I

Wall Fission Spectrum

Fast Reactor Spectrum (FERMI)

Graphite Moderated Spectrum

Light-Water Moderated Spectrum IETRI

(Spectra Normalized to Equal Flux Greater

than 0.0674 MeV)

/ \ / \

(0.06741

г 3

Lethargy, u

TECHNICAL REPORTS SERIES No 107

Neutron Fluence Measurements

I N T E R N A T I O N A L A T O M I C E N E R G Y A G E N C Y , V I E N N A , 1 9 7 0

NEUTRON FLUENCE MEASUREMENTS

The following States are Members o f the International Atomic Energy Agency :

AFGHANISTAN GREECE NORWAY ALBANIA GUATEMALA PAKISTAN ALGERIA HAITI PANAMA ARGENTINA HOLY SEE PARAGUAY AUSTRALIA HUNGARY PERU AUSTRIA ICELAND PHILIPPINES BELGIUM INDIA POLAND BOLIVIA INDONESIA PORTUGAL BRAZIL IRAN ROMANIA BULGARIA IRAQ SAUDI ARABIA BURMA IRELAND SENEGAL BYELORUSSIAN SOVIET ISRAEL SIERRA LEONE

SOCIALIST REPUBLIC ITALY SINGAPORE CAMBODIA IVORY COAST SOUTH AFRICA CAMEROON JAMAICA SPAIN CANADA JAPAN SUDAN CEYLON JORDAN SWEDEN CHILE KENYA SWITZERLAND CHINA KOREA, REPUBLIC OF SYRIAN ARAB REPUBLIC COLOMBIA KUWAIT THAILAND CONGO, DEMOCRATIC LEBANON TUNISIA

REPUBLIC OF LIBERIA TURKEY COSTA RICA LIBYAN ARAB REPUBLIC UGANDA CUBA . LIECHTENSTEIN UKRAINIAN SOVIET SOCIALIST CYPRUS LUXEMBOURG REPUBLIC CZECHOSLOVAK SOCIALIST MADAGASCAR UNION OF SOVIET SOCIALIST

REPUBLIC MALAYSIA REPUBLICS DENMARK MALI UNITED ARAB REPUBLIC DOMINICAN REPUBLIC MEXICO UNITED KINGDOM OF GREAT ECUADOR MONACO BRITAIN AND NORTHERN EL SALVADOR MOROCCO IRELAND ETHIOPIA NETHERLANDS UNITED STATES OF AMERICA FINLAND NEW ZEALAND URUGUAY FRANCE NICARAGUA VENEZUELA GABON NIGER VIET-NAM GERMANY, FEDERAL REPUBLIC OF NIGERIA YUGOSLAVIA GHANA 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 ob ject ive is " t o acce lerate and enlarge the contribution of atomic energy to peace , health and prosperity throughout the wor ld" .

© I A E A . 1970

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 May 1970

TECHNICAL REPORTS SERIES No. 107

NEUTRON FLUENCE MEASUREMENTS

INTERNATIONAL ATOMIC ENERGY AGENCY VIENNA, 1970

NEUTRON FLUENCE MEASUREMENTS IAEA, VIENNA, 1970

STI /DOC/10 /107

FOREWORD

For research reactor work dealing with such subjects as radiation e f -fects on solids and such disciplines as radiochemistry and radiobiology, the radiation dose or neutron fluence is an essential parameter in evaluat-ing results . Unfortunately it is very difficult to determine. Even when the measurements have been accurate, it is difficult to compare results obtained in different experiments because present methods do not always ref lect the dependence of spectra or of different types of radiation on the induced p r o c e s s e s .

After considering the recommendations of three IAEA Panels, on Чп-pile dosimetry ' held in July 1964, on 'Neutron f luence-measurements ' in October 1965, and on Чп-pile dosimetry ' in November 1966, the Agency established a Working Group on Reactor Radiation Measurements. This group consisted of eleven experts f rom ten different Member States and two staff members of the Agency.

On the measurement of energy absorbed by materials f r om neutrons and gamma-rays , there are various reports and reviews scattered through-out the l iterature. The group, however, considered that the time was ripe for all relevant information to be evaluated and gathered together in the f o rm of a practical guide, with the aim of promoting consistency in the measurement and reporting of reactor radiation. The group arranged for the material to be divided into two manuals, which are expected to be u s e -ful both for experienced workers and for beginners. The present manual was edited by Dr . J. Moteff, of the General Electr i c Company, Cincinnati, Ohio, USA, and the companion volume, on 'Determination of absorbed dose in reac tors ' , will be published by the Agency shortly.

The authors who contributed to the present manual are:

Chapter 1 : Chapter 2:

Introduction Neutron spectra

J. Moteff S . B . Wright (Atomic Energy Research Establishment, UKAEA, Harwell, United Kingdom) Y. Droulers (Commissariat à l 'Energie atomique, Centre d'études nucléaires de Grenoble, France)

Intermediate neutrons W. L . Zi jp (Reactor Centrum Nederland, Petten NH, Netherlands)

Fast neutrons R . E . Dahl and H. H. Yoshikawa (Pacif ic Northwest Laboratories , Battelle Memorial Institute, Richland, Wash. , USA) A. Keddar (Division of Nuclear Power and Reactors, IAEA, Vienna)

Chapter 3: Thermal neutrons

Chapter 4:

Chapter 5:

Index:

The original contributions have been changed somewhat during edit-ing in order to make the manual consistent. The editor and contributing authors wish to express their appreciation to Dr. S. Sanatani (IAEA), who was the Scientific Secretary of the Working Group during the early stages and to Dr. A . K e d d a r (IAEA), the present Scientific Secretary, for their help in assembling the manual. Special thanks are also due to Dr. W. KOhler of the Agency f or his comments and help during the initial planning of the book and during the final editorial work.

CONTENTS

CHAPTER I. INTRODUCTION 1

CHAPTER II. NEUTRON SPECTRA 7

II. 1. Introduction 7 II. 1 .1 . Maxwellian spectrum II. 1. 2. F iss ion neutron spectrum II. 1 .3 . Intermediate neutron spectrum II. 1 .4 . Reactor spectrum II. 1. 5. Physical p r o c e s s e s encountered in irradiation

experiments И. 1. 6. Division of the neutron spectrum f o r the monitoring

of irradiation experiments

II. 2. Thermal neutron region 19 II. 2 . 1 . Conventional thermal flux densities II. 2. 2. True thermal flux spectrum II. 2 . 3 . Theoret ical spectra II. 2 . 4 . Variation of the spectrum a c r o s s a reactor lattice

II. 3. Fast neutron region 26 II. 3 . 1 . Calculation of fast neutron spectra II. 3. 2. Variations in the fast neutron spectra in a

heterogeneous reactor II. 3. 3. Comparison of experimental and theoretical spectra II. 3 .4 . Comparison of theoretical spectra II. 3. 5. The effect of the neutron spectrum on experimental

measurements

Re ferences to Chapter II 43

CHAPTER III. THERMAL NEUTRONS 45

III. 1. Theory of detector response 45 III. 1 .1 . General method III. 1 .2 . Westcott ' s notation III. 1 .3 . Formal i sm of Horowitz and Tretiakoff

III. 2. Measurement of thermal neutron flux density and fluence . . . . 52 III. 2 . 1 . Relation between flux density, f luence and

detector activity III. 2 .2 . Measurement of low flux densities III. 2 . 3 . Measurement of high f luences

1П. 3. Measurement of thermal neutron spectra 65 III. 3 . 1 . Principle of the method III. 3. 2. Example of the experimental procedure

III. 4. Other measuring methods 69 III. 4. 1. Measurement of flux density III. 4. 2. Fluence measurements

Re ferences to Chapter III 75

CHAPTER IV. INTERMEDIATE NEUTRONS 77

IV. 1. Introduction 77 I V . 1 . 1 . General IV. 1. 2. Response of a resonance detector IV. 1 .3 . Fluence and spectrum measurements IV. 1 .4 . Types of resonances

IV. 2. Spectrum characterist ics 81 I V . 2 . 1 . The 1 /E spectrum IV. 2. 2. Other spectrum representations

IV. 3. Detectors 88 IV. 3 . 1 . Resonance integral c r o ss - se c t i on I V . 3 . 2 . Detector response IV. 3 . 3 . Self-shielding effect

IV .4 . Fluence measurements I l l I V . 4 . 1 . Experimental details IV. 4. 2. Data reduction and treatment IV. 4 . 3 . Data reporting

IV. 5. Spectrum measurements 130 IV. 5 .1 . Experimental details

IV .6 . Other methods 132 IV. 6 . 1 . Recent developments with resonance detectors IV. 6. 2. Other fluence measurement methods IV. 6. 3. Other spectrum measurement methods

IV. 7. Concluding remarks 135

References to Chapter IV 136

CHAPTER V. FAST NEUTRONS 141

V . l . Introduction 141

V . 2 . Neutron spectrum 142

V . 3 . Detector response ; 145

V . 4 . Fluence measurements 147 V . 4 . 1 . Experimental details V . 4 . 2 . Data reduction V . 4 . 3 . Summary of f luence measurements

V . 5 . Spectral determination f r o m monitor activations 176

V . 6 . Other methods of fluence measurements 178

References to Chapter V 179

INDEX 183

CHAPTER I

INTRODUCTION

For reactor research work in the f ields of radiation chemistry and radiation damage to solids, the neutron energy distribution and the neutron fluence 1 are essential parameters f o r the evaluation of the experimental results. The accurate determination of these factors in research reactors , and particularly in materials test reactors is extremely difficult. In many reactors there is pronounced variation of the power level and, therefore, of the flux density with t ime. Moreover , the various experimental a s s e m -blies, control-rod positions a:nd local burn-up will tend to perturb the flux density throughout the reactor core and re f lector regions. These pertur-bations make it almost impossible to estimate doses and f luences f r om measurements made on cold, clean reactor co res . For this reason it is desirable to measure the^ radiation environment in exactly the same core geometry as would be used f or each experiment.

At the present time it seems feasible to recommend speci f ic methods, at least f o r neutron fluence measurements. However, even if unified ex -perimental techniques could be suggested, there still exist serious problems in the presentation of experimental results. An example may be found in the field of graphite irradiations. Radiation damage has, on different occasions , been expressed as a function of thermal neutron fluence, f iss ion neutron fluence, or fast neutron fluence. Without more detailed information on the reactor neutron environment, an intercomparison of results cannot even be attempted in any valid manner.

For many years , ' in the field of radiation effects to metals, it was assumed that significant changes in properties will only occur for high neutron fluence levels (> 1018n-cm"2) and with neutron energies above 100 keV. As a result, neutron dosimetry in support of metals research was limited to the measurements, in the fast neutron region. Now it is shown that at elevated temperatures severe embrittlement can occur in metals exposed to neutron f luences of less than 1014n-cm"2 and this embrittlement is sensitive to thermal neutrons^ This serious change in the ductility of structural and cladding materials has been attributed to the presence of trace quantities of boron and to the resulting formation of helium gas by the 10B (n, c/¡ 7 Li í reactions. Changes in physical properties produced by transmutations by, for example,(n, 7 ) , (n, p) and (n, a ) reactions require a cofnplete know-ledge of neutron spectrum before theory and experiment can be better corre lated. There is a further fundamental difficulty; the present in-complete understanding of the damage-producing mechanism hampers attempts to correlate a fluence parameter with the radiátion damage caused in

1 Neutron f luence, previously referred to as the neutron dose, is defined as the t ime integral of the neutron flux density.

1

2 INTRODUCTION

the specimen. Consequently.it would be difficult to compare results ob -tained in different types of reactors even if fluence measurement methods, units and constants were standardized.

The absence of standard procedures for reactor neutron measurements is a serious handicap. Even for thermal neutron measurements, for which units are well defined, there is still no proper guarantee of compatibility of measurements made at different reactors , for instance, if different effective c ross - se c t i ons are being used.

As a result of all these uncertainties, it is at present difficult to compare experimental results obtained in the same field of research at different research centres . Therefore , there is definite need for a con-certed effort f o r a unified approach to the problem of in -reactor fluence measurements.

Measurement of the activity induced in an activation monitor, or the measurement of absorbed dose at a particular time, may not be difficult to accomplish. However, because of the characterist ics of reactors , there is no assurance that such measurements are necessar i ly pertinent to the experiment f o r which these measurements were made. Part of the uncertain-ty is caused by the known temporal and spatial variations in the reactor radiation f ield. The spatial perturbations are of three types: gradients, shielding ef fects and perturbations. The gradients are inherent in all reactors , whereas shielding ef fects and perturbations may be introduced by the experimental assembly.

Gradients are generally small in the large c o res of heavy-water and graphite reactors , at least in positions which are not very close to the fuel elements. In the case of l ight-water-moderated or sodium-cooled fast breeder reactor cores , which are relatively compact, gradients are much more pronounced. For irradiation in such reactors , an uncertainty of only a few mi l l imetres in the sample location may give* r ise to appreciable e r r o r s in dose or fluence estimates.

Shielding effects are noticeable f o r all samples that are introduced into a reactor in a container or in experimental assemblies of any kind. For thermal neutrons and for gamma radiation this effect usually means an attenuation of the fluence rate (dose rate), while for fast neutrons it may even cause an increase in fluence rate, especial ly in re f lector positions in light-water reac tors .

An irradiation sample generally causes a thermal flux depression within its own volume and in its immediate vicinity due to the absorption of neu-trons in the sample and a displacement of the moderator within that volume. In fast spectrum reactors the same irradiation sample may cause a local increase in the low-energy neutrons. A sample may also be placed where the flux perturbation caused by other experiments is noticeable. Generally, however, these perturbations are less pronounced with gamma radiation than with thermal neutrons and may even be reversed f or irradiations made in a fast neutron flux.

All these ' sources of spatial variation of flux make it necessary to p e r -f o r m neutron measurements at the exact location of the sample to avoid the danger that the conditions measured by the detector will not apply to the conditions at the experiment.

In addition to the spatial variations in flux density, variations with time also occur . Such time variations are always present since they are mainly caused by factors connected with the control and operation of the reactor .

3 INTRODUCTION

During the f irst 50 hours of steady-state operation in any cycle , the core will reach equilibrium poisoning requiring up to 5% Дк/к of the excess reactivity built into the clean cold core . This will, in many cases , require considerable change of the shim-rod positions to maintain a stable power level which will, in turn, alter the flux density patterns in the core and r e -f l ec tor . The burn-up of reactor fuel will, in the long term, also cause this same effect during the operating lifetime of the core . During the initial phase of the operating cyc le , the gamma radiation will also build up with the establishment of an approximately equilibrium concentration of the f iss ion products .

These ef fects are largely due to changing reactivity requirements with consequent need for changed shim and control-rod positions which will change flux density and dose^rate patterns continuously while the reactor is operating. They are severe in pool reactors of more than a few mega-watts power level but are noticeable also in the heavy-water and graphite reactors . The altered flux density patterns may also cause changes in the leakage flux densities and may thus, through interaction on the neutron sensing control channels, cause disturbances in the operating power level . This may make the ef fect even more serious or, in some cases , reduce it for speci f ic experimental locations.

To these inherent effects should also be added ef fects that are caused by fuel pattern changes, new experiments, etc. that may be introduced during the runs with speci f ic experimental equipment. Moreover , there are the c y c l e - t o - c y c l e variations which would be encountered in any long-term experiment.

Since such time variations occur, there is an obvious need to measure the actual fluence or dose that an experiment has received. This may be done ( l ) by a continuous measurement of the flux density or the dose rate during an experiment by means of ca lor imeters , ion chambers, gas loops, se l f -powered detectors, etc. ; or (2) by the time integration of fast, inter-mediate or thermal neutron flux densities by means of activation detectors or f ission fo i l s . These techniques involve making measurements at the full power of the reactor and under the ambient conditions prevailing at the location of the experiment. Additional problems, however, are posed by irradiations for very long t imes and to very high f luences.

The purpose of this manual is to describe in some detail the techniques of neutron fluence measurements by using activation detectors. Some general mention, however, will be made on techniques suggested in item (1) above.

The organization of this manual may be better appreciated by making a careful study of the neutron differential energy spectrum, one typical of a well-moderated reactor being as shown in Fig. I. 1. There is f i rst the overall neutron spectrum which is generally divided into three components designated as the thermal, intermediate or fast neutron energy regions. There are- many reasons for the establishment of these speci f i c regions and most of these can, in some manner, be related to the relative neutron energy dependence of the scattering and absorption c ross - se c t i ons of the reactor materials , and of the activation detectors. And secondly, there is the broad range of the differential neutron flux density, ' (E), which can ' be greater than thirteen orders of magnitude enveloping a neutron energy, E, range which is greater than ten orders of magnitude.

4 INTRODUCTION

10'1 10' 10J NEUTRON ENERGY, E(eV)

FIG. 1 1.1. Typical neutron differential spectrum in a well-moderated reactor showing the various components generally used in the literature to describe neutron energy regions. - ,

Accordingly, as introductory information, chapter II deals with the general nature of the overall reactor neutron spectra. Supporting examples of calculated or measured neutron spectra in actual reactors are shown. A general review of the mathematical relationships coupling the three components (thermal, intermediate and fast) of the spectra is presented. In addition, the materials research, unique to the three regions of the spectra are brief ly discussed.

Thermal neutrons are discussed in ChapterlH. Specifically, the methods associated with the measurement of thermal neutron fluence and the effective temperature of the thermal neutrons are presented. The types of -activation detectors unique to the measurement of neutron fluence in the low-energy range are described, together with associated problems such as flux perturbations and foi l handling techniques. Mathematical relationships which were mentioned in Chapter II are further developed, and some new equations are presented.

Intermediate energy neutrons, which are the source for the thermal neutrons and have, in turn, the fast neutrons as their source, are discussed in Chapter IV. The c ross - sec t i ons , for many of the elements, in this energy region are quite complex in that they have pronounced resonance structure. The measurement of neutron fluences in this energy region is not as well developed as that in the thermal and fast neutron regions.

5 INTRODUCTION

Sandwich foi l techniques, as a means of separating the more prominent resonance activation which would occur over a small interval of a speci f ic neutron energy f r om that due to neutrons of all energies, are given special attention. For such work self -shielding becomes a source of important problems and the treatment of appropriate correct ion factors are also presented. Due to the nature of the c ross - se c t i ons , and in view of the importance of this energy region in radiation damage studies and transmu-tation reactions in both fast and intermediate energy spectrum reactors , the mathematical treatment of neutron fluence in this chapter is c o r r e s p o n -dingly much more detailed than that presented in either Chapters III or V. It is also felt that some duplication of those equations common to the thermal and to the fast neutron energy regions is needed to ensure continuity of presentation in this chapter.

Fast neutron measurements are discussed in Chapter V. Particular attention is paid to the c lass of threshold detectors normally used f or fluence measurements in this energy region. Since the fast neutron energy region is important to the study of radiation damage to sol ids, this subject is amplified beyond the stage reached in Chapter II. The selection of c o m -pounds with higher melting temperature than the pure element itself is mentioned in this chapter. Data reduction, especial ly the treatment of those reactions with high burn-up c r o s s - s e c t i o n s of i s omer i c states with short half - l ives, is d iscussed.

In concluding, it should be pointed out that the general method of f luence measurements using radioactive fo i l techniques as presented in this manual has not changed significantly f r o m that developed by the pioneers in the f ield of reactor neutron physics . The important changes, f o r example, are in the availability of better reaction c r o s s - s e c t i o n s as a function of neutron energy, more accurate hal f - l i fe and disintegration data f or the radioactive isotopes, and materials of higher purity. Therefore , those individuals responsible f o r neutron fluence measurements must review the current literature so as to include this new information in their analyses as the data become available. It is for this reason that the present manual does not recommend sets of standard c r o s s - s e c t i o n s , but does show how the c r o s s -sections are used in neutron fluence measurements . Of course , there is also the continuing improvement in the performance of the different types of counters used in the determination of detector activities. Finally, the increasing use of high-speed computers in the calculation of reactor spectra f or the case of complex test geometries , in the reduction of experimental data and in spectrum unfolding methods should contribute to improve reactor neutron spectra and fluence determinations.

CHAPTER II

NEUTRON SPECTRA

II. 1. INTRODUCTION

The neutrons produced in the f iss ion process have an average energy of approximately 2 MeV. In a thermal reactor these neutrons are slowed down by col l is ions with the moderator atoms until they are in thermal equilibrium with the moderator and have an average energy of approximately 0. 025 eV. There exists therefore , in a thermal reactor , a spectrum of neutron energies covering a range of m o r e than eight decades.

Because of this large range of neutron energies, neutrons in a reactor can take part in many different physical processes . In analysing irradiation experiments, therefore , it is essential that the neutron energy spectrum is taken into account. F o r some processes , such as radioactive capture, it is sufficient to use a simple approximation f o r the neutron spectrum; f or other p rocesses a detailed knowledge of the energy spectrum is required.

The neutron energy spectrum is, for convenience, often divided into three energy regions: the thermal region, consisting of neutrons in thermal equilibrium with the moderator , the fast or f ission region in which the neutrons f r om f iss ion are produced, and the intermediate or slowing-down region which joins these two. It must be emphasized that this division is purely arbitrary and the neutron energy spectrum in a reactor is a continuous function of energy with no c learly defined boundaries.

This arbitrary division into three regions is useful, however, when considering the general f o rm of the neutron spectrum in a thermal reactor , and it will be adopted in the following sections.

II. 1. 1. Maxwellian spectrum

When neutrons reach thermal equilibrium with the moderator , their energies are determined by the thermal energy distribution of the moderator atoms and the neutron energy spectrum becomes a Maxwellian distribution at the temperature T°K of the moderator material. This spectrum is c o m -monly expressed in several different f o rms which must be c learly distin-guished. Either the neutron flux or the neutron density is quoted and either may be given as a function of neutron velocity or of neutron energy.

The neutron density as a function of velocity n(v) is given in Eq. (И. 1) and the neutron flux as a function of energy cp(E) in Eq. (II. 2). Both of these equations are normalized to unit area (Fig. II. 1).

n(v) = ^ ( - ^ y / 2 v 2 e x p ( - m v 2 / 2 k T ) (II. 1)

Ф(Е) = e x p ( - E / k T ) (II. 2)

7

8 CHAPTER III

a) MAXWELLIAN VELOCITY DISTRIBUTION

NEUTRON ENERGY (»V)

b) M AXWELLIAN FLUX DISTRIBUTION

FIG. И. 1. Maxwellian distribution at 20 .4 °C .

F r o m Eq. (II. 1) the mos t probable ve loc i ty can be derived as

/ 2kT V / 2

which c o r r e s p o n d s to an energy of

1 /2 m v j = kT

F o r many purposes it is adequate to define a conventional flux cp0 as

Ф0 = n v0 , (II. 3)

.where n is the total neutron density and v0 is an arbi trary ve loc i ty usually taken as 2200 m / s e c [1]. This ve loc i ty is chosen because it is the mos t probable ve loc i ty of a Maxwell ian density distribution at 20. 44°C. It c o r -responds to an energy of 0. 025 eV. The c r o s s - s e c t i o n s required f o r use with this conventional flux a re the c r o s s - s e c t i o n s at 2200 m / s e c and these are the values l isted in m o s t tabulations.

NEUTRON. SPECTRA 9

In the equations for the space and time dependent problems the fluxes •which appear are the true fluxes and it is necessary in these problems to use values f o r cross -sect ions properly averaged over the neutron spectrum.

The average velocity of a Maxwellian spectrum can be found from Eq. (II. 1) and is (2/sTtt) (m/2kT)i . The average velocity is, therefore, greater than the most probable velocity by a factor of 2/Jtt or 1. 128. F o r a material with a 1 /v c ross -sec t ion the average cross -sec t ion in a Maxwel-lian spectrum at 20. 44°C will be smal ler than the 2200 m / s e c c ross - sec t i on by a factor of 1. 128.

By differentiating Eq. (II. 2), the peak of the flux is found to occur at an energy kT. This is not the most probable energy, however. This occurs at 1/2 kT and the average energy at 3/2 kT.

Although the Maxwellian spectrum is a good approximation for many positions in a thermal reactor , neutrons only reach equilibrium with the moderator in regions where neutron absorption is small such as graphite or heavy-water thermal columns. In regions where neutron absorption is significant it takes place over the whole thermal spectrum. F o r steady-state conditions the intensity of the thermal flux must be maintained by the neutrons slowing down f rom the intermediate spectrum. This constant source of neutrons into the high-energy end of the thermal spectrum means that the average energy is higher than for the case of thermal equilibrium with no absorption and results in a thermal neutron spectrum which is Maxwellian in f orm, but at a higher temperature than the moderator. This situation will be discussed further in section II. 2.

II. 1. 2. Fiss ion neutron spectrum

The energy distribution of neutrons produced in the f ission process is known as the f ission spectrum and has been measured, for all the common f iss i le elements. Severed empirical relations -have been fitted to the experi-mental results within the accuracy of the measurements, of which one of the most commonly used is that due to Watt [2].

s'(E) = Ae~E sinh n/~2e (11,4)

where E is the energy in MeV; S(E) the number of neutrons per unit energy interval'and A is the norriializing'constant 2/(ire) = 0. 484. Modifications to this formula have been made by including fitted constants in the two terms [3].

s(E) = Ae_bE sinh s/ cE

where, f or Z35U, A = 0.4527; b = 1. 036; and с = 2. 29. This expression differs only slightly f rom that due to Watt.

An alternative form of the f ission spectrum which is also frequently used is the Maxwellian form

S(E) = а \Ге exp ( - E / e ) (II. 5)

where e is the characteristic energy of the process and a = 2/\/(теЗ). F o r 235U the best value for e is 1. 290 MeV which gives a = 0. 770.

Figure II. 2 shows the f ission spectrum S(E) in the Maxwellian form.

10 CHAPTER III

ENERGY (Mev) a) ENERGY SPACE

3 2 LETHARGY, и

S) LETHARGY SPACE FIG. II. 2 . Fission spectrum.

The energy spectrum in a r e a c t o r is in general not a f i s s i on spectrum owing to the e f fects of the m o d e r a t o r and to neutron leakage. However , the f i s s i on spec trum is often taken as a f i r s t approximation to the neutron spectrum, c l o s e to the neutron s o u r c e , f o r energies above about 1. 5 MeV. It should be noted that although S(E) is the number of neutrons emitted p e r unit energy p e r unit t ime , the flux is not obtained by multiplying S(E) by the neutron ve loc i ty . This can be shown as f o l l ows .

If the cube 6A6r shown in F ig . II. 3 is cons idered , with 6A perpendicular to r , then the number of neutrons entering the cube in unit t ime is N(E) = (6А/4тгг2) S(E).

The length of t ime each neutron remains in the cube, assuming no absorpt ion, is given by

т(Е) 6r V

where v is the ve loc i ty corresponding to the neutron energy E. the neutron density n(E), at the vo lume 6A6r i s given by

T h e r e f o r e ,

n(E) = N(E) t ( E ) 6A6r 1 S(E)

4ят2 v

NEUTRON. SPECTRA 447

Sv • ÍA$r

FIG. II. 3. Source-receiver geometry.

Therefore , the flux density cp(E) is

Ф(Е) = n(E) v = g g

Thus apart f r om the geometric attenuation factor the flux at the volume 6A6r is the f iss ion spectrum S(E). This is only true f or a medium in which leakages, absorption and moderation can be neglected. It is , however, a reasonable approximation c lose to the fuel, for energies above 1. 5 MeV, and is frequently used as an approximation to the high-energy neutron spectrum in a reactor .

II. 1. 3. Intermediate neutron spectrum

The general case of neutron slowing-down in a moderating medium is too complex to be treated here. However, the general f o rm of the spectrum can be derived by considering the simple case in which neutron absorption is neglected. The number of col l is ions per second per cm2 at energy E is given by

D(E) = ф(Е) N as

where N is the atomic density, ф(Е) the flux, and CTs the scattering c r o s s -section. If ДЕ is the average energy loss per col l is ions, the number of neutrons slowing down past energy E per second per cm.3 is

q (E) = D(E) ДЕ = q>(E) N as ДЕ

The average change in the logarithm of the energy is a constant f or all energies well above the thermal region, i. e.

Ç = AlnE

= Д Е / Е

t h e r e f o r e q ( E ) = 9 ( E ) N C T S § E . But if there is no absorption q(E) must equal the total source density q0

and hence

- i s k F (п. 6)

12 CHAPTER III

Thus provided the scattering cross -sect ion is a constant, as is usually the case over the energy range being considered, then Eq. (II. 6) yields the familiar l / E spectrum.

The distances covered by neutrons during the slowing-down process are long compared with the spacing of the fuel in the reactor. The inter-mediate flux, therefore, shows little smal l -sca le variation across the lattice cell. This is in marked contrast to the thermal and fast neutrons, both of which show considerable variations c lose to the'fuel.

II. 1. 4. Reactor spectrum

It is difficult to handle the full range of the reactor neutron spectrum when energy is used as the variable since a range of nine decades must be covered. It is , therefore, convenient to introduce a dimensionless variable known as the lethargy in place of the energy. The lethargy 'u1 is defined by the equation

du. = -d (ünE) = - .^г- (II. 7)

(i. e. u = in E q / E )

where E0 is the constant of integration. A value of 10 MeV is usually taken f o r E0 since there are few neutrons produced in fissions with energies higher than this value. As neutrons slow down, their lethargy increàses. The • lethargy corresponding to the thermal energy 0. 025 eV is ln(4 X108) = Í9. 8.

The neutron flux; can now be expressed as a function of lethargy. If ф(и) is the flux per unit lethargy interval, the flux in an infinitesimal range du is cp(u)du. This must be equal to the flux expressed as a function of energy cp(E)dE, i. e.

cp(u)du = -cp(E)dE (II. 8)

The negative sign is necessary to take account of the fact that u increases as E decreases. Equations (II. 7) and (II. 8) yield , •

• ф(и) = Еф(Е) : (II. 9)

Substituting this in Eq. (II. 6) f or the intermediate neutron spectrum, the following expression is obtained

(11.10)

Thus in the slowing-down region the flux spectrum as a function of lethargy is a constant.

The complete reactor spectrum is composed of some combination of the three parts of the spectrum we have discussed so far. The exact form of the spectrum depends markedly on the particular reactor and the position in the reâctor and it is impossible to obtain the form of the spectrum for the general case. We can, however, derive a spectrum for an infinite hydrogen moderated reactor and we will use this to illustrate the influence of the neutron energy spectrum on irradiation experiments.

NEUTRON. SPECTRA 13

NEUTRON ENERGY. E . (iv)

FIG. 11,4. Neutron spectrum typical o f a l ight-water-móderated reactór (normalized T 0 (m) du = 1).

It can be shown [4] that the f o rm of the fast and intermediate spectrum for a hydrogen moderated reactor is

Es(u) 'ф(и) = S(u) + j S(u')du' (11.11)

where Es(u) is the macroscop i c scattering c ross - se c t i on . This spectrum will be valid down to the region of the Maxwellian spectrum. 1 In the thermal region the.spectrum will be approximately Maxwellian, but the prec i se f o rm will depend on the moderator temperature and the absorption c r o s s - s e c t i o n of the medium. .Figure II. 4 shows a typical neutron spectrum for a homo-geneous hydrogen,moderated reactor, with a moderator temperature of 20°C, obtained using Eq. (II. 11) f o r the high-energy spectrum and a SOFOCATE calculation f or the thermal spectrum. •

F o r comparison, a Maxwellian spectrum and a f iss ion spectrum have been superimposed on this spectrum. The Maxwellian .can be seen to be a very good fit to the thermal spectrum although the neutron temperature of this spectrum is approximately 27 degC higher than the moderator t emper -ature. The f iss ion spectrum fit was obtained by normalizing to the reaction rate in the 58Ni (n,p)5 8Co reaction. It сапЛэе seen f r om the diagram that the f iss ion spectrum is not a good approximation to the homogeneous hydrogen

14 CHAPTER III

moderated reactor even in the region above 2 MeV. In a heterogeneous reactor and c lose to the fuel elements there will be a larger proportion of uncollided f iss ion neutrons and the neutron spectrum above 2 MeV will be a better fit to the f ission spectrum.

II. 1. 5. Physical p rocesses encountered in irradiation1 experiments

Because of the wide range of neutron energies that exist in a reactor , a number of fundamentally different physical p rocesses can occur in an irradiation experiment. Each of these processes will depend on neutron energy in a different manner and so each will have its own response function in a given reactor neutron spectrum. It is , therefore , essential to take account of the neutron spectrum if a full understanding of an irradiation experiment is to be achieved.

The response function for neutron capture reactions is simply a product of the neutron c ross - se c t i on and the neutron flux cr(u)cp(u). F o r displace-ment reactions this function is the product of the flux, the scattering c r o s s -section of the material and a damage parameter which gives the ef fect ive-ness of a neutron col l ision at lethargy u in pro'ducing damage.

In-the following examples the response functions for the different reactions considered are calculated for the infinite hydrogen reactor spectrum given in Fig. II. 4. These functions, plotted in Fig. II. 5, illustrate the energy range of the neutron spectrum which contributes most to each, reaction.

Neutron capture

One of the most commonly encountered reactions is the radioactive capture, or (n, 7) reaction. In this reaction the binding energy of the capture neutron is l iberated and no energy need be supplied in the form of kinetic energy of the neutron to enable the reaction to proceed. The reaction is , therefore , poss ible with the lowest energy neutrons encountered in a reactor. Most reactions of this type have c ross - sec t i ons which are approximately inversely proportional to the neutron velocity. They, therefore , respond mainly to the thermal region of the neutron spectrum. Curve I of Fig. II. 5 shows the response function of a l / v detector in the reactor spectrum shown in Fig. II. 4. The response function for this reaction corresponds exactly to the neutron density distribution in the reactor .

Some neutron capture reactions exhibit very large peaks in their c r o s s -sections at energies just above thermal. At these energies the wave nature of the neutron manifests itself and results in very strong absorption lines at energies corresponding to the energy levels of the compound nucleus formed by the neutron capture. This so - ca l led resonance capture often results^in neutron capture c ross - sec t i ons which are many times larger than the physical s ize of the nucleus involved. The response function of a typical resonance reaction in the light-water reactor spectrum is shown in curve II of Fig. II. 5. In this curve only the f irst two resonances of the 197Au (n, 7) reaction have been shown and these have not been taken to their peak value which would be well off the page. The resonances account f or approximately half the total response function in the light-water reactor spectrum and the majority of this is due to the f irst resonance peak at 4. 9 eV.

NEUTRON. SPECTRA 15

Rupture of chemical bonds

In radiation chemistry experiments the main process of interest, apart f r om transmutation by (n, y) reactions and subsequent radioactive decay, is the rupture of chemical bonds. This process requires the transfer of a few electron volts to the molecule to overcome the bond energy. Both gamma rays and neutrons will produce this effect in a reactor but only the neutron-induced events are relevant to this discussion. The gamma reaction will, therefore , be neglected here but it must always be taken into account in practical cases . Thermal neutrons do not carry sufficient energy to break molecular bonds by direct col l ision with atoms and so most bond rupture is produced by the intermediate and fast regions of the neutron energy spectrum. After a col l ision, the reco i l atom will usually rece ive considerably m o r e energy than is required to break the chemical bonds. This energy will be dissipated by further col l is ions with atoms in the material , producing further bond rupture until all the displaced atoms reach thermal equilibrium. The total number of molecular bonds ruptured in this way will be proportional to the total energy transferred to the atoms by the neutron scattering events. If it is assumed that the neutron scattering c r o s s - s e c t i o n f o r the material is a constant function of energy, the response function will be of the f o rm shown in curve III of Fig . II. 5.

When a neutron is captured by a nucleus, approximately 8 MeV of binding energy is released in the f o rm of gamma rays and in the process the nucleus reco i l s with an energy of the order of 100 eV. Both these reco i l atoms and the capture gamma rays will produce chemical effects indistinguishable f r o m those considered above. They will be a direct result of the absorption of thermal neutrons and will, therefore , have a response function appropriate to the neutron capture reaction involved.

3 10"' to" NEUTRON ENERGY (ïV)

IP IP2 IP3 IP4 IP5 IP6 IP7

b-1 '/v DETECTPR П Au'«7 (r>*) Ш NEUTRON HEATING E CARBON ATOM DISPLACEMENT 2 Ni58

16 CHAPTER III

Usually in a reactor c o re the effect of this p r o c e s s will be small compared with the effect of neutron scattering reactions induced by high-energy neutrons. This may not be true, however, in the re f lector of a reactor or in the thermal column.

Nuclear heating

A large fraction of the energy liberated in all neutron-induced events in a material will* eventually appear as heat and so all neutron interactions must be considered in heating calculations. Most of the heating produced by neutrons is due.to the energy transferred to the material by elastic scattering events. Exceptions to this are materials with higher neutron capture c r o s s - s e c t i o n s such as cadmium and ;boron where considerable heating can be produced by the radiation emitted as a result of thermal neutron, capture. The heating in a material is , again, proportional to the total energy liberated and, therefore , the response function will be that -shown in curve III of Fig. II. 5.

Radiation damage

In radiation damage to crystall ine and polycrystall ine materials the main process of interest is the displacement of atoms f r om their normal lattice sites to interstitial sites. To do this the atom must be. given energy of the order of 25 to 50 eV. Gamma rays cannot contribute significantly to this p r o c e s s in a reactor co.re and the displacements are mainly produced by fast neutron col l is ions. Once a pr imary displacement atom (primary knock-on atom) is produced, it will cause secondary displacements by col l is ions with other atoms in the lattice until its energy falls below twice the displacement energy. Thus the total number of displacements will be approximately proportional to the total energy received by the primary knock-on atom. If, however, the energy of the knock-on atom is high enough, some energy will be lost by ionization. The energy spent in ionization will not produce 'atomic displacements in the crystal and the result is a saturation in the number of displacements produced at higher energies [5-7] . The response function f o r damage in graphite, calculated assuming the Thompson and Wright damage function [7], is shown in curve IV of Fig. II. 5.

The simple model of radiation damage outlined above ignores any effect of annealing and of the clustering of defects by their migration. In the energy range of most importance in radiation damage experiments each col l is ion can produce a large number of displacements and the distribution of these defects will not change much f r om one neutron spectrum to another. Annealing of defects will alter the amount of damage remaining in the material , but it is unlikely that this will alter the resulting effects of i r rad i -ations carr ied out in different neutron spectra. As a f irst approximation, therefore , the effect of annealing can be neglected in estimating the depend-ence of radiation damage on neutron energy spectrum.

In solids where the mean f r e e path of moving atoms is of the same order as the lattice spacing the damage will be produced as local ized disordered regions, or 'displacement spikes ' . The s ize and distribution of these spikes will depend on the energy of the knock-on atom and hence on the neutron

NEUTRON. SPECTRA 17

energy. This type of radiation damage will have a different dependence on the neutron energy' spectrum f rom that used above f or the calculation of the graphite damage response function and hence the response function f o r this process may be1 very different f rom the curve shown in Fig. II. .5. Unfor -tunately, theories of radiation damage are not yet sufficiently well developed f o r a reliable response function f o r this type of damage to be quoted.

The reco i l energy of an atom after.neutron capture is sufficient to produce atomic displacements and this effect can be appreciable under s o m e irradiation conditions [9]. F o r most irradiations in the core of a nuclear reactor the number of displacements produced by the process will be small compared with that-produced by fast neutron col l is ions, but outside the c o r e of a thermal reactor, this may not be the case and thermal neutron capture effects may be appreciable. The reco i l atom produced by thermal neutron capture only has sufficient energy to produce a small number of displacements and cannot produce the large displacement c lusters which can arise f r om -fasit neutron scattering. The displacement distribution and hence the "radiation damage observed may, therefore , be different f o r these two processes .

Not all.the changes in physical properties observed when a material is irradiated in a reactor and-that are re ferred to by the general title of radi -ation damage are due. to the production of vacancies and interstitials in the lattice. F o r example; .the embrittlement of austenitic steels at high tem-peratures is caused-by the production of helium ,in the lattice [10]. The response function in this . ease l s that of the helium-producing reactions, mainly the 10B (n, a) reaction, rather than the function f or atomic, d isplace-ments. : It is thus-essential to ensure that an effect is correlated with a particular neutron .energy band, be fore interpretation of that effect is made in terms of that, energy-band,.

Threshold reactions -,

When a nucleus captures a neutron, reactions other than the (n, 7) reaction already considered are possible . The most common of these are the (n, a), (n, p) and (n, 2n) reactions. F o r these reactions the response function is governed by the energy, released or absorbed in the reaction. If the. target nucleus plus the incident neutron are heavier than the, reaction products, .then energy, is. re leased in the process and, in theory at least, the reaction is possible with zero -energy neutrons. Examples of this type of reaction are the e Li (n , a)3H and the 10B (n, a)7Li reactions. If, on the other hand, the target nucleus plus the incident neutron are lighter than the.reaction products, then the extra mass must be supplied in the f o r m of kinetic energy of the incident neutron. The reaction will, therefore , only be possible with neutrons with m o r e than this minimum energy.

In addition to satisfying the basic energy balance of the reaction, a charged particle has to penetrate the electrostatic potential barr ier around the nucleus be fore an (n, p) or an (n, a) reaction can proceed. The probabi l -ity of the particle doing this increases as the energy of the partic le is increased and since the extra energy has to be supplied by the incident neutron energy, the apparent threshold energy of the reaction will be higher than the theoretical minimum [11]. In pract ice , the majority of reactions ' of this type have apparent thresholds greater than 1 MeV, even if the

18 CHAPTER III

reaction is exothermic. The response function of two typical threshold reactions, 5SNi (n, p)58Co and 21AL (n, a)24Na, are shown in curves V and VI in Fig. II. 5.

F r o m the curves it can be seen that the response functions of the dif -ferent physical p rocesses differ widely and it is not possible to relate one p r o c e s s to another without some consideration of the neutron spectrum. In particular, the threshold reactions are confined mainly to the energy range above 2 MeV, whereas the radiation damage processes have a large fraction of their response below this energy. As the neutron spectrum in the range 100 keV to 2 MeV varies considerably f or different irradiation positions, it is not possible to relate the reaction rates f o r threshold r e -actions to irradiation damage results without some analysis of the effect of the neutron spectrum.

II. 1. 6. Division of the neutron spectrum for the monitoring of irradiation exp erim ents

Although there are no distinct boundaries in the neutron spectrum in a reactor , the response functions do fall into two'broad c lasses : those with the maximum response in the thermal neutron range; and those with maximum response in the MeV range. It is , thèrefore , convenient to divide the neutron spectrum into two regions: the thermal and epithermal region; and the fast region. This pract ice is almost universally adopted f o r monitoring of irradiation experiments in thermal' reactors .

Thermal neutron fluence measurements have sometimes been used for experiments on radiation damage because of the simplicity of long-term measurements with cobalt and this pract ice may be revived by the advent of pr imary emission flux detectors based on rhodium. When using this technique, it must be remembered that although the fast neutron spectrum at any point depends mainly on the source density c lose by and the m o d e r a -ting medium, the source density itself depends on the product of the thermal flux and the fuel content. In the interpretation of thermal f luence measure -ments f o r this purpose, therefore , the burn-up of the fuel must be taken into account. By careful calibration of the experimental position and the effect of fuel burn-up on the spectrum, this technique can be useful f o r f luence measurements under some conditions. The thermal flux density also depends strongly on the neutron absorption of the medium c lose to the monitor positions, and even small changes in experimental conditions can ser iously alter the neutron spectrum. Techniques of this kind must, there -f o re , be used with extreme caré.

Neutrons in the energy range 100 eV to 1 keV in a thermal reactor do not contribute material ly to either the thermal' or the fast neutron reactions and so it is reasonable to consider the spectrum divided somewhere in this range. A l so over this energy range the 1 / E f o r m of the spectrum should be a good approximation f o r most thermal reactor conditions since what di f ferences there are in reactor spectra are found outside this energy range.

It must be emphasized that this division is purely arbitrary and may not be applicable to all conditions. A l so f or some effects such as defect production there may be significant contributions f r o m both spectrum regions.

NEUTRON. SPECTRA 1 9

II. 2. THERMAL NEUTRON REGION

II. 2. 1. Conventional thermal flux densities

The results of neutron fluence and flux density measurements in the thermal neutron region are almost always interpreted using a spectrum model . The most widely used models make use of the fact that f o r a large number of materials the neutron absorption cross^sect ion varies approx-imately as the inverse of the neutron velocity [12], i. e.

. W . 2 Û

where

20 CHAPTER II

OOOI OOI 01 . . I 10 100 E (eV)——

FIG. II. 6. Neutron spectrum in a graphite exponential stack at 20°C (from Ref . [16]) .

1.4,

1-2

10

0-8

/

•f + е й .

~ x — -X-

®

E 0 (E)

0-2

/

i

о ®

2 0 ° C

1 6 0 ° С 1

T

i

X 244 °C 1

T

i

+ 321 ° C

f . 1 1

IO 15 Е/кты

2 0

FIG. il. 7, Joining region spectra (Ref. [ 1 6 ] ) .

NEUTRON. SPECTRA 21

T A B L E I I . I , ' F L U X D E N S I T Y A N D S P E C T R U M P A R A M E T E R S F O R A

T Y P I C A L L I G H T - W A T E R R E A C T O R S P E C T R U M

Parameter ' Parameter value •

Total flux density Ф 1 . 0 n - c m " 2 - sec" 1

Westcott flux density 0 . 2 6 n - c m " 2 - "sec"1

Neutron density below 0. 5 èV X 2200 m / s ë c 0 .25 n- c m " 1 ' sec" 1

Fitted Maxwell ian 0. 24 n - c m " 2 - sec " 1

Westcott V . • . 0. 07 .

Neutron temperature 47"C

Equivalent fission flux measured by 58Ni(n, p)5 8Co 0. 32 n- c m " 2 - sec" 1

Equivalent fission flux measured by 2 7Al(n, a)MNa ; 0 .48 n* c m " 2 ' s ec " 1

f o r absorption o r f iss ion reactions and is usually used f o r this purpose without any further re ference to the true neutron spectrum in the reactor . Extensive compilations of . c ross -sect ion data are available f o r the m o r e common f o rms of the convention [14, 15] and the thermal c ross - se c t i ons normally l isted are those corresponding to a neutron velocity of 2200 m / s e c .

These conventional fluxes must not be confused with the fluxes predicted by reactor calculations which are "true fluxes defined by the integral

1 О - j n(v) v dv . (11.13)

Comparisons between these two types of flux presentations can be made either by comparing the predicted and measured reaction rates of some suitable reactions, o r by converting the conventional flux into a true flux spectrum.

II. 2. 2. True thermal flux spectrum

There are good theoretical reasoris ' for assuming that in a wel l -'moderated'thermal reactor syëtem the neutron spèctruminthe thermal region can be represented by a function of the f o rm

• ' ; ; „ + j x / E } , . : . . . . . ' ; ш . н )

where Ф is the thermal flux density; T the temperature of the neutrons; ' X the ratio of the slowing-down flux to the thermal flux density; and J is a-joining function. .. Several measurements of the energy spectrum of a beam of neutrons extracted f r om moderating and multiplying media have shown that f o r wel l -moderated systems based on the m o r e common m o d e r a -tors ' such as wâtér and graphite Eq. (II. 14) gives a véry good representation of the spectrum ; [15-17] . F igure II. 6 shows a typical spèctrum obtained-by Coates and Gayther f or the 'moderator of a graphite-moderated exponential stack with natural uranium fuel.

22 CHAPTER III

. By fitting the Maxwellian and the slowing-down spectrum to the spectrum measured f o r dif ferent stack temperature, Coates and Gayther a lso produced a curve of the joining function J f o r their lattice. This is shown in F ig . II. 7.

It is not always poss ib le to extract a beam f r o m a reac tor to allow t ime -o f - f l i gh t measurements to be made and so if the true flux spectrum is required , it has to be obtained by other means. This is probably best achieved by calculating the spectrum with one of the computer codes avai l -able f o r this purpose [15]. The computed spectra may be compared direct ly with integral measurements by comparing the computed and measured react ion rates . ' Such direct compar isons have the advantage that they do not re ly on the assumption of a wel l - thermal ized spectrum used f o r c o n -ventional flux determination.

If it is not poss ib le to compute the flux spectrum, a reasonably good approximation may be obtained f o r wel l - thermal ized systems f r o m the measured conventional flux and spectrum parameters . The contribution of the s lowing-down flux and the joining function to the conventional flux must f i r s t be subtracted f r o m the measured conventional flux. The fract ion of the conventional flux remaining is then the flux due to the Maxwellian part of the spectrum only. T o convert this to a true Maxwellian flux at a t e m -perature T , • it must be multiplied by the ratio of the average ve loc i ty in a Maxwellian to the 'most probable veloc i ty , i . e . 2/\Гж = 1. 128, and by the ratio of the average ve loc i ty of a Maxwellian at temperature T to that at temperature T 0 , i. e. s /T /T 0 . This then gives us

Фмах = 1- 128 N/T/T0 Ф0 ' (11.15)

where ФМах' is the Maxwellian thermal flux; ф0 is the conventional flux c o r r e c t e d f o r the epithermal contribution; T 0 is 20. 44°C, and T is the neutron temperature.

The flux spectrum in the thermal and epithermal region can now be obtained by.substitution in Eq. (II. 14).

II. 2. 3. Theoret i ca l spectra

A full treatment of the theoret ical methods of determining thè thermal neutron spectrum is beyond the s cope of this book. However , s o m e know-l e d g e of the results of such calculations is useful in the understanding of flux density measurements and so the m o r e c ommon methods will be reviewed br ie f ly .

The full equation f o r the slowing down of neutrons can only be treated in v e r y general t e r m s and most theoretical-methods treat the steady-state space independent prob lem. In this c a s e the neutron balance equation can be written as

[ста(Е) + ctS(E)] p(E)dE = [ ^ с т ( Е ' -» E) ф ^ ' ^ Е ' Ы Е + S(E)dE (II. 16)

where cra(E) is the absorption c r o s s - s e c t i o n ; CTS(E) is the scattering c r o s s -section; ст(Е'-» E) is the c r o s s - s e c t i o n f o r scattering neutrons of energy E* into the energy band (E, E + dE); ф(Е) is the neutron flux; and S(E) is the neutron source .

NEUTRON. SPECTRA 23

Even now the equation can only be solved if some simplifying assump-tions,are made. If, f o r example, the absorption i s assumed to be zero , then the solution is a Maxwellian distribution in thermal equilibrium with the moderator atoms.

Wigner and Wilkins [18] have considered the case of scattering by a monatomic gas and in particular have obtained a solution f or moderation by hydrogen treated as a monatomic gas of unit mass . They assume that the scattering c ross - sec t i on is constant and that the absorption c ross - se c t i on • is inversely proportional to the neutron velocity. With these simplifying assumptions a differential equation is obtained which can be solved to give the neutron spectrum. This theory forms the basis of the computer program SOFOCATE [19] which is in common use f or calculating the thermal spectrum in a l ight-water-moderated reactor .

Although results obtained using this theory are adequate f o r many purposes, serious discrepancies can occur. These discrepancies are usually largest in the joining region of the spectrum and are most serious in systems with plutonium fuel. They ar ise because at energies below 1 eV a scattering col l is ion does not break the bonds of the water molecule and so the col l ision takes place with the molecule as a whole. When this occurs , • the rotational and vibrational energy states of the water molecule have to be taken into account. Nelkin [20] has constructed a model f o r slowing down in light water, based on experimental measurements of the scattering of neutrons, which takes these energy states into account. Spectra ca l -culated using the Nelkin model f o r a plutonium-fuelled system have been found to give results in good agreement with foil measurements and reason-able agreement with t ime-of - f l ight measurements [21].

F o r a crystalline moderator such as graphite or zirconium hydride the neutron col l is ions near thermal energies will take place with bound atoms rather than f r ee atoms. This will impose the crystal energy level structure on to the scattering process and it is essential to take this into account in spectrum calculations. In certain cases the crystal atoms may be considered as either Einstein or Debye osci l lators and an approximate solution of the slowing-down equation obtained. The alternative approach is to determine the scattering law of the moderator experimentally and to use this in the slowing-down calculations.

An alternative theoretical technique which is extensively used f o r ca l -culating neutron spectra is the Monte Carlo calculation. In this technique, instead of solving the neutron balance equation, individual neutron histories are follbwed both in space-and in energy as the neutrons slow down. At each neutron col l ision the probabilities of all the possible results of the col l ision are determined f r om the physical p rocesses involved and the neutron fate is determined according to these probabilities on thé basis of chance. By following several thousand neutron histories in this way, suf -ficient data on the neutron density distribution can be built up to f o r m an accurate neutron spectrum. With the Monte Carlo technique the distribution can be determined as accurately as the knowledge of the scattering laws allows provided sufficient computing capacity is available.

The theoretical solutions of the slowing-down equations predict that the neutron temperature is related to the moderator temperature by an equation of the f o rm

Tn =T0 (1 + C m f ) (II. IV)

24 CHAPTER III

f o r 0 < m сга/ст5< 0. 5, where :Tn is the neutron temperature; T0 is the m o d -erator temperature; m is the mass of the moderating atom; aa is the . absorpt ion-cross -sect ion at the velocity corresponding:to T0 ; CTs is the . scattering c ross - se c t i on ; and С is a constant.

The value of the constant С is not well established, but the values obtained l ie in the range 0. 6 to 1. 1. Coveyou, Bate arid Osborn [22] obtained a value of 0. 9 using a Monte Carlo method to solve the monatomic gas approximation. They found that this fitted results f o r moderators with masses between 1 and 25 to within 5%. •

The t ime-of - f l ight measurements made by Coates and Gayther. f o r a graphite lattice give neutron temperatures which are rather higher than predicted by theory [16], but this discrepancy decreased as the moderator temperature was raised. They attribute this discrepancy to the 'ef fects of the lattice bonds producing an apparent increase in the mass of the carbon atom in the scattering events. At higher energy (E > 1 eV) the effects of the crystal binding become less important and the heavy gas model is consistent with experiment at these energies [ 16]. • .

« II. 2. 4. Variation of the spectrum across a reactor lattice

The thermal neutron flux density in a reactor is very sensitive to absorbing media such as control rods, or fuel elements. This produces quite substantial variations in the thermal flux across the lattice cel l in a heterogeneous reactor . Figure II. 8 shows the thermal flux variation measured across the diagonal of a square D2O lattice with concentric tube fuel elements. No attempt was made in these measurement's to determine the hyperfine structure through the fuel plates.

FIG. II. 8. Westcottf lux distribution along the diagonal 'o f the lattice in a DzO reactor (Daphne).

NEUTRON. SPECTRA 25

In a l ight -water r e a c t o r the thermal flux variat ions take p lace over much shor ter distances than in either a heavy -water o r graphite r e a c t o r and cons iderab le flux peaking can o c c u r in the water gaps between the fuel plates in this type of r eac to r .

Epithermal neutrons are much l e s s sensi t ive than thermal neutrons to changes in med ium and, consequently, the epithermal flux distribution does not show much f ine s tructure a c r o s s the latt ice ce l l . The epithermal flux density m e a s u r e d under the s a m e conditions as/the thermal flux distribution shown in F ig . II. 8 is shown in F i g . II. 9. ;

F i g u r e s II. 8 and II. 9 show that there is a< cons iderab le variation in the neutron spec trum a c r o s s the latt ice ce l l due mainly to the variation in the thermal component of the f lux. In fact , in this experiment the epithermal index (Westcott r) var ied b y a lmost a f a c t o r of 2 f r o m 0. 085 to 0. 145, although the epithermal flux changed by l e s s than 10%.

The neutron temperature in a heterogenous r e a c t o r is a lso sensi t ive to the variat ions in the med ium and i n c r e a s e s marked ly inside the fuel e lements . F i g u r e II. 10 shows the variat ion in the neutron temperature a c r o s s the latt ice ce l l in DAPHNE m e a s u r e d under the s a m e conditions as the thermal f lux measurements shown in F i g . II. 8. v ,

It is obvious f r o m Figs . IL 8, Ц. 9 and II. 10 that in the neighbourhood of a neutron a b s o r b e r the neutron spec trum changes rapidly and quite smal l e r r o r s in thé locat ion of measur ing fo i l s can lead to signif icant e r r o r s in the resul ts . This i s espec ia l ly true f o r m e a s u r e m e n t s involving the use of cadmium boxes and in these measurements c a r e must b e taken to ensure that the p r e s e n c e of the cadmium box does not inf luence the final m e a s u r e d spectrum.

RADIAL . DISTANCE (CMS) .

FIG. II. 9. Epithermal flux-distribution along the diagonal df the lattice o f а Е>гО reactor (Daphne). ' -

26 CHAPTER III

FIG. II. 10. Neutron temperature distribution along the diagonal o f the lattice of a DzO reactor (Daphne).

II. 3. FAST NEUTRON REGION

As discussed in Chapter V, the experimental techniques f o r measuring fast neutron spectra that are available at present are very limited in their scope and most of the experimental information currently available has been obtained by using threshold activation detectors. This method suffers f r om -poor energy resolution and a shortage of suitable detectors with thresholds below 1 MeV and hence really only gives a broad indication of the general shape of the spectrum above about 1 or 2 MeV. F r o m Fig. II. 5 it can be seen that the energy range 10 keV to 1 MeV is important f or the analysis of p ro cesses such as radiation damage and in this energy range the neutron spectra dif fer markedly f r o m reactor to reactor . It is , therefore , essential that some information in this region is obtained if radiation damage in reactors is to be fully understood. Spectrometers suph as the hydrogen-f i l led spherical proportional counter [23] and the semiconductor sandwich counter [24, 25] show promise of measuring the neutron spectrum over at least part of this energy range under ideal conditions, but it is unlikely that either will be applicable to high-flux research reactors . We, therefore , have to re ly , to some extent at least, on theoretically determined neutron spectra.

II. 3. 1. Calculation of fast neutron spectra

The basic slowing-down equation which must be solved is the same as that used f or the calculation of the thermal neutron spectrum, but there are two very important di f ferences in the problem. First ly , the energies being considered are much higher than the thermal energies of the moderator atoms or the binding energies of the molecules or crystals and so both these

NEUTRON. SPECTRA 27

effects may be neglected. The reactor may, therefore , be regarded as a stationary monatomic gas. The second di f ference is in the source distribution used f or the calculation. In the.fast neutron region the neutron spectrum changes considerably with the distance f r o m the fuel elements and it is essential to take the heterogeneous source distribution into account as well as the material distribution in the reactor if real ist ic spectra are to be obtained.

The basic methods of diffusion theory, transport theory or Monte Carlo techniques cah all be applied to the calculation of high-energy neutron spectra, but all suffer f r om disadvantages. Simple diffusion theory breaks down c lose to boundaries in a heterogeneous system and cannot be applied to voids. As the experimentalist is usually interested in the spectrum in a void, o r near void region and as the assumption of a moderator filling the void alters the neutron spectrum appreciably, diffusion theory is only of l imited use in heterogeneous reactor systems. The h igher -order spherical harmonic approximations, P3 and P5, overcome the objection to diffusion theory at boundaries to some extent but are still not valid in void regions and so again have to be used with caution in experimental situations. Monte Carlo methods are not limited by boundaries or voids in the system but because of their basical ly statistical nature, a large amount of computing is required to obtain rel iable results. Transport-theory also over comes the objections to diffusion theory and spherical harmonics and is not subject to the statistical e r rors associated with Monte Carlo technique. This method, however, requires both a large amount of computing t ime and also a very large computer store to obtain results f o r a real ist ic geometry.

Two other methods developed f o r calculating neutron spectra at a distance f r o m the source of neutrons arethe moments method and the removal c r o s s - s e c t i o n method. These two methods break down c l ose to the source of neutrons but are suitable f o r the calculation of the neutron spectrum in shields.

The Computing methods mentioned above are all well developed and give rel iable spectra in situations f o r which they are applicable. However, they do re ly on a relatively large computer being available. A method of predicting the neutron spectrum when a computer is not available has been developed by Genthon [26] based on fitting empirical functions to spectra calculated by normal theoretical methods f o r different reactor types and positions.

Genthon considers reactors with heavy-water, l ight-wáter and graphite moderators . He assumes the neutron spectrum above á few hundred electron volts is made up of two parts: the homogeneous part which depends mainly on the reactor type; and the heterogeneous part which depends on the proximity of the fuel. The spectrum Ï (E) is then represented by

' ï ( E ) = K [ ï 0 ( E ) + h Y e ( E ) ] (11.18)

where (E) is the homogeneous part of the spectrum; ¥ (E) is the hetero -geneous part of the spectrum; and К and h are fitted constants. ; The homogeneous part of the spectrum has the f o r m

?„ (E) = u(E0 - E) exP g + u(E0 - E)F Np (E) ' (II. 19)

28 CHAPTER III

where u(E0 - E) is the unit step function, i. e.

u(E0 - E) = 1, E < E0

0, E > E0

E0 is a threshold energy depending on the moderator ; and N0(E) is the fission, spectrum. . . ..

The value of the constant b is 0. 22 f o r heavy water, 1. 45 f or light water and 0. 28 f or graphite. F is a constant making ¥(E) continuous at E = E0 .

The simplest f o rm of the heterogeneous component given by,Genthon is

= ( l . 15 -3L - F ) N , > (E) (11.20)

and фг is chosen such that oo oa

E)dE = J cp(E)dE E0 E0

where ф(Е) is the true spectrum. In pract ice фг would be measured using a suitable threshold detector, such as 3 2S(n, p). Better fits to calculated spectra can be obtained by replacing the f iss ion spectrum N0(E) by empir i -cally fitted functions and Genthon gives functions he has obtained for l ight-water, heavy-water and graphite reactors .

This model gives reasonably» good representations of the neutron spectrum in the co re of wel l -moderated reactors and slightly l ess good spectra in the re f lec tor region c lose to the core . Because the functions which are fitted are essentially smooth functions, the model does not show up irregularit ies in spectra caused by resonances in the c ross - se c t i ons or , f o r example, the peak which appears in graphite reactor spectra at ap-proximately 4 MeV. This .is usually not important f or the analysis of effects such as radiation damage and the model would be used f or conditions which do not deviate too much f r om those used in the original fitting analysis. Two examples of the spectra obtained for a l ight-water reactor are shown in Fig. II. 11, i •...•:•

At quite small distances f r o m the fuel elements it becomes, poss ible to treat the neutron spectrum as a s imple sum of contributions f r om individual fuel elements in the reactor . The neutron flux density as a function of energy and distance f r o m a single fuel element in a moderating.medium may then be.calculated, using one of the calculation techniques previously des -cr ibed, and from.this the spectra at different points in the reactor can be constructed. Some typical spectra calculated f o r a single fuel element in D 2 0 by the Monte Carlo method [29] are shown in Fig . II. 12 and a spectrum calculated f r om these results f o r an empty fuel element position in the reactor PLUTO is compared with a direct calculation of this spectrum in Fig. II. 13. The flux densities obtained by these two methods of calculation are s o m e -what different, due probably to the fact that iñ the direct calculation the spectrum was calculated in a void, whereas f or the lihe source calculation the spectra are calculated in a continuous moderator ' region. 'The spectrum shapes are , however, c losely s imilar . In this case where the c losest fuel is 15 cm f rom the experimental position, the fact that the void is replaced by heavy water has no effect on the spectrum shape. ' This is , however, not

NEUTRON. SPECTRA 29

always true.and,, as we shall s ee in the next sect ion , the f i l l ing of a void with m o d e r a t o r can change the spec trum shape signif icantly. This type of spectrum calculation should only be used when there can be no loca l d i s -tortion of the neutron spectrum by mater ia l s c l o s e to region of interest , but under suitable conditions it prov ides a good way of obtaining neutron spec t ra in r e f l e c t o r s and shields with a min imum of t ime spent on the initial computer calculation.

lOOeV 10 ЮО NEUTRON ENERGY a) CORE CENTRE

I MeV 2

SPECTRUM TO BE FITTED Еф(Е) --FITTED SPECTRUM Ey(E) : —

lOOeV IKeV Ю IOO IMcV 2 NEUTRON ENERGY

b) REFLECTOR, 20cm» FROM.THE CORE EDGE

FIG. II. 11. Spectra in a light-water reactor (from Ref. [ 2 6 ] ) .

30 CHAPTER III

J

О о: < x i— Ш 2 Z Z) IK Ш CL X => I U-

0

»l - 2 - 1 0 1

NEUTRON ENERGY(MeV)

FIG. II. 12. Neutron spectrum as a function of distance from a line source in heavy water (10 000 n-cm"z-sec_1).

II. 3. 2. Variations in the fast neutron spectra in a heterogeneous reactor

The flux of uncollided f iss ion neutrons will be greatest in or near the fuel elements in a heterogeneous reactor and will fall rapidly as the distance f r o m the fuel element is increased. The flux at a few keV, however, r e -mains m o r e or less constant across the lattice cel l of the reactor and changes only as the average power density across the reactor changes. Thus the neutron spectrum above a few keV changes f r om a hard spectrum in or near the fuel elements to a much softer spectrum at the centre of the lattice cell .

The variation of the flux at a few keV across a lattice cell will fol low the flux distribution at 5 eV quite c lose ly and a typical example of this variation is shown in Fig . II. 9. The variation of.the fast neutron component as measured by the 3 1P(n, p)31Si reaction f o r the same lattice is shown in Fig. II. 14. If one looks m o r e c lose ly at the energy spectrum above 2 MeV, by comparing the reaction rates of several threshold detectors, it is found that f o r positions further f r om the fuel the spectrum deviates m o r e f r om that of a f iss ion spectrum.

The neutron spectra across the lattice cel l , corresponding to these flux distributions, have been calculated by a Monte Carlo technique and are shown in Fig. II. 15. The spectrum in the centre region is subject to rather large statistical e r r o r s , but these spectra show the change in spectrum across the lattice quite c learly . The reaction rate of 31P(n, p) has been c a l -culated f o r these spectra and is superimposed on the measurements in Fig . II. 14.

NEUTRON. SPECTRA 31

J 1

-

"Ifl и \r Jtf' s

H.

1

V NEUTRON ENERGY (MeV)

( a )

1 Jll JL

- l -

NEUTRON ENERGY (MeV)

(b)

FIG.II. 13. Spectrum in an empty lattice position in a heavy-water-moderated reactor (Pluto): (a) spectrum from a reactor calculation; (b) spectrum calculated from line source results.

32 CHAPTER III

о EXPERIMENTAL POINTS MEASURED BY THE P31 (np) SI3' REACTION.

— REACTION RATE CALCULATED FROM THE MONTE CARLO SPECTRA.

5 5 5 5

K a z VOID FUEL REGION MODERATOR s g MODERATOR

1 II 1 1 "1 RADIAL DISTANCE (cms)

FIG. II. 14. Variation of the fission flux along the diagonal of the lattice of a heavy-water reactor (Daphne).

-1

L J - X -A n_ l r .1 .IV

- I NEUTRON ENERGY(MeV)

(a>

NEUTRON. SPECTRA 33

Г1 "1 " r1" 1 I

pi r-l

J Д

-I L

L 1

I

NEUTRON ENERGY (MeV) (b)

L IT fl 4. 4

~LJ 1

\ I.

NEUTRON ENERGY(MeV) (c)

FIG. 11.15. Variation o f the neutron spectrum across a heavy-water reactor lattice: (a) spectrum in centre void; (b) spectrum at the outer edge o f the fuel, 4 . 5 c m from the centre; ( c ) spectrum in the moderator, 7. 3 c m from the centre.

34 CHAPTER III

SPECTRUM IN AVOID SPECTRUM IN HEAVY WATER

т Г I-

г

j u 1 i J IJ

j ; r-

: 1 1 Í Рч 1

• - J tr / 1 ч

i

V О

NEUTRON ENERGY (MeV)

FIG. II. 16. Spectra in the central experimental region of a hollow fuel element with and without heavy water in the experimental thimble.

NEUTRON. SPECTRA 35

0 1 500

5 зоо

J1 Г гЧГ

Ml ' I 5 I 0 1 NEUTRON ENERGY (MeV)

(с) FIG. II. 17. The effect o f a 10-cm experimental hole and a typical rig on the fast neutron spectrum in the " moderator of a graphite-moderated, natural-uranium reactor (BEPO): (a) normal lattice; (b) 10 - cm experi -mental hole; (c) typical graphite and aluminium rig in experimental hole. - ' •».'

36 CHAPTER III

A heavy-water reactor with a s ix- inch lattice pitch has been used to illustrate the variation of the neutron spectrum across the lattice, but any other heterogeneous reactor will give s imi lar variations. In a l ight-water reactor , which approximates much m o r e c lose ly to a homogeneous system, the fast spectrum variation across the c o r e will be small . However, at places of experimental interest, such as the c o r e re f l ec tor boundary or experimental positions in the c o r e where the homogeneity is destroyed, variations in the neutron spectrum will occur .

Another factor which can significantly alter the neutron spectrum is the replacement of a moderator region by a void, particularly c l ose to a fuel region. Many experimental irradiations are per formed in voids or nearly void regions and so this effect can be important in the analysis of experimental results. F igure II. 16 shows the effect of substituting heavy water f o r the void in the experimental region at the centre of an annular fuel element in a heavy-water reactor on the spectrum in that region.

Similar but smal ler variations can be observed in experimental holes mid-way between the fuel elements. As an example, Fig. II. 17 shows the spectra in a 10 -cm diameter region, mid-way between four fuel elements in BEPO both void and fi l led with graphite. In this position the effect of the void on the neutron spectrum is small and could probably be neglected in all but the most accurate work. At greater distances f r om the fuel this effect becomes even l e s s important.

The presence of experimental equipment can also affect the fast neutron spectrum. Many experimental r igs consist largely of aluminium because of its low thermal neutron absorption c ross - se c t i on . In the high-energy range aluminium has some pronounced scattering resonances and these show up in the neutron spectrum. The effect of these on experimental results will , however, be small. F igure II. 17 also shows the effect of a typical exper i -mental r ig on the neutron spectrum in a graphite reactor . The effect on the neutron spectrum of materials with high inelastic scattering c r o s s -sections, such as iron, will be much m o r e serious and their presence must be taken into account in spectrum calculations.

In general, therefore , fast neutron spectra do change with alterations in the reactor loading and with position in the reactor and these changes can be sufficiently large to cause observable di f ferences in measurements of propert ies such as radiation damage under different experimental conditions. Even changes such as the substitution of a graphite sample c a r r i e r f o r an aluminium c a r r i e r can producé a discontinuity in the results when radiation damage measurements are plotted against the activation of a threshold detector.

II. 3. 3. Comparison of experimental and theoretical spectra

Al l the neutron spectra quoted so far as examples of spectra in reactors have been theoretical spectra and no experimentally determined spectra have been mentioned. This is because the experimental information currently available is rather limited both inr energy range covered and the reactor environments which have been measured. At present it is impossible to obtain adequate experimental information on the neutron spectrum in a thermal reactor and the usual pract i ce is to rely on theoretical spectra,; comparing these with experimental information where possible .

NEUTRON. SPECTRA 37

The most commonly used method of obtaining experimental information on fast neutron spectra is by means of threshold activation detectors. Only a small number of suitable detectors are available f o r this technique and even these are limited in usefulness by uncertainties in differential c r o s s -section curves. In principle it is poss ible to obtain a neutron spectrum directly f r om threshold detector measurements f or energies above 1 MeV and several methods of analysing these results have been developed (Chapter V). Unfortunately, the e r r o r s in the experimental measurements combined with the uncertainties in the c ross - se c t i ons often result in un-stable solutions and even in the best cases spectra obtained directly f r om threshold detector measurements with no initial assumption of a spectrum shape are not very satisfactory. It is usual, therefore , to use threshold detectors in conjunction with some theoretical f o rm of the neutron spectrum. This can be done in two ways: either an analytic model of the spectrum is assumed and the measurements used to fit some constants; or the measured reaction rate is compared directly with the reaction rate calculated using a theoretical spectrum. In this latter case some direct experimental check on computed spectra in the range above 1 MeV may be obtained. A typical comparison of this type, obtained by Kôhler [28], is shown in Fig. II. 18. In

E'(M«V)

FIG. II. 18. Comparison of the calculated and measured flux above threshold energies for a light-water reactor (calculated distribution is normalized by the measured values) (Ref. [ 2 8 ] ) .

TABLE II. II. THRESHOLD DETECTOR DATA USED BY KÔHLER [28]

Reaction о E e f f (barn) , (MeV)

232Th(n, f) . 0 .14 1 .40

" 8 U(n, f) о. 6o: 1. 55 31P(n,p) 0 .140 • 2 . 9

32S(n,p) •:' .0 .350 1 ' 3 . 2 '

2 4Mg(n,p) ; 0 .060 6 . 3

œ Fe(n,p) - i 0 .110 7 . 5 21Al(n, a) 1 ,o.ii3 С 8.1

38 CHAPTER III

Ен Й H S w j H J и ¡э fe

О J J о к <

tí M

tí я ь 4 Й

3 w

й-.

тэ

NEUTRON. SPECTRA 39

this f igure the curves show the calculated distribution of the integrated flux above an energy E for a l ight-water reactor and the experimental points" are the threshold detector measurements of the integrated flux plotted at the effective threshold energy of the detector in the calculated spectrum. The threshold detectors and c ross - se c t i ons used in this comparison are l isted in Table II. II.

The uncertainties in the absolute value of the threshold c r o s s - s e c t i o n s can be eliminated f r o m these comparisons by considering the ratios of reaction rates f o r several detectors in two different reactor spectra. This has been done f or a hollow fuel element at the centre of the heavy-water reactor PLUTO compared with the same position in the reactor but with the fuel element removed; the results are shown in Table II. III. A three-dimensional Monte Carlo calculation was used to obtain the theoretical reaction rates and the. e r rors quoted f or the calculated ratios are those due to the statistics of the calculation only. The calculated spectra f or the two situations are shown in Fig. II. 13(a) and the solid spectrum of Fig. II. 16.

The calculated reaction rate ratios are all lower than the measured ratios by m o r e than would be expected f r o m the e r r o r s . This could be due to a di f ference between the calculated source distribution and the actual source distribution in the reactor in the two cases , leading to an e r r o r in the spectrum intensity. To eliminate this effect the ratios are also shown normalized to unity f o r the thorium f iss ion reaction.

Although these ratios show that there is no large inconsistency between the calculated and measured reaction rates in this case, they also show the small changes to be expected in the reaction rates of theshold detectors f o r two markedly different reactor environments with quite different neutron spectra. This demonstrates the limitations of threshold detectors f o r neutron spectrum measurement in a reactor . The method can give useful information in the energy range above 2 MeV and will demonstrate the deviation of the spectrum f r o m the f ission spectrum in this range, but it ' gives no information in the region below 1 MeV where the main di f ferences in fast neutron spectra occur .

Proton reco i l techniques using photographic plates have been used f o r neutron energy spectra f o r many years . Although tedious to use, these are capable of measuring neutron spectra down to approximately 500 keV and an example of a spectrum measured by Gore [29] is shown in Fig. II. 19. The drop in the measured spectrum at low energies is probably due to the l oss of low-energy proton tracks. In the last few years the proton reco i l technique has been extended to l ower energies using gas- f i l led proportional counters [23]. These are only applicable over a limited range of energies and rely on measuring the neutron energy spectrum above this energy range by other methods. F igure II. 20 shows a spectrum obtained by Benjamin et al. [23] f o r a fast reactor assembly using a combination of gas- f i l led counters and photographic plates. The measure