain only second and higher order terms

in Sy The mathematical connection between optimisation and symmetry or equivalently that

between the symmetry related vanishing of the first order correction terms and the

optimisation equations (1.2) will not be discussed here. We shall limit ourselves to the use of

geometrical and physical reasonings which would lead, in a natural way, to the adoption of a

specific type of spatial symmetry for the optimised (within practical limits) measurement of

a given physical quantity. In fact, the ideal symmetry suitable for a number of measurements

can be determined in a quite straight forward manner as indicated below.

The line SD joining the effective centres of the source S and the detector D can be taken as

the symmetry axis of a measurement system in which the source has no preferred direction

of emission (Fig. 1). The natural choice for one of the angular coordinates is the angle of

scattering Θ, which is the angle between the lines joining a typical point Τ on the target with

S and D. The second angular coordinate is the angle 0, the azimuthal angle, which measures

the rotation of the plane STD around the axis SD, as measured from a fixed standard

position.

S =the effective centre of the source,

D =the effective centre of the detector,

SO=the symmetry axis of the system,

Τ = typical point on the targete,

Ρ = fixed reference plane through SD,

θ =the angle of scattering and

φ =the azimuthal angle.

Fig. 1 Angular coordinates in a typical radiation physics

experiment.

When the phenomenon or the entity under study is independent of the angular coordinates θ

and 0, the inherent geometry of the system is obviously spherically symmetric. This is the

basis of the wellknown sphere transmission technique (Fig. 2), which has been used rather

extensively to measure such entities. In its ideal form the spherically symmetric measuring

system consists of a point isotropic source, surrounded by an infinitesimally thin concentric

spherical target and an omnidirectional (at least within the tangent cone from D to the target

surface S) detector. The spectral response of the detector is determined by the quantity