Table Of ContentOTHER TITLES IN THE SERIES
Vol. 101 FERZIGER AND ZWEIFEL—The Theory of Neutron Slowing Down in Nuclear
Reactors
Vol. 102 SIMMONS—Radiation Damage in Graphite
Vol. 103 EVANS—Fast Breeder Reactors
Vol. 104 YEMEL'YANOV AND YEVSTYUKHIN—The Metallurgy of Nuclear Fuel
Vol. 105 CEMBER—Introduction to Health Physics
Vol. 106 EGELSTAFF AND POOLE—Experimental Neutron Thermalization
Vol. 107 CAMERON AND CLAYTON—Radioisotope Instruments, Volume I
Calculational Methods for
Interacting Arrays of
Fissile Material
A. F. THOMAS, B.SC.
AND
F. ABBEY, M.A., A.R.I.C,
U.K.A.E.A.
PERGAMON PRESS
O X F O RD • N EW Y O RK • T O R O N TO
S Y D N EY • B R A U N S C H W E IG
Pergamon Press Ltd., Headington Hill Hall, Oxford
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Copyright © 1973 A. F. Thomas and F. Abbey
All Rights Reserved. No part of this publication may be
reproduced, stored in a retrieval system, or transmitted, in any
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recording or otherwise, without the prior permission of
Pergamon Press Ltd.
First Edition 1973
Library of Congress Cataloging in Publication Data
Thomas, Alan Francis, 1928-
Calculational methods for interacting arrays of fissile material.
(International series of monographs in nuclear energy, v. 108)
Bibliography: p.
1. Criticality—Nuclear engineering.
2. Nuclear engineering—Safety measures.
I. Abbey, Frank, join* author. II. Title. III. Series.
TK9153.T45 1973 621.48'35 73-8604
ISBN 0-08-017660-7
Printed in Great Britain by A. Wheaton & Co., Exeter, Devon
Foreword
THE problem of assessing the criticality safety of interacting arrays of
fissile materials is one that is both of practical importance and of
theoretical interest. From the time that man first assembled a nuclear
reactor and demonstrated that it was capable of producing energy at a
rapid rate the question was raised as to what precautions need to be
taken to prevent such a nuclear reactor being produced inadvertently
by the chance assembly of sufficient fuel. It was always appreciated that
this could be prevented by ensuring that the fuel was at all times dis-
persed through a sufficiently large volume and, in the early days, there
was so little fissile material about that such a solution could easily be
applied.
With the breeding and concentration of new and more reactive iso-
topes, however, and the introduction of nuclear power programs,
demanding the manufacture and processing of ever-increasing amounts
of fissile materials, it has been necessary to steadily reduce the dispersal
volume from its initial large oversafe size to what is today approaching
its minimum value. The pressure has come, not only from those who
manufacture reactor fuel and process spent fuel but also from the
reactor operators, to define the minimum areas required for fuel storage
and even from civil authorities who are concerned with the hazards that
may arise in transport.
It is clearly not practicable to assess the safety of more than a small
number of arrays of fissile materials by direct experimentation. Such
experiments are slow, costly and specific. It is necessary to combine
a careful analysis of the experiments that can be done with a general
knowledge of neutron physics to obtain a deep and detailed understand-
ing of the factors involved which can then be used to assess theoretically
and accurately the safety of the practical arrays that arise.
vii
viii FOREWORD
There are many pitfalls in this subject and extrapolations from experi-
ence with non-interacting systems can easily prove misleading. In
particular the effect of an external reflector round an array can easily be
underestimated and that of inserting materials between the elements can
be important.
Messrs. Thomas and Abbey have had long experience in these prob-
lems as a result of which they are well qualified to write an authoritative
monograph on the subject. They describe in detail the main theoretical
methods and include, in graphical and tabular form, many useful
results which can be used by those faced with practical problems of this
sort.
They show that the behaviour of neutrons in an array divides into two
parts which can, to a large extent, be treated independently. These are
the neutron multiplication within units of the array and the transmission
of neutrons between units. For the former the usual methods of neutron
physics are applicable but used so as to place emphasis on the neutrons
entering and leaving the unit. The latter is mainly a geometrical problem,
being entirely so for an air-spaced array.
As a conclusion of the proceedings of the Livermore Array Sym-
posium! (1968) it was stated that "the papers and discussions clearly
indicate the wide range of calculational techniques used to solve criti-
cality problems (of arrays). These techniques range in complexity from
the semi-empirical density-analogue methods to the detailed and sophis-
ticated Monte Carlo models. The attitudes towards elementary, non-
Monte Carlo methods were diverse. These attitudes ranged from the
belief that elementary methods should be discarded entirely in favour of
rapid, accurate, Monte Carlo methods to an attitude of considerable
reliance on elementary methods. . . . We believe that analytical tech-
niques are sometimes capable of providing some physical insight into
otherwise poorly understood interaction problems."
This monograph is mainly concerned with the so-called elementary
methods but developed in such a way as to give both the required results
and the physical insight into the mechanisms involved. Monte Carlo
methods are mentioned but from the point of view of analysing the
results to give the maximum information on the size of critical arrays.
The subject of interacting arrays is thus one both of practical import-
fUSAEC Report CONF-680909 (1968).
FOREWORD ix
ance and of mathematical interest. The reader will find herein not only
the concepts and methods described which will enable him to ensure
criticality safety but also the foundation of some interesting mathe-
matical developments if he is so inclined.
Risley E. R. WOODCOCK
Introduction
CRITICALITY control is of particular importance in the safe design and
operation of chemical and metallurgical plant processing or fabricating
fissile materials, in the handling and storage of enriched fuel for reactors,
and in the associated transport operations. Assessment of the effects of
neutron interaction between different parts of the system is an extremely
important element in this control. It has been shown(1) that the energy
yield from a critical excursion in an interacting array of several fissile
units will be higher than from a simple homogeneous system of the same
initial excess reactivity.
In a few instances separate items of plant may be spaced far enough
apart for it to be obvious that neutron interaction must be negligible,
but in most practical cases such an arrangement will prove grossly
uneconomic and a minimum safe spacing, or maximum size and number
of units at fixed spacing, must be determined. Experimental determina-
tion, ideal in principle, is seldom feasible in practice for interacting
systems, if for no other reason than that items of plant, etc., may not
exist at the design stage. Hence, the criticality adviser uses calculational
methods, supported by experiments on a few reference arrays, and it is
these methods which are the subject of this monograph.
The treatment is aimed at the intending criticality specialist. It does
not set out to provide a critical review of the considerable literature
which exists on neutron interaction, or to draw comparisons between
the many possible methods of calculation, each of which has advantages
in its own special field. Rather it is intended to describe the basic prin-
ciples involved as illustrated by a number of methods of calculation
which have proved their worth in daily use at major establishments.
xi
CHAPTER 1
The General Nature of the Interaction Problem
AN INTERACTING array may be defined as a system of two or more
bodies incorporating fissile material which are close enough together
for some neutrons to induce fissions in bodies other than those in which
they had their origin. If the bodies concerned are individually net
neutron sources this possibility of neutron exchange will clearly give rise
to an increase in reactivity and may result in the whole system becoming
critical, even though each body would be well subcritical in isolation.
Formally at least, the general reactor equations derived from the
Boltzmann transport equation may be applied to any system of fissile
material, however complicated. Hence, the description of a fissile system
as an interacting array is, in Avery's words,(2) "a statement only of how
one wishes to consider the system but implies, of course, that it is
believed advantageous to do so".
Many methods of calculation have been formulated: for any but the
simplest systems an exact analytical treatment by transport theory or
diffusion theory seems out of the question and a variety of alternative
approaches have been developed! ranging from highly detailed attacks
on few body problems to simple correlations of experimental results
with little or no physics. For nuclear safety purposes the aim is always
to provide answers which are as realistic as possible but demonstratably
conservative. Evidently the more detailed the analysis of the problem and
the more thorough the consideration of the many factors entering into
the exchange of neutrons in a given method the more realistic will the
answers be. At the same time, however, the more rigorous a method the
tWe are not concerned here with the theory of the lattice type of interacting array
commonly used in reactors, interest being confined to non-reactor situations.
1
2 THE GENERAL NATURE OF THE INTERACTION PROBLEM
less easy will it be to apply and the more restricted will it be, in general,
in its range of application. A real need exists for the simpler methods,
therefore, especially in circumstances where the expense of complex
calculations cannot be justified or where considerations other than
nuclear safety (e.g. ease of access to equipment) dictate minimum
spacings.
A recent development accompanying the introduction of high-capacity
computing machines has been the application of Monte Carlo methods
of calculation to interacting arrays. The advantages of the Monte
Carlo technique for systems of complex geometry are now being
exploited in programs which give results of high accuracy for the expen-
diture of relatively modest amounts of machine time and with few
limitations on the types of array which may be studied. This approach
to interaction problems is unquestionably the most powerful currently
available but, of course, it can be taken advantage of only where ade-
quate computing facilities exist.
Following a further introductory chapter on practical aspects of the
problem the remainder of the monograph is divided into two parts:
Chapter 3, broadly, deals with what may be termed simple "hand"
methods of calculation and Chapter 4 with the Monte Carlo method.
CHAPTER 2
The Problem in Practice
THE types of interaction problem which are amenable to other than ad
hoc treatment are limited to those involving separable fissile systems.
All methods of calculation must take into account either explicitly or by
implication two fundamental properties of such an array—the extent to
which the individual systems are able to multiply neutrons and the
extent to which the neutrons emitted by the various systems are reflected
back or impinge on other systems in the array, thus avoiding loss by
escape or absorption in the intervening media. Most methods deal with
well-defined arrays in which the bodies are fixed in their relative position
one to another and the density and composition of moderating and
reflecting material can be defined fairly precisely, e.g. chemical plant,
where the relative positions of vessels and their most reactive loading
can be specified with some confidence.
Within this general framework four main approaches to interaction
problems may be distinguished:
(i) Consideration of the equilibrium conditions of neutron fluxes in a
critical array, leading to relations between the neutron economy of the
individual bodies with surface sources due to interaction and their
spatial relations one to another and to any reflecting and moderating
material. Methods based on this approach assume some simplified
model for the neutron flow in the array, enabling values to be assigned
to parameters from measurements of neutron multiplication, from other
experiments, or from calculations on allied systems. If the results
obtained are to be conservative the significant physics of the situation
must be well enough understood to ensure that the simplifications intro-
duced increase rather than decrease the apparent reactivity of the array.
3
