T E C H N I C A L R E P O R T S SERIES No. 1 4 3

Developments in the Physics of Nuclear Power Reactors

INTERNATIONAL A T O M I C ENERGY AGENCY, V IENNA, 1973

DEVELOPMENTS IN THE PHYSICS OF NUCLEAR POWER REACTORS

The following States ate Members of the International Atomic Energy Agency:

AFGHANISTAN ALBANIA ALGERIA ARGENTINA AUSTRALIA AUSTRIA BANGLADESH BELGIUM BOLIVIA BRAZIL BULGARIA BURMA BYELORUSSIAN SOVIET

SOCIALIST REPUBLIC CAMEROON CANADA CHILE CHINA COLOMBIA COSTA RICA CUBA CYPRUS CZECHOSLOVAK SOCIALIST

REPUBLIC DENMARK DOMINICAN REPUBLIC ECUADOR EGYPT, ARAB REPUBLIC OF EL SALVADOR ETHIOPIA FINLAND FRANCE GABON GERMANY, FEDERAL REPUBLIC OF GHANA GREECE

GUATEMALA HAITI HOLY SEE HUNGARY ICELAND INDIA INDONESIA IRAN IRAQ IRELAND ISRAEL ITALY IVORY COAST JAMAICA JAPAN

"JORDAN. ' , ' KENYA , . . KHMER REPUBLIC KOREA, REPUBLIC OF KUWAIT LEBANON LIBERIA LIBYAN ARAB REPUBLIC LIECHTENSTEIN LUXEMBOURG MADAGASCAR MALAYSIA MALI MEXICO MONACO MOROCCO NETHERLANDS NEW ZEALAND NIGER NIGERIA NORWAY PAKISTAN

PANAMA PARAGUAY PERU PHILIPPINES POLAND PORTUGAL ROMANIA SAUDI ARABIA SENEGAL SIERRA LEONE SINGAPORE SOUTH AFRICA SPAIN SRI LANKA SUDAN SWEDEN SWITZERLAND SYRIAN ARAB REPUBLIC THAILAND TUNISIA TURKEY UGANDA UKRAINIAN SOVIET SOCIALIST

REPUBLIC UNION OF SOVIET SOCIALIST

REPUBLICS UNITED KINGDOM OF GREAT

BRITAIN AND NORTHERN IRELAND

UNITED STATES OF AMERICA URUGUAY VENEZUELA VIET-NAM YUGOSLAVIA ZAIRE, REPUBLIC OF ZAMBIA

The Agency* s Statute was approved on 23 October 1956 by the Conference on the Statute of the IAEA held at United Nations Headquarters, New York; it entered into force on 29 July 1957. The Headquarters of the Agency are situated in Vienna. Its principal objective is "to accelerate and enlarge the contribution of atomic energy to peace, health and prosperity throughout the world".

(C) I A E A , 1973

Permission to reproduce or translate the information contained in this publication may be obtained by writing to the International Atomic Energy Agency, Karntner Ring 11, P.O. Box 590, A-1011 Vienna, Austria.

Printed by the IAEA in Austria March 1973

TECHNICAL REPORTS SERIES No.143

DEVELOPMENTS IN THE PHYSICS OF NUCLEAR POWER REACTORS

BASED ON LECTURES PRESENTED AT THE THIRD INTERNATIONAL ADVANCED SUMMER SCHOOL

IN REACTOR PHYSICS, HERCEG-NOVI

Edited by J. POP-JORDANOV

Boris Kidri6 Institute of Nuclear Sciences, Vinfia, Belgrade, Yugoslavia

INTERNATIONAL ATOMIC ENERGY AGENCY VIENNA, 1973

DEVELOPMENTS IN THE PHYSICS OF NUCLEAR POWER REACTORS IAEA, VIENNA, 1973

STI /DOC/10 /143

FOREWORD

This book is based on lectures presented at the Third International Advanced Summer School in Reactor Physics held atHerceg-Novi , Yugoslavia f rom 31 August to 10 September 1970. The School was organized by the International Atomic Energy Agency, in cooperation with the Government of Yugoslavia, within the framework of the NPY Agreement. (The NPY-Pro jec t was a joint research program in reactor physics between Norway, Poland, Yugoslavia and the IAEA.) Ninety-nine special ists , with at least Master of Science degrees , f rom 28 countries in 5 continents, took part in the School.

The purpose of the School was to review the latest developments and problems in the theoretical and experimental physics of thermal nuclear power reactors . An introductory lecture on the role of reactor physics in the design and operation of power reactors was followed by a ser ies of lectures devoted to crit icality calculations, burnup physics , reactivity coef f i c ients , control , operation, fuel management and dynamics of power reactors . All these lectures were concerned with thermal reactors ; an ad-ditional lecture on the physics of fast power reactors was presented at the end.

In addition to the various lecturers whose papers appear in this volume, the Agency wishes to thank J. Berg , P. Blomberg, E .Cri toph, R.L. Crowther, B. Kolbasov, T . Kulikowska, J.E. Lunde, J. Pop-Jordanov (Chairman), D. Popovic , N. Raisic , F. Storrer and J.G. T y r o r , who formed the con-cluding Panel of the School.

Having in mind the world-wide interest in the subjects presented at the School, the N P Y - P r o j e c t Joint Committee decided to recommend publica-tion to the IAEA. In the course of preparing the final version of their con-tributions for publication, most lecturers have modified their original texts to bring them up to date.

As research in reactor physics is an important precondition for the development of nuclear power engineering, the present book is expected to be of interest both to scientists f rom countries with highly advanced r e -actor physics research , as well as to those f rom the countries where such research is just starting to develop. The book is suitable for special ists engaged in research and development as well as in design and operation of nuclear power reactors . It can be used as a re ference book in the field and can also serve as an advanced textbook for postgraduate study.

CONTENTS

The role of reactor physics in the design and operation of power reactors 1 E. C r i t o p h

Some comments on criticality calculations in thermal neutron reactors 15 M . C . E d l u n d

Reactor burnup physics 47 J. G. T y r o r

Analysis of reactivity coef f ic ients for power reactors 95 J . E . L u n d e

Control of power reactors 135 R. L. С r o w t h e r

Reactor physics problems concerning the startup and operation of power reactors 183 P . E . B l o m b e r g

Fuel management in power reactors : Fuel management and refuelling schemes 201 E. C r i t o p . h

Fuel management in power reactors : A simplif ied treatment of reactivity and reactivity l i fetime transients due to fuel management in power reactors 221 J . E . L u n d e

Dynamics of power reactors : Hydrodynamic models of boiling channels 235 J . E . L u n d e

Dynamics of power reactors : The ef fect of reactivity feedback on hydrodynamic stability in a heavy-water-moderated, boil ing-l ight -water -coo led reactor 241 E. C r i t o p h

Introduction to the physics of fast power reactors 247 F. S t o r r e r

THE ROLE OF REACTOR PHYSICS IN THE DESIGN AND OPERATION OF POWER REACTORS

E. CRITOPH Chalk River Nuclear Laboratories, Chalk River, Ont. , Canada

1. INTRODUCTION

This summer school will deal in detail with most of the areas in the power reactor f ield in which reactor physicists play a dominant ro le . It is appropriate that this be preceded by a general description of the role of reactor physics in the design and operation of power reac tors , so that the significance of these particular areas with respect to an overal l project is appreciated. This is an example of the whole being greater than the sum of its parts. 'The training which a reactor physicist rece ives tends to p r o -duce an understanding of basic principles over a wide range of topics , and as a result reactor physicists are often called upon to co-ordinate effort in more than one area and f rom more than one discipline.

The reactor physicist must resist the tendency to become so immersed in day-to-day problems as to lose sight of the overal l role of reactor physics. It is his responsibility to f orm an independent opinion of the o v e r -all requirements of a project with respect to reactor physics and to try to make sure they are met.

The rest of this presentation gives a general picture of the areas in which reactor physics has a role and some indication of the way in which this role can be successful ly played. Many of these areas will be treated in detail elsewhere in this book.

2. THE SCOPE OF REACTOR PHYSICS

Reactor physics is a relatively young subject and it has as yet no extremely well -def ined boundaries. The emphasis, if not the scope , has already shifted considerably over its short l i fe . Nevertheless there is a consensus as to what does constitute " reac tor physics" in that there is usually general agreement as to whether a speci f i c topic can be labelled " reac tor physics" or not. I will try to use this generally accepted range.

The best I can do for a general definition of the scope of reactor physics is to quote f r om the terms of re ference of the European-American Committee on Reactor Physics (EACRP). In this document the scope of the committee is defined by the following:

"Questions relating to the space, time and energy distributions of neutrons and radiations in different media fall within the scope of the Committee but the determinations of differential nuclear c r o s s -sections do not. "

1

2 CRITOPH

No doubt there is room for discussion as to the adequacy of this as a definition of the general scope of reactor physics but it does serve to define reasonably well the scope which I will assume.

It is c lear that valid reactor physics activities extend f rom research through applied research and development to engineering. It is difficult to be exact in the application of these terms but it is very apparent that over the last twenty years the emphasis has been shifting away f r om the research end of this continuous spectrum towards the engineering end. This i s , of course , to be expected and no doubt the trend will continue. As more becomes known regarding the physics of reactors and as satis-factory methods become available for predicting their character ist ics , the emphasis is increasingly on applying this knowledge rather than at-tempting to expand the basic knowledge — i. e. on the engineering aspect rather than the research aspect.

This paper deals with the role of reactor physics f rom two extreme points of view — an engineering one and a research one.

The "engineering point of view" assumes that definite pro jects have been authorized, and considers what reactor physics answers are required and the basis on which these answers can be supplied within a short - term schedule.

The " r e s e a r c h point of v iew" , on the other hand, considers the reactor pro jects which have been, are being, or might be undertaken; what the shortcomings in reactor physics information were , are , or might be; and how to remove these shortcomings in the longer term.

(There is another point of view which might be called the "pure research point of view" which would advocate the removal of any uncertainty in reactor physics regardless of the practical significance of the uncertainty or the ef fort involved in eliminating it. )

3. INTERACTION OF REACTOR PHYSICS WITH OTHER DISCIPLINES

Before going on to discuss the role of reactor physics f rom these two points of view, I should like to mention briefly the interaction of reactor physics with other disciplines.

Over the whole range f r o m research to engineering the reactor physicist must interact with people in other f ields. The most important of these f ields are:

Nuclear Physics Computer Technology Electronics Fluid Flow Materials Fuel Design Heat Transfer and Transport Control Economics .

The problem of how to accomplish appropriate interactions is often difficult. It is c lear that in order to establish satisfactory communication the reactor physicist must be familiar with his own field right up to the interface with the other discipl ine, and probably somewhat beyond. Even

ROLE OF REACTOR PHYSICS 3

with this condition fulfilled much iteration back and forth is often required before satisfactory solutions to complex problems are found. Without satisfactory communication, which requires some overlap of knowledge and jargon, the situation would be hopeless .

The EACRP has recognized this problem and has suggested the following sentence to be added to the scope which was mentioned previously:

"Since reactor physics is only one component of integrated reactor per formance , some consideration of adjoining fields such as engin-eering, materials , and computer technology will be necessary to ensure that the implications for reactor physics are understood and appropriately covered. "

I would like to give a couple of examples of what I mean. Figure 1 shows a prototype Douglas Point fuel bundle before and after

irradiation in NRU. (During the irradiation the bundle delivered 140 MWd of heat, or the same amount of heat as can be derived f r om 400 t of coal. ) The performance represented by this figure is truly a credit to the fuel de-sign engineers. However let us probe a little more deeply into what it also shows. This particular design has evolved as a result of a cooperative ef fort among not only fuel design engineers but also others involved in the f ields of fluid f low, materials , reactor physics , engineering design, c o m -puter technology and economics . In particular, reactor physicists have been involved, not only in experiments to check the physics characterist ics and in developing methods of calculation to predict the reactivity of lattices made up of such bundles, the effects of irradiation on isotopic composition and reactivity, and the distribution of power output through the bundle, but

F I G . l . Prototype Douglas Point fuel bundle (a) before and (b) after irradiation in NRU.

4 CRITOPH

also in the evolution of the design itself . The design is the result of many iterations among the various disciplines involved and the success implied by the figure re f lects on all of them. In particular, reactor physicists have been involved in setting the overal l dimensions (length and diameter), selection of materials , and speci f ic geometrical arrangement.

Another type of example which comes to mind is illustrated in Fig. 2. This shows the gain of a f irst azimuthal mode disturbance in a CANDU-BLW reactor design both with and without a zonal control system. This type of information i s , of course , very important in designing a zonal control system to ensure spatial stability in a power reactor . The major disciplines which are involved, in addition to reactor physics, are fluid flow, computer technology and control engineering. In this case , which is a boi l ing- l ight-water-cooled, heavy-water-moderated power reactor , the change in coolant density distribution in a reactor channel with channel power, which is the concern of the fluid flow engineer, is intimately con-nected with changes in reactivity due to changes in coolant density distri -bution. The calculations become complex so that efficient use of c o m -puters must be made. Therefore there must be a contribution f rom several disciplines in order to solve the design problem.

The attitude toward interdisciplinary problems tends to be different f r om the engineering point of view and the research point of view. From the engineering point of view there is usually an attempt to assign problems to the discipline most heavily involved, but few problems involve only one discipline and people f r o m different f ields must of necessity work together to accomplish what is important; namely, getting the problem solved. F r o m

OPEN LOOP GAIN B L W - 2 5 0 WITH CONTROLLER OF THE FORM

ak К лП

UNCONTROLLED REACTOR \

1.0

0.8

z

0.4

0.2

10" 10 FREQUENCY ( c / s l

FIG.2. First azimuthal gain in a CANDU-BLW reactor, with and without zonal control,

ROLE OF REACTOR PHYSICS 5

the research point of view the effort is more strongly directed towards a single discipline. In this case interdisciplinary activity o c curs mainly in a serv i ce sense; e. g. the latest techniques in computing and e lectronics are used where possible to increase accuracy and eff ic iency of experiments or calculations.

4. ROLE OF REACTOR PHYSICS FROM AN ENGINEERING POINT OF VIEW

As has been mentioned, the engineering point of view comes into play when power reactor projects have been authorized and it is important to have certain information in a relatively short time. Even f rom an en-gineering point of view reactor physics has a rather different status than many other disciplines associated with power reactor design and operation. This stems f r om the fact that reactor physics deals with matters which are relatively new and different and most of the things which are new or dif-ferent involve reactor physics. Certainly the part played by reactor physics in reactor design and operation is diverse and important.

Pro jects usually run through three distinct stages — conceptual design, design (and construction), and operation — and reactor physics has a s o m e -what different role to play in each of these stages.

4. 1. Conceptual design stage

During the early days of power reactors most projects were initiated with the aim of establishing a concept which seemed feasible and which seemed to have the potential to compete with non-nuclear power sources . Nowadays the interest is more in other nuclear competition and during the conceptual design stage a large number of alternatives are usually examined with an eye to choosing the most attractive on which to base a design. During this stage the emphasis is not on absolute accuracy in determining the exact character ist ics of any one system, but on relative accuracy and the ability to examine a very large number of major alternatives in a reasonable t ime.

In general the factors which it is important to determine for each alternative are:

General feasibility Capital cost Fuelling cost Operating cost Safety Reliability Development requirements.

The reactor physics which is required at this stage is that which is relevant to the determination of these factors .

However much more might be expected of the reactor physicist than the supply of reactor physics information. It can be expected that he also co-ordinate all the relevant data on thermodynamics, hydraulics, civil design, mechanical and fuel design, mater ia ls , costing, and ground rules into an evaluation procedure. During this stage the reactor physicist is

6 CRITOPH

often a key f igure, not only for his knowledge of reactor physics but for his ability to co-ordinate the contributions f r o m a number of different disciplines.

In the past the choice of a general concept has often been more of an art than a sc ience and acceptance has frequently been largely based on the " sa les ability" of its supporters. This is rapidly changing and now " o b -ject ive assessment" is generally accepted. Nevertheless there are no hard and fast rules which are fol lowed during a conceptual design. A certain amount of imagination is extremely valuable in arriving at a concept which is competitive. Reactor physicists , of course , have no monopoly on ima-gination, but neither are they deficient in it. Their special ist 's knowledge of those factors which make power reactor projects different f rom conventional systems should give them a decided edge in suggesting major innovations and they should exploit this edge to the utmost.

Requirements in reactor physics terms

The areas of reactor physics which are most involved in assessing the relative meri ts of alternative concepts (in approximate order of importance) are:

(1) Criticality estimates, reactivity coef f ic ients , burnup predictions (2) Fuel management (3) Dynamics and control , startup prob lems, shielding.

First and foremost the reactor physicist must be familiar with the general methods available in these areas and exerc ise good judgement in choosing the most suitable ones. As has already been pointed out the emphasis during this stage is on speed and relative accuracy rather than rigour and absolute accuracy. Calculations by the most sophisticated methods available are usually out of the question and approximations are an absolute necessity . For example during this stage of assessment for our D2O reactor concepts one assumes bidirectional fuelling, for which the approximation of actual core properties by uniform irradiation-averaged properties is quite acceptable.

This particular choice of important areas of reactor physics and their order is dictated largely by procedures adopted for intercomparing various alternatives in the conceptual stage. Usually the main factors (mentioned previously) are relatively insensitive to many parameters of the system. Often a particular design — the " r e f e r e n c e " design — is looked at in fairly fine detail and then used as a model on which to evaluate reasonably similar concepts. Thus many features may be approximately characterized on the basis of this model and a few parameters on which they depend to the f irst order . For example the whole shield structure may be characterized, for the purposes of sizing and costing, by the model plus the two parameters of reactor core radius and height. The external system of bo i lers , turbine, etc. may be similarly determined by the parameters reactor power, primary coolant flow and inlet and outlet temperatures and pressures of the primary coolant.

It turns out that the most relevant parameters are usually those con-nected with core size and number of fuel channels — for which criticality calculations are important; fuel life and cost — for which criticality and burnup estimates are important; and general safety features of the system — f o r which reactivity coeff ic ient estimates are important.

ROLE OF REACTOR PHYSICS 7

During this stage the reactor physicist must also remain aware of the possibility of doing a few key experiments which would help to settle i m -portant points of doubt. If t ime is available for such experiments he should make recommendations as to their suitability.

4 . 2 . Design (and construction) phase

Once a definite concept has been chosen and a spec i f i c project started, the ro le of reactor physics undergoes a marked change. There will still be a large number of prob lems, but these will now deal with minor modi -fications and detailed alternatives. The main requirement f r om an engineer-ing point of view is clearly the answers to a host of questions dealing spec i f i c -ally with what should be built. Prec i se specif ications must now be made on dimensions, materials and behaviour.

The emphasis is now on absolute accuracy — consistent with effort and expense. There is seldom time for drastic innovation with respect to theory, methods, or fundamental experimentation. The reactor physicist assosicated with design must live largely with what is available and be very select ive and discriminating in what he asks f or in the way of short - term theory and experiments (research point of view). Obviously then it is i m -portant that he be up-to-date in his knowledge of what is available and that he exerc ise much common sense and have a feel for what is reasonable. A pr ior training in reactor physics research i s useful in this regard.

Requirements in reactor physics terms The initial job is usually to f i rm up, in cooperation with other disc ipl ines ,

the reactor parameters f rom the conceptual phase; e. g. pitch, channel diameter, fuel design, vesse l s ize , r e f l e c tors , etc. A decision as to whether a mock-up lattice experiment is required should be made early in the design so that it can be done in time to give information which is still useful f o r design. Other experiments may also be planned but these are almost in-variably of a very speci f ic nature directly related to a particular design problem.

Another important job for the reactor physicist during the design stage is the estimation of flux and power distributions. These estimates must be made for a large number of conditions, e. g. insertion of booster and control rods , fuel configurations arising during fuel management, moderator level changes, etc. These estimates are important in design of control mechanisms, selection of fuel management schemes , positioning of ion chambers , shield design, fuel design, etc.

Burnup estimates must be made in connection with the choice of fuel management scheme, fuel design, fuelling machine design, etc.

Reactivity coef f ic ients must be estimated in support of safety analyses and control system design.

The reactivity ef fects of absorbers must be estimated in connection with control rod design. In this particular f ield, initial estimates are often made on rough designs which the reactor physicists initiate and these are then used by engineers to arr ive at a more detailed design. This must then be checked and perhaps modif ied and the whole process repeated again.

Reactor dynamics estimates must be made in support of safety and control system designs.

8 CRITOPH

Shielding calculations must be done in support of shield design. The problem of studying fuel management and proposing alternative

schemes is usually left pretty much entirely to the reactor physicists although there may well be feedback and iteration required with the channel, fuel and fuelling machine designers as well as with the future operators.

In fact reactor physics has a role to play in practically every aspect of design associated with the reactor system itself . Also it has a major role in safety analyses and plans f or operation; in particular the plans for the approach to equilibrium. E r r o r s in any of these estimates can lead to serious problems in the performance of the reactor .

The final task of the reactor physicist in the design stage is to take a ma jor ro le in the planning of startup, and in advising on experiments for the startup and commissioning phase.

4. 3. Operation

The period during which reactor physicists are most in evidence around an operating power reactor is during the startup and commissioning phase. It is during this phase that the groundwork is laid for the future safe and efficient operation of the plant. It is the duty of reactor physicists to plan and help carry out the physics experiments required to ensure such opera-tion. These usually involve much more than the well-known approach-to-cr i t i ca l , and include the measurement of as many key parameters as can be justified by the above l imited target. The results of these measure -ments must then be interpreted in terms of performance during operation and any required changes in design o r operating procedures determined.

Once this phase of the work is over the reactor physics involvement usually drops o f f , and f r om an engineering point of view assumes relatively minor importance. However, even f r om this point of view, there are still a number of legitimate reactor physics activities.

(a) If there are any changes due to unusual c ircumstances ( e . g . downgrading of heavy water, extensive fuel fai lures) the reactor physicist is called upon for advice.

(b) Responsibility f o r following fuel management during the approach to and early stages of equilibrium is usually given to the reactor physicist and his comments and advice are usually well rece ived.

(c) Improved performance is often contemplated on the basis of experience and here again the reactor physicist becomes involved in such areas as fuel design, power distribution, and fuel management.

(d) Minor problems in shielding and criticality ( e . g . in storage bays) often arise and necessitate consultation with reactor physicists.

(e) Operators and supervisors must be trained and this generally requires an elementary understanding of reactor physics, which is usually supplied by a reactor physicist.

(f ) Reactor physicists are often called upon to help diagnose the cause of unusual observations (e. g. excess ive noise on certain instruments).

F r o m an engineering point of view, however, a well -designed station should operate smoothly as designed and while it does so the reactor physics effort required is not considered particularly vital to the operation of a single station.

ROLE OF REACTOR PHYSICS 9

5. ROLE OF REACTOR PHYSICS FROM A RESEARCH POINT OF VIEW F r o m the research point of view the ro le of reactor physics in the design

and operation of power reactors is twofold:

(a) To anticipate the type of reactor physics problems which will arise f o r existing concepts and to accumulate and document the knowledge required to provide a sound basis on which these problems may be handled.

(b) To think ahead to new promising concepts and to provide the reactor physics basis on which the potential of these concepts may be soundly evaluated.

5. 1. Existing concepts

It is obvious f r om what has been said that, even f r o m the research point of view, in o rder to do a proper job reactor physicists must be well aware of the state-o f - the-art as applied to design and operation. It is only then that they can judge what is required and plan a sensible program to obtain the basic knowledge to fulfil the requirements. From a research point of view it is important that reactor physicists associated with the actual design and operation make known the weaknesses they see in existing in-formation and methods.

The general objectives of reactor physics research related to design and operation of existing concepts can be summarized as fol lows:

(a) To provide f i rm basic sets of reactor parameters over the full range of possible interest for the concepts. In the past " f i r m " has implied "experimental" but with the extensive availability of large computers it is now broadening to include calculations by extremely sophisticated and accurate methods. Nevertheless the point has not yet been reached where integral experiments can be completely eliminated s ince, at the very minimum, they are required as spot checks. Also in some cases where there is a choice between calculation and experiment, the ex -periment wins out on the basis of time and cost.

(b) To provide f i rm basic information on the ef fects , over the full range of interest, of perturbations to the reactor system.

(c) To provide methods for estimating these parameters and ef fects on a variety of levels of sophistication (and hence accuracy) . These methods should be suitable for applications which vary in the amount of detail that can be provided and in the amount of time and effort that can be expended. Evaluations of the accuracy to be expected f rom any set of these methods also f o rm part of the objective. These methods essentially provide means for interpolating between the " f i r m " parameters .

(d) To provide this information as inexpensively and efficiently as possible . This objective leads to research in such areas as experimental techniques, design of experiments, and application of computer technology.

(e) To provide basic information and development on the means of measuring flux, reactivity etc. and monitoring for failed fuel.

(f ) To provide a pool of expertise which is available for consultation on speci f i c problems related to reactor physics. It should be noted that the opportunity for direct consultation with a reactor physics research group is extremely valuable to a design o r operations team.

1 0 CRITOPH

Of course these objectives are never reached in an absolute sense, in that there are always qualifications with respect to accuracy, speed, and available tools . Thus it might be felt at a particular time that a speci f ic concept is in "good shape" and that no further reactor physics research is justified in its support, only to have the situation change later. The change could be caused by a variety of factors including:

(i) Variations in the concept — such as larger unit s i zes , or new ideas in fuel design, etc.

(ii) Changes in competitive factors — such as increased installed capacity leading to increased incentive for improvement, or poor performance relative to another concept.

(iii) Developments in research techniques or tools which improve the c o s t / benefit ratio for research.

(iv) Changes in power station requirements brought about by changing con-ditions — e . g . introduction of load following as a requirement as the nuclear fraction of power generating capacity increases .

Bearing these factors in mind it appears that there will be a continuing requirement for reactor physics research on presently existing concepts for some time in the future.

One area of reactor physics research which seems to be neglected is that of research in the operating reactors themselves. This is due partly to the difficulty of doing controlled experiments and partly to the disruptions such experiments would cause in operating schedules. Consideration is seldom, if ever , given during the design of power reactors to including ex-perimental reactor physics faci l it ies. Yet this appears to be about the only way to measure some of the parameters , under operating conditions, for which one cannot yet rely solely on calculation.

Examples of spec i f i c problem areas requiring research

5. 1 .1 . Detailed uniform lattice measurements

While there are buckling measurements which cover quite well the range of interest for most concepts, there is still a shortage of reliable, accurate detailed lattice measurements over the full range. This is par-ticularly true of some of the s impler lattices which were studied before all the techniques f or detailed measurements were developed. In many cases this information would be very valuable.

5. 1 .2 . Non-uniform lattice

There is a shortage of relatively simple methods f or treating non-uniform lattices reasonably accurately — especially when there is a high degree of non-uniformity.

5. 1. 3. Handbook-type design information

More work in reducing reactor physics information to a simple engineering f o r m which could safely be used over defined ranges by engineers with little reactor physics training would appear to be justified.

ROLE OF REACTOR PHYSICS 1 1

5. 1. 4. Spatial r e a c t o r dynamics

E f f o r t is required in this f ie ld part icularly in the area of optimum posit ioning of flux detec tors and contro l dev i ces f o r spatial contro l of flux and power in large r e a c t o r s .

5. 1. 5. Reactivity coe f f i c i ents

There is a need f or measurements of fuel t emperature , coolant t e m p e r a -ture , and coolant void coe f f i c i ents o v e r a wider range of lattice des igns , fuel m a t e r i a l s , and temperatures . These measurements are very dif f icult and a faci l ity in an operating h igh-power r e a c t o r in which detailed physics m e a s u r e m e n t s could be made would be useful .

5. 1. 6. Single rod exper imental techniques

The l e s s fuel required f o r lattice measurements the cheaper they b e c o m e . This accounts f o r the p r o g r e s s i o n f r o m f u l l - s c a l e exper iments to few rod exper iments . Single rod techniques are currently used but m o r e r e s e a r c h is required to establ ish f i rmly their potential and their l imitations.

5 . 1 . 7 . Se l f -powered detectors

There i s cons iderable s c o p e f or r e s e a r c h aimed at a quantitative understanding of s e l f - p o w e r e d neutron and Y-ray detec tors . This should lead to development of better detec tors and should also help in determining the best way to use such detectors to i m p r o v e fuel management and permit c l o s e r approach to fuel rating l imits .

5. 1. 8. Modular coding

Reac tor phys ics c odes are becoming very c omplex and often quite inf lexib le . Methods of handling such c o d e s , keeping them up - to -date , and adapting them to the solution of spec i f i c pract i ca l p rob l ems are becoming increasingly dif f icult . The use of the latest developments in computer technology f o r these p r o g r a m s is imperat ive . Modular coding i s the latest development which appears applicable and work is proceed ing in severa l countr ies on its use .

5. 1. 9. Non-destruct ive burnup measurements

Methods of non-destruct ive determination of fuel burnup are required f o r sa feguards applications and also in s o m e r e a c t o r concepts f o r determining whether or not to r e m o v e fuel f r o m the r e a c t o r .

5. 2. New concepts

Probably the mos t excit ing area in r e a c t o r phys ics is r e s e a r c h dealing with new concepts . Here a l so , famil iarity with the present s t a t e - o f - t h e - a r t with respec t to design and operation of existing concepts is useful s ince it g ives a f ee l f o r the type of p r o b l e m s which wil l likely ar i se and the areas in which to look f o r improvements .

1 2 CRITOPH

The latest concepts are the molten-salt and the high-temperature gas - coo led reactors . Currently there is also much emphasis on Pu recyc le and speculation regarding use of Th fuel cyc les in D aO reactors . Almost enough reactor physics research has been done to permit a sound evaluation of the potential of these concepts but much remains to be done to fully satisfy the eventual requirements for detailed design and operation. In some cases decisions have been made to proceed with this work. (It should be mentioned that such decisions are often made difficult by other important factors which are almost impossible to tie down, e. g. the future Pu market. Also potential gains, while significant, do not always justify required development expenses. )

At least for Pu recyc le and use of Th fuel the additional research is required mainly in the areas of lattice measurements, fuel management, and reactivity coef f ic ients .

6. COMPATIBILITY AND CONFLICT BETWEEN THE TWO POINTS OF VIEW

The engineering and research points of view are basically quite compatible. To have a successful overal l program both points of view must be represented. The engineering point of view is essential to getting on with the job and getting something done. Without it projects would drag on indefinitely while better and better information was accumulated and used. The research point of view, on the other hand, is essential to en-sure that a satisfactory basic level of information is available at all t imes , in o rder that no serious flaw occurs in the design o r operation.

What is important is that these two points of view be recognized and that communication be good between research groups and those associated directly with design and operation. Feedback in both directions is neces -sary — f rom the research people to design and operating people to ensure that the best information is being used, and in the opposite direction to ensure that emphasis is being put on the most important long-term problems.

When confl ict does arise it is usually the result of di f ferences of opinion on priorit ies . Two common areas of conflict will now be briefly discussed.

6. 1. Experiments in operating reactors

This conflict usually o c curs during the planning of startup experiments. The desire for early operation often leads to the following criterion being adopted to decide whether or not a given experiment should be done.

"Only those experiments having a direct bearing on the safe and efficient operation of the particular station under consideration should be done unless they in no way interfere with, or add to the time for , commissioning. "

This removes an important source of information for future pro jects . Perhaps research-minded reactor physicists are r e m i s s in not putting enough emphasis on deriving more information f rom operating reactors . They could go as far as to support the design of reactor physics research faci l it ies into some power reactors .

ROLE OF REACTOR PHYSICS

6 .2 . Mock-up versus fundamental experiments

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F r o m the research point of view mock-up experiments are often of little value relative to that of more fundamental experiments requiring the same expense and ef fort . From the engineering point of view the m o c k -up experiment is often the only one which will give the information r e -quired in time.

Obviously there can be requirements which are best met by both types of experiment. A recognition of both points of view helps in resolving the conflict.

It is only when one point of view is held in the extreme to the complete exclusion of the other that serious confl ict develops. Fortunately most people recognize both points of view and can see the c ircumstances of validity of each.

Possible co-ordination by means of ambitious simulation program

It has been suggested that the effort of research groups on the one hand and design and operations groups on the other, not only in reactor physics but in control engineering, heat transfer , fluid f low, fuel design and mechanical systems design, could be co-ordinated by means of an ambitious simulation program.

Under this scheme the groups would f i rs t cooperate in drawing up an agreed block diagram of an ultimate target in reactor plus external circuit simulation. Once this was done the next step would be to make simplif ications to the diagram to fit a reasonable short - term target for a working system. This simplif ied target would then be carr ied out with each group providing input for the areas in which it was knowledgeable.

At this stage a decision would have to be made as to which sections should have analogue and which sections digital simulation. Certain areas could be arranged so that actual hardware could be tested during develop-ment. Also it would no doubt be advisable to include alternative simula-tions, to suit different types of prob lems, in many sections.

Once this initial simulation was complete the design and operations group would use it to solve some of their problems, while the research groups would work to improve the content of the various blocks, having due regard to an agreed set of pr ior i t ies , and also to extend the simula-tion in the direction of the ultimate target.

At present anything approaching the ultimate target would be out of the question, but in 5 - 10 years ' t ime equipment could well be available to permit such a simulation on a useful scale . To be truly useful the simulation would have to include in each area a substantial fraction of available knowledge regarding the behaviour of that area.

Such an approach is not too different f r om approaches presently used except it ties together the information f r om one area with that f rom another. The disadvantage of this is that it requires very large capacity to run the whole simulation together, although it could be run in sections if necessary . The advantage is that it formally f o r c e s co-ordination among a number of disciplines and ensures that the methods being used in various areas are compatible, at least as regards input and output.

Even though the final goal of such a scheme could not be reached for many years it would be useful to start on it now.

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7. SUMMARY AND CONCLUSIONS

Reactor physics has a diversif ied and important role to play in the design and operation of power reactors . In the initial stages of a project the reactor physicist plays a key role in formulating the concept on which a design will eventually be based. His skills in analysis and experimental technique provide the basis f o r the sound evaluation of the physics of alternative concepts and he is often called upon to coordinate information f r o m many fields into an evaluation procedure. During detailed design the reactor physicist is rel ied upon to supply vital information on the physics aspects of almost every area of reactor design. He also contri-butes to safe and efficient operation of the reactor when it is operational.

The main message which I want to leave is that the reactor physicist should not be satisfied with being only a good specialist in a narrow field. He should also take an interest in the problems of neighbouring areas to an extent which will enable him to contribute more to any reactor endeavour. In this day and age one of the main problems is co-ordinating the knowledge in many different disciplines to solve real problems.

Also the reactor physicist should be capable of adopting both short-and long- term views, depending on the situation. He should always be prepared to support long-term effort for the solution of important problems, making the best of existing knowledge to meet important short - term targets.

SOME COMMENTS ON CRI TICA LIT Y CALCULATIONS IN THERMAL NEUTRON REACTORS

M . C . EDLUND Virginia Polytechnic Institute and State University, Blacksburg, Virginia, United States of America

1. INTRODUCTION

The earliest criticality calculations for thermal neutron chain r e a c -tors used the "age" model for neutron slowing down developed by Fermi in the early 1930s in connection with his experiments on transuranic elements. Age theory proved entirely adequate for criticality calculations of the f irst large graphite-moderated, natural-uranium-fuelled reactors and indeed served as a basis for criticality calculations for a considerable range of reactor types for a number of years . The early multigroup diffusion theory work of Brooks, Hurwitz and Ehrlich which was applied particularly to small intermediate neutron spectra reactors also employed age theory.

With the development of l ight-water reactors , the United States naval reac tors and the homogeneous reactor at Oak Ridge, the need for an alternate neutron slowing-down model was obvious, and from 1950 to 1960 most of the theoretical models and techniques which are used today in calculations for large water power reactors were developed. These rec ipes centre around multigroup diffusion theory and extensions of the Greuling-Goertzel model for neutron slowing down.

Neutron transport in power reactors is accurately portrayed by the well-known Boltzmann equation. However, because of the mathematical complexity of this equation, it was necessary to use approximations. This was particularly true in the early years of reactor development, when digital computers were also in their infancy. However, the development of high-speed, large-capac i ty digital computers during the past twenty years has led to a marked improvement in the calculation of resonance absorption and thermal neutron spectra in lattices and has made the solution of larger problems, such as three-dimensional fuel burnup in reactors , a matter of routine. The large-capaci ty computers also enable the use of the d is -crete-ordinates or Sn method in the solution of geometrical ly complex problems. Its general use for problems in thermal reactors , however, is still limited in pract ice .

This presentation outlines the development of the principal features of reactor calculations used today in the design of water power reactors in the United States of Amer i ca . I shall begin with a discussion of the general multigroup diffusion equations and their solution on digital computers. This will be followed by a discussion of the second part of the problem, i . e . the calculation of group constants in the fast spectrum region with and without resonance absorption and thermal neutron spectra. Finally, I shall make some comments on the use of variational principles for calculating

1 5

1 6 EDLUND

functionals which ar ise in three-dimensional flux synthesis problems and the use of the stationary functional as a Lagrangian for developing approxi-mate theories .

2. MULTIGROUP DIFFUSION THEORY

The multigroup diffusion equations can be written in the following form

О.УФ. - (2 . + E .) Ф. + У 1..Ф. + vx . yX . í ' . i i 4 ai s i i j i j J

for N groups of neutron flux, Ф1. The f irst term gives the leakage, Eai^i is the absorption rate in the i-th group, EsiO¡ is the rate at which neutrons are removed f rom the i-th group by elastic and inelastic scattering and Eji is the rate at which neutrons are transferred f rom the j-th to the i - th group by elastic and inelastic scattering. The last term is the f ission source in which Xi is the fraction of the f ission neutron spectrum in the i - th group and for simplicity is written only for one fissionable isotope. The "group constants" and the are functions of space. A number of rec ipes for the group constants have been used and I shall consider these rec ipes in the next section.

I shall now turn to the problem of solving these equations on digital computers. F irs t , I shall put these equations into matrix form by defining a column flux vector, Ф, consisting of the group fluxes and the following operator matr i ces

L = (—V D . V 1 X . + ai 2 , )6 . . s i l j

F = * i S f j

I = £ . . 1Д

The operator L is diagonal, F fills only the f irst few rows and the last few columns in a thermal neutron reactor and if all thermal neutrons are lumped into the N-th group, £ is lower diagonal, i . e . no up-scatter . The multigroup equations then have the following matrix form

ЪФ - ХФ = уРФ (2. 1)

The boundary conditions on Ф are the usual continuity of flux and cur -rent on all interfaces inside a reactor , and Ф or a linear combination of Ф and V $ vanish on an extrapolated boundary which is taken to be convex. (For the mathematically m o r e meticulous student, I note that these continuity conditions may not be entirely satisfied at corner points inside a reactor where the material properties of the reactor undergo a

COMMENTS ON CRITICA LIT Y CALCULATIONS 1 7

discontinuity. However, the set of discontinuities at these corner points in practical problems has a Lebesgue measure of zero and thus makes no contribution to integral properties of the chain reacting system. )

The multigroup diffusion equations can be cast into the form of an eigenvalue problem. It is known from elementary reactor theory that the neutron flux must be everywhere positive and that it assumes asymptotically with time a dominant or persistent spatial mode . For example, L- i /F for a one-group diffusion model is a self -adjoint Helmholtz operator which has a basis of real orthogonal eigenfunctions, Z n ( r ) , with real , discrete eigen-values in any bounded region. In this case, the time dependent equation

1 i f - (L-vFH

has the solution oo

V2 > P3 etc . to

As t approaches infinity, only the dominant mode Z j ( r ) will persist . For the steady state or the just critical reactor and one can cast

the eigenvalue problem into a different form which will be more useful in obtaining a solution to the multigroup equations. Starting with

L = vF

for the one-group case one notes that the inverse of L is mere ly the integral of its Green's function over the reactor which is certainly greater than zero . Thus,

= vL_1Fí

and

Мф = Хф (2.2)

where

M = L -1F

The criticality problem is thus equivalent to finding the maximum value of the eigenvalue of M. In practice, X is written as l/vc where vc is a

1 8 EDLUND

f ict it ious neutron regeneration factor which would make the r e a c t o r just c r i t i ca l a c cord ing to Eq. ( 2 . 2 ) . (An operational definition of the static e f fec t ive multipl ication constant is к eff ~v/vc . )

Returning to the mult igroup Eqs (2. 1), one f i r s t notes that L, which is now a diagonal matr ix , has e lements which are integrals of the Green ' s function f or each neutron group flux over the r eac to r , G¡ (r1 -» r ) . The operator , L - E , thus has an inverse given by

(L -S )T ! G i ( r ' н Reactor

r)dr '

(L-E) . î = 0 4 for (i < j)

4 - i for (i > j)

The iterative numer ica l methods which are used to so lve the mult igroup equations lead naturally to solving for the neutron source or power d i s t r i -bution in the r e a c t o r . I shall there fore work with S = F0 , the f iss ion rate, as the unknown v e c t o r . Multiplying Eq. (2 .1 ) by (L-E)" 1 f i r s t and then by F , one obtains

MS = XS (2 .3 )

where

M e F(L-Z) - 1

Habetler and Martino [1] show, using the Hi l le -Yos ida Theorem [2] and the work of Krein and Rutman [3], that Eq. (2. 3) has a posit ive rea l dominant eigenvalue, which is s imple and co r responds to an eigenfunction, ф о , which is posit ive everywhere within the r e a c t o r . F u r t h e r m o r e , | Л. 0| < 1 and M is convergent .

The mult igroup Eqs (2. 3) can be solved using the source iteration method (this method is s o m e t i m e s cal led the power method) . Starting with a guess of the f i s s i on s o u r c e distribution in both space and energy ( f i ss ion distribution among the neutron energy groups) , S ( 0 ' , one f i r s t computes the group f luxes using the equation

(Ь-Е)ф = S ( o )

Labell ing the neutron energy groups in ascending order with decreas ing neutron energy , one obtains solutions of N one -group diffusion equations, one at a t ime , starting with the f i r s t or highest energy group. Each of these is an inhomogeneous equation having a known s o u r c e distribution, and

COMMENTS ON CRITICA LIT Y CALCULATIONS 19

the boundary value problem can be solved by numerical iteration. The solution of each of the N one-group boundary value problems is called an inner iteration. Before discussing their solution, I shall outline the ca l -culation of the eigenvalue (which is called the outer iteration).

Assuming for the moment that one can compute the group fluxes, given a source distribution, one proceeds with the calculation of the crit ical eigenvalue. F r o m Eq. (2. 3) the f irst iterate of the f ission source dis-tribution is

.(1) = M S (о)

and the approximate eigenvalue is

A (o) _ (S (o)

(S (o) m s ( o ) I S (°>)

(S (o)

(S (o) s ( 1 ) l S)

(2.4)

Other expressions for the estimated eigenvalue can be used. However, the Rayleigh quotient given here is fairly simple and accurate. If M were se l f -adjoint, it would be the obvious choice, since the functional would be stationary with respect to small changes from the dominant eigenfunction of M and e r r o r s in X would be of second order for f irst order e r r o r s in S. The procedure is continued by using S ( 1 ' to obtain the group fluxes f rom the inner iteration and a new estimate of X calculated. If we continue the iteration procedure, S'k> and X(k) converge to the dominant mode and eigenvalue in the limit as к goes to infinity. One can see this by expanding S (°) in the eigenfunctions, фп , of M

, (o) = I A ip L. n n 11=1

The k-th iterate of the f ission source is

(к) = M k (O) = J к _ к £ L* n Tn rt L n=l n=l

А ф n n

Since X0 > Xn for all n, for large к

S (k> * AkA ф О О О

i . e . S . . . A ( k >

2 0 EDLUND

The X(n) are not apprec iab ly di f ferent f r o m Xo and the mult ipl ier of А0ф0 should be of o r d e r unity, o r if the original source guess is bad, at mos t a fac tor of 10_1 to 10.

T h e r e a r e two types of convergence c r i ter ia used in terminating the iteration. The f i r s t is a c r i ter ion on the s u c c e s s i v e est imates of the eigenvalue. One can set

(a) 1 -. 0 0 , ( k - 1 )

where e can be made quite smal l , a po int -by-po int convergence of

-10 if one wishes ; or one can require

(b) 1 -S ( k ) ( r )

S ^ r ) < &

f o r al l r (or points in the mesh) where 6 might be ~ 10"2 . Depending on the cho i ce of e and the nature of the prob lem, either cr i ter ia can be used. If detailed power distributions are des ired , the s o u r c e convergence cr i ter ion is the mos t accurate . On the other hand, if only the eigenvalue is des ired , c r i t e r i on (a) is usually suf f ic ient . The number of i terations in this case can be s ignif icantly reduced by solving the adjoint problem at each iteration and estimating the eigenvalue f r o m a stationary functional obtained f r o m variational ca lculus . I shall d i s cuss this in the last sect ion.

I shall turn now to the solution of the one -group boundary value prob -l e m s which constitute the inner i teration. In general , mater ia l propert ies vary cons iderab ly throughout a r e a c t o r and des igner -manufac turers do not build infinite slab or spher i ca l ly s y m m e t r i c r e a c t o r s . Thus, one must r e s o r t to non-analyt ic techniques . The method of approximating derivatives by finite d i f f e rences and solving high-order sys tems of l inear a lgebra ic equations is a natural procedure when using digital computers . Fortunately, the propert ies of the matr ix operator of the mult igroup equations al low the use of theorems concerning non-negative m a t r i c e s , to assure that a dominant eigenfunction and eigenvalue exist and that the iterative procedures used wil l converge to the des i red solution [4, 5].

One must so lve N equations of the f o r m

(k)

= Si oo - 1 v 1 v ,00 + I ' 1=1 3

(2 .5 )

with the boundary conditions given e a r l i e r . R e m e m b e r i n g that the ca lcu la -tion starts with the f i rs t group, the last t erm on the right-hand side of E q . ( 2 . 5 ) is known, and defining

3(10 E s

COMMENTS ON CRITICA LIT Y CALCULATIONS 2 1

the diffusion equation (Eq. (2. 5)) becomes

L. . ф . = s . i l i l

which, suppressing the indices, is mere ly

- V • D Уф + £ф = s

This equation can be written in finite dif ference f o rm by dividing the reactor by a finite number of coordinate planes, depending on the geometry, and approximating the derivatives by the appropriate central di f ferences . For example, in one dimension,

dx dx x=x. 1 = a. ф . ,.. + Ь . ф . . l i + l l i - i с . ф. i l ( 2 . 6 )

where the subscript is the label of the mesh point counted from a boundary or plane of symmetry . The coeff ic ients depend on the value of the diffusion coeff ic ient at x¡.x , x ¡ and x w and the di f ferences, A x ¡ and A x ¡ . i . If the di f ferences are equal and if the diffusion coefficient is constant, V • DV0 assumes the simple form

V . в7ф = — 2 — [ф - 2ф + ф. ] (Дх)2 1 + 1 . 1 1 - 1

A set of dif ference equations having slightly different coeff icients,- a ¡ , b¿ and Cj can be derived by integrating the differential operator f rom x i " x i - l / 2 to x j + AXj /2 and substituting the same differences for the derivatives. In any event, the form of the dif ference approximation is the same as given in Eq. ( 2. 6) which is called a three-point dif ference approximation and which relates the flux at a given mesh point to the flux at its nearest two neighbours.

Similarly, di f ference equations for two- and three-dimensional geo -metry can be written. These equations will involve five and seven points, again relating the flux at a point to its four and six nearest neighbours.

The dif ference equations representing Eq. (2, 5) in one dimension have the f o rm

а1ф1+1 + V i - 1 " ( c i + ' V * ± = s i

The number of equations is equal to the number of mesh points. In a typical criticality calculation the boundary points are not numbered and the ф are set equal to z e r o . If the more general boundary condition аф +bV = 0 is used, the boundary points are included in the mesh and the equations modif ied to satisfy the boundary condition.

The one-group diffusion dif ference equations used for the inner iteration have the general matrix form

Аф = s (2 .7 )

2 2 EDLUND

where the components of the ф vector are values of the flux at the mesh points. The number of equations is Nx • Ny - N z where Nx is the number of mesh points in the x direction, etc.

These equations can be solved by a number of iterative methods. Two of the most commonly used in reactor calculations are

(a) Successive over-relaxation (method of Young) — SOR (b) Chebyshev polynomial extrapolation

The SOR method is a linear extrapolation of success ive iterates in which the extrapolation or relaxation factor is chosen to minimize the spectral radius of the iteration matrix. The Chebyshev method is a semi- i terat ive method in which the norm of the sum of the e r r o r vectors of all iterates including the iterate being computed is minimized. The two methods are widely used and have comparable convergence rates.

A standard method for the solution of Eq. (2. 7) by iteration (since it cannot be iterated as it stands) is to transform the equation into the form

Ф = M + f (2.8)

where M = I - D"1 A a n d f = D " 1 s . The matrix is split into parts, the sum of a matrix of its diagonal elements, D, and the non-diagonal elements. M is called the Jacobi matrix of A and Varga [6] shows that the eigenvalues of M are real and have absolute value less than unity and thus M is convergent. This is the algebraic analogue of the Habetler-Martino theory for the continuous operator mentioned earl ier and used in the discussion of the source iteration calculation of the crit ical eigenvalue.

The most direct method, sometimes called simultaneous relaxation, simply iterates Eq. (2. 8) by using

,ф ( к + 1 ) = Мф ( к ) + f (2 .9)

The e r r o r for any iterate is

E(k> = MkE(°>

where is the e r ror in the guess in the original flux distribution. Expanding E ( 0 ) in the eigenfunctions of M, фп , the e r ror in the k-th iterate becomes

CO CO

E = Mk У А ф = J ÀkAi|! 0 as k + « ил n n n n n n=l n=l

Although Eq. (2. 9) will converge, the rate of convergence may be slow. The convergence rate can be improved simply by noting that we can use the

COMMENTS ON CRITICA LIT Y CALCULATIONS 2 3

values of ф ( к + 1 ) with the lower triangular part of M during the (k + 1) i ter -ation. Thus, letting M = L + U where U is upper and L lower triangular (M has no diagonal elements), one can improve the (k + 1) iteration by using ^(k+l) w i th L as fol lows,

ф(к+1) = Ь ф (к+1) + и ф (к) + £

or

ф(к+1) = ( 1 _ ъ ) -1иф (к ) + ( x . L ) - ^ (2 .10)

Geiringer [7] demonstrated the convergence of this method and showed that the eigenvalues of (I - L)"1 U are the squares of the eigenvalues of the Jacobi matrix . The rate of convergence (which is proportional to the negative logarithm of the spectral norm of the matrix operator (absolute value of the eigenvalue with largest magnitude)) is thus twice as fast as the simultaneous relaxation method. This method is sometimes called a Gauss-Seidel iteration.

Since a large number of computations are required to solve a typical reactor crit icality problem, there is a correspondingly great incentive to reduce the number of arithmetic operations by improving the rate of con-vergence of the iteration method. The dif ference between two success ive iterates using Eq. (2 .10) is

Ф _ ФИ0 = + и ф (к) + f _ ф(к)

The rate at which approaches zero can be increased by multi-plying the right-hand side of this equation by a number greater than unity. Thus, multiplying the right-hand side of the equation by u, called the r e -laxation factor, the iteration equation for the success ive over-relaxation method becomes

ф(к+1) = ( X - C Ú L ) — + ( l - O O ) L ] ф ( к ) + o o d - u D ^ f (2.11)

Young [8] showed that the best choice of the relaxation factor is

2 (JL) - -

1 + / l -y z (M)

where (M) is the spectral norm of the Jacobi matrix of the original equations.

The optimum value of u obtained by Young minimizes the spectral , radius of the success ive over-relaxation matrix in Eq. (2 .11) . When the optimum value of to is used, the number of iterations is reduced to about the square root of the number required for the Gauss-Seidel iteration. (The latter is the same as choosing и = 1. )

The relaxation factor can be estimated by using either the simultaneous or success ive relaxation methods, which can be made an automatic part of

2 4 EDLUND

a digital computer program for solving the multigroup diffusion equations. Experience indicates that optimum values of и be in the range 1 . 5 - 2 . 0 for a wide range of reactor criticality problems.

In the Chebyshev polynomial method, a linear combination of all the previous iterates is used as the source for the current iteration; i . e . , the source for the (k + 1) iteration is

S ' ( k > = I A.S J-0 J

The sum of the e r ror vectors is

where

F ( k > = I j - 0 J

E ( j ) . S (J ) _ s* = M j E ( ° )

S* is the c o r re c t solution of Eq. (2. 8) and E ( 0 ' is the e r ror in the original source guess. Thus,

F ( k ) = У A mJE ( o )

Expanding E ( 0 )

in the eigenfunctions of Mi//m = Х т ф т ,

N » - , _ T mm У ъ ф ь mm m=l

and one obtains

n\ N к . F = У Ъ ф I А.Х3

m=l m m j=0 m

The e r r o r vector , , can be minimized by selecting the A j such that the polynomial

к P. (X) = I A.AJ

k J-0 J m

has its least maximum value subject to the condition P k ( l ) = 1.

COMMENTS ON CRITICA LIT Y CALCULATIONS 2 5

Flanders and Shortly [9] applied Chebyshev minimax theory to this problem, by noting that the least maximum value of Pk(X) in the interval -1 to 1 is given by

T (x/y(m)) P (xl = — v x ; Tk(l/v,(m))

where T k (y ) = cos (kcos" 1 y) are the Chebyshev polynomials and /J(M) is the spectral radius of M. Recurrence formulae using the recurs ion formulae for Chebyshev polynomials are used as discussed by Wachspress . In general, calculations are performed in cyc les of m iterations and at the end of each cyc le the process is repeated with new estimates of the eigen-values. Hence, the term semi- i terat ive is used to describe the method.

The two iteration methods described have comparable convergence rates . The Chebyshev method can also be used to reduce significantly the number of outer iterations.

The PDQ-7 program (Argonne Code Abstract 275) is an example of one of the latest neutron diffusion-depletion codes for 1, 2 and 3 dimensions available for use on a Control Data 6600 Computer. The one-dimensional group equations are solved by Gauss elimination and the two- and three-dimensional equations by the Chebyshev semi- i terat ive method. The outer iterations are accelerated by Chebyshev polynomial extrapolation.

The total number of neutron groups is limited to f ive. The product of the number of groups and mesh points is limited to 300 000, with the additional condition that no plane can contain more than 8000 points. Typical running times in hours are reported to be roughly equal to the product of the number of groups and mesh points divided by 150 000.

3. MULTIGROUP CONSTANTS

Asymptotic reactor theory with the extended Greuling-Goertzel model of neutron slowing down is widely used to calculate fast neutron group constants for multigroup diffusion equations. In each region of a reactor having uniform material properties, a geometr ic buckling is estimated and used to compute a fast neutron spectrum of a bare, homogeneous reactor having the same buckling. The resulting spectrum is then used as the weight function to obtain average values of the desired quantities for each neutron group for the region.

This recipe for obtaining fast group constants is not entirely satisfactory if the number of neutron groups is small . Neutron spectra show consider-able variation f rom the asymptotic spectrum near region interfaces in the reactor and particularly so at the co re ref lector interface. Also the problem of estimating a geometr ic buckling for a region is more a matter of guess -work than rational calculation. Fortunately, in many practical cases , the crit icality eigenvalue using a 3 or 4 neutron group calculation does not depend strongly on the buckling used in the fast spectrum calculation. Starting with the Boltzmann equation I shall derive the consistent PL approximation for the asymptotic fast neutron spectrum using the extended Greul ing-Goertzel model for neutron slowing down.

2 6 EDLUND

The one-dimensional energy-dependent Boltzmann equation for a homo-geneous multiplying medium with no extraneous sources is

u + Г т(и)ф(х,и,ц) = 2ir [ du' [ d u ' c í u ' ^ y ' H - p,u ' - u)(x,u\p') Í ] (3.1)

where ф(х, u,/u) is the neutron flux distribution in angle and lethargy, i . e . the number of neutrons cross ing unit area normal to a direction having a direction cosine, ц, with respect to the x -ax i s in unit time and having lethargy in unit range about u. For simplicity it is assumed that the flux distribution depends only on one space coordinate, x — this will not limit the generality of the results . The coll ision distribution function f n,u'-> u) gives the probability that neutrons having direction and lethargy д 1 and u' be in unit range about ц and u after col l is ion, с (u1) is the average number of secondary neutrons produced per col l is ion having lethargy in unit range about u.

Since the Boltzmann equation for a homogeneous infinite medium with uniform isotropic sources is invariant to translation of the coordinate system, a solution of Eq. (3.1 ) is

Ф(х,и,р) • F(k,u,y) e ± 1 k x (3 .2)

where к is a real positive number. Indeed, as demonstrated by Davison [10], this gives the asymptotic solution for a multiplying medium, provided к is the largest positive real root of the characterist ic equation obtained by substituting Eq. (3. 2) into Eq. (3 .1 ) . In a bare homogeneous reactor , setting k 2 equal to the geometr ic buckling gives an approximate condition for crit icality.

An outline of the development of the PL approximation of the Boltzmann equation will now be presented. The mechanics of neutron-nuclei interac-tions leads to a natural division of the neutron diffusion problem into two energy regions, slow and fast. In the fast region (above about 1 eV), the motion of the nuclei can be ignored and the elastic scattering distribution function is completely determined by the differential scattering c r o s s -section and the angle of scattering. On the other hand, for slow neutrons, the "thermal motion" of the nuclei dominates the mechanics of the scatter-ing process , and the coll ision distribution function cannot be obtained f r om simple kinematic considerations.

The col l is ion distribution function for fast neutrons is the sum of three terms corresponding to the three fundamental neutron-nuclei interactions :

(a) Elastic scattering by stationary nuclei (b) Inelastic scattering (c) F iss ion .

To simpli fy the presentation, the heterogeneity of the fuel -moderator lattice structure will be ignored. Also , it will be assumed that the medium has only one type of moderator atom, one type of inelastic scatterer , and that all atoms except the moderator have infinite m a s s . The effect of lattice

COMMENTS ON CRITICA LIT Y CALCULATIONS 2 7

heterogeneities, primarily on resonance absorption and inelastic scattering, will be considered later. The extension of the resulting equations to m o r e than one type of moderator and inelastic scatterer will be obvious by inspec-tion of the results .

The angular distribution of f iss ion neutrons and of neutrons inelastically scattered by heavy elements is isotropic in the laboratory coordinate sys tem 1 . Thus, their contribution to c ( u ' ) f ( / u ' u' -» u) becomes simply

[ v ( u ' ) S , ( u ' ) s ( u ) + I . ( u ' ) f . (u' + u ) ] (3 .3) 4ir f i n 1 n

where s(u) is the f ission neutron energy spectrum, v (u ' )E f (u1) is the m a c r o s c o p i c f ission c r o s s - s e c t i o n multiplied by the total f ission neutron yield, E i n ( u ' ) is the m a c r o s c o p i c inelastic scattering c ross - sec t i on , and f i n (u1 -> u) is the probability that an inelastically scattered neutron will be scattered f rom lethargy u1 into unit range about u.

Since the elastic scattering of neutrons by stationary nuclei is equi-valent to the interaction of a neutron with a central f orce field in which energy and momentum are conserved, the energy lost by the neutron depends only on the angle of scattering and the mass of the target nucleus. From elementary kinematics, the cosine of the angle of scattering in the laboratory system, ц 0 , is a single-valued function of the lethargy gain, U= u - u ' , given by

U0CU) = \ [(A + 1) e-U/2 - (A - 1) eU/2] (3.4)

where A is the ratio of the mass of the target nucleus to the mass of the neutron. There fore , the elastic scattering distribution function is a func-tion of only two variables — the initial neutron energy and the angle of scattering; the energy after scattering is determined by Eq. (3 .4) . It is a distribution in u - ц space which (a) is proportional to the differential scattering c ross - se c t i on (до, u') in the laboratory system, and (b) vanishes except on the curve given by Eq. (3 .4) . Thus,

c ( u ' ) f e ( i i ' 4- y ,u ' + u) = Е̂ Сл .u'JiDT - u'(u ) ] (3.5)

where u ' ( /n 0 ) satisf ies Eq. (3. 4). I now turn to the general analysis of the Boltzmann equation without

further transformation of the elastic scattering distribution function, since the most useful form of this function is determined by the general method used to obtain approximate solutions of the Boltzmann equation.

The spherical harmonics method [11] applied to the Boltzmann equation has proved to be the most useful tool in understanding neutron diffusion in reac tors . It is central to reactor theory, providing a powerful technique

1 This is not precise — the angular distribution o f an inelastically scattered neutron will depend on the state in which the target nucleus is left after collision. However, for incident neutron energies of about 2 MeV or greater in heavy nuclei, there are many excited states available to the target nucleus, and averaging over many states tends to make the distribution isotropic.

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for developing physical concepts as well as practical design methods. It is particularly useful in asymptotic reactor theory for those cases in which the angular distribution of the flux is nearly isotropic . The spherical harmonics expansion of the flux then converges rapidly, and one need retain only the f irst two terms in the expansion. This is the basis of almost all reactor design calculation codes used today. Accordingly, one expands the Four ier t r a n s f o r m of the flux distribution in spherical harmonics

*(k,u,y)= L| 0 ^ j i - L Ф1(к,и)Р|_(м) (3 .6)

where P l ( m ) a r e the Legendre polynomials. Per forming these operations, Eq. (3 .1) becomes , for 0 á u S u 0 ,

P T ( U ) - iky] z -

COMMENTS ON CRITICA LIT Y CALCULATIONS 2 9

FIG. l . Direction cosines.

Referr ing to Fig. 1, ¡л0 , ц and ju 1 are related by the addition theorem for Legendre functions:

PL (y o ) - PL (v)PL (u ' ) + F[cos т(ф - ф ' ) ]

Terms containing F [ cos m (ф -ф ' ) ] vanish upon integration over the az i -muthal angle ф' because of symmetry, and E q . ( 3 . 8 ) reduces to

SI = 1 —1 J. (k,u)P. (p) (3.9) L=0 4ir L L

where

JL (k,u) = J du'B°(u' ,и)ф1 (к,и ' ) (3.10) U - Ç

Differential scattering c ross - se c t i ons for light elements are isotropic in the centre -o f -mass coordinate system over most of the energy range of interest. Hence it is convenient to work with c ross - sec t i ons in the centre-o f - m a s s system. Following the notation of Zweife l and Hurwitz [12] in their work on transformation of c ross - se c t i ons , I shall express the B£(u ' , U) in terms of the differential scattering c r o s s - s e c t i o n in the centre -o f -mass system, £^ ( ju c , u ') , where ц с is the cosine of the angle of scattering in the CMS. Thus,

# v u ' ) = j 0 ^ B í ( u , ) W (3.11)

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Since the differential c r o s s - s e c t i o n s in the CMS and LS are related by

E s ( v u ' ) d u c = z ° ( v u ' ) d u 0

+1

bJ(u'.U) = ЙГ f « V e ( V u ' ' ü , p i c 4 ) -1

+1

= 2тг J dp0sj(w0,u,)e[u' - " ' ( w ^ l V V -1

+ 1 dp - 2n J dV[u- - и'(У0)]Рк(Р0) ̂ Z 2Jir Bf(«')PL(yc)

-1 Noting that

6[U" - U'(w0)] - 6[y0 - P 0 (U)] d¿-= 6[u0 - p0 (U)] ( - J n )

one obtains the desired result

B°(u',U) = г t KL ( Ü ) B L Í U , ) (3.12)

and the transformation matrix

- V 1 w w ( - ( З Л З ) where and ц с are evaluated at U = u - u 1 . F r o m the kinematics of the scattering col l ision,

The Fourier - transformed Boltzmann equation is

t l T (u) - 1кц]ф(к,и,у) • J JL(k,u)PL(y) + S(k,u) (3 .15)

where "o

S(k,u) = f du ' [vCu' )Z , (u ' )s (u) + £ i n ( u - ) f 1 n ( u ' - и)]Ф0 (к,и' ) 4ir I T

о

+ f V f th*th o < «

COMMENTS ON CRITICALITY CALCULATIONS 3 1

J L (k ,u) U f du' Î T | l f (U )B®(u ' ) í L (k ,u ' ) J K = 0 K u-ç

(3.16)

Although Eqs (3. 15) and (3. 16) can in principle be solved numerica l ly with high-speed digital computers , it is much m o r e instructive (and economica l of machine t ime) to consider methods of approximating the integral (3 .16) . A l l of the methods widely used in reac tor design calculations can be obtained by a Tay lo r ' s s e r i e s expansion of B £ ( u ' ) 0 L (k, u1) about u,

с 00 f i\n „ ЭП[В^(и)ф. ( k , u ) ] (u1 )Ф. ( k , u ' ) = S Щ - U " (3 .17)

* L n=0 Эи

Defining

Tu^(duuVu) G"(U)= Ê T"KB^(U) (3 .19)

K=0

. э п [ е " + 1 ( и ) ф , ( к , и ) ] q ( k . u ) = - £ - (3 .20)

L Эи"

Eq. (3. 16) b e c o m e s

О, ч , ЭЯ| (к,и) J (k,u) = G|_ " - " S u — ( 3 ' 2 1 )

where q L ( k , u) is the L - t h Legendre component of the s lowing-down density. The age approximation is obtained by using only the f i rs t term for

q 0 (k , u) and ignoring all q L f or L > 0 in the expansion (3. 20). By the usual method of generating equations for the Legendre components of the neutron flux, Eq. (3 .15) gives for the age approximation

£ т (и)Ф о (к,и) - 1кф1(к,и) = J o (k ,u ) + S o (k ,u )

Г т (и )ф ] (к ,и) - j 1кф0(к,и) = J|(k,u)

. 3q0(k,u) J0(k.u) = G°o{uHo(k,u) - S —

J ^ k . u ) = 6°(и)ф 1 (к,и)

q (k,u) = 6 (и)ф (к,и) и 0 0 ( 3 . 2 2 )

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For isotropic scattering in the CMS, G§ =ES , G j = - , and G? = м0 Es . One obtains immediately the well-known results of age theory; namely, the slowing-down density q= Ç E8ф and the conservation equation

Ж 3" Га + 3 ( E T - v 0 Z s )

+ S

The extended Greuling-Goertzel approximation is a straightforward extension of their method to the general anisotropic scattering law [13]. Formally, the method replaces the expansion (3. 20) for q L (k , u) by a f irst-order linear differential equation in u. The linear combination of 9qi, (k, u)/9u and qL(k, u) is

3q,(k,u) со ЭП[б[1(и)ф|(к,и)] « эП[б"+ 1(и)Ф.(к,и)] X L ( u ) я ц + * " X L { U ) 2 ñ Ln L L au L L n-1 a„n n=0 n=l 3U 3un

where X(u) is a function of lethargy which one can choose to give the best approximation. Collecting terms and rearranging, one obtains

3q. (k,u) i 2 . 1 dXi 1 00 x l ( u ) + q L ( k ' u ) = * Ь ( x l g l \ + g l V + GL*L dT * M l + ¿ 2

(3.23)

The expansion which contains second and higher derivatives of is neglected, and XL(u) is chosen to make the f irst derivatives of б " ф ь vanish. Thus, choosing

V") - - 1 (3.24) the differential equation becomes

XL(u) 3qL(k,u)

3u + qL (k,u) = 6^(и)ф1(к,и) 4 du

- 1 (3.25)

For hydrogen, this equation is exact since the sum in Eq. (3. 23) vanishes. The consistent P - l approximation is obtained by including all terms

in the equations for , JL , and q L through L = 1. The set of consistant P - l Greuling-Goertzel equations is then

Г т(и)фо(к,и) - 1'кф^к.и) = JQ(k,u) + SQ(k,u) (3.26)

Ет(и)ф1(к,и) - рф 0 ( к ,и ) = . y k . u ) (3.27)

COMMENTS ON CRITICA LIT Y CALCULATIONS 3 3

J o ( M , . G ; V k . H , . M ^ l (3 .28,

3qi(k,u) J j ik .u) = G f y f k . u ) - (3.29)

3q0(k,u) i I < M • + q 0 (k ,u) - -G j (u ) $ 0 (k ,u ) 1 (3.30) о 3U

3q,(k,u) . / d M — 1 ^ — + q - , ( M ) = -б { (и)ф 1 (к .и) 1 - dU1) (3.31)

S 0 (k,u) = I d u ' [ v ( u ' ) r f ( u ' ) s ( u ) + E i n ( u , ) f 1 n ( u ' - " ) ]ф 0 (к .и ' ) 0 + s ( u ) v t h s f t ^ t h 0 (k ) (3.32)

These equations can be extended to the case of more than one moderator by noting that each moderator will contribute a term given by Eq. (3 .16) . Thus, Jo and J^ in Eqs (3.26) and (3.27) are replaced by sums that contain, for each moderator , terms that satisfy Eqs (3. 28) through (3. 31).

Several available computer codes can be used to solve the finite dif-ference form of the fast spectrum equations outlined in Section 3.1 .

The neutron energy range f rom 10 MeV to the thermal cutoff that defines the division between fast and slow neutrons is' divided into a large number of groups called microgroups . Equations (3.26) through (3. 32) are integrated over each lethargy interval, Auj =Uj - U j . j , and integrals of the product of c ross - sections and the Legendre components of the flux are r e -placed by the product of the average value of the c ross - se c t i on in Auj and the group flux,

Lj - J L(u')du' U j -1

i . e.

where

Uj J G j (u 'H L (u ' )du ' - (3.33)

U M

UJ g" f G"(u')du' Lj Auj J L

" j - l

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Except for resonance reactions, and near the thresholds of inelastic scatter-ing reactions, the cross -sect ions vary slowly enough so that the approxima-tion (3. 33) is highly accurate for group widths no smaller than 0. 25 to 0.50 in lethargy.

However, the c ross - sec t i on and neutron flux vary extremely rapidly with lethargy near a resonance, and the truncation of the Taylor 's series expansion of В k(u ') ф l (k, u1) about u used in developing the fast spectrum equations is a poor approximation. Since the resonances are narrow com-pared with the maximum logarithmic energy decrement (Ç) per scattering coll ision in hydrogen and deuterium, the number of collisions in the resonances is small compared with the number of total collisions over a lethargy increment (?)• Thus, the flux at lethargies outside the resonance is well represented by the fast spectrum Eqs (3. 26) through (3. 27), pro-vided one accounts for the neutrons absorbed in the resonances to obtain a correct neutron balance. Let pj be the resonance escape probability for the resonances in the j-th microgroup. Then the number of neutrons absorbed in the j-th group resonances is (1 - P j ) q o ( u j - i ) where q о (u j-i ) is the slowing-down density at the lower lethargy bound of the j-th m i c r o -group. The absorption term in Eq. (3. 26) becomes

where £ a j is the average value of the slowly varying (nonresonance) ab-sorption cross - sec t ion .

Resonance absorption in the Ет1 term is neglected since it does not affect the neutron balance. It can make only a small contribution to the diffusion coefficient by virtue of the above arguments with respect to the relative number of col l is ions. I shall discuss later the calculation of resonance absorption, which in this method is decoupled from the spatial neutron diffusion calculation.

Resonance scattering, which occurs only in elements other than hydrogen and deuterium, does not have a large influence on the spectrum outside the resonances, since most of the slowing down of neutrons resulting f rom scattering is merely included in the average macroscop ic c ross - sec t ions , except in the calculation of the resonance escape probability.

Inelastic scattering is treated by the use of neutron group transfer matr i ces . The rate at which neutrons are inelastically scattered into the j-th microgroup is given by

V i г . (и ' )ф (u ' )du' - (1 - P. )q 0 (u. ) + £ V j a 0 3 0 j - l a j о (3.34)

( 3 . 3 5 )

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Defining the inelastic transfer matrix by

U, Uj I . , J _ f du' f duf. (u1 u) (3.36)

i j ди, J j in v 1

which gives the probability that neutrons scattered inelastically in group i be emitted in group j, and replacing the integral of products with products of averages , one obtains

M !1n j - ^ £1п l V l J < 3 - 3 7 )

where E in i and фOÍ are the average values of the inelastic c r o s s - s e c t i o n and neutron flux in the i - th group.

With the definitions and approximations discussed above, and using a superscript (m) to re f e r to the m-th isotope and subscript (j) to re fer to the j-th neutron group, the microgroup equations are

^ a j + j + » Х > > о о + Вф10

= Г т [ ( Г 0 ) > ^ ( 1 - Р а . ) ] Ч > ^ ) + 5 г . + 1 . п , (3.38) J

• • • „ • [ . „ - « . t , ( 3 - 3 9 '

• i ' V • f - ' « А ' " |З И |

where

(3.41)

r a j - Z m E a j ; E Tj " V f j (3.42)

T1 = r frm - ?m Im ) r i n j V r i n j r i n j I j j ) (3.43)

m m ' i n j E sm i V o i (3.44)

w"! = - (Gbm J о J ( 3 . 2 2 )

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(О? =

хТСд) - x f r ^ )

i ж .

0 j (X )m + -U 1 ' J 2

.m 1 — T ^ T

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The group constants are then obtained from

D = Í L . (3.53) п. B ^

1 Г Гап1 = лГ ra (u)$0 (u)du ( for smooth (3.54)

™ ¡J absorption only) m-1

= t f v l f (u )* 0 (u )du

"m-l

(3.55)

"ill = I X(u)du (3.56)

um-l

? ? Ji3m= J du J du' "V"

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TABLE I. NON-LEAKAGE PROBABILITIES COMPUTED BY MOMENTS AND BY FOURIER TRANSFORM METHODS

Data from Paschall:

7 = 156.30 ± 1. 92 c m ¡ 3 = 9. 55± 0 .6 X 104 cm11 rs = 1 .49 ± 0 . 2 X 10« c m '

P(B2)

ВНЮ"» c m 2 ) Moments o f

experimental data Fourier transform

of experimental data P - l calculation

1 0.9747 0.9724 0.9753

2 0.9508 0.9479 0.9517

4 0 .9065 0.9024 0.9074

6 0 .8656 0.8612 0.8664

8 0 .8269 0.8237 0 .8285

10 0.7888 0.7893 0.7932

20 0.5010 0.6516 0.6483

1.0

0.9

0.8

ГЧ CD

0.6

0.5 0 4 8 12 16 20

B 2 (10~3 c m 2 )

FIG.2. Fourier transform of slowing-down kernels for H ¡ 0 .

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The absorption of fast neutrons can be divided conveniently into the following four energy regions:

(a) Low energy (smooth l / v for some isotopes) (b) Resolved resonance (c) Unresolved resonance (d) High energy .

The absorption c ross - se c t i on in the f irst and the fourth energy regions is smooth and contributes to the absorption and total c ross - se c t i ons , as mentioned previously. The low-energy region extends from the thermal neutron cutoff 1 eV) to the energy of the f irst resolved resonance for each isotope. The resolved resonances extend up to energies of about 1 keV. The unresolved region is fairly narrow, usually ranging f rom 1 keV to tens of keV, above which, in the high-energy region, the smooth c ross - se c t i on can be obtained directly f rom measurements of a (n, y) and a (n, f ) .