Neutron Fluence Measurements

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Wall Fission Spectrum

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Graphite Moderated Spectrum

Light-Water Moderated Spectrum IETRI

(Spectra Normalized to Equal Flux Greater

than 0.0674 MeV)

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(0.06741

г 3 Lethargy, u

TECHNICAL REPORTS SERIES No 107

Neutron Fluence Measurements

INTERNATIONAL ATOMIC ENERGY AGENCY, VIENNA, 1970

NEUTRON FLUENCE MEASUREMENTS The following States are Members of 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 objective is "to accelerate and enlarge the contribution of atomic energy to peace, health and prosperity throughout the world".

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 effects on solids and such disciplines as radiochemistry and radiobiology, the radiation dose or neutron fluence is an essential parameter in evaluating 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 reflect the dependence of spectra or of different types of radiation on the induced processes. After considering the recommendations of three IAEA Panels, on Чп-pile dosimetry' held in July 1964, on 'Neutron fluence-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 from ten different Member States and two staff members of the Agency. On the measurement of energy absorbed by materials from neutrons and gamma-rays, there are various reports and reviews scattered throughout the literature. The group, however, considered that the time was ripe for all relevant information to be evaluated and gathered together in the form 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 useful both for experienced workers and for beginners. The present manual was edited by Dr. J. Moteff, of the General Electric Company, Cincinnati, Ohio, USA, and the companion volume, on 'Determination of absorbed dose in reactors', will be published by the Agency shortly. The authors who contributed to the present manual are:

Chapter 1 : Introduction J. Moteff Chapter 2: Neutron spectra S.B. Wright (Atomic Energy Research Establishment, UKAEA, Harwell, United Kingdom) Chapter 3: Thermal neutrons Y. Droulers (Commissariat à l'Energie atomique, Centre d'études nucléaires de Grenoble, France) Chapter 4: Intermediate neutrons W. L. Zijp (Reactor Centrum Nederland, Petten NH, Netherlands) Chapter 5: Fast neutrons R.E. Dahl and H. H. Yoshikawa (Pacific Northwest Laboratories, Battelle Memorial Institute, Richland, Wash., USA) Index: A. Keddar (Division of Nuclear Power and Reactors, IAEA, Vienna) 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.Keddar (IAEA), the present Scientific Secretary, for their help in assembling the manual. Special thanks are also due to Dr. W. KOhler of the Agency for 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. Fission neutron spectrum II. 1.3. Intermediate neutron spectrum II. 1.4. Reactor spectrum II. 1. 5. Physical processes encountered in irradiation experiments И. 1. 6. Division of the neutron spectrum for 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. Theoretical spectra II. 2.4. Variation of the spectrum across 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 References 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. Formalism of Horowitz and Tretiakoff III. 2. Measurement of thermal neutron flux density and fluence .... 52 III. 2.1. Relation between flux density, fluence and detector activity III. 2.2. Measurement of low flux densities III. 2.3. Measurement of high fluences 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 References to Chapter III 75

CHAPTER IV. INTERMEDIATE NEUTRONS 77

IV. 1. Introduction 77 IV.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 characteristics 81 IV.2.1. The 1/E spectrum IV. 2. 2. Other spectrum representations IV. 3. Detectors 88 IV. 3.1. Resonance integral cross-section IV.3.2. Detector response IV. 3.3. Self-shielding effect IV.4. Fluence measurements Ill IV.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 fluence measurements V.5. Spectral determination from 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 fields of radiation chemistry and radiation damage to solids, the neutron energy distribution and the neutron fluence 1 are essential parameters for 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 time. Moreover, the various experimental assemblies, control-rod positions a:nd local burn-up will tend to perturb the flux density throughout the reactor core and reflector regions. These perturbations make it almost impossible to estimate doses and fluences from measurements made on cold, clean reactor cores. For this reason it is desirable to measure the^ radiation environment in exactly the same core geometry as would be used for each experiment. At the present time it seems feasible to recommend specific methods, at least for neutron fluence measurements. However, even if unified experimental 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, fission 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 fluences 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 knowledge of neutron spectrum before theory and experiment can be better correlated. 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 fluence, previously referred to as the neutron dose, is defined as the time integral of the neutron flux density.

1 2 INTRODUCTION the specimen. Consequently.it would be difficult to compare results obtained 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 cross-sections 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 for 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 characteristics of reactors, there is no assurance that such measurements are necessarily pertinent to the experiment for which these measurements were made. Part of the uncertainty is caused by the known temporal and spatial variations in the reactor radiation field. The spatial perturbations are of three types: gradients, shielding effects and perturbations. The gradients are inherent in all reactors, whereas shielding effects and perturbations may be introduced by the experimental assembly. Gradients are generally small in the large cores of heavy-water and graphite reactors, at least in positions which are not very close to the fuel elements. In the case of light-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 millimetres in the sample location may give* rise to appreciable errors in dose or fluence estimates. Shielding effects are noticeable for 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, especially in reflector positions in light-water reactors. An irradiation sample generally causes a thermal flux depression within its own volume and in its immediate vicinity due to the absorption of neutrons 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 for irradiations made in a fast neutron flux. All these 'sources of spatial variation of flux make it necessary to per- form 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 first 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 reflector. 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 cycle, the gamma radiation will also build up with the establishment of an approximately equilibrium concentration of the fission products. These effects 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 effect even more serious or, in some cases, reduce it for specific experimental locations. To these inherent effects should also be added effects that are caused by fuel pattern changes, new experiments, etc. that may be introduced during the runs with specific experimental equipment. Moreover, there are the cycle-to-cycle 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 calorimeters, ion chambers, gas loops, self-powered detectors, etc. ; or (2) by the time integration of fast, intermediate or thermal neutron flux densities by means of activation detectors or fission foils. 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 times and to very high fluences. 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 first 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 specific regions and most of these can, in some manner, be related to the relative neutron energy dependence of the scattering and absorption cross-sections 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 briefly 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 foil 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 cross-sections, 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 foil techniques, as a means of separating the more prominent resonance activation which would occur over a small interval of a specific neutron energy from 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 correction factors are also presented. Due to the nature of the cross-sections, and in view of the importance of this energy region in radiation damage studies and transmutation reactions in both fast and intermediate energy spectrum reactors, the mathematical treatment of neutron fluence in this chapter is correspon- 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 class of threshold detectors normally used for fluence measurements in this energy region. Since the fast neutron energy region is important to the study of radiation damage to solids, this subject is amplified beyond the stage reached in Chapter II. The selection of compounds with higher melting temperature than the pure element itself is mentioned in this chapter. Data reduction, especially the treatment of those reactions with high burn-up cross-sections of isomeric states with short half-lives, is discussed. In concluding, it should be pointed out that the general method of fluence measurements using radioactive foil techniques as presented in this manual has not changed significantly from that developed by the pioneers in the field of reactor neutron physics. The important changes, for example, are in the availability of better reaction cross-sections as a function of neutron energy, more accurate half-life and disintegration data for the radioactive isotopes, and materials of higher purity. Therefore, those individuals responsible for 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 cross-sections, but does show how the cross- 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 for 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 fission process have an average energy of approximately 2 MeV. In a thermal reactor these neutrons are slowed down by collisions 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 more 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. For some processes, such as radioactive capture, it is sufficient to use a simple approximation for the neutron spectrum; for other processes 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 fission region in which the neutrons from fission 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 clearly defined boundaries. This arbitrary division into three regions is useful, however, when considering the general form 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 commonly expressed in several different forms which must be clearly distinguished. 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/2v2exp(-mv2/2kT) (II. 1)

Ф(Е) =exp(-E/kT) (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.

From Eq. (II. 1) the most probable velocity can be derived as

/ 2kT V/2 which corresponds to an energy of

1/2 mvj = kT

For 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 arbitrary velocity usually taken as 2200 m/sec [1]. This velocity is chosen because it is the most probable velocity of a Maxwellian density distribution at 20. 44°C. It corresponds to an energy of 0. 025 eV. The cross-sections required for use with this conventional flux are the cross-sections at 2200 m/sec and these are the values listed in most 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 for cross-sections 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. For a material with a 1/v cross-section the average cross-section in a Maxwel- lian spectrum at 20. 44°C will be smaller than the 2200 m/sec cross-section 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. For steady- state conditions the intensity of the thermal flux must be maintained by the neutrons slowing down from 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 form, but at a higher temperature than the moderator. This situation will be discussed further in section II. 2.

II. 1. 2. Fission neutron spectrum

The energy distribution of neutrons produced in the fission process is known as the fission spectrum and has been measured, for all the common fissile elements. Severed empirical relations -have been fitted to the experimental 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, for Z35U, A = 0.4527; b = 1. 036; and с = 2. 29. This expression differs only slightly from that due to Watt. An alternative form of the fission 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). For 235U the best value for e is 1. 290 MeV which gives a = 0. 770. Figure II. 2 shows the fission 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 reactor is in general not a fission spectrum owing to the effects of the moderator and to neutron leakage. However, the fission spectrum is often taken as a first approximation to the neutron spectrum, close to the neutron source, for energies above about 1. 5 MeV. It should be noted that although S(E) is the number of neutrons emitted per unit energy per unit time, the flux is not obtained by multiplying S(E) by the neutron velocity. This can be shown as follows. If the cube 6A6r shown in Fig. II. 3 is considered, with 6A perpendicular to r, then the number of neutrons entering the cube in unit time is N(E) = (6А/4тгг2) S(E). The length of time each neutron remains in the cube, assuming no absorption, is given by 6r т(Е) V where v is the velocity corresponding to the neutron energy E. Therefore, the neutron density n(E), at the volume 6A6r is given by N(E) t(E) 1 S(E) n(E) = 6A6r 4ят2 v 447 NEUTRON. SPECTRA

Sv • ÍA$r

FIG. II. 3. Source-receiver geometry.

Therefore, the flux density cp(E) is

Ф(Е) = n(E) v = gg

Thus apart from the geometric attenuation factor the flux at the volume 6A6r is the fission spectrum S(E). This is only true for a medium in which leakages, absorption and moderation can be neglected. It is, however, a reasonable approximation close 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 form of the spectrum can be derived by considering the simple case in which neutron absorption is neglected. The number of collisions 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 CTsth e scattering cross- section. If ДЕ is the average energy loss per collisions, 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 for all energies well above the thermal region, i. e.

Ç = AlnE

= ДЕ/Е

therefore q(E) = 9(E)NCTS§E.

But if there is no absorption q(E) must equal the total source density q0 and hence

- isk F (п. 6) 12 CHAPTER III

Thus provided the scattering cross-section 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 intermediate flux, therefore, shows little small-scale variation across the lattice cell. This is in marked contrast to the thermal and fast neutrons, both of which show considerable variations close 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 Eq/E) where E0 is the constant of integration. A value of 10 MeV is usually taken for 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) for 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 of a light-water-móderated reactór (normalized T 0 (m) du = 1).

It can be shown [4] that the form 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 macroscopic scattering cross-section. 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 precise form will depend on the moderator temperature and the absorption cross-section of the medium. .Figure II. 4 shows a typical neutron spectrum for a homogeneous hydrogen,moderated reactor, with a moderator temperature of 20°C, obtained using Eq. (II. 11) for the high-energy spectrum and a SOFOCATE calculation for the thermal spectrum. • For comparison, a Maxwellian spectrum and a fission 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 temperature. The fission spectrum fit was obtained by normalizing to the reaction rate in the 58Ni (n,p)58Co reaction. It сапЛэе seen from the diagram that the fission 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 close to the fuel elements there will be a larger proportion of uncollided fission neutrons and the neutron spectrum above 2 MeV will be a better fit to the fission spectrum.

II. 1. 5. Physical processes encountered in irradiation1 experiments

Because of the wide range of neutron energies that exist in a reactor, a number of fundamentally different physical processes 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 cross-section and the neutron flux cr(u)cp(u). For displacement reactions this function is the product of the flux, the scattering cross- section of the material and a damage parameter which gives the effectiveness of a neutron collision 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 liberated 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, possible with the lowest energy neutrons encountered in a reactor. Most reactions of this type have cross-sections 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 cross- 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-called resonance capture often results^in neutron capture cross-sections which are many times larger than the physical size 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 first 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 for approximately half the total response function in the light-water reactor spectrum and the majority of this is due to the first resonance peak at 4. 9 eV. NEUTRON. SPECTRA 15

Rupture of chemical bonds

In radiation chemistry experiments the main process of interest, apart from 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 collision with atoms and so most bond rupture is produced by the intermediate and fast regions of the neutron energy spectrum. After a collision, the recoil atom will usually receive considerably more energy than is required to break the chemical bonds. This energy will be dissipated by further collisions 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 cross-section for the material is a constant function of energy, the response function will be of the form 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 form of gamma rays and in the process the nucleus recoils with an energy of the order of 100 eV. Both these recoil atoms and the capture gamma rays will produce chemical effects indistinguishable from 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.

NEUTRON ENERGY (ïV) 3 10"' to" IP IP2 IP3 IP4 IP5 IP6 IP7 1 '/v DETECTPR b- П Au'«7 (r>*) Ш NEUTRON HEATING E CARBON ATOM DISPLACEMENT 2 Ni58

22 20 18 16 14 12 IO 8 6 4 2 О LETHARGY

FIG. II. 5. Reaction functions for some typical reactions in the typical light-water reactor spectrum. 16 CHAPTER III

Usually in a reactor core the effect of this process will be small compared with the effect of neutron scattering reactions induced by high-energy neutrons. This may not be true, however, in the reflector 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 cross-sections 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 crystalline and polycrystalline materials the main process of interest is the displacement of atoms from 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 process in a reactor co.re and the displacements are mainly produced by fast neutron collisions. Once a primary displacement atom (primary knock-on atom) is produced, it will cause secondary displacements by collisions 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 for 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 collision can produce a large number of displacements and the distribution of these defects will not change much from 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 irradiations carried out in different neutron spectra. As a first approximation, therefore, the effect of annealing can be neglected in estimating the dependence of radiation damage on neutron energy spectrum. In solids where the mean free path of moving atoms is of the same order as the lattice spacing the damage will be produced as localized disordered regions, or 'displacement spikes'. The size 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 from that used above for the calculation of the graphite damage response function and hence the response function for this process may be1 very different from the curve shown in Fig. II. .5. Unfor- tunately, theories of radiation damage are not yet sufficiently well developed for a reliable response function for this type of damage to be quoted. The recoil energy of an atom after.neutron capture is sufficient to produce atomic displacements and this effect can be appreciable under some irradiation conditions [9]. For 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 collisions, but outside the core of a thermal reactor, this may not be the case and thermal neutron capture effects may be appreciable. The recoil atom produced by thermal neutron capture only has sufficient energy to produce a small number of displacements and cannot produce the large displacement clusters which can arise from -fasit neutron scattering. The displacement distribution and hence the "radiation damage observed may, therefore, be different for these two processes. Not all.the changes in physical properties observed when a material is irradiated in a reactor and-that are referred to by the general title of radiation damage are due. to the production of vacancies and interstitials in the lattice. For example; .the embrittlement of austenitic steels at high temperatures is caused-by the production of helium ,in the lattice [10]. The response function in this .easels that of the helium-producing reactions, mainly the 10B (n, a) reaction, rather than the function for atomic, displacements. : It is thus-essential to ensure that an effect is correlated with a particular neutron .energy band, before 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. For 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. released in the process and, in theory at least, the reaction is possible with zero-energy neutrons. Examples of this type of reaction are the eLi(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 form of kinetic energy of the incident neutron. The reaction will, therefore, only be possible with neutrons with more than this minimum energy. In addition to satisfying the basic energy balance of the reaction, a charged particle has to penetrate the electrostatic potential barrier around the nucleus before an (n, p) or an (n, a) reaction can proceed. The probability of the particle doing this increases as the energy of the particle 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 practice, 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. From the curves it can be seen that the response functions of the different physical processes differ widely and it is not possible to relate one process 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 for different irradiation positions, it is not possible to relate the reaction rates for threshold reactions 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 classes: 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 practice is almost universally adopted for 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 practice may be revived by the advent of primary 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 close by and the moderating medium, the source density itself depends on the product of the thermal flux and the fuel content. In the interpretation of thermal fluence measurements for 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 for fluence measurements under some conditions. The thermal flux density also depends strongly on the neutron absorption of the medium close to the monitor positions, and even small changes in experimental conditions can seriously alter the neutron spectrum. Techniques of this kind must, therefore, be used with extreme caré. Neutrons in the energy range 100 eV to 1 keV in a thermal reactor do not contribute materially to either the thermal' or the fast neutron reactions and so it is reasonable to consider the spectrum divided somewhere in this range. Also over this energy range the 1/E form of the spectrum should be a good approximation for most thermal reactor conditions since what differences 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. Also for some effects such as defect production there may be significant contributions from both spectrum regions. NEUTRON. SPECTRA 19

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 for a large number of materials the neutron absorption cross^section varies approximately as the inverse of the neutron velocity [12], i. e.

.W.2Û where

R = / a(v) cp(v)dv

o.v. -2-3 n(v) v dv V

= CT0nv 0 (II. 12) where n is the total neutron density integrated over all velocities. From Eq. (II. 3) the product nv0 is the conventional flux cp0. Thus the quantity which is measured using a 1/v detector is the total neutron density and not the neutron flux density. Although this density includes the high-energy neutrons, mo'st of the neutrons are concentrated in the thermal region. In a typical case approximately 90% of the neutron density in a thermal reactor is in the Maxwellian peak and a further 5 to 7% is below 0. 5 eV. It is common practice to treat spectra of this type as if the whole of the neutron density occurs in the thermal region but in some models of the spectrum only the sub-cadmium fraction of the neutron density is used as a measure of the thermal flux density [13]. This second convention is closer to the truth but it still includes some neutrons in the region of the slowing-down spectrum together with those in the Maxwellian. It also introduces difficulties associated with an exact analysis of the cadmium cut-off. • In models of this type two parameters are used to characterize the spectrum. These are the neutron temperature corresponding to the flux distribution and a parameter that defines the intensity of the slowing-down spectrum. Both will depend to some extent on the model used for the spectrum and in using data of this kind it is essential to ascertain which model was used for the original measurements. The spectrum parameters and flux densities for the spectrum shown in Fig. II. 4 are listed in Table II.I. This type of conventional flux is only applicable to well-moderated thermal reactor systems and leads to some difficulties in highly under- moderated systems where the spectrum is largely epithermal. Where applicable, the convention is very convenient for calculating reaction rates 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, •f + ей. 1-2

/ ~x— -X- 10

0-8 ®

E 0 (E) / о 20°C ® 160° С i 1 X 244 °C T + 321 °C

if . 0-2 1 1

IO 15 20

Е/кты

FIG. il. 7, Joining region spectra (Ref. [16]). NEUTRON. SPECTRA 21

TABLE II. I, ' FLUX DENSITY AND SPECTRUM PARAMETERS FOR A TYPICAL LIGHT-WATER REACTOR SPECTRUM

Parameter ' Parameter value •

Total flux density Ф 1.0 n-cm"2- sec"1

Westcott flux density 0.26 n-cm"2- "sec"1

Neutron density below 0. 5 èV X 2200 m/sëc 0.25 n- cm"1' sec"1

Fitted Maxwellian 0. 24 n-cm"2- sec"1

Westcott V. • . 0. 07 .

Neutron temperature 47"C

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

2 1 Equivalent fission flux measured by 27Al(n, a)MNa ; 0.48 n* cm" ' sec"

for absorption or fission reactions and is usually used for this purpose without any further reference to the true neutron spectrum in the reactor. Extensive compilations of .cross-section data are available for the more common forms of the convention [14, 15] and the thermal cross-sections normally listed are those corresponding to a neutron velocity of 2200 m/sec. 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, or 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 well- 'moderated'thermal reactor syëtem the neutron spèctruminthe thermal region can be represented by a function of the form

•';;„+ jx/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 from moderating and multiplying media have shown that for well-moderated systems based on the more common moderators' such as wâtér and graphite Eq. (II. 14) gives a véry good representation of the spectrum;[15-17]. Figure II. 6 shows a typical spèctrum obtained- by Coates and Gayther for 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 for different stack temperature, Coates and Gayther also produced a curve of the joining function J for their lattice. This is shown in Fig. II. 7. It is not always possible to extract a beam from a reactor to allow time-of-flight 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 available for this purpose [15]. The computed spectra may be compared directly with integral measurements by comparing the computed and measured reaction rates. ' Such direct comparisons have the advantage that they do not rely on the assumption of a well-thermalized spectrum used for conventional flux determination. If it is not possible to compute the flux spectrum, a reasonably good approximation may be obtained for well-thermalized systems from the measured conventional flux and spectrum parameters. The contribution of the slowing-down flux and the joining function to the conventional flux must first be subtracted from the measured conventional flux. The fraction of the conventional flux remaining is then the flux due to the Maxwellian part of the spectrum only. To convert this to a true Maxwellian flux at a temperature T, • it must be multiplied by the ratio of the average velocity in a Maxwellian to the'most probable velocity, i.e. 2/\Гж = 1. 128, and by the ratio of the average velocity of a Maxwellian at temperature T to that at temperature T0, i. e. s/T/T0. 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

corrected for the epithermal contribution; T0 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. Theoretical spectra

A full treatment of the theoretical methods of determining thè thermal neutron spectrum is beyond the scope of this book. However, some knowledge of the results of such calculations is useful in the understanding of flux density measurements and so the more common methods will be reviewed briefly. The full equation for the slowing down of neutrons can only be treated in very general terms and most theoretical-methods treat the steady-state space independent problem. In this case the neutron balance equation can be written as

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

where cra(E) is the absorption cross-section; CTS(E) is the scattering cross- section; ст(Е'-» E) is the cross-section for 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 assumptions,are made. If, for example, the absorption is 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 for moderation by hydrogen treated as a monatomic gas of unit mass. They assume that the scattering cross-section is constant and that the absorption cross-section • 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 for calculating the thermal spectrum in a light-water-moderated reactor. Although results obtained using this theory are adequate for 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 arise because at energies below 1 eV a scattering collision does not break the bonds of the water molecule and so the collision 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 for slowing down in light water, based on experimental measurements of the scattering of neutrons, which takes these energy states into account. Spectra calculated using the Nelkin model for a plutonium-fuelled system have been found to give results in good agreement with foil measurements and reasonable agreement with time-of-flight measurements [21]. For a crystalline moderator such as graphite or zirconium hydride the neutron collisions near thermal energies will take place with bound atoms rather than free 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 oscillators 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 for calculating 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 collision the probabilities of all the possible results of the collision are determined from the physical processes 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, sufficient data on the neutron density distribution can be built up to form 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 form

Tn =T0 (1 + Cmf) (II. IV) 24 CHAPTER III

for 0 < m сга/ст5< 0. 5, where :Tn is the neutron temperature; T0 is the moderator temperature; m is the mass of the moderating atom; aa is the . absorption-cross-section at the velocity corresponding:to T0; CTs is the . scattering cross-section; and С is a constant. The value of the constant С is not well established, but the values obtained lie 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 for moderators with masses between 1 and 25 to within 5%. • The time-of-flight measurements made by Coates and Gayther. for 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'effects 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 cell 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. Westcottflux distribution along the diagonal'of the lattice in a DzO reactor (Daphne). NEUTRON. SPECTRA 25

In a light-water reactor the thermal flux variations take place over much shorter distances than in either a heavy-water or graphite reactor and considerable flux peaking can occur in the water gaps between the fuel plates in this type of reactor. Epithermal neutrons are much less sensitive than thermal neutrons to changes in medium and, consequently, the epithermal flux distribution does not show much fine structure across the lattice cell. The epithermal flux density measured under the same conditions as/the thermal flux distribution shown in Fig. II. 8 is shown in Fig. II. 9. ; Figures II. 8 and II. 9 show that there is a< considerable variation in the neutron spectrum across the lattice cell due mainly to the variation in the thermal component of the flux. In fact, in this experiment the epithermal index (Westcott r) varied by almost a factor of 2 from 0. 085 to 0. 145, although the epithermal flux changed by less than 10%. The neutron temperature in a heterogenous reactor is also sensitive to the variations in the medium and increases markedly inside the fuel elements. Figure II. 10 shows the variation in the neutron temperature across the lattice cell in DAPHNE measured under the same conditions as the thermal flux measurements shown in Fig. II. 8. v , It is obvious from Figs.IL 8, Ц. 9 and II. 10 that in the neighbourhood of a neutron absorber the neutron spectrum changes rapidly and quite small errors in thé location of measuring foils can lead to significant errors in the results. This is especially true for measurements involving the use of cadmium boxes and in these measurements care must be taken to ensure that the presence of the cadmium box does not influence the final measured spectrum.

RADIAL . DISTANCE (CMS) .

FIG. II. 9. Epithermal flux-distribution along the diagonal df the lattice of а Е>гО reactor (Daphne).'- 26 CHAPTER III

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

II. 3. FAST NEUTRON REGION

As discussed in Chapter V, the experimental techniques for 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 from -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. From Fig. II. 5 it can be seen that the energy range 10 keV to 1 MeV is important for the analysis of processes such as radiation damage and in this energy range the neutron spectra differ markedly from 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- filled 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 rely, 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 for the calculation of the thermal neutron spectrum, but there are two very important differences in the problem. Firstly, 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 difference is in the source distribution used for the calculation. In the.fast neutron region the neutron spectrum changes considerably with the distance from 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 realistic 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 from disadvantages. Simple diffusion theory breaks down close to boundaries in a heterogeneous system and cannot be applied to voids. As the experimentalist is usually interested in the spectrum in a void, or near void region and as the assumption of a moderator filling the void alters the neutron spectrum appreciably, diffusion theory is only of limited use in heterogeneous reactor systems. The higher-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 basically statistical nature, a large amount of computing is required to obtain reliable results. Transport-theory also overcomes the objections to diffusion theory and spherical harmonics and is not subject to the statistical errors associated with Monte Carlo technique. This method, however, requires both a large amount of computing time and also a very large computer store to obtain results for a realistic geometry. Two other methods developed for calculating neutron spectra at a distance from the source of neutrons arethe moments method and the removal cross-section method. These two methods break down close to the source of neutrons but are suitable for the calculation of the neutron spectrum in shields. The Computing methods mentioned above are all well developed and give reliable spectra in situations for which they are applicable. However, they do rely 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 for different reactor types and positions. Genthon considers reactors with heavy-water, light-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[ï0(E) + hYe(E)] (11.18)

where (E) is the homogeneous part of the spectrum; ¥ (E) is the heterogeneous part of the spectrum; and К and h are fitted constants. ; The homogeneous part of the spectrum has the form

?„ (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 for heavy water, 1. 45 for light water and 0. 28 for graphite. F is a constant making ¥(E) continuous at E = E0. The simplest form 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 practice фг would be measured using a suitable threshold detector, such as 32S(n, p). Better fits to calculated spectra can be obtained by replacing the fission spectrum N0(E) by empiri- cally fitted functions and Genthon gives functions he has obtained for light- water, heavy-water and graphite reactors. This model gives reasonably» good representations of the neutron spectrum in the core of well-moderated reactors and slightly less good spectra in the reflector region close to the core. Because the functions which are fitted are essentially smooth functions, the model does not show up irregularities in spectra caused by resonances in the cross-sections or, for example, the peak which appears in graphite reactor spectra at approximately 4 MeV. This .is usually not important for the analysis of effects such as radiation damage and the model would be used for conditions which do not deviate too much from those used in the original fitting analysis. Two examples of the spectra obtained for a light-water reactor are shown in Fig. II. 11, i •...•:• At quite small distances from the fuel elements it becomes, possible to treat the neutron spectrum as a simple sum of contributions from individual fuel elements in the reactor. The neutron flux density as a function of energy and distance from a single fuel element in a moderating.medium may then be.calculated, using one of the calculation techniques previously described, and from.this the spectra at different points in the reactor can be constructed. Some typical spectra calculated for a single fuel element in D20 by the Monte Carlo method [29] are shown in Fig. II. 12 and a spectrum calculated from these results for 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 somewhat different, due probably to the fact that iñ the direct calculation the spectrum was calculated in a void, whereas for the lihe source calculation the spectra are calculated in a continuous moderator'region. 'The spectrum shapes are, however, closely similar. In this case where the closest fuel is 15 cm from 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 see in the next section, the filling of a void with moderator can change the spectrum shape significantly. This type of spectrum calculation should only be used when there can be no local dis- tortion of the neutron spectrum by materials close to region of interest, but under suitable conditions it provides a good way of obtaining neutron spectra in reflectors and shields with a minimum of time spent on the initial computer calculation.

lOOeV 10 ЮО I MeV 2 NEUTRON ENERGY a) CORE CENTRE

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. [26]). 30 CHAPTER III

О <о: x Шi— 2

Z ) IK Ш CL X => I U-

»l -2-10 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 fission neutrons will be greatest in or near the fuel elements in a heterogeneous reactor and will fall rapidly as the distance from the fuel element is increased. The flux at a few keV, however, remains more or less constant across the lattice cell of the reactor and changes only as the average power density across the reactor changes. Thus the neutron spectrum above a few keV changes from 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 follow the flux distribution at 5 eV quite closely and a typical example of this variation is shown in Fig. II. 9. The variation of.the fast neutron component as measured by the 31P(n, p)31Si reaction for the same lattice is shown in Fig. II. 14. If one looks more closely at the energy spectrum above 2 MeV, by comparing the reaction rates of several threshold detectors, it is found that for positions further from the fuel the spectrum deviates more from that of a fission spectrum. The neutron spectra across the lattice cell, 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 errors, but these spectra show the change in spectrum across the lattice quite clearly. The reaction rate of 31P(n, p) has been calculated for 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)

JL 1 Jll

-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 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 LJ - X n_l r .1-A.I V

-I NEUTRON ENERGY(MeV)

(a> NEUTRON. SPECTRA 33

r-l 1 pi Г1 "1 " r1" I L J -I Д L 1

NEUTRON ENERGY (MeV) (b)

L IT fl 4. 4

~LJ 1 \ I.

NEUTRON ENERGY(MeV)

(c)

FIG. 11.15. Variation of the neutron spectrum across a heavy-water reactor lattice: (a) spectrum in centre void; (b) spectrum at the outer edge of the fuel, 4.5 cm from the centre; (c) spectrum in the moderator, 7. 3 cm from the centre. 34 CHAPTER III

SPECTRUM IN AVOID SPECTRUM IN HEAVY WATER г I- тuГ j 1 i J IJ j; : 1 r- 1 Í Рч 1

/ •-J tr 1 ч

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

гЧГ 0 J1 1 500 Г

5 зоо

Ml ' I 5 I 0 1 NEUTRON ENERGY (MeV) (с) FIG. II. 17. The effect of 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 experimental hole; (c) typical graphite and aluminium rig in experimental hole. - ' •».' 36 CHAPTER III

A heavy-water reactor with a six-inch lattice pitch has been used to illustrate the variation of the neutron spectrum across the lattice, but any other heterogeneous reactor will give similar variations. In a light-water reactor, which approximates much more closely to a homogeneous system, the fast spectrum variation across the core will be small. However, at places of experimental interest, such as the core reflector boundary or experimental positions in the core 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 close to a fuel region. Many experimental irradiations are performed in voids or nearly void regions and so this effect can be important in the analysis of experimental results. Figure II. 16 shows the effect of substituting heavy water for 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 smaller 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 filled 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 from the fuel this effect becomes even less important. The presence of experimental equipment can also affect the fast neutron spectrum. Many experimental rigs consist largely of aluminium because of its low thermal neutron absorption cross-section. 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. Figure II. 17 also shows the effect of a typical experimental rig on the neutron spectrum in a graphite reactor. The effect on the neutron spectrum of materials with high inelastic scattering cross- sections, such as iron, will be much more 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 differences in measurements of properties such as radiation damage under different experimental conditions. Even changes such as the substitution of a graphite sample carrier for an aluminium carrier 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

All 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 practice 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 for this technique and even these are limited in usefulness by uncertainties in differential cross- section curves. In principle it is possible to obtain a neutron spectrum directly from threshold detector measurements for energies above 1 MeV and several methods of analysing these results have been developed (Chapter V). Unfortunately, the errors in the experimental measurements combined with the uncertainties in the cross-sections often result in unstable solutions and even in the best cases spectra obtained directly from 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 form 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. [28]).

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

Reaction о Eeff (barn) , (MeV)

232Th(n, f) . 0.14 1.40

"8U(n, f) о. 6o: 1. 55

31P(n,p) 0.140 • 2.9

32S(n,p) •:' .0.350 1 ' 3.2 '

24Mg(n,p) ; 0.060 6.3

œFe(n,p) - i 0.110 7.5

21Al(n, a) 1 ,o.ii3 С 8.1 38 CHAPTER III

Ен Й H S jw H J и ¡э fe

О J J о к <

й-.

tí M

тэ тз <и •о О ев ч) сз 3 3 О <3 о *«5 «U и 2 и tí я ь 4 Й

3 w NEUTRON. SPECTRA 39 this figure the curves show the calculated distribution of the integrated flux above an energy E for a light-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 cross-sections used in this comparison are listed in Table II. II. The uncertainties in the absolute value of the threshold cross-sections can be eliminated from these comparisons by considering the ratios of reaction rates for several detectors in two different reactor spectra. This has been done for 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. errors quoted for the calculated ratios are those due to the statistics of the calculation only. The calculated spectra for 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 more than would be expected from the errors. This could be due to a difference between the calculated source distribution and the actual source distribution in the reactor in the two cases, leading to an error in the spectrum intensity. To eliminate this effect the ratios are also shown normalized to unity for the thorium fission 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 for two markedly different reactor environments with quite different neutron spectra. This demonstrates the limitations of threshold detectors for 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 from the fission spectrum in this range, but it ' gives no information in the region below 1 MeV where the main differences in fast neutron spectra occur. Proton recoil techniques using photographic plates have been used for neutron energy spectra for 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 loss of low-energy proton tracks. In the last few years the proton recoil technique has been extended to lower energies using gas-filled 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. Figure II. 20 shows a spectrum obtained by Benjamin et al. [23] for a fast reactor assembly using a combination of gas-filled counters and photographic plates. The measured spectra are compared with a 32-group Carlson transport theory calculation. Another technique which has shown promise in recent years is the semiconductor neutron spectrometer based on the measurement of the reaction products of the ?Li(n, a)3He reaction [30] or the 3He(n, p)3H reaction [31]. These counters suffer from severe interference from thermal and epithermal neutrons in a thermal reactor, but nevertheless'have been used to measure high-energy neutron spectra [31]. They also show promise of an extension of their range down to approximately 50 keV [32]. A spectrum 40 CHAPTER III obtained by Silk [24] in the moderator of a graphite reactor is shown compared with a Monte Carlo calculation in Fig. II. 21. Although techniques for measuring the fast neutron spectrum in a thermal reactor haye developed considerably in the past few years, they have not yet reached the stage where satisfactory neutron spectra can be measured in reactors and one still has to rely, to a large extent, on theoretical spectra. The experimental results which are available agree reasonably well with the calculations, although some disagreement can be observed particularly with threshold activation detectors. These discrepancies are not in general very large and the calculations are adequate for most purposes, at least until more reliable experimental data is available.

о 0 < 4 MONTE CARLO CALCULATIONS

1 3

EXPERIMENTAL RESULT

0-2 OA 0-6 I 2 4 NEUTRON ENERGY (MCV) FIG, II. 19. Comparison of neutron spectra measured with photoplates and a Monte Carlo calculation, for a hollow fuel element in a heavy-water reactor (Ref. [29]).

700 - Ф 4 ATMOSPHERE GAS COUNTER I 600b 4.Я J I ATMOSPHERE GAS COUNTER i LF-, PHOTOPLATES L--| THEORETICAL SPECTRUM 5 500- (32 GROUP) <ОС X • Ш 400 • • t § ЭОО • к pXXJSJ й ti 200 S 11. loo • 1.. ..I- 1 . . ..1 L A •xL L 002 OOS 01 0-2 O S I O 20 S O IO O .10-2 Ю-5 ENERGY, M«V

FIG. II. 20. Experimental and theoretical spectra in a 'Vera' assembly. NEUTRON. SPECTRA 41

— CALCULATED SPECTRUM MEASURED SPECTRUM

H \

5 I I 0 NEUTRON ENERGY IMeV)

FIG. 11.21. Comparison of a spectrum measured with semi-conductor neutron spectrometers and a Monte Carlo calculation in the moderator of a natural-uranium, graphite-moderated reactor.

II. 3. 4. Comparison of theoretical spectra

All the available methods of calculating neutron spectra have limitations which make them difficult or impossible to apply in some circumstances1 and this means that no one method of calculation can be recommended as a universal method. Situations can be found where two or more of the methods of calculation are applicable and this allows one to compare results of the different methods. This has been done for several of the methods discussed here and it has been found that the results obtained by the different methods compare very well. A typical example is shown in Fig. II. 22, in which a P3MG spherical harmonic calculation is compared with a Monte Carlo calculation for a hollow, fuel element filled with heavy water in the heavy- water reactor PLUTO. There is, therefore, no reason for believing that any of the highly developed theoretical techniques is to be preferred, provided that the theory is applicable to the system under consideration. Here it mugt be remembered that the theory on which a calculation is based may break down for some cases, of practical interest. Most calculations of the neutron spectrum for practical reactor environments involve some approximations, especially in the geometry of the system. These can have a noticeable effect on the calculated spectrum and on any reaction rate integrals based on it. The consequences of any approximations made must, therefore, be taken into consideration whenever theoretical spectra are used in the analysis of experiments. However, the agreement between the different theoretical techniques and between theory 42 CHAPTER III

< X 250 if

5 1 5 1 0 I NEUTRON ENERGY (MeV)

FIG. 11.22. Monte Carlo and PI calculations of the spectrum inside a hollow fuel element in a heavy-water reactor, filled with heavy water.

and the experimental results available is sufficient to give reasonable con- fidence in the theoretical spectra. The techniques available to date can be used for almost all reactor situations and give sufficiently good neutron spectra for many experimental applications.

II. 3. 5. The effect of the neutron spectrum on experimental measurements

In the preceding sections it has been shown that the neutron spectrum varies considerably from reactor to reactor and can vary significantly for the same position in a reactor if the conditions are changed, for instance, by the introduction of cooling water. These changes are not well monitored by the current methods of neutron fluence measurements involving threshold detectors. Fluence measurements based on reactions such as 54Fe(n,p)54Mn, although adequate, provided all the irradiations are performed under identical reactor conditions, are not sufficient if results of different irradiation experiments are to be compared or if results are to be extrapolated to other reactor positions. Simmons [33], for instance, found that there could be a difference of as much as a factor of two in the relationship between graphite irradiation damage and the reaction rate of the 5SNi(n, p)58Co monitors for different reactor positions. Similar results have been obtained for radiation damage in many other materials. In this type of experiment it is essential, therefore, that some estimation of the neutron spectrum is used if the results are to be properly analysed. The differences in neutron NEUTRON. SPECTRA 43 spectra encountered in work of this kind are usually quite large and theoretical spectra are sufficiently accurate for the analysis. Experimental results on radiation damage in materials such as graphite and steel have been treated by this method with satisfactory results [7, 34, 35].

REFERENCES TO CHAPTER II

[1] WESTCOTT, C.H., The Specification of Neutron Flux and Effective Cross-Section in Reactor Calcu- ' lations, Rep. RPI-11, (1 Nov. 1955). [2] WATT, B. E. , Energy spectrum of neutrons from thermal fission of 235U, Phys. Rev. 87 (1952) 1037. [3] ZUP, W. L., Review of Activation Methods for the Determination-of Fast Neutron Spectra, Rep. RCN-37 (1965). [4] MURRAY, R. L., Nuclear Reactor Physics, Prentice Hall, Inc. (1957). [5] KINCHIN, C.H., PEASE, R. S., Reports on Progress in Physics, 18 (1955) 1. [6] SNYDER, W. S., NEUFELD, J. S,, Vacancies and displacements in a solid resulting from heavy corpuscu- lar radiation, Phys. Rev. 103 (1956) 862. [7] THOMPSON, M. W., WRIGHT, S. B,, New damage function for predicting the effect of reactor irradiations of graphite in different neutron spectra, J. nucl. Mater. 16 (1965) 146. [8] BEELER, J.R., Primary damage state in neutron irradiated iron, J. appl. Phys. 34 (1963) 2873. [9] Coltman, R. R., "Reactor irradiation studies at 4°K", Radiation Damage in Solids (Proc. Symp. Venice, 1962) 2, IAEA, Vienna (1962) 205. [10] HARRIES, D.R. et al., "Irradiation behaviour of steel as a structural and cladding material", Int. Conf. peaceful Uses atom. Energy (Proc. Conf. Geneva, 1964) 9, UN, New York (1964) 232. [11] HUGHES, D.J., Pile Neutron Research, Addison Wesley (1953). [12] WESTCOTT, С. H., WALKER, W. H., ALEXANDER, Т.К., "Effective cross sections and cadmium ratios for the neutron spectra of thermal reactors". Int. Conf. peaceful Uses atom. Energy (Proc. Conf. Geneva, 1958) 16, UN, New York (1958) 70. [13] DELATTRE, P., PROSDOCIMI, A., "L'activité du Groupe de travail «Dosimétrie» d'EURATOM", Neutron Dosimetry (Proc. Symp. Harwell, 1962) 2, IAEA, Vienna (1963) 11. [14] WESTCOTT, С.H., Effective Cross-Section Values for Well-Moderated Thermal Reactor Spectra, Rep. CRRP-960 (1 Sep. 1960). [15] Reactor Physics Constants, Rep. ANL-5800 2nd ed. (July 1963). [16] COATES, M.S., GAYTHER, D. B., Time of Flight Measurements of Neutron Spectra in a Graphite Uranium Lattice at Different Temperatures, Rep. AERE-R-3829. [17] AMSTER, H.J., The Wigner-Wilkins calculated thermal neutron spectra compared with measurements in a water moderator, J. nucl. Sci. Engng 2 (1957) 394. [ 18] WIGNER, E. P., WILKINS, J. E., Effects of the Temperature of the Moderator on the Velocity Distri - bution of Neutrons with Numerical Calculations for H as the Moderator, Rep. AECD-2275 (14 Sep. 1944). [19] AMSTER, H., SUAREZ, R., The Calculation of Thermal Constants Averaged Over a Wigner-Wilkins Flux Spectrum Description of the SOFOCATE Code, Rep. WAPD-TM-39 (Jan. 1957). [20] NELKIN, M., Scattering of slow neutrons by water, Phys. Rev. 119 (1960) 741. [21] COATES, M. S., DAY, D. H., POOLE, M.J., SMITH, J. С., TRIP, D.J., Measurements of Neutron Spectra in Bare Aqueous Plutonium Reactors, Rep. AERE-R-4668 (Aug. 1964). [22] COVEY OU, R.R., BATE, R.R., OSBORN, R. K., Effect of moderator temperature upon neutron flux in infinite, capturing medium, J. nucl. Energy 2 3 (1956) 153. [23] BENJAMIN, P.W., KEMSHALL, C. D., REDFEARN, J., The Use of a Gas-Filled Spherical Proportional Counter for Neutron Spectrum Measurements in a Zero Energy Fast Reactor, Rep. AWRE NR2/64 (July 1964). [24] SILK, M. G. , Attempts to Determine the Fast Neutron Spectrum in a Thermal Reactor by Means of 6Li and 3He Semiconductor Spectrometers, Rep. AERE-M-1590 (May 1965). [25] WINDSOR, Margaret E., WRIGHT, S. B., Measurement of the fast neutron spectrum in a fast reactor by a 6Li semiconductor spectrometer, J. nucl. Energy 20 (1966) 465. [26] GENTHON, J., Formulation des Répartitions Spectrales Energétiques de Flux Neutroniques en Pile, Rep.CEA-R-2403 (1964). [27] WRIGHT, S. В., Calculation of High Energy Neutron Spectra in Heterogeneous Reactor Systems, Rep, AERE-R-4080. [28] KOHLER, W., Die Bestimmung der Energetischen Verteilung desschnellen Neutronenflusses in Reaktoren, Atomkernenergie 9 (1964) 81. 44 CHAPTER III

[29] GORE, J.E., The Fast Neutron Energy Spectrum Inside a Mark III.Fuel Element in the DAPHNE Reactor - A Nuclear Emulsion Determination, RD/B/N387. [30] LOVE, T.A., MURRAY, R. B., MANNING, J.J., TODD, H.A., "A silicon surface-barrier fast-neutrón spectrometer", Nuclear Electronics (Proc.Conf. Belgrade, 1961) IAEA, Vienna (1962) 415. [31] LEE, M.E., AWCOCK, M.L., "A helium-3 filled semiconductor counter for the measurement of fast neutron spectra", Neutron Dosimetry (Proc. Symp. Harwell, 1962) 1^, IAEA, Vienna (1963) 441. [32] HUBER, R.J., In core experiments with a lithium-6 'sandwich' fast neutron spectrometer, Trans. Am. nucl. Soc. 7 (1964) 368. • [33] BELL, J.C. et al., Stored energy in the graphite of power producing reactors, Phil. Trans. R. Soc. London Al043 (1962) 254. [34] HARRIES, D.R., BARTON, P.J., WRIGHT, S. B., Effects of neutron spectrum and dose rate on radiation hardening and embrittlement in steels, J. Brit. nucl. Energy Soc. (Oct. 1963) 398. [35] SHURE, K., Neutron Spectral Effects on Interpretation of Radiation Damage Exposure, Rep..WAPD-T-1622 (Rev. ) (Sep. 1963). CHAPTER III

THERMAL NEUTRONS

III. 1. THEORY OF DETECTOR RESPONSE

III. 1. 1. General method

Spectrum characteristics

In an infinite diffusion non-absorbing medium all neutrons, whatever their energy, finally reach a state of thermal equilibrium with the atoms of the medium. The kinetic theory of gases leads to the supposition that in this ideal state the velocity distribution obeys the Maxwell-Boltzmann laws and the neutron density per unit interval of velocity can be written

n(v) = nth4T (2Ír)3/2v exp (-mv2/2kT) (HI. 1) oo where nth =^n(v)dv is the total neutron density; m is the neutron mass; a к is the Boltzmann constant; and T is the absolute temperature of the medium. Equation (HI. 1) is equivalent to Eq. (П. 1) when n is normalized to unity. Actually, in a reactor system the medium is always a finite, absorbing medium.. Some of .the neutrons escape or are captured before they are thermalized. In addition, the capture cross-sections, which often have a 1/v dependence, result in a selective weakening of the neutron spectrum at low. energy. This circumstance has two consequences: (a) the neutron spectrum can only be regarded as approximately Maxwellian; and (b) the '•temperature' of this pseudo-Maxwellian spectrum may greatly exceed the temperature of the medium. Nevertheless, the thermal neutron flux can be defined as

Ф = J n (v)v dv (HI. 2) о and in cases where the distribution is approximately Maxwellian and, as pointed out in Chapter H, one has: most probable velocity

vq = (2kT/m)* = 2200 m/sec

45 46 CHAPTER III at 20. 44° C; with a corresponding energy [maximum of the distribution cp(v) = n(v)v]

ET = kT = 0. 0253 eV mean velocity

v v 128 v t = ¿ o = o

Detector response - pure thermal spectrum

The reaction rate R of a material generally conforms to the expression:

R = К J n(v) a (-v)v dv (III. 3)

The activation reaction rate of a 1/v detector irradiated in a thermal neutron flux can thus be written:

R = К J n(v) ~ vdv = KK' Jn(v)dv = K" n(h (III. 4) о о

The activity of the detector is, therefore, proportional to the total thermal neutron density, not to the thermal neutron flux density, and so is independent of the spectrum and of the velocity or energy distribution of the therm ail neutrons. However, the flux density cannot be determined by measuring the activity of 1/v detectors. An additional measurement of the spectral distribution of the neutrons is necessary (e. g. measurement of Maxwell' s parameter T). This information, together with the previous measurement, defines the integral and differential values of the thermal neutron flux. One, therefore, adopts-a conventional flux defined by the expression cp0 = nthv0, namely the product of the total thermal neutron density and the velocity v0 of 2200 m/sec, which is the most probable velocity for a Maxwellian distribution at the ordinary neutron temperature (293. 6°K). Although this concept of ' conventional flux' has no physical significance, it can be used directly for calculating the reaction rates of materials whose cross-section varies as 1/v in the thermal energy range. The cross- section that should be used is that for a neutron velocity of 2200 m/sec, ст0:

Rth(l/v) = cp0a0 (III. 5)

This restrictive use of the conventional thermal flux is correct for any spectral distribution of thermal neutrons. If the thermal neutrons are distributed according to Maxwell' s law, the integral value of the thermal neutron flux, cpth, after determiningT, can be derived from the value of the conventional flux, cp0, usingthe expression:

(Ш.6)

Detector response - real case

A reactor spectrum is composed of slowed-down neutrons and neutrons in thermal equilibrium with the medium. This spectrum is regarded as correctly, described by the superposition of the following two components: (1) A component characteristic of the thermal energy range (2) A component characteristic of the epithermal energy range. The activity induced by the epithermal neutrons may represent a considerable share of the total activity of the detector. Discrimination between the two ranges is possible using cadmium covers. The reaction rate can thus be expressed as the sum of two ternis representing the neutron contributions of the two energy ranges separated by the lower energy threshold of the resonance integrals; this threshold is related to the special property of cadmium to act as an almost perfect high-pass filter. The concept of ' cadmium ratio' , R^, is defined as the ratio of the activity of a bare detector, i. e. a detector without cover, to the activity of the same detector (or an identical detector) irradiated under a cadmium cover. The bare, irradiated detector is exposed to neutrons of all energies.

Consider the case where the total activity is represented by Atot. The detector irradiated under a cadmium cover is only subject to reactions with the so-called epicadmium neutrons, which have not been absorbed in the cover. If the activity of this detector is A ¡, one can write:

(III. 7)

The difference between the activities Atot and Aepi corresponds to that part of the activity of the bare detector which is induced by the neutrons that are captured in the cadmium in the case of irradiation under a cover:

where (Ш. 8)

These neutrons are regarded as having an energy lower than the effective cut-off of the cadmium coyer, Eçd. The energy band between 0 and includes practically all the thermal neutrons and a small variable contribution of epithermal neutrons. In the case of detectors with a 1/v activation cross-section extending beyond the effective cadmium cut-off, the difference between the activities of a bare irradiated detector and a detector irradiated with cadmium cover is proportional to the total density of the neutrons n(0,Ecd) whose energy is less than the effective cadmium cut-off Cd

Effective cadmium cut-off

The ' effective cadmium cut-off' is the energy associated with a perfect filter (infinite absorption below the cut-off energy; zero absorption above it) under which an irradiated material would have the same reaction rate as under a cadmium cover. 48 CHAPTER III

In principle, this energy depends on: (a) the shape and dimensions of the cadmium box; (b) the angular and energy distribution of the neutrons; and (c) the absorption characteristics and geometry of the detector. In the case of a 1/v detector in most reactor spectra the relative strength of the epithermal neutrons has no appreciable effect on the value of the effective cut-off; moreover, by choosing a sufficient cadmium cover thickness the value can be made almost independent of the temperature of the Maxwellian component of the spectrum. The effective cadmium cut-off energy values, EClj, for different measuring conditions are presented in Chapter IV. The values are in the energy interval 0. 5-0. 7 eV. Summing up, the conventional concept of thermal neutron flux described above must be modified to accommodate the concept of effective cadmium cut-off. The definition adopted states that the thermal flux is the product of the total density of neutrons in the energy band 0-Ecd and the neutron velocity 2200 m/sec and may be expressed as the following equation:

cp0 = n(0, Ecd)v0' (III. 9)

III. 1.2. Westcott's notation

Characteristics of the spectrum

The neutron spectrum is regarded as the superposition of a Maxwellian spectrum of temperature T and a 1/E spectrum terminated at lower energies by a cut-off function Д. The simplest Д function in normal use is the step function in which

Д = 1 for e>aj KT

Д = 0 for E< ц KT where ц varies with the type of reactor in question (for D^O reactors ц = 5). It is clear that this type of step-function relation introduces a discontinuity at energy ц KT (slight in view of the. fact that the Maxwellian component at this energy is already much greater than-'the 1/E component) (see Fig. Ш. 1). Westcott [ 1 ], therefore, proposed various Д functions centred about ц KT varying continuously from 0 to 1, with or without a peak.

Limitations of the method

For the Maxwell spectrum hypothesis to be valid the reactor must be well moderated. If, as a result of neutron captures in the fuel, the quantity £a/?£s becomes too great and if the neutron temperature Tn becomes too high in relation to the temperature of the moderator TM< it is necessary to calculate the true spectrum. As a result of these limitations Westcott established the following reasonable limits:

Ea/fEs< 0.1 T„/TM< 1.07 (III. 10) THERMAL NEUTRONS 49

ÍP'(E) MAXWELL SPECTRUM

<Г (E) 176LU

NEUTRON ENERGY (eV)

FIG. III. 1. Spectra and typical 1/v and non-l/v cross-sections in the thermal neutron region.

Detector response

The effective cross-section <7 is defined as the cross-section of a 1/v detector which, in the spectrum in question, would give the same reaction rate as the detector actually employed. The reaction rate R may therefore be written as

R = nv05 (Ш. 11)

where 30 ¡I a (v) n (v) v dv (III. 12)

v0/n(v) dv 0 50 CHAPTER III and OO i n = J^n (v) dv

It can be shown that one may Write:

1 f Y- -4 exp [ - (v/vT)2] a (v)dv ' (III. 14) VqCTO J V7T Vf 0

( 4Ty s — о-(v) -^gffo Д(Е)^ (111.15) . Сто ЧтТЬ/ о

The calculation,., especially for the parameter s, is only valid for thin detectors since it neglects self-shielding effects. The value of r is determined by measuring the cadmium ratios R^ and,,is also more valid for thin detectors. • . •• , .. .1, • •. . In the calculation of detector responses several different conditions maybe considered: <•••••-.••'

(a) For the case of a perfect cadmium filter with a cut-off energy Ecd and a 1/v detector it can easily be shown that

WT/T0 = —^-j- ч/тгЕса/кТо, (Ш. 16)

(b) For the case of a perfect cadmium filter and,a.detector response of 1/v at energies below Ecd but with resonance structure above,, energy Ecd the following relationship holds:

• - Rcd[s0-4b07^]-s0 (Ш- 17)

(c) For the case of a real cadmium cut-off where the cadmium response is calculated from its true cross-section, a similar expression may be obtained by replacing is/7rEcd/kT0 by the coefficients К calculated in Ref. [ 2] for different cadmium thicknesses. THERMAL NEUTRONS 51

The main conclusion that can be drawn from these relationships is that measurement of the cadmium ratio gives r vT/T0 and not simply the parameter r. This subject will be further discussed in section III. 3.

III. 1. 3. Formalism of Horowitz and Tretiakoff [3]

Using the reduced velocity variable x = v/vT (where vT = -ДкТ is the most probable velocity of a neutron in thermal equilibrium with the medium), the neutron density is given by the relation:

n(x) = M(x) + 2rHE(x) (III. 18)

By convention, the following normalizations are adopted:

J n(x)dx = 1;J M(x)dx = 1; J E(x)dx ï 0 (III. 19) .о о о . where M(x) is the Maxwellian spectrum corresponding to ambient temperature T; and E(x) is a function behaving like 1/v2 at infinity and dépends on the thermalization model adopted. The coefficient rH is an index character- izing the relative strength of the epithermal neutron component. For an infinite homogeneous medium one has:

rH n/T0/T [£a/fi; ] :

Westcott' s formalism can be retained for defining the effective cross- sections: . , .

(7 (x) M (x)xdx

S = 5 = ч/Т/То J и (x) n (x)xdx (III. 20)

0 xчQ / n(x)dx 0 . _

The ratio S = ст/a can be used to indicate the deviation from the 1/v law; for a 1/v absorber one has â = a0 and S'= 1. • ..-•.' Tables listing the ratio S have been calculated [3] for different thermalization media, in particular the graphite model and the heavy-gas model. Other studies are under way to determine the characteristics of a model that can be used for heavy-water and light-water moderators. The cross-sections can be written in the form

ô = °o(gH+rHSH> with

gH = Л7т^ /^М(х)х dx 0 52 CHAPTER III and CO

SH = f ^ E(x)2x dx d CO 0

If the same temperature is used, g has the same value as in Westcott' s formalism; the same applies to S, to within a factor \lя/4, when the only resonances are at high energy. For the factor r one has the relation:

rH = rw s/T/7

This new formalism has the advantage that it uses a spectrum which takes into account the properties of the moderator, i. e. an almost true spectrum. It does not involve the cut-off energy, the form of cut-off or the neutron temperature, which cannot be very accurately defined. In addition, this method also applies to a heterogeneous medium, so it can be directly used in reactor calculations. For media in which absorption is not slight, i. e. for which the parameter r is greater than 0. 1, the function E(x) and hence SH are not independent of r. The factor S is then obtained from tabular values but tables giving SH can also be calculated and the effective cross-section determined.

Ш. 2. MEASUREMENT OF THERMAL NEUTRON FLUX DENSITY AND FLUENCE

III. 2.1. Relation between flux density, fluence and detector activity

General formulae

It has been shown that the reaction rate of a detector irradiated for a time t¡ and counted at the end of a post-irradiation period td is

A = N ф a f(t¡, id) (Ш. 22) where A is the absolute disintegration rate of the detector; N is the number of parent nuclei; a is the cross-section in barns; ф is the mean flux density in the detector; t¡ is the irradiation time; ta is the decay time after irradiation; and

f(ti, td) = ( 1 - exp(-Xtj)] exp(-Xtd) for constant flux density during irradiation. The quantity required, in general, is not

(III. 23) 0 but

(Ш. 24) о the flux density of thermal neutrons, which is not perturbed by the detector. THERMAL NEUTRONS 53

• Two correction coefficients are therefore introduced, Fcd as given in Eq. (Ш. 8) and Fj given as:

Fi = (III. 25) Фо

For neutrons of energy less than Ecd the general formula (III. 22) then becomes

AFcd = Ncffe a Fj f(t¡,td) (Ш. 26) and the flux density will be given using the conventional flux defined in section III. 1.1

'o = = (IIL 27)

It should be noted that this formula is only valid if the number of initial nuclei can be regarded as constant and if the flux density remains constant during irradiation.

If the product Xt; is very much smaller than 1, formula (III. 21) can be written . .

A'= N a 9mXti exp(-Xtd) (III. 28) and formula (III. 26) can be written

Ф0 tj = Ф0 = ñ á^P ГТТ-\ (HI. 29) 1 " N a0 Fj X exp(-Xtd) v ' which gives the relation between fluence and activity. . In practice, in the example with cobalt, one shall use this formula with the following three provisos: (a) For irradiation periods longer than two months a correction of more than 1% has to be made for replacing [ l-expi-Xtj)] by Xt¡ (b) For fluences greater than 3 X 1020 n- cm-2 depletion of the 59Co nuclei needs a correction of more than 1% (c) For flux densities'greater than 1014 n- cm"2- sec-1 and times greater than two months a correction also has to be introduced for burh-up of 60Co nuclei which are captured with a cross-section of six barns.

Determination of the coefficient Fcd

It is shown that 'the coefficient Fcd is obtained by measurement of the cadmium ratio Rcd;. This measurement can be made in two ways, the choice again depending on practical considerations. The bare detectors or detectors under a cadmium cover will be irradiated either (a) successively at the same point, or (b) simultaneously at different points. (a) Detectors successively at the same point: During each of the two irradiations the reactor is brought to a power level which is accurately monitored using an appropriate detector (Au, Cu, Co, etc.); by monitoring it is possible to ensure that the activities of the two irradiated detectors - bare and under cadmium cover - correspond to the same irradiation conditions. 54 CHAP TER.Ill

(b) Detectors .simultaneously at different points:. In carrying out simultaneous irradiations great care should be taken to ensure that the flux perturbation (principally thermal flux depression) caused by the cadmium cover does not affect the response of the-bare irradiated detector. A minimum distance has to be adopted depending on the nuclear characteristics of the irradiation medium, especially the moderator, and the dimensions of the covers. The minimum distances in centimetres necessary between the irradiation point of a detector in one of the three types of cadmium box described in this manual and that of a bare detector to ensure that the thermal flux depression at right-angles with the bare, detector does not exceed .0. 5% are given in Table III. I for the four commonly used moderating media. Obviously, in practice the case of irradiation in a pure moderating medium will seldom arise. However, in all the other cases it can be assumed that the presence of absorbers tends to reduce the minimum distances required. , • . ,

TABLE III. I. BARE TO CADMIUM COVERED DETECTOR SEPARATION DISTANCES3 ' ,..••-

. Type of cadmium box Moderator material

• ' H20 DjO С Be

14 mm diam. for 10 mm foil 7.5 22 17 16

,7 mm diam. for 3 mm foil 5.1 . 6.8 5.8 6.3

For wire 5.1 6.8 5.8 6.3

3 The mimimum distances (in cm) between the irradiation point of a detector in a cadmium box and that of a bare detector to ensure that the thermal flux depression normal to the bare detector does not exceed 0.37».

Flux perturbation factor. Fx [4] • -.

The introduction of an.absorbing detector or sample into a thermal neutron flux gives rise to a self-shielding of the detector, and .a flux depression in the medium, which surrounds it. If one designates the mean flux in the sample

The factor Fx can be divided into two designated G and H (see Fig. III. 2): , (a) The coefficient of self-shielding G of the sample placed in the. medium. It is the ratio of:the. mean flux, in the detector ф to the flux at the surface ф5 :

G = ф/фэ (III. 80)

G can be calculated, assuming an isotropic mono-energetic flux and using the,theory of collision probabilities [5]. . It is the probability that neutrons entering, the .sample will not be captured in it. This term can be measured if the detectors are very small or in a medium with.a large mean free p^th. In this case it can be assumed that H = 1; THERMAL NEUTRONS 55

(b) The coefficient of flux: depression H, which depends on the external medium and can be defined as the ratio of the fluxes at the surface of the detector placed in the medium cps and the flux prior to the insertion of the detector, cp0:

H = cps/cp0 (Ш.31)

The factor Fi" is thus the product of the two terms:

; F1 = ф/фр = (ф/ф5 ) n,(9s /Фо) = GH (III. 32)

Thermal neutron self-shielding coefficient G

To establish the general formula for self-shielding one considers only the case of an isotropic flux of mono-energetic neutrons, which is in fact true for most of the measurements made in a thermal spectrum. The self-absorption factor G of a sample is thë probability that the neutrons entering the sample will not be captured in it. In the case of a pure absorbing sample, each collision is a capture and this probability is . P0 = 1 - P, . In the general case, where the sample is also diffusing, one has

^ i-i^VAvr.) - <ш-зз>

Pc is the collision probability as tabulated by Placzek [6]; Ec is the macroscopic capture cross-section; and £t is the macroscopic total cross-section. The collision probability has been-calculated and tabulated by Placzek for plates,' cylinders arid spheres;1 - Here, the study will be'limited to plates and cylinders. 56 CHAPTER III

ía! JPLaies In the case of plates, one has

1-PC =¿ [1 -2E3(x)] (III. 34)

where x = Et a; Et is the macroscopic total cross-section of the sample; and a = 2V/S, the mean chord of the sample, V and S denoting respectively the volume and surface of the sample. It should be noted that the mean chord tends to the value of the sample thickness when this is much less than the other dimensions.

En(x) = J e"xu u~ndu 1 the function tabulated by Placzek [ 6]. The table of values of Pc is given not as a function of sample thickness but as a function of x. It is useful to be able to calculate Pc, both for intermediate values of x and so as to obtain the exact value corresponding to a given thickness, especially for the E3 table, which is not complete enough; use is made of the following series expansions:

E3(x) = i [ exp(-x) (l-x)-fx2 Ej(x) ] (EI. 35)

exp(-x) = (l.x) + f(l-|)+^(l-|)+--

x + + Ei(x) = - Y if - lnx - y

For x « 1 Eq. (III. 34) becomes

1 - Pc = 1 - fx - ^ + + § (In x +7) (III. 36) with 7 = 0. 577216.

(bj_ _Cylinders_ In the case of cylinders we have

1 - Pc = fx {2x[ k^ + K0I0] - 2 + - KQIl + KlI0} (Ш. 37)

In, Kn are Bessel functions for the value x. The mean chord a = 2V/S is equal to the radius r of the cylinder when the radius is much less than the length of the sample. THERMAL NEUTRONS 57

As in the case of plates, it is of interest to be able to calculate Pc for a given radius. Using the Bessel function expansions, for values of x much less than unity one obtains

1-PC =l=|x-(ln|+7-|)| (III. 38) or with a more complete expansion

1 p. - 1 - f* - f С1" I + ï - !) - f (b 1 + T - D

Using the formula a = 2V/S to define the mean chord, it is possible, in a first approximation, to take account of edge effects in the self-absorption factor; these effects may be considerable when the detectors are of small diameter.

Flux depression coefficient H

In the medium surrounding an absorbing sample there is a flux depression and it is customary to regard the sample as a negative source of neutrons whose intensity is determined by the total number of neutrons absorbed per unit area per second. In a first approximation the diffusion theory can be used to give the flux distribution in the neighbourhood of the negative source. When the dimensions of the detector are comparable to the diffusion or transport lengths of the medium, diffusion theory cannot be applied and more refined methods, such as the transport theory or a Monte Carlo method, must be considered. The only case that has been studied theoretically and experimentally (by Ritchie and Eldridge) [7] was related to plane circular detectors of small diffusion Xs . These authors state that the flux depression can be written in the form

H = (III. 40) 1 + g£caG 1 +

The g factor depends on the calculation methods and is defined differently by different authors. It depends on the dimensions of the detector and the neutron properties of the medium^ i. e. on the mean free paths of diffusion and transport, Xs and Xt, the diffusion length L and the ratio y between the diffusion and total cross-sections of the medium. Several authors have proposed formulae for calculating g. Mention should be made in particular of the formula of Skyrme, modified by Ritchie and Eldridge [7], which in principle is valid for any radius of detector:

4 r_ ЗТГ £ (Ш. 41) 7Г Xt 16 L - T

When r is the detector radius and К is a coefficient determined from the curves drawn as a function of 2r/X for different values of y. 58 ..-CHAPTER III

Since these corrections are always very small/ g can be calculated from the following approximate formula, which is valid for detectors with a diameter less than 10 mm ••. i

gs.l: 05 r/Xt

This approximate formula giving g shows that for the most common media, such as air, graphite and heavy water, g is very small so that H is approximately unity and the flux depression is negligible; however, this .correction must be taken into account for light water or hydrogenous media. These results have been confirmed through measurements carried out by Spernol et al. [8], that show that to a first approximation g can be taken as 0, except in the case of hydrogenous media, and then too when the detectors are very small. Table III. II gives the coefficients F calculated for three types of cobalt detector in different media.

TABLE III. II. FLUX PERTURBATION FACTOR (F) FOR COBALT DETECTORS

Type of detector Moderator material

H^O dzO . c Be

Foil: 10 mm diam. ; 0.912' 0.929 " " 0.929 0.925 0.1 mm thick ''

Foil: 3 mm diam. ; 0.926 0.93Ï 0.931 0.930 0.1 mm 'thick

Wire: 0.125 mm diam. ; 0..975 0.975 0.975 0.975 , 10 mm length

III. 2. 2. Measurement of low flux densities

For the types of study in which these measurements have to be made the quantity required is most usually the flux density nv0. The corresponding fluences are generally'less than the value Ф = 1014 n- cm"2, corresponding to a few hours' irradiation in fluxes less than or equal to 1010 n- cm"2- sec"1. These measurements are generally carried, out in low-power reactors, critical assemblies or training reactors, or, even in standard piles consisting simply of neutron sources placed'in a moderator block or graphite, light water or heavy water. The most typical types of. study involving measurement of low thermal neutron flux densities are: Core configuration studies . , • • . Determination.of the power of a core from fission rate calculations Determination of the flux peaks in the neighbourhood of the control rod Fuel burn-up studies Calibration of a standard pile Measurement of flux depression Flux distribution imirradiation facility models ,, . Buckling measurement in critical or exponential experiments. : • : THERMAL NEUTRONS 59

Choice of detectors

A complete study of all the isotopes usable for measuring low flux densities will not be attempted in this manual, however those most normally used will be mentioned and some indication of the criteria involved in making a choice will be presented. Table III. Ill gives a summary of the nuclear characteristics of these normal detectors. Some discussion regarding each of these detectors, indicating the criteria applied in their selection, is presented in this section. It will be noted at once that the detectors have been arranged in order of decreasing sensitivity. Dysprosium and indium are obviously the best detectors for determination of very low flux densities. Dysprosium - Dysprosium is used in one of the following three forms: metallic dysprosium, aluminium-dysprosium alloys or dysprosium ojdde, Dy203, deposited for example on an aluminium support. In spite of the rather strong resonances in the eV region, the epithermal activation of the detector is low as compared with the thermal activity. This makes the detector particularly useful for measuring low fluxes in locations where it is difficult or impossible to determine the cadmium ratio. Indium - The useful activation is obtained from the isotope indium-115. It can be seen that for irradiations which are very short but sufficient to give a suitable 116mln activity there will be practically no }14mIn activation. The parasitic activities of 114In and 116In disappear a few minutes after irradiation. Detectors of pure indium must be at least 1/100 mm in thickness. For smaller thickness deposits mùst be evaporated onto supports. The main disadvantage of indium results from the fact that the strength of the resonances in the eV region causes considerable epithermal activation. Manganese - Manganese is a good thermal neutron detector with a strictly 1/v cross-section in the thermal range. It is generally employed in the form of pure metal or as a Mn-Ni alloy. Copper - This detector possesses the same advantages of 1/v capture as manganese, but it is less sensitive owing to its smaller cross-section and longer half-life. Thin foils, wires or tapes having very good mechanical properties can easily be made. This detector is therefore often used for flux mapping in the cores of low-power reactors. Gold - This detector is used particularly for precision measurements for absolute calibration. Its decay scheme is such that the absolute activity can be determined by |3Y-coincidence counting. It can be used in the form of thin metal foils or deposits evaporated under vacuum.

Preparing for measurement and irradiation

Having selected the type of detector, one must now deal with the different stages in making the measurements. The detectors are made from spectroscopically pure metal foils either by stamping out or, for greater thickness, by turning on a lathe. Care should be taken to ensure that the tool does not contaminate the detector. In general, it is advisable to clean the detectors in pickling mixture after fabrication and to keep them in a container (for example quartz or polyethylene test tubes). Very accurate weighing is required when it is necessary to determine the absolute activity of the detector. In series production of detectors for 60 CHAPTER III THERMAL NEUTRONS 61 relative measurements, this weighing is not necessary provided that: (1) the detectors are of identical geometry; (2) in the case of/3-counted isotopes, the thickness is sufficient for constant /3 self-absorption. For making the detector supports the choice will fall upon materials which are easy t-o work and do not become highly activated, or whose activity decreases quickly after irradiation. For the measurement of low fluences, the materials are plexiglass (Lucite) or aluminium of quality A5 to A9. There are obviously as many types of support as there are types of irradiation position. The detectors can be fixed to the supports in different ways:. Using radiation-resistant adhesive paper (Scotch tape, type ET 56 F) for low-flux irradiations at ambient temperature in water or in any other medium The aluminium or cadmium boxes containing the detectors can be stuck on with Araldite (allowing sufficient time for drying) The box can be simply fixed into a recess in the support, using thin easy-to-cut aluminium wires. The irradiation times will be determined so as to give adequate activity at the moment of counting. The flux density should be kept constant during irradiation so that the saturation activity can be easily.deduced from the counted activity. This obviously requires: Accurate time measurement (irradiation and waiting period respectively) No unduly short irradiations (preferably at least 1/2 hour) If possible, the detector supports should be positioned after the reactor has reached the power level in order to avoid corrections for activation during the approach to criticality The irradiation must be terminated either by abrupt stopping of the reactor or by withdrawal of the detector supports at an exact moment, insofar as this is compatible with reactor safety. Great attention must also be paid to appropriate geometrical positioning of the detectors and supports in order to avoid all doubts as to the flux received by the detector. This is especially important in the case of studies relating to flux distributions with a steep spatial gradient, e. g. measurement of flux peaks between fuel plates, measurements of flux depression etc. Proper positioning will generally require supports of definite geometry and adequate rigidity. For the low activities dealt with here the detectors can generally be recovered manually without the need for a hot cell. It will generally be sufficient to have a decontaminable surface lightly protected by lead or glass wall. In addition, it will usually be necessary to clean the detectors so as to remove traces of impurities which might have been deposited during fitting or irradiation.

Counting

Details of the electronics used for counting will not be considered, since there is abundant literature available on this. Relative counting of the detectors is used quite frequently. In low flux measurements of this type, generally involving simultaneous irradiation of a large number of detectors, the latter will often be put in an automatic sample changer. Two counting systems are possible: 62 : CHAPTER IV1

(a) A single counter whose stability will be controlled by using a source with a long half-life (e. g. uranium);' the detectors may pass under this counter once or several times- (b) Several counters of the same type; each detector passes under each counter in succession to allow for the possibility of failure of one of them. This gives the relative activities of the К detectors a<). . . . áj, ак. • • ; ' Absolute calibration of the series is also important. Three methods commonly used are: (a) By determining the absolute activity of one or more detectors of the series, using a coincidence method (this is possible for simple decay schemes like that of 198Au) or a 4ît counter method (possible for thin detectors with negligible self-absorption) (b) By measuring the absolute activity of a detector (generally gold) irradiated in the same box as one of the detectors of the series and calculating the corresponding fluence (c) Using a standard pile of known flux density. A detector of the same type as the series is irradiated in this known flux density and its activity then compared with that of the others; this gives directly : the thermal fluences corresponding to each detector of the series. The first two methods confront one with the problem of how the absolute activity of a detector es related to the non-perturbed thermal flux density in the detector location (section III. 2. 1). With the third method, the fluences can be determined directly by simply correcting the total activity of the detectors for the contribution due to epithermal neutrons.

Ш. 2. 3. Measurement of high fluences

The measurement of high fluences is generally involved in controlling long-term irradiations of fissile or non-fissile materials, in'research or power reactors. • The main studies in which these measurements are required are: (a) Fuel tests requiring determination of the number of fissions or the burn-up (for. this purpose the thermal-neutron fluence must be known); and (b) certain radiation-damage tests on structural materials in which the thermal neutrons can give rise to defects, e. g. embrittlement of steel by the (n, a) reaction on boron traces in the steel.

Choice of detectors • . 1

As in section III. 2. 2, this section will be limited to a consideration of the methods most generally employed, all of which are based on the use of cobalt-59 in one form or another. The nuclear parameters of the reaction 59Co (n, y)60Co are given in the Appendix listing the characteristics of all these detectors. The element cobalt exists as the isotope 59Co in 100% abundance. The isotopés 60mCo and 60Co are formed by (n, y) reaction in accordance with the transition diagram shown; these two states have about the same probability of formation. Th'é isotope' 60nto is unstable with a half-life of 10. 5 minutes and it is almost completely transformed into 60Co, which has a half-life of 5. 26 years. The diagram shown in Fig. III. 3 represents the formation of 60Co. FIG.III.4. Modified reaction and decay scheme for 5to.

In practice, with an irradiation time of longer than 1-2 hours in a relatively stable..neutron flux (necessary for the saturation of 60mCo: 99% in 70 min and 99. 9% in-110 min) and a cooling time of several hours before the sample is counted (necessary for decay of the 60mCo formed), the simplified transitions shown in Fig. III. 4 can be adopted. The main-choice which has to be made is whether to use pure or alloyed cobalt. - The advantages and disadvantages of these two possibilities are considered in turn. 1.ai. ,-FHr_? _cobalt_ Advantages: First, it is easy to make wires or thin discs of pure cobalt. Secondly,! since the melting point is high, these detectors can be used at high temperatures. Disadvantages:. The main disadvantage of cobalt arises from its very high specific activity in case of long irradiations (illustrated by Table III. IV). It can be seen that above 1020 n- cm"2 the activities and dose rates become prohibitive for simple handling of the irradiated sources. Even if the detectors are recovered in a*hot cell) the counting system must be placed in a shielded area. The natural consequence is to use cobalt in the alloyed form. (bi JA11oyed_coba.lt Advantages: The specific activity is reduced by a factor of 100-1000 for the same detector geometry. Disadvantages and difficulties: The metal alloyed with'the cobalt should not produce short-lived Isotopes and should contain no cobalt impurities. Moreover, the cobalt content should be homogeneous, otherwise it is necessary to pre-irradiate each detector for calibration. Actually, three types of alloy are m'use or on trial. They are: (i) aluminium cobalt in the proportions 1/100 or l/lOOO; (ii) copper cobalt; and (iii) zirconium cobalt. Most suppliers of pure metals can provide the alloys, but it is generally necessary to check their homogeneity. The Central Bureau of Nuclear Measurements at Geel has developed an Al alloy with 1% Co 64 : CHAPTER IV1

TABLE III. IV. INDUCED ACTIVITY OF COBALT DETECTORS

Type of detector Thermal neutron fluence Specific activity Dose rate ! (n'cm' ) (mCi) mrad(tissue)h"' at 30 cm

Disc • 10 mm diam. 1016 3.04 x 10"2 4.26 x 10"1 0.1 mm thickness 10" 3.04 x 10"1 4.26 X 10°

1018 3.04 x 10° 4.26 x 101

1019 3.04 x 101 4.25 x 102

1020 3.03 x 102 4.25 x 103

10 21 2.97 x 103 4.16 x 104

10 22 2.45 x 104 3.43 x 10s

Disc 3 mm diam. 1016 \ 2.74 x 10"3 3.83 x 10"2 0.1 mm thickness 1017 2.74 x 10"2 3.83 x 10"'

1018 2.74 x 10"1 3.83 x 10°

1019 2.74 x 10° 3.83 x 101

lO20 2.73 x 101 3.82 x 102

10 21 2.67 x 102 3.75 x 103

10 22 2.21 x 103 3.Ó9 x 104

Wire 0.125 mm diam. iols 4.76 X 10~4 6.65 x 10"3 10 mm length 1017 ' 4.76 x 10"3 6.65 x 10"2

1018 4.76 x 10"2 6.65 X 10"1

1019 4.76 x 10"1 6.65 x 10®

1020 ~ 4.74 x 10° 6.63 x 101

1021 4.64 X 101 6.51 X 102

10 22 3.84 x 102 5.36 x 10s

whose content is guaranteed to within ± 1% so that precalibration is not necessary.1 The copper and zirconium alloys, which are on trial in various laboratories, have better properties at high temperàture.

Preparing for measurement and irradiation

What has been said in section III. 2. 2 also applies here. However, it should be pointed out that it is perhaps easier to manufacture wires of strictly identical geometry than thin plates. Cobalt will almost always be used with covers to avoid contamination. The supports will be made according to the particular circumstances of the case in such a way as to get the monitor as close as possible to the test sample. The materials employed depend mainly on the temperature conditions.

1 This material is available through the IAEA Laboratory Seibersdorf. THERMAL NEUTRONS 65

At T < 550°С aluminium is the best material for the covers and supports, at T>550°C stainless steel or refractory materials such as silica can be used for the covers. As pointed out in section III. 2. 2, great attention must be paid to the problem of detector positioning. Since the spatial flux gradients are very steep, especially in the case of fuel irradiation, the detectors should always be placed very close to the samples and if possible inside them. Under the conditions for which cobalt acts as a perfect integrator (see section III. 2. 1) the difficulties involved in measuring the irradiation time and the problem of ensuring constant flux density can be disregarded. The detectors will usually be recovered in a hot cell at the same time as the samples. Isolation of the detector will sometimes be facilitated by its magnetic properties by using a magnet. In some cases the detector can be removed from its cover either mechanically or by dissolving the cover chemically (e. g. aluminium covers can be dissolved in a concentrated solution of caustic soda). In any event, the detectors should be decontami- nated before counting to remove traces of impurities deposited during irradiation or recovery. Reweighing may be necessary to check the weight of cobalt before counting.

Counting

In general, the counting proceedure will be similar to that discussed previously for measuring low fluences. With cobalt counting is generally by direct comparison of the gamma activity of the source, in an ionization chamber or by a scintillation counter with that of a standard source calibrated by a specialized laboratory.

Ш. 3. MEASUREMENT OF THERMAL NEUTRON SPECTRA

For most practical applications it is sufficient to regard the thermal neutron spectrum as the spectrum of a gas of neutrons in thermal equilibrium with the nuclei of the moderator. In other words, one assumes a distribution function of the Maxwell-Boltzmann type, whose most probable energy (or temperature) is that of the moderator. In reality the presence of absorbers perturbs the theoretical distribution, and hence the concept of neutron temperature is no longer strictly applicable. This circumstance does not prevent one from continuing to assign a Maxwellian distribution to the true spectrum, which makes it possible to determine an ' effective neutron temperature1 that is different from that of the moderator temperature.

IH. 3.1. Principle of the method

Choice of detectors

The thermal neutron spectrum generally has a steep slope in the range between 0. 05 and 0. 5 eV. In this energy region, therefore, a resonance detector is very sensitive to the position of the thermal spectrum in the whole energy range, i. e. to the spectrum temperature. This is particularly true for the case with lutetium-176, which has an important resonance for slow neutrons at 0. 142 eV. 66 : CHAPTER IV1

The activity of such a lutetium detector irradiated in a reactor depends on various factors: (1) the spectrum temperature; (2) the thermal flux intensity; and (3) the ratio of epithermal to thermal flux density. The latter two factors can be determined by irradiating, together with a bare and cadmium covered lutetium detector, a material with-a 1/v cross- section in the thermal range and a resonance cross-section in the epithermal range. The resulting activity and cadmium ratio of the detectors are such that the proportion of epithermal flux and the thermal neutron density can then be measured for any neutron spectrum; • In principle, therefore, absolute measurements can be used for deducing the temperature of the thermal spectrum regardless of any assumptions as to the moderator temperature. In practice and owing to the uncertainty of the knowledge regarding the cross-section of 176Lu and the decay scheme of 177Lu, one must rely on relative measurements, assigning a temperature to a reference spectrum. The isotope 175Lu constitutes 97. 4% of the natural mixture. It has a resonance at 2. 61 eV and a half-life of 3. 7 hours and .seems eminently suitable for comparison with 176Lu (6. 72 d half-life). Farinelli [9] has studied the relative behaviour of the two isotopes at different temperatures for different spectra characterized by Westcott' s index r, representing the ratio of epithermal to thermal neutrons. The sensitivity of such a method depends on r. In most cases, the proportion of epithermal neutrons is too great to use the curves calculated by Farinelli. Gold is particularly suitable for this type of measurement. Silice it is not found mixed with 176Lu, as is the case with 175Lu, an additional irradiation is necessary. However, it can be counted separately.

Relation between detector activity and neutron temperature

The following notation will be used in this section:

Ax : activity.of a lutetium detector irradiated at position x ax : thermal neutron activity of a gold detector irradiated at position x К : coefficient including the time and relative counting constants Rcd: cadmium ratios of gold .corrected for thermal and epithermal self- absorption i • Tx : neutron temperature in degrees Kelvin at position x. Westcott' s notations are used in the calculation. Therefore in a spectrum of index r, the activity of one mg of 176Lu can be, written as:

ALu = nv0[Kff0(g + r s)]Lu (Ш.42)

The dependence of the lutetium activation integral on the spectrum temperature T is expressed in the functions g(T) and s(T). Values are tabulated by Westcott [1] for thin detectors. The same parameter values are assumed for the detectors used in view of the. low isotopic content of 176Lu. It can be seen that if an experimental-value is obtained for the group (g + r s) from an activity measurement, .the temperature T can be determined by comparing this value with the function g(T) + rs(T). The following approach is considered: i THERMAL NEUTRONS 67

(a) Measured values, of (g + r s) are obtained by making a series of relative activity measurements in different experimental positions x. In reference to some position designated 1, one has

Ax = ím¿MIA (III. 43)

Ai (nv0)i (g + r s)j

The moderator temperature, say T1;. is assigned to position 1, which is chosen well away from the core. Then g(Tx) and s(Tx) are known. The value of the group (g + r s)x can be determined from the ratio of measured activities if the ratio (nv0 )x/(nv0)j and the different spectrum indices r are also known. These parameters will be determined by .the irradiation of the gold detector. The activity produced in one m g of gold under the influence of the thermal neutrons can be written:

AAu = nv0[Kcr0g]Au (III. 44) where g is practically independent of temperature, so that

(nvp)x = (AAU)X (III. 45) (nvo)i (aAU)I

In addition, the cadmium ratio Rcd measured for gold in position 1 and given its theoretical value by correcting for flux depression, can be written, similar to that shown in Eq. (III. 17), as

•' g

ri " R¿d(S0 + 4^kTo/îrEcd) - s0 ''!."'.

For each position x one, therefore, has .

(g + rs^ + rshf^; "...

(b) Calculated values of g(T) + r s(T) are determined in the following manner. The values obtained for (g + r s)x must not be compared with a network of calculated curves g(T) + rx s(T) for the different measured values of rx. However, measurement of the cadmium ratio gives r ч/Т/T0, not r. Obviously, r cannot be deduced from r n/T/T0, without a knowledge of T, which is the parameter to be measured. . The values of the expression: ' •

g + [rx(Tx/T0)i] f s(T0/T)i ] are calculated as a function of T for different values of [rx(Tx/T0)i] determined from the cadmium ratios of gold measured in the different positions. The g and s factors of 17eLu are taken from Ref. [ 1] for the Д4 joining function.

It is clear that such a function is only identical with g'+ rxs when T = T , but this equivalence at a single point is sufficient and yields the temperature required. 68 : CHAPTER IV1

Fuel element MTR type

Ш. 3. 2. Example of the experimental procedure

As an example, the measurements made in the core and reflector of the MELUSINE reactor operating at low power are considered. The positions are indicated in Fig. III. 5.

Characteristics of the detectors

The Johnson-Matthey gold detectors were in the form of discs, 2. 5/100 mm thick and 10 mm in diameter (average weight 43 mg). The lutetium detectors were made by mixing lutetium oxide as homogeneously as possible in the powdered silica and, thereby, forming the support; the silica does not become highly activated with most of the activity being due to isotopes of very short half-lives. A vegetable glue is used for grain cohesion. After pressing and sintering, the detectors are in the form of discs 1 mm thick and 10 mm in diameter and containing an average of 2 mg of the isotope 17eLu.

Irradiation facilities

The detectors were positioned on stringers made of plexiglass or aluminium, depending on the irradiation position. The plexiglass stringers were slid between the seventh and eighth plates of the fuel elements, these plates being 2. 4 mm apart. The aluminium stringers were used in the reflector positions. The vertical flux gradient between positions 0 and +10 (denoting 10 cm above the median plane of the fuel elements) is not constant, so the specific activity of a bare gold detector in position + 5 is deduced from the average of the spécifie activities of bare gold detectors at positions 0 and +10, together with a correction coefficient of 1. 019 ±0. 002 (correction determined from vertical readings). The lutetium detectors were irradiated in position +5. Since it was necessary to protect them from humidity, they were placed in sealed covers of plexiglass with thin walls (0. 5 mm). Two irradiations (half an hour and one hour) were carried out in MELUSINE at 1 kW. THERMAL NEUTRONS 69

Counting

It was only necessary to make relative measurements of the activity of the individual gold and the individual lutetium detectors, respectively. The gold detectors were counted using a scintillation counter with constant geometry. One of the detectors was used for reference between each detector and background count, respectively. This enabled one to check the apparatus for any possible slow drift and also to determine the reproducibility; the latter is limited by the maximum fluctuation of the counting electronics, which in most cases is far greater than the statistical error at 68%. This maximum fluctuation was adopted (± 0. 8%) to define the precision for all but two detectors. These two detectors had low count rates and owing to the background the statistical error at 68% is greater than this fluctuation. A preliminary gamma-spectrometry study was made to show that the activity due to activation of 175Lu (half-life 3. 7 h) had disappeared. Counts of the 210-keV peak were subsequently made with a photomultiplier counting system one week after irradiation. Although gamma counting is less efficient than beta counting with a G-M counter (gamma de-excitation only involves 10% of the activity), it has the advantage of eliminating self-absorption during counting. Moreover, the plexiglass cover can be kept intact, thereby avoiding any loss of material during handling. To eliminate errors which might arise through non-uniform distribution of the lutetium inside the disc (cracks had appeared after sintering), each disc was counted on both faces in two opposite directions. The standard deviation of these four measurements yielded the counting error which is of the order of 0. 5%. Finally, no correction was required to take into account the natural radioactivity of the 176Lu (half-life 2. 2 X 1010 yr). Calculations showed that this activity per disc did not exceed 14 counts/sec as compared with approximately 10 д Ci for the measured activity at the moment of counting. On analysing the results shown in Table III. V, it can be seen that the neutron temperatures measured in the reflector are within the limits of applicability of Westcott' s conventions. For position 35 Tn/TM is still equal to 1. 05. On the other hand, the measurements made in the core are well outside the safe limits (Eq. (Ш. 10)) defined by Westcott. The capture effect due to the highly enriched uranium and the metal/water ratio of the core certainly lead to a non-Maxwellian distribution of the thermal neutrons.

The core results giving Tn/TM = 1. 25 should, therefore, be treated with caution, especially since the semi-empirical formulae of Brown and Coveyou lead to a Tn/TM ratio of approximately 1. 12 for a homogeneous medium with the same capture cross-section per atom of hydrogen.

III. 4. OTHER MEASURING METHODS III. 4. 1. Measurement of flux density

BF3-counters, boron ionization chambers, 235 и fission chambers

These well-known measuring techniques are used in particular for reactor control. Without going into detail, we shall refer briefly to studies now being carried out in order to miniaturize these counters using semi-conductors. 70 : CHAPTER IV1 THERMAL NEUTRONS 71

Gas loops [10]

A continuous gas circulation is set up with'a constant flow. The circuit consists of two main parts: inside the reactor there is an irradiation coil surrounding the experimental facility and outside the reactor there is a counting coil surrounding a scintillator crystal coupled to a photomultiplier. The counting circuit is regulated so as. to count the gamma radiation ' characteristic of deactivation of the product formed in the chosen reaction. A gas in normal use is chosen so that the open cycle system can be employed. This makes it much easier to maintain a constant flow without regulation. The time spent passing through the reactor must be short in order to get a fast response, so use is made of nuclear reactions having the following characteristics: Large cross-section High isotopic concentration of the parent nucleus Short half-life of-the product formed. Finally, for obvious reasons of safety, the gas must not be of a corrosive or toxic nature either before or after passing through the radiation environment of thé reactor core. For measuring thermal neutron flux, pure argon conforms well to the requirements mentioned above. The characteristics of the reaction 40Аг(п,-т)41Аг are as follows: Cross-section a = 0. 65 b Isotopic concentration = 0.996 1 • ' 41Ar decay: half-life - 1. 85 h • r The energy of the gamma radiation associated with 41Ar deactivation is 1. 3 MeV. Reactions of-the (n, y) type on the isotopes 36Ar and 38Ar give rise to products having no gammaactivity. The argon employed (quality U) contains 50 ppm nitrogen' and 10'ppm oxygen. The activity of the nitrogen-16 formed in the 160(n¿ p)16N and 15N(n, p)16N reactions is negligible compared with that of the 41Ar formed. This is also true of the oxygen-1'9 activity produced in the lsO(n, 7)190 reaction. The irradiation volume-provided must be such that the activity óf the gas can be counted with good statistical accuracy. However, the fact that the selected counting volume may be very large compared- with the irradiation volume allows considerable flexibility in selecting the latter. The choice will finally be decided by space requirements in respect of the position and configuration of the experimental facility. . 1 - The activity counted by the scintillation counter canbe expressed in.the form:

. . A.;,= KNcVc фа [ 1 - ехр(Л, К V /Q)]exp (-> NdVd/Q) . (III. 46) where К is the geometrical factor; Nis.the density in atoms/cm3; V is the volume; ф is the flux in n- cm"2- sec"1; and Q is the gas flow in atoms/sec. The suffixes i, d and с denote the quantities for irradiation, transit and counting, respectively. , ..... For a given circuit exposed under a constant flux density the activity A of the gas will only be. a,function of the flow. Figure III. 6 illustrates .the general trend of the curve of A. It can be seen that the .curve passes through a maximum at the optimum flow value; At this value slight variations in flow do not affect the measurement to á troublesome extent. 72 : CHAPTER IV1

Ar activity

Argon flow

FIG.III.6. Argon induced activity plotted against gas flow.

The gas flow is stabilized by using a two-stage expander. The flow- meter is of the rotating type. Since the time taken by an atom of the fluid detector to pass through the reactor is very small compared with the half-life of the product formed, formula (III. 46) can be written as

A = kNcVc Фа (X N¡ Vi /Q) exp (-X NdVd /Q) (III. 47) where A can be seen to vary in proportion to the density of the gas in the irradiation volume. It will, therefore, be necessary to introduce into the sensitivity term a correction factor taking account of temperature variations in this volume; temperature variations in the outlet volume and in the counting volume are regarded as negligible. As an example of the gas loop application, two facilities of this type have been constructed on loops in the MELUSINE reactor for studying fission-product diffusion in sintered uranium oxide irradiated at high temperature. The U02 samples are of small dimensions (diameter 3 mm, height 15 mm). The irradiation coil is in the form of a solenoid with five turns and a length of 5 cm; it is fixed on the outer envelope of the loop in contact with the cooling water. The part inside the reactor is all made of stainless-steel tube with an inner diameter of 1 mm. At its ends there are two valves "firmly attached to the connection head. Calibration is carried out by placing a cobalt detector in the position intended for the U02 sample. The detector is in the form of a 3-mm diameter disc cut from a foil 0. 1 mm thick. This measurement gives the sensitivity of the facility at the temperature reached inside the oven with nuclear heating alone (about 150°C). Subsequently, the effect of oven temperature on the readings of the counting circuit was measured at power level: this effect is quite considerable, reaching 5%, and corresponds to a temperature rise of 13 degC in the irradiation coil.

Collectrons (self-powered detectors)

Consider a material A which is activated with a cross-section cr and a time constant X to produce a material B. After time t, the activity will be

XN = N^CT[1 - exp(-Xt)] (III. 48) THERMAL NEUTRONS 73 where Np is the number of nuclei in material A; and cp is the neutron flux density in n- cm"2- sec"1. After exposure for several half-lives

XN = Npcpcr (III. 49) i. e. the activity is proportional to the flux density. Consider a material'A such that the product material В emits a /Tparticle in the device shown in Fig. III. 7. Material A constitutes the emitter. The latter is surrounded by dielectric, which itself is surrounded by a conducting material.

Insulator

FIG.III. 7. Sketch showing circuit and material arrangement of a typical collectron.

Some of the j3~ particles emitted by material A are stopped by the collector, where they give rise to negative charges. Simultaneously, positive charges appear on the emitter with a unit increase in the atomic number. If the emitter is connected to the collector, an electric current i will flow:

i = К X N = К Npcpo- (III. 50) where К is a constant depending on the nuclear decay scheme and on the proportion of electrons (j3~) captured by the collector. The device, therefore, acts as a current injector.

In formula (III. 50) the term Np is the number of nuclei of the material A at the instant t=0 before introduction into the reactor. Therefore,

Np = N0 exp (-crcpt) so that formula (III. 50) becomes

i = К N0 ф a exp (-acpt) (Ш. 51)

An advantage of this formula is that it indicates the consumption of material A, arid this should serve as a guide in choosing the emitter. Based on the above discussion it is seen that the emitter should have the following properties : It should be a conductor Its thermal-neutron cross-section a should be small enough to ensure low consumption of nuclei, and large enough for the current produced to be easily measured The product of the reaction should emit ^-particles (positive or negative) whose maximum spectrum energy should be great enough (of the order of 1 MeV) for the particles to be not all stopped in the emitter itself or in the dielectric. 74 : CHAPTER IV1

TABLE III. VI. NUCLEAR DATA ON Rh AND V EMITTERS

Emitter Cross-section Isotope Maximum 6 energy Per cent consumption 2200 m/sec half-life (MeV) of initial nuclei (b) (fluence-5 x 1019 n-cm"!)

103Rh 100$ 156 4.3 m 2.44 0.75 <7o

44 sec 98%

51V 99.76% 4.5 3.75 m 2.73 0.02271) .

МОЧЬ

The main properties of two emitters in current use are listed in Table III. VI. The dielectric material should be such that it does not deterior- ate under radiation. For long-term use polyethylene is not suitable. In

practice, А12Оз or MgO is chosen. For the collector, any conducting material of sufficient thickness to stop the j3-particles will be satisfactory. These detectors are used in various countries (see Ref. [11] in particular). Their main advantages are: Their sensitivity is such that they can operate in very high fluxes in reactor cores or in conjunction with irradiation capsules They can be made with the techniques used for thermocouples and have similar dimensions Simple instrumentation (galvanometer or recording millivoltmeter). Tests are being made to try to explain the parasitic currents observed and to determine the conditions for high-temperature use.

HI. 4. 2. Fluence measurements

Two other methods of fluence measurements, although not used much, will be discussed. These methods with certain improvements might be used in addition to the cobalt method.

Depletion of gold nuclei

The (n, 7) reaction for thermal neutrons on 197Au produces 198Hg which accumulates, being stable and having a very small cross-section. By measuring the amount of gold which has disappeared (e. g. by subsequent reactivation of the detector).or the amount of mercury formed (by spectro- scopic analysis), it is possible, in principle, to determine the integrated total fluence to which the detector was exposed. The method is well suited for very high neutron fluence, i. e. above 1021 n- cm"2. It may, therefore, be an interesting alternative to the cobalt method in this fluence range. Many tests will still have to be made, however, to ascertain that the impurities in gold do not obscure the measurement.

Isotopic analysis of irradiated fuel

Thermal neutron fluence measurements within the fuel are of great interest to metallurgists, and can be carried outin various ways. All these methods have the disadvantage of requiring careful preliminary chemical preparation. Four techniques are briefly discussed. THERMAL NEUTRONS 75

(a) Measurement of 235U depletion [ 12]: Isotopic analysis of the fuel gives the 235U content and comparison of this with the initial content indicates the number of nuclei burned by the thermal fluence. (b) Measurement of plutonium content [12]: The plutonium content of • an irradiated fuel is directly related to the number of neutrons captured in the 23SU. Measurement of the Pu/238U ratio by double isotopic dilution is a method currently employed for measuring the fluence within the fuel. (c) Measurement of variation in the isotopic content of boron: For studying plutonium fuels the above methods ((a) and (b)) cannot be used since the formation of the different plutonium isotopes with their strong resonances complicates interpretation of the results. Use is, therefore, made of natural boron, which is introduced into the sample in small amounts during manufacture. Under irradiation the content of boron-10 decreases and the thermal fluence is obtained directly by measuring the change in isotopic content of the boron-. (d) Analysis of fission products [ 13]: For highly enriched fuel subjected to long irradiations measurement of the fission products, in particular of 148Nd, is often employed to indicate the number of fissions.

REFERENCES TO CHAPTER III

[1] WESTCOTT, C.H., Effective Cross-section Values for Well Moderated Thermal Reactor Spectra, Rep. AECL 1101. [2] WESTCOTT, C.H., WALKER, W.H., ALEXANDER, Т.К., "Effective cross-sections and cadmium ratios for the neutron spectra of thermal reactors", Int.Conf. peaceful Uses atom. Energy (Prof. Conf. Geneva, 1958) 16, UN, New York (1958) 70. [3] Calcul des sections efficaces effectives et thermalisation des neutrons, Note CEA No.438 (1963). [4] MARTINEZ, J.S., Rep. UCRL-6526 (1961). [5] CARRE, J.C., ROULLIER, F., VIDAL, R., Rep. MIN 76 (1965). [6] CASE, K.M., de HOFFMANN, F., PLACZEK, G., Introduction to the Theory of Neutron Diffusion, Los Alamos Scientific Laboratory (1953). [7] RITCHIE, R.H., ELD RIDGE, H.B., Nucl. Sci. Engng 8 (1960) 300. [8] SPERNOL, A., VANINBROUKX, R., GROSSE, G., "Thermal flux perturbation by cobalt detectors", Neutron Dosimetry (Proc.Symp. Harwell, 1962) 1., IAEA, Vienna (1963) 547. [9] FARINELL1, U., Neutron Dosimetry (Proc.Symp. Harwell, 1962) 1, IAEA, Vienna (1963) 195, 211. [10] PLEYBER, G., Boucle à gaz pour mesure continue de flux de ijeutrons thermiques, Rep.CEA No.R 2541. [11] HILBORN, J. W., Self-powered neutron detectors for reactor flux monitoring, Nucleonics 22 2 (1964) 69,74. [12] BIR, R., CHENOUARD, J., LUCAS, Monique, "Méthodes utilisées pour l'analyse isotopique de l'uranium et du plutonium", Nuclear Materials Management (Proc.Symp. Vienna, 1965), IAEA, Vienna (1966) 707. [13] HAECK, W Л., Proposed Determination of Nuclear Fuel Burn-up Based on the Ratio of two Stable Fission Products of the Same Elements, Rep. IDO-14642 (Sep. 1965).

CHAPTER IV

INTERMEDIATE NEUTRONS

IV. 1. INTRODUCTION

IV. 1.1. General

Materials which are activated by exposure to neutrons can serve in principle as neutron detectors. In the low-energy region the cross-section for absorption and activation generally varies inversely proportional to the neutron velocity. This energy region is therefore sometimes called the 1/v region. Between about 1 eV and a few keV, especially for elements with intermediate and high mass numbers, there often are particular energies for which the rate of interaction (e. g. activation) is exceptionally large. The differential cross-section curve for such an element shows in this energy region narrow peaks which are called resonance peaks. Neutrons whose energies are in the region where such peaks occur have sometimes been called resonance neutrons. In the newer terminology neutrons of kinetic energy between the energies of slow and fast neutrons are called intermediate neutrons. In the neutron metrology field this range is taken from cadmium cut-off to say 0. 1 MeV. Epithermal neutrons are defined as neutrons of kinetic energy greater than that of thermal agitation. But often the term is restricted to energies just above thermal, i. e. energies comparable with those of chemical bonds. Therefore, we have avoided the term epithermal and we use in the following the terms intermediate flux density and intermediate neutrons instead of the older terms epithermal flux and epithermal neutrons. Classical examples of nuclides having resonance peaks are indium (peak at 1.46 eV) and gold (peak at 4. 9 eV). The reactions taking place in the intermediate energy range are usually of the radiative capture type (i.e. n, 7 reactions). Other reactions such as (n, p) and (n, ct) reactions are generally only possible with fast neutrons with energies above about 1 MeV. Such reactions are used in the activation technique with threshold detectors (see Chapter V). Since resonance detectors and threshold detectors in practice consist of different materials, a resonance detector is not sensitive for fast neutrons. The resonance activation detectors can provide experimental data on the intermediate flux density spectrum in and around nuclear reactors, which data are required for reactor physics calculations, shielding experiments and radiation damage studies [ 1-3]. However, in reactor experiments the main emphasis still lies on measurements of thermal and fast flux fluences as the measurement of intermediate neutrons is a far more complicated procedure. The difficulties met in practice are related to the following points:

77 78 : CHAPTER IV1

(a) The response of an activation detector to intermediate neutrons has always to be separated (by the use of foil covers, such as cadmium, or by a 1/v subtraction procedure) from the response to thermal neutrons (b) The response corresponding to the resonance peak in the cross- section curve is in general perturbed by the response corresponding to the 1/v part of the cross-section (c) The resonance integral cross-sections are often insufficiently known . . (d) The correction for self-shielding is much larger and more complicated for intermediate neutrons than for thermal neutrons (e) The use of extremely thin foils necessary to reduce the self- shielding correction requires refined laboratory techniques. These points will be considered in detail in the following sections. In this Chapter more attention is given to the theoretical background, the reactions of interest and the data treatment than to fabrication techniques for detectors and to normalization procedures. The reason for this layout here lies in the fact that, because of the difficulties mentioned above, there are as yet no well-established and generally accepted techniques to recommend for direct use to newcomers in the field; any newcomer is supposed to collect experience with the techniques for thermal and fast flux density measurements prior to starting the application of resonance detectors.

IV. 1. 2. Response of a resonance detector

The flux density distribution of neutrons slowing down in a medium with small resonance capture is given by (see, e. g. , Ref. [4])

. . (IV. 1,

where cp(E) is the intermediate flux density per unit energy interval; q0 is the source density; Ç is the average logarithmic energy decrement per collision; N is the number of slowing-down atoms per unit volume; and crs is the microscopic scattering cross-section. The flux density of the intermediate neutrons can often be approximated with a 1/E distribution. The. equation given above can then be written as

Ф (E)dE = вЩ- (IV. 2)

The number of interactions of a given type per second is given by

Ef E2 P = N J (E)dE.= N 0 J^CT (E) ' (IV. 3) Ej E i where N is the total number of target nuclei. To suppress the activation by thermal neutrons, the detector is normally enclosed in a cadmium cover so that neutrons with energies below a certain value in the neighbourhood of 0. 55 eV are not transmitted through the cadmium. INTERMEDIATE NEUTRONS 79

The integration limits E and E are in practice often put equal to 0. 55 eV (the cadmium cut-off) and oo or 2 MeV, • respectively. The quantity ; ' '-/.wf' . E. where the integration is extended over a resonance region, is called resonance integral cross-section and will be abbreviated with the symbol I. The total intermediate flux density, equal to the energy integral of the flux density per unit energy interval is therefore equal to

The quantity в can therefore be given the interpretation of an intermediate flux density per unit lethargy. In the case of a single resonance peak in the a(E) curve the cross- section in the resortancê region can be divided into two parts: A contribution crr(E), ideally givert by the Breit-Wigñer formula A'contribution cr!/v(E), due to thé faét that when no resonances are present the cross-section generally varies as 1/v. The equation

00 со t«

fa(E)-§ =/a1/v(E).f + f ar(B) f (IV. 5)

Ecd ECd Ecd

is abbreviated as I = Ii/V + I' . Only the fraction I' of the quantify I in the expression for the response of a resonance detector is a response from a narrow energy region. For good spectrum measurements the Ii/V contribution should be as small as possible.

IV. 1. 3. Fluence and spectrum measurements

The methods for measuring- intermediate neutron fluences and the corresponding interpretation can be divided in two categories. (a) A known 1/E spectrum (at least between cadmium cut-off and a few keV) given by -

9(E)dE = e§ (IV. 6)

The quantity в, divided by .the 2200 m/sec flux density, or in the Westcott formalism the spectral index r, is measured either by the,cadmium ratio technique or by the two-foil technique employing a non-l/v detector and a 1/v detector. (b) The spectrum is unknown. In this case one can start testing the constancy of 0 (or r) as measured with several resonance detectors with different resonance energies. 80 : CHAPTER IV1

If the experimental values for в or r are not constant, it is sometimes possible to draw general conclusions about the trend in these values as function of resonance energy. Generally one can then try to modify the 1/E distribution by introducing a parameter assuming a l/E(1 + m' distribution. In the case of an unknown spectrum the в value must preferably be determined from the detector response in a very narrow energy region. One therefore tries to reduce the 1/v contribution to the response and possibly also the peak contributions from other resonances. Fastrup and Olsen [5] introduced the method of the semi-empirical differential index r. The triple-foil technique can be very useful, or in other cases the application of appropriate neutron filters. The experimental data give information on the slowing-down density at particular energy values.

IV. 1.4. Types of resonances

Sometimes the total cross-section in the resonance region contains a non-negligible contribution from scattering. When scattering occurs, the neutron may or may not remain in the resonance region where they have an appreciable chance to contribute to activation. This chance depends upon the resonance width. И E j and E2 denote the energy of the neutron before and after scattering, then the following relationship exists:

, _ A2+ 2A cos 6 + 1 . . Ez/El (ÂTïjs (IV"7)

.where A is the atomic mass of the scattering nucleus; and в is the scattering angle. Therefore, when the scattering is an isotropic process,

P /F A2+l 2/1 - (A+ If

so that the average energy loss in a scattering process is given by

Neutrons with energies Ei equal to the resonance energy Er can, after being scattered, contribute to the activation if

2A Er/(A+l)2 < Г

where Г denotes the resonance width. As A»1 this condition is normally written as

2Er/AT< 1

Multiple scattering can lead to activation in the resonance region if

2Er/AГ« 1. INTERMEDIATE NEUTRONS 81

Taking into account the relations

CT0(capture) = a0,r = сто.т Гу/г and

cr0(scattering) = cr0 n = стот Гп/г where the subscript zero refers to the maximum values reached at the resonance energy Er, and Г = Гу+ Гп + , one can distinguish the following types of resonances: (a) Гу^>Гп, i.e. the resonance is predominantly a capture (activation) resonance

(i) 2Er/AT>l: in this case, the scattered neutrons give no contribution to the activation (ii) 2Er/AT< 1: scattered neutrons have a chance to contribute to the activation; multiple scattering will not be important and can generally be neglected as the scattering cross-section is small and consequently the mean free path for scattering large

(b) ГГ«ГП, i.e. the resonance is predominantly a scattering resonance

(i) 2Et/AT>l: in this case, there is no contribution to the activation from the scattered neutrons (ii) 2Er/AT< 1: scattered neutrons give a contribution to the activation; as the scattering cross-section is large corresponding with a scattering mean free path which is small, multiple scattering cannot be neglected.

IV. 2. SPECTRUM CHARACTERISTICS

IV. 2. 1. The 1/E spectrum

In many cases a 1/E spectrum is assumed for the intermediate neutrons. In the expression

ф (E)dE = 0 ^r the quantity в denotes the intermediate flux density per'unit lethargy, the lethargy u being defined by the relation u =ln(E0/E) where E0 is 10 MeV. From theoretical considerations it can be derived that the intermediate neutrons have a 1/E spectrum under the following conditions: (a) The medium in which the fast neutrons are being slowed down is homogeneous and infinite (i. e. no leakage) (b) The fast neutron sources are homogeneously distributed in space (c) The slowing-down power fEs does not depend on energy (d) During the moderation process there is no absorption (e) The moderating atoms behave as free particles and have the same mass as the neutron. Although these conditions are not satisfied in practice, so that deviations from the 1/E spectrum might be expected, it is very often not 518 : CHAPTER IV1 worthwhile to take possible deviations into account. Moreover, it is a difficult task to give reliable experimental evidence for small deviations. For many purposes the assumption of al/E form of the intermediate neutron spectrum will satisfy the requirements. In cadmium ratio measurements the following simple spectrum model is often used: A thermal flux density cpM(E), given by a Maxwellian distribution An intermediate flux density cp(E) = 0/E for energies above a certain cut-off energy ¿¿kT, which lies below the cadmium cut-off Ec

The formalism of Westcott

In some publications there is a preference for the Westcott [6,7] formalism, which has been derived for well-moderated systems. This formalism has been discussed in Chapter III. Summarizing the Westcott formalism for describing activation reaction rates in well-moderated systems, the following statements may be made: (a) The reactor neutron spectrum, in terms of neutron density, is expressed as ,

4 2 Л n(v)dv = J- n {(1-br) ^-3 exp (- (v/vT )2) + r vT dv (IV. 9)

where b is a factor, dependent on the choice of the joining function Д. The epithermal index r is a measure of the relative contribution of the intermediate flux density to the total spectral distribution. The neutron density is n and vT is the velocity of a neutron having energy kT (i. е. E = mv2 = kT). The distribution is normalized in such a way that

n = / n(v) dv

(b) The neutron spectrum, in terms of the neutron flux density,' is expressed as '

Ф (E)dE = n ( ^г У {(1-r/^ A dv) exp (-E/ET) + r |} dE 4 0/ « • T ' (IV. 10)

ф0 = n v0 (IV. 11) INTERMEDIATE NEUTRONS 83

(d) The reaction rate (activity at saturation) is here given by

A = N ф0 a . (IV. 12) where 8 = a0 (g + rs).

Iri this expression ст is an effective- cross-section and ст0 is the cross-section for 2200 m/sec neutrons. The effective cross-section ст becomes nuclide in a Maxwellian flux density at temperature T, divided by the reaction rate in a 2200 m/sec flux density, or

S(kT) = exp (-vVvTV(v) dv and

, . 1 ( 4T f r , , л v0 , . dE , or '

•• • ' s - :

ст0 \ttTo /

For the case of a pure 1/v absorber g = 1 and s = 0. Sometimes a modified parameter s0 = s(T0/T)^ is introduced, since s0 is practically independent of the neutron temperature. ' (e) The neutron spectrum is characterized by: A cut-off function Д (a simple step function or modified form) connected with a 1/E spectrum for the non-Maxwellian component A neutron temperature T for the Maxwellian component A conventional flux density ф2200 m/,sec An epithermal index r. For a discussion of several cut-off functions see Ref. [7] which gives tables for g and s functions and â values.

The unified formulation of Nisle • -

In the proposed unified formulation [8-10] it is assumed that: (a) The neutron flux density distribution -can be represented by two major components, the Maxwellian and the 1/E components, joined by a transition region ; (b) Each component can be characterized by a single parameter, ET (= kT) for the Maxwellian and F for the 1/E component (c) The transition region is related to the Maxwelli'an part by a parameter ¡л which is relatively constant for a large class of reactors. The proposed formalism unifies the concepts of ' conventional flux density', 'effective cross-section' and ' resonance integral'. It is based 84 : CHAPTER IV1 on integral reaction rate, is applicable to any nuclear reactant but abandons essentially the cadmium ratio concept. According to the assumptions, the flux density function cp(E) is defined by

cp(E) = FM + F, + F1/e (IV. 13) where Fm is the normal Maxwellian component

FM = 2тг"*Ет"3/2Е exp(-E/ET) for Д<Е<оо

F, is the transition, or joining, component expressed by

Fj = 0 for 0

Fj = ii [(a-W^E-Ex) for Ex

1 Fj = *k [ hET J" for aET

Fj = 0 for hET

F1/e is the 1/E component expressed by

F1/e = FjE"1 for hET

Here the quantity Fj is equal to the quantity в introduced before. The parameters a and h used in the definition of Fj are derived from time- of-flight measurements on a reactor flux density. Nisle takes a = 36 and h = 5. 0 (corresponding to а ц of 2. 82). The conventional flux density has the value nv0 independently of the values of Fa and T. But the Maxwellian and the non-Maxwellian, which includes 1/E and transition components, are not independent of F and T:

ф (Maxwellian) = nv0 [ 1 + 2Fj {ц ET)"* ]_1

Ф (non-Maxwellian) = nv0 2Гх(дЕт)"^ [ 1 + 2Fa (Et)"*]"1

For a material the reaction rate is given by

reaction rate = N nvQa P(Fa, T) where P(F1; T) is a dimensionless effective cross-section factor and equal to the reaction rate per second per target nucleus per unit cross-section per unit conventional flux density. Tables of P^, T) values for some important materials are given in Nisle1 s publications. Expressions for the Maxwellian reaction rate, the non-Maxwellian reaction rate and the resonance integrals I and I' can easily be derived. These expressions contain such factors as P(F1( T) and Fj^Ej). The factors PfFj.T) can be calculated from cross-section data or can be derived from measurements, provided that one reference value of P(Fj, T) of a reference material is available at known values of and T. Activation, reactivity or radiochemical methods or chamber type detectors can be used to measure P(F1; T). Nisle suggests that his formalism provides the basis for the development of a universal flux density detector capable of measuring with one INTERMEDIATE NEUTRONS 85

irradiation all three flux spectrum parameters (Fj, T, nvQ). Two elements must be used to obtain these three parameters: a lutetium/copper, lutetium/manganese or lutetium/cobalt system. For successful application to this purpose, it is required that the nuclear data are better known. The unified formalism of Nisle is very interesting and promising, but it has not yet been used in other publications up till now. The importance is such that it certainly will require careful attention and further study.

IV. 2.2. Other spectrum representations

The l/E1+m spectrum representation

As mentioned above (section IV. 2.1.), the 1/E spectrum can only be expected under certain conditions (called a. . . . e) that are not always satisfied in practice. Recently, Schumann and Albert [11] have given a qualitative description of the deviations from the 1/E shape in the energy region from 1 to 500 eV that can be expected when the assumptions are not valid any more. Their arguments are repeated here. A violation of requirement (a) discussed in section IV. 2. 1, that the medium is homogeneous and infinite, has no strong effect. For a finite heterogeneous slab-type reactor the neutron spectrum in the moderator can be written as

(IV. 14)

where i denotes the linear dimension of the fuel region, D the diffusion coefficient and ?£s the slowing-down power. If the fast neutron sources are not homogeneously distributed (as assumed under (b)) but localized, then for large distances to the sources the spectrum decreases steepèr than corresponds with the 1/E shape. When the moderator atoms do not behave as free particles, e. g. by the effect of chemical bonds or crystal structure, we have in general the case that the slowing-down power of the moderator decreases towards lower energies, so that the spectrum in the lower-energy region increases for lower energies with respect to a 1/E spectrum. This effect becomes significant below 1 eV, but is still present in the region of dissociation energies at 5 eV. In the presence of 1/v absorbers lower-energy neutrons are absorbed preferentially and the modified spectrum falls off in the lower- energy region towards lower energies with respect to a 1/E spectrum. Both effects (1/v absorption and chemical bonds) have been described by the relations

with

a = 3£a(ZI (kT)i

and

b = kT [Ь(2^Ш)2+ С] 86 : CHAPTER IV1 where ay is the scattering cross-section of the free atom, T is the moderator temperature, к the .Boltzmann-constant, and С a constant that is equal to unity for free moderator atoms but taking larger values as the effect of chemical binding becomes stronger. • Assuming that the flux density depressions in the neighbourhood of the resonance peaks can be described qualitatively in this way, and that the energy distribution of the intermediate neutrons in.the resonance region falls off weaker than 1/E, the influence of the absorption can be observed in various isolated resonances. The flux density just below the resonance peaks does not reach the same high value as in the case without resonance, absorptions. When the moderating atoms have a mass number greater than unity, then there is a violation of assumption -(e) (section IV. 2. 1)'. There is, however, no effect in the lower energy region, but only in the region above 100 keV. Schumann and Albert [ 1.1] conclude that no significant deviations from the 1/E distribution occur in the region 1 to 500 eV, even when assumptions (a)...... (e) are violated strongly. . The . deviations can be described by writing cp(E)- = l/E1+m,. where m is a constant f 0j except perhaps for the lowest energy part of the intermediate region. • The l/E1+m spectrum has been used by.several authors. Bigham and Pearce [ 12] performed experimental and theoretical work for the ZEEP reactor in Canada, on the same line as mentioned above. Their conclusions are: (1) that resonance capture causes depression in the flux density near the uranium; and (2) that the spatial non-uniformity of the fission source distribution causes an excess of high-energy neutrons close to the fuel rod and a deficit at the cell boundary. This effect increases with rod spacing and is particularly pronounced in the case of a vacant site which shows a deficit of high-energy neutrons. In a reflector, the spectrum is expected to fall below 1/E at high energies because of the absence of fission neutron sources. This deviation from the 1/E form should increase with increasing distance. The results of Bigham and Pearce from measurements with indium, gold and manganese detectors in the graphite reflector of ZEEP are consistent with a spectrum of the form 1/E1'13 in the outer region. Connolly et al. [ 13] remarks that the value of the reduced resonance integral for higher energy resonance is quite sensitive to the form of the intermediate neutron spectrum. Values of I' for gold, manganese and copper have been calculated, assuming a neutron flux density distribution given by l/E1"™, for values of'm from zero to 0. 05 in steps of 0. 01. These values, normalized to the value of I' for m = 0, are given in Table IV. I.

Semi-empirical spectrum representation

Recently Genthon [14,15] proposed a semi-empirical spectrum representation for the intermediate energy region. He mentions that below 300 keV the relation cp(E) = в/E can be checked experimentally and that above about 5 keV this relation does not hold any longer. In the fast region, say above 2 MeV, the spectrum can be represented with the formula

N(E) = С E0-5 exp (-J3E) (IV. 16) INTERMEDIATE NEUTRONS 87

TABLE IV. I. VARIATION OF THE REDUCED RESONANCE INTEGRAL WITH THE-SLOPE OF THE INTERMEDIATE NEUTRON DISTRIBUTION FUNCTION [13] : ' '

• . m mAu ®Mn _ . . 63Cu

0 1.0. 1.0 1.0

0. 01 . -1.018 -1.062. ; 1.072

0. 02 1.037 1.128 1.147 '

0. 03 1.056 1.198 1.228

0. 04 1.075 1. 272 1. 311

0. 05 1. 096 1.351 ; 1.405 '

where ¡3 = 0. 775 in case of a pure fission neutron spectrum. The constant С is a normalization constant. Between about 500 eV and about 1 MeV there is no good.detector for measuring the spectrum distribution. In general, the extrapolation of the 0/E relation to higher energies is insufficient. The approximation cp(E) = (0/E) [ C?(E0)/ES(E)]'can be used for a hydrogeneous medium with monoenergetic fast neutron source. Genthon considered therefore the expression (0/E)exp(WÊ), where the factor ex approximates very well the ratio. Es (Eo)/£s (E). This correction is, however, not sufficient in the region between 500 eV and 2 MeV, so an additional term was introduced which is determined by the fast neutron distribution. He used the following spectrum representation:

с (К) - 0/E • for 1 eV<Ë<5 eV

eex b ф (e) = p( ^) + (ф{ _ №) n(e) for 5 eV

ç(E) ф; N(K) for Ej

• .. ф (E) = фо(Е) +; фе(Е) '. ' (IV. 17) where 0

. - . (ф0(Е) ;= — exp(bNfe) for 5 eV

фо(Е) = 0F N(E) for Ej

фе(Е) = (фf - 6F) N(E) for 5 eV,< E 88 : CHAPTER IV1

He calls cpo(E) the homogeneous'component of the distribution. It corresponds in the case of a homogeneous medium with homogeneous sources. It depends on the parameter b and therefore on the type of moderator. The function cpe(E) represents a heterogeneous component. It is positive or negative and accounts for additional contributions (e.g. a flux converter) or for depressions (e. g. reflector) with respect to fission sources. Within this frame work only the quantities в and ф{ have to be determined experimentally (with a resonance detector and a threshold detector) while the general shape of cpo(E) and cpe(E) is related with the type of moderator.

IV. 3. DETECTORS

IV. 3. 1. Resonance integral cross-section

Theory of resonance integrals

According to the Breit-Wigner single level formula the radiative capture cross-section ay and the elastic scattering cross-section anin the vicinity of an isolated resonance are given by

ay(E) = (|)* (Е.Ег)^(г/2)» (IV" 18)

°n№) = 4 A' g I (Е-Е.У+ ir/2 + ¥ I + ^-e) 2

where 2^rXr is the neutron wavelength at resonance energy in centre-of-mass system; Гп is the probability per unit time interval for the compound nucleus to emit a neutron; Гу is the probability per unit time interval for the compound nucleus to de-excitate by emitting one or more photons; Г is the total probability per unit time interval to emit a photon or a particle (ris the total width on half the height of the resonance peak; it is therefore called the half-width and is equal to the sum of the partial half-widths); g is a statistical factor depending on the angular momentum J of the compound nucleus and the spin I of the original nucleus and that of the interacting neutron; it can be written as g = 1/2 (2 J + 1)/(2I + 1),J taking the values J = 1 ±i ; and R' is radius of a sphere where scattering is possible and penetration is impossible. It is of the same order of magnitude as the geometrical radius of the nucleus. Relation (IV. 19) for the elastic scattering cross-section may be written as

2 Г2 EE 2 *n(E) = **rg (E_Er)a l (r/2y¡ + 4^lgrnR- (E_Er)2 (r/2)a + 4ttR- (IV. 20)

In this form an interference term appears originating from resonance scattering represented by the first term on the right-hand side and the potential scattering which causes the constant term 4îrR|2, resulting from the scattering at a sphere (radius R' ) impossible to penetrate. If the process INTERMEDIATE NEUTRONS 89 of interaction of a neutron with a nucleus is limited to elastic scattering and radiative capture, which is nearly the case, the total cross-section is simply the sum of relation (IV. 18) giving ay and relation (IV. 20) giving a„. Assuming that in the neighbourhood of the resonance (Er/E)^ = 1, it follows that

crT(E) = an(E) + стг(Е) = тА2 g + (r/2)2

+ 4^2g TnR- (Е.Ег)2Е;Е(Гг/2)2 + ^R'2 (IV. 21)

Denoting the maximum values reached at energy Er with the subscript zero:

/I v2 rn E ff0 = Sp % = ст0 Г ' CT°n - a0 "f

it follows that

= 1+{2(ЕСТ-Ег)/Г}2

(E) - + 2°0R' 2(Е-Ег)/Г o + 4^R,2 l+{ 2(Е-ЕГ)/Г}2 + JtRi 1+{ 2(Е-Ег)/Г}2

and

.. сто ; 2ct0R' 2(Е-Ег)/Г g ат(Е) • 1 +{ 2(E-Er)/Г}2 + *R- 1+{2(Е-Ег)/Г}2+ 47rK

crtotal = a(resonance) + a(interference) + a(potential)

All the expressions above are derived with the assumption that the nucleus is in rest. This is not always justified and therefore in some cases one has to apply a correction which, in analogy with the procedures in acoustics and optics, is called thé Doppler correction. References [16-18] should be considered for further discussion on the Doppler correction. Assuming a neutron spectrum represented by cp (E)dE = <9dE/E the number of interactions of a given type per second is given by

N jfcr(E) ep(E)dE = N в Дт (E) ^ (IV. 22)

kl Г dE

where J cr(E) — is a resonance integral cross-section designated by the

symbol I. To calculate I, an expression in known variables must be used. Changing cr(E) into the radiative capture cross-section for a given activation ffy(E) yields the activation integral, which need not to be equal to the absorption resonance integral. The contribution from the 1/v contribution of the cross-section will be taken separately. In the case of a narrow resonance Г is small and the function 1/E may be considered as a constant 90 : CHAPTER IV1 and brought therefore before the integral sign. Integration yields the well- known Breit-Wigner approximation of the resonance integral for isolated, narrow resonances ! = Д №) f =£1+{2(E%)/T}i f = I - 4w (IV. 23)

If Г is greater, i. e. if the function 1/E cannot be considered as a constant, we have to account for the variation of 1/E during the integration procedure. If in this case the integration is performed between the limits Ecd and oo, one obtains after some elementary calculations the following final expression

2 a 2 2 ti = t' n a-b , 2a ri , ^ ?, u, (IV. 24) 1 1BW + { а _7T r+Ti? m — ГТ4Г2 Ï ^(а-ьп where a = Er/17and b = Eqj/Г.

Calculation of resonance integrals

From the general single level Breit-Wigner formula it follows that

a°y ~ г ст°'

n. °Ьп " T CT0t

o-On = 47rX?g S-

Using'the relations ... , 2 4тг(кг)2 = = 2-60370 barn

I'aot = ^ = ™-0trr/2Er '

the following expression is obtained

I'act = (4. 10 X 106)g ГпГу/Ег2 (IV. 25)

where Er is the resonance energy, expressed in eV, and the T's are the level widths in eV. Another, less accurate, formula for the calculation of the resonance integral is given by Dresner [19].

I' = 27гаьг(Е0Ег)7Г (IV. 26)

where crot in the absorption cross-section for .2200 m/sec neutrons, E0 is 0. .025 eV and Er and Г are the neutron resonance energy and width of the INTERMEDIATE NEUTRONS 91 resonance. This formula is easily derived from the- general single level Breit-Wigner formula

Г„Г ay(E) = nkh E (E-Er)2 + (Г/2)2

Substitution of the values Er and E0, gives directly the ratio

aoy(Er) /Ед V (Ед ~jEr )2 + (Г/2)2 _ /EqV E* ffy(Eo) (Г/2)2 -\eJ (Г/2)2.

Substitution of ff0y(Et) from this expression in the relation

I act =™0уГ/2Ег gives directly the expression mentioned above.

The 1/v contribution to the resonance integral cross-section

The expression given above for the epicadmium resonance integral for one resonance peak does not contain the contribution of the 1/v part of the cross-section curve. If cr °cl/v, one can put a = OqV0/v (the subscript zero referring to the value for 2200 m/sec), giving

ff(E) /ЕЛ* dE ЕЛ* EflV = (ct 2a„ . fr / oV^fr e lW' E2 J J EJ EJ EI

a (Ej) - a (E2) (IV. 27)

For E9>>E, then

a(E)^f = 2cr (Ej) (IV. 28)

The lower limit Ex of the integration can be Ecd = 0. 5 eV, 0. 55 eV and 0. 68 eV. or E = 5 kT.

rf^F1 a(E) — = 0.90 a0 5 кт

cr(E)^ = 0.45 aQ

0.5 eV jp a (E) — = 0. 43 a0 0.55 eV

A(E) ^ = 0.39 CT0 0.68 eV 92 : CHAPTER IV1

Contributions to I from higher unresolved resonances

Sometimes it is necessary to evaluate also the contributions for all resonances at higher energies. Dresner [ 19, 20] has investigated the estimation of resonance capture integrals from measured average resonance parameters using average ratios of level widths to level spacings. Then the resonance integral is computed by summing over all the resolved levels and further for the unresolved levels by integrating over the higher- energy region (E >E' ). According to Dresner the integral over the energy region with unresolved resonances is given by

я I (E>E- ) = ME)f '(1.3X 106/E' ) (27гГу /D) (/3-ln(l + /3)//3) i E (IV. 29) where

/3 = (ГГ/ГП°)Е'*

and Гу and T°a denote the average radiative capture and reduced neutron widths, respectively, and D denotes the average energy spacing of resonances of the same spin and parity. Stoughton and Halperin [21] do not recommend this method of Dresner when the resonance parameter data are completely lacking. While the method of Dresner can be used to estimate resonance integrals under such circumstances to within a factor of three, the present use is to compare calculated resonance integrals with experimental values when the error . is expected to be relatively much smaller. An alternative method of determining the total resonance integral, if differential cross-section data are available, is simply to integrate graphically under the curve. In principle the graphical integration should give a more accurate result since the resonance parameters come from the same data and since approximations are used in calculating the integral from the resonance parameters.

The effective resonance integral

So far the following quantities have been introduced: "Л® f

ECd

^ =/«*№)§ -2oo(£f ECd ^^ satisfying the relation INTERMEDIATE NEUTRONS 93

These equations apply strictly only to 'infinitely1 thin detectors, i.e. thin detectors for which the self-shielding effect can be neglected. For detectors with finite thickness a s elf-shielding correction factor has to be applied. Moreover, the effect of the Doppler broadening of the resonance peak has to be taken into account. The self-shielding correction function depends on the total cross-section at(E) and the position X within the detector; the Doppler effect depends on the resonance energy Er, the temperature T and the mass number A. Taking both effects into account one obtains the effective resonance integral Ieff which depends on the size, shape and temperature of the materials as well as on its microscopic cross- section. Baumann [ 22] defines, therefore, the effective resonance integral averaged over the volume as

Zeff ^//С(ЕД<Т'А)°а'1Г X (IV-30)

vol Ecd

This equation is sometimes written as Ieff = Ge{{ I.

Measurement of resonance integrals

The resonance integral can be determined experimentally by two methods: the oscillation and the foil activation methods. The first method implies reactivity measurements with a reactor oscillator, while the second method implies foil activations to compare cadmium ratios for foils of unknown materials to the cadmium ratio of foils of a reference material, for which in general gold is employed. A brief discussion of the merits and difficulties of the different methods is given by Jirlow and Johansson [23] and in the Reactor Handbook [24]. In the second method the following expression may be Used for the reference foil

where the subscripts sub and epi refer to subcadmium and epicadmium quantities. With the superscripts r and x denoting the reference and the sample materials we obtain

YiV / i Y (Rçd-i)r

WJ ~WJ (Rcd-l)x

When thick foils are used the equation has to be modified to take into account correction factor for the finite thickness 94 : CHAPTER IV1

where Gepi and Gtt, are the epicadmium and subcadmium correction factors for thick foils; and F is a correction factor to account for the intermediate neutron absorption by the cadmium. Correction factors F for gold and indium for various thicknesses of cadmium have been published [ 25, 26]. Drake and Brown [27] described a modification of this method (the vanadium-subtraction technique, or 1/v subtraction technique) in which no cadmium covers are required. The captures due to the 1/v component of the cross-section of the unknown sample are determined by auxiliary activation of a 1/v absorber and are subtracted from the total capture to determine the net captures, due to the peak component of the cross-section. The activation of each nuclide is normalized by making exposures in a Maxwellian spectrum (standard pile) in addition to the spectrum of interest. They choose vanadium, which has a cross-section that approaches a 1/v dependence on energy to within a few per .cent, to determine the 1/v captures. The resonance integral in the material of interest is then measured relative to that for gold. This technique, therefore, requires measurements of the activation rate for three separate detectors (gold, vanadium and the sample) in each of the two neutron spectra.

IV. 3. 2. Detector response

The expression for the saturation activity induced in an activation detector is

Asat, = cr(Ë) cp(E) dE (IV. 31) о where N is the total number of target nuclei in the detector; a (E) is the differential cross-section for the reaction under consideration; and

a = A^/N = f a (E) ф (E) dE (IV. 32) 0

whére a denotes a specially defined specific activity (i.e. the saturation activity of the nuclide of interest in the detector, divided by the total number of target atoms in the detector). The expression at the right-hand side denotes the integral of the response function ст(Е)ф(Е) and is therefore called the response integral. For the gold and cobalt detectors the functions a(E) and а(Е)ф(Е) are shown in Figs IV. 1-4. In fact, it is this response integral which results directly from measurements with activation detectors.

Single foil without cadmium cover

When considering the case of one uncovered activation foil in a well-moderated reactor spectrum, the Westcott formalism can suitably be applied. One then obtains (see section IV. 2. 1)

a = Ajat /N = ф0ст INTERMEDIATE NEUTRONS 95

10'

2 3 45678 910' 2 3 4 5 6 78910* 3 4 5 6 76910 NEUTRON ENERGY (eV)

FIG. IV. 1. Cross-section curve for the 197Au(n, y)18,Au reaction.

where cp0 denotes the 2200. m/sec flux density; and

â -

For somewhat thicker foils corrections have to be applied for self-shielding. In this case, the above equation becomes

a = cp0o-o(Gth g+ г Gint s) (IV. 33) where Gth and G¡nt denote the self-shielding correction functions for thermal and intermediate neutrons respectively. The response due to intermediate neutrons can only be separated from the response due to thermal plus intermediate neutrons when special techniques are applied, such as the cadmium- ratio technique and the two-foil technique to be discussed later. , 96 : CHAPTER IV1 INTERMEDIATE NEUTRONS 97

Ь 2

s- 10- 89 S

3 i 5 E 7 8910' 2 3 ( 5 6 7 8910* 3 4 5 6 7 69103 NEUTRON ENERGY (eV)

FIG. IV. 4. Response curve for the 59Co(n, y)60Co reaction.

Single foil under cadmium cover

When considering the case of a cadmium-covered activation foil in a reactor spectrum where the intermediate neutrons with energies above the cadmium cut-off energy obey a 1/E spectrum, one-obtains with the assumption

ф (E) = 0/E for E>Ecd

the expression

« = Asat/N = в Jo (E) ~ = 01 = 0 (IVv + I- ) Ecd

. where I is the total resonance activation integral; I1(/y is the 1/v contribution to I; and I' is the peak activation integral. In general, for thick foils,

a = G 0 (Gl/vV+G'1') (IV. 34) 98 : CHAPTER IV1

Definition of cadmium ratio

If an activation foil is irradiated in a reactor neutron spectrum, the induced activity will not only be caused by thermal neutrons, but also partly by intermediate neutrons. To distinguish between these two activations, the detector material is often enclosed in a cadmium cover. The curve of the absorption cross-section for cadmium depends in such a way on neutron energy that at a certain thickness of the cadmium (about 0. 75 mm) nearly all neutrons below a cut-off energy Ecd (about 0. 5 eV) are absorbed, while nearly all neutrons above this energy pass the cadmium without appreciable capture (see also the section on cadmium cut-off). An uncovered foil will be activated by both thermal and intermediate neutrons, while a foil covered with cadmium will be activated only by intermediate neutrons. The cadmium ratio Red is defined as

response of foil without cadmium cover RCd response of same foil with cadmium cover

д _ reaction rate in foil without cadmium cover cd reaction rate in same foil with cadmium cover

Using a simple step function with the step at E = /лкТ for the joining of the intermediate flux density cpint to the Maxwellian flux density cpj^^E) one can write now

f Ф (E) a (E)dE J фШх(Е) a (E)dE + f 9¡nt (E) a (E)dE

Red Л =° ^

J ф (E) a (E)dE J q}nt (E)or (E)dE (IV. 35) Ecd Ecd

Ecd

JФмах(Е) a (E)dE + f

q5int(E)

ECd

If one considers a material which obeys the 1/v law below the cadmium cut-off the intermediate neutron activation under a cadmium cover can be neglected with respect to the total activation. For, the ratio of the second term in the numerator to the first term can be shown to be equal to 0.45 б/фо (with ц = 5) and in the core of a reactor 0 is always much less than ф0. Here it was assumed that фш, (E)dE = 0dE/E. It follows that

Rcd_i = y°gg° (IV. 36)

ECd INTERMEDIATE NEUTRONS 99

The EANDC1 recommends the value of 0. 55 eV to be used as lower limit in the epicadmium resonance integral cross-sections [28]. It should be noted that this integral cross-section cannot be derived accurately from measurements with a cadmium filter unless the energy dependence of cr (E) is known in the neighbourhood of 0. 5 eV. If it is not known, the experimenter will usually have to assume 1/v variation. In principle, the cadmium cut-off energy depends on (a) the form and the. dimensions of the cadmium box; • (b) the angular and spectral distribution of the neutrons; and (c) the absorption cross-section and the geometrical shape of the detector.

Definition of cadmium cut-off

Cadmium is widely used as a filter for thermal neutrons in the measurement of reaction rates for intermediate reactor neutrons. There- fore, the lower limit of the epicadmium resonance activation integral is an effective cut-off energy that is in general dependent on the thickness and shape of the cadmium cover and also on the extent of anisotropy of the neutron flux density. Following Dayton and Pettus [29], the effective cut-off energy is defined as the cut-off of a perfect (infinitely sharp) filter that allows the same total number absorptions by the filtered sample as does the given cadmium filter. Suitable cadmium filters terminate the intermediate neutron energy spectra at about 0.55 eV. This is sufficiently high to exclude most of the low-energy deviations of

Cadmium ratio method with 6/E formulation

In a previous section (see Eq.IV.36) the following formula was derived

ECd where g = 1 if the cross-section obeys the 1/v law below the cadiyiium cut-off. Often one follows Dancoff [30] in his modification of this

1 European-American Nuclear Data Committee. 100 : CHAPTER IV1

TABLE IV. II. EFFECTIVE CADMIUM CUT-OFF FOR 1/v ABSORBERS [28]

Cadmium thickness Collimated (mm) neutron beam Isotropic neutron flux density

Small Small Foil in sample in sample in cadmium spherical cylindrical Plane filter sandwich shell shell (eV) (eV) (eV) (eV)

0.76 0.473 0. 62 0.476 0.50

1. 02 0. 512 0. 68 0. 518 0. 55

1.52 0.567 0. 77 0.583 0.62

3 m

1.0 1.5 CADMIUM THICKNESS(mm)

FIG. IV. 5. Effective cadmium cut-off energies for thin 1/v detectors (based on data from Ref. [ 6]).

expression, introducing a parameter a denoting the peak-to-residual ratio and given by

1/v

Since

one obtains the relation

E Cd Y 1 (IV.37) E 2(Rcd- l)(l+o) V E INTERMEDIATE NEUTRONS 101

It must be kept in mind that these relations apply only to thin detectors for which the self-shielding effect can be neglected. For thick foils cr0 and I have to be multiplied with appropriate correction functions G. In the following a detailed consideration of the application of correction functions for the case of thick foils is given. Although the absorption cross-section of cadmium decreases appreciably in the neighbourhood of the cadmium cut-off (which normally is situated between 0.4 and 0.6 eV), the neutrons of energies above this cut-off have a certain probability to be absorbed in the cadmium cover used in cadmium ratio determinations. The count rate Repi of a cadmium covered foil is therefore not exactly equal to Rjnt, the count rate of the bare foil induced by intermediate neutrons. For this reason one sometimes introduces a correction factor Fcd, which is defined by the relation

For an extremely thin foil, for which there occurs no flux density depression, no self-shielding and no self absorption, the following relation holds:

Aih (0) Rth (0) Rth(°) y0crm(kT)

Aint(0) Rint(0) Repi(0)F¿d(0) = Г (E dE

The symbol 0 between brackets indicates that the quantity of interest is considered for the case of extremely thin foils ('zero thickness1 case). In many cases the foils cannot be considered as having this zero thickness and then corrections have to be applied. For these thicker foils one can define two ratios (both less than 1):

t) r^ m ( j r m Repi Ct)

These ratios are, actually, correction functions dependent upon the thickness t of the foil. In these correction functions G are now included the

(a) Correction factor for the flux density depression (which is dependent upon the surrounding material) (b) The correction factor for self-shielding (c) The correction factor for self-absorption of the radiation under consideration (especially in (3-counting).

With aid of the relation

/о \ - _ Rth W _ Rth

^ Cd' " Repi (t) = R ePi(t) " Repi « Cd 102 : CHAPTER IV1 and the.expressions mentioned above one obtains the expression:

Rth(O) = Rthffl G epi w i r„ ... 1 G epi (t) i

Rint(0) " Repi(t) Gth(t) F¿d - \KCd"*Cdj Gth (t) ' F¿d

Д Gepi(t) F1 Cd

Gth (t) (IV. 39) Zktf Gepi (t) F' / Cd Ecd

If , Gth/Gepi and (Rcd) are known or have been determined, one can determine the parameter 6. The formula above using the G functions can only be used if experimental data on these functions are available. Walther [31] and Hohmann [32] give graphs showing the picture of the G functions for gold foils in water. For indium foils these data are given by Greenfield et al. [33]. Manner and Springer [34] have shown that these experimental Gepi(d) functions correspond very well with the self-shielding correction function G (т,(3). This method is not generally used for lack of sufficient experimental data. In many cases preference is given to a detailed consideration of the values of the separate correction factors constituting together the G correction function. There is, however, one promising application of this method.which is used, for instance, in the centre at Mol. Using a medium where the neutron spectrum is stable and very well known (e.g. a calibrated standard, graphite pile), one can determine an experimental conversion factor К given by

к q Gth(t)

о which is dependent on the detector material and the detector dimensions. For detectors calibrated once in this manner there is a simple relation between 0/<¿>o and the cadmium ratio: '

Under these conditions this method does not require exact-knowledge of the cross-section Value, the. resonance integral or the correction functions G. INTERMEDIATE NEUTRONS 103

Cadmium ratio relation between the ratio 0/cpoand rvT/T0

From the results of cadmium ratio measurements one can derive either the ratio 0/фоог the epithermal index r. The relation between these two quantities can easily be derived from the formulae for the cadmium ratio. In the Westcott notation one has

R = ÎK±SÊl = CTo(g + rs) (IV. 41) Cd , i oo ar* T

ECd

where К = i(?r Ecd/E0)^, from which it follows

EL Fl. ffo(g"K VT0 (Rcd-D = f

On the other hand, one has the expression

From the last two equations one finds the relation

1 Ta-i (IV. 42) г{фУ "К/ J-o '

Approximatively, one has g = 1 and К = 2 (for Eca = 0. 5 eV) so that

For the determination of the ratio of the intermediate neutron flux Two-foidensity lt ometho the conventionad without cadmiul neutromn coveflux rdensit y one normally uses the cadmium ratio for one foil. The cadmium depresses the thermal flux density, but gives also a perturbation in the intermediate neutron flux, density. In an alternative technique two foils with different cross-section curves are used, the.first with a more or less pure 1/v absorption, and the second with a large resonance peak so that its resonance integral is large compared to the thermal cross-section. 104 : CHAPTER IV1

The saturation activities per target atom for the materials 1 and 2 can be written as

= Ai/Ni = Si Gthi CToi Фа. + 9 GintA

«2= A2/N2 = §2 Gth2 a02 ФШ + 9 Gint2I2 or in the preferred Westcott notation as

ai = {gi Gthi + rj^ Gînti soi} (IV- 43)

°-2 = ФоCT 02{s i Gthl + rJ~Y0 Gint2 s02} (IV- 44)

In these equations ст0 denotes the 2200 m/sec cross-section, в the inter - mediate flux density per unit lethargy, g the Westcott g-factor r\ÍT/T0 and s0 the modified Westcott expressions. Corrections for flux density depression and self-shielding of thermal and intermediate neutrons can be accounted for by multiplying the ст0 and I or s0 values with appropriate correction factors (Gth and Gint respectively, derived from theory or experiment). Two linear equations in the two unknowns, cp0 and r \lT/T0 срц, can be solved giving

0l(Gint2 S02g02) -^(Glntl S01CT01>

ф° (giG,hl OT01) (G"Int2 s02 ст02) - (g2 Gth2 CT02) (G|ntl s01CT01)

G g о-г( ё i G,th l cr01j' - Q-!(g2 th2 02) Фо ~ ( g G g)(G- (g; thl u01' ^iiint 2 02 02' эг^шг^ог' intl 01 01'

The absolute determinations are in some cases difficult, or at least associated with large inaccuracies. This method can be modified when a well-known reference flux density is available. Let for instance a thermal column or a standard pile be available where the intermediate flux density is negligible with respect to the thermal neutron flux density. The quantities referring to the reference flux density are denoted with the superscript r

ï= n^oi {si GJhl + G-n'ti s01} (IV. 45)

0r = nff02{g^Gïh2 + soi}

These four equations can be solved very easily. For the two-foil method it is not essential but, however, very practical if one of the two detectors is a pure 1/v detector, as in that case the corresponding s0 is equal to zero. In practice the detector couples Cu-Au or Mn-Au have been applied. The main advantage of the two-foil method is the abandonment of cadmium boxes around the detectors, and in this way the difficulties with the perturbation by the cadmium and the cadmium cut-off energy are avoided. However, the correction for self-shielding remains important and also the proper choice INTERMEDIATE NEUTRONS 105 of the joining function. The accuracy of the two-foil method is comparable to the cadmium ratio method. Recently Hart and Cabell proposed the combination of the two reactions 59Co(n, 7 )60Co and 109Ag(n, 7 )110mAg, leading to two nuclides with a long half- life, especially for the determination of the thermal and intermediate fluences during long irradiations [35, 36]. To minimize self-shielding corrections, the use of Co-Al and Ag-Al alloys, containing about 1% Co and Ag respectively, is suggested. To apply this method directly the activities in the two detectors have to be determined absolutely. Because of the complex decay scheme of 110mAg and the related complex gamma-ray spectrum, the standardization of 110mAg sources is still not easily performed. There- fore Hart and Cabell performed relative measurements comparing the counting results obtained in an unknown spectrum with the counting results from a known spectrum, which latter spectrum is determined by separate experiment in which other detectors (e. g. gold) are also used. Standard- ization of no^Ag might perhaps be accomplished with specialized counting techniques (sum-coincidence method, semi-conductor counting technique).

Two-foil method with cadmium cover

The double-foil technique can also be formulated without using the Westcott formalism by introducing cp(E) = tf(E)/E. A 1/v detector and a resonance detector are irradiated simultaneously under a cadmium cover. In the resonance peak proper one may readily assume that the quantity 0(E) can be approximated by the constant value 0(Er). For the l/v detector one obtains the relation:

« 00

e1= / a(E)^pdE = a0E0i/ f§ dE

ECd ECd which, by virtue of the relation ст(Е) I 1 i/v - S iH^VE^ ECd can be written as

^I^/ffdE

ECd For the resonance detector one obtains

.2./a(E)^dE.0(Er)I¿+I2,1/v^/ ffdE

ECd Ecd

When the resonance integrals X//v, I2 jyv and ï2 are known and when the absolute specific saturation activities al and a2 are determined, the quantity

0(Er) may easily be found from the relation

0(Er) 1 2 J-2 J-l.l/v 106 : CHAPTER IV1

Triple-foil method

In this method three resonance detectors with the same thickness of the same material are irradiated under a cadmium cover. The neutrons having resonance energies are absorbed predominantly in the outer foils, while the l/v contribution to the activation is equal for all three foils. In the triple-foil techniques one eliminates the 1/v contribution to the activation by determining experimentally the difference in activation between an outer foil and the inner foil, in this way obtaining an activation contribution which is proportional to the pure resonance activation. The self-shielding however has to be taken into account. In practice the difference is taken between the average activation in the two outer foils and the activation in the inner foil; in this way one can eliminate a possible effect of a flux density gradient over the three detectors in the cadmium box. As this difference in activation is a response from the narrow resonance region, this method is particularly suitable for obtaining information on the spectral distribution of the intermediate neutrons when several resonance reactions are applied.

IV. 3. 3. Self-shielding effect

Flux density depression

The following remarks are taken from the article of Judd [37]. Flux density depression is important when dealing with activation by thermal neutrons since a thermal neutron that is absorbed in passing through the foil is not available to diffuse back into it. This gives rise to a depression within a distance from the foil comparable with the neutron mean free path in the diffusing medium. The situation is, however, different when resonance activation is considered. For a cross-section that consists only of a single narrow resonance (r«Ç Er) in the slowing-down part of the spectrum there is no effect of flux density depression. A neutron absorbed in the resonance peak region as it passed through the foil would not have been available for absorption on passing through a second time since to do so, it must suffer at least one scattering collision which would remove it from the resonance energy range. Therefore the flux density at, and below, an energy of about Er(l-§) will be perturbed, but not the flux at Е0 itself. There is a finite, however small, probability for neutrons to suffer a large number of small angle scatterings and to return with small total energy decrement, but this probability is itself small. The depression effect of one resonance on another at lower energy is small, provided the first resonance is narrow and the resonances are separated by 3 or 4 ÇEr. In practice, one usually finds that t(E) = tE(E) is small except at a resonance energy. At the resonance energy there is a small flux density depression effect due to the small, but non-zero, value of Tat energies Er(l+f) and above. Thus, as an ap- . proximation, it seems best to calculate the correction factor f from Tat an energy immediately above the resonance. Often one has f = 1 except at thermal energies. INTERMEDIATE NEUTRONS 107

S elf-shielding factor

The activation in a thin layer dx at depth x of an activation foil by neutron incident from one hemisphere with angles between 9 and 0+d0 is given by

dA=NS oact(E)cp(E) e"£t(E)x/cos0 IsinSdSdEdx (IV. 47) where N denotes the number of target atoms per unit volume; S denotes the surface area; o"act denotes the activation cross-section; ctj. denotes the total cross-section; and cp(E) is the isotropic spectral flux density. Let t denote the total thickness of the foil. If there is no self-shielding we have the following equation:

dA = NS aact(E)cp(E) |sin6dedE dx

The s elf-shielding factor G for an activation detector is defined as the ratio of the experimental activation to the theoretically expected activation without self-shielding, the latter activation being determined by the appropriately chosen activation cross-section and the flux density of neutrons at the surface of the detector. The s elf-shielding factor defined in this way is always less than unity. Assuming an isotropic flux density, one has generally

. t ti/2

J f f

/ J CTact(E)cp(E)sinededxdE E x=0 0=0

For monoenergetic neutrons, isotropically distributed, the following expression for the s elf-shielding factor can be derived [38]

1-(1-т)е-т-т2Е (т) 1-2Е,(т) ф) G= Гт (IV-49)

where En(x) = f(e"xt/tn)dt(general definition of exponential integral) and T = Et = t/X(the foil thickness, in units of mean free path). The calculation of self-shielding factors for resonance activation detectors is possible when some simplifying assumptions are made. The most common assumptions are the following: (a) The resonance activation detector is a thin slab or a very thin circular disc (b) The neutron flux density incident on this detector is either mono- directional or isotropic (c) The resonance peak is narrow, i. e. the flux density is practically constant in the resonance peak region (d) The 1/v activation contribution can be neglected with respect to the activation due to the peak' in the cross-section curve 108 : CHAPTER IV1

(e) The Doppler effect can be neglected, i. e. the Doppler width given by Д = 2(ЕгкТ/А)г is small with respect to the total peak width Г; in practice the condition is Д/Г <0. 5 (f) The resonance peak is not too close to the cadmium cut-off, so that the lower energy integration limit can be taken equal to -oo (g) The resonance peak is isolated, i. e. not perturbed by possible other resonance peaks (h) The scattering cross-section in the resonance region is much lower than the activation crossvsection. Taking into account these assumptions for the case of resonance neutrons, the formulas for the self-shielding factor in foils with thickness t are given in Table IV. III.

TABLE IV. III. FORMULAS FOR THE SELF-SHIELDING FACTORS, G

Case Normal incidence Isotropic incidence

Constant cross- section

г = Noabs t

1/v cross-section

r = Not(Eth/El)

E h = 0.025 eV

E1 = cut-off energy

B-W resonance cross-section

r = No t G(t)= 1 7 Hw^lfl 4 ot т/2 J

If the effect of self-absorption of the induced beta-activity must be taken into account, one can define a correction function G(t, /3) which corrects for both s elf-shielding and s elf-absorption and in which the parameter /3 is related to the absorption coefficient for the beta radiation. This case has been treated by Beckurts and Wirtz [38, 39] and by Manner and Springer [34]. If the induced activity in the detector can be measured with a gamma counter, there is no need to correct for self-absorption. For some of the above-mentioned cases the function G(t) is shown in Figs. IV. 6-8 (reproduced from Ref. [22]) where the function is denoted with the symbol F. INTERMEDIATE NEUTRONS 109

THERMAL ISOTROPIC X no- 4 a-CONSTANT JOE,ma* (RESONANCE)

' 0.886 0*k( MAXWELLIAN THERMAL) FIG. IV. 6. Self-shielding factors for thin 0.7 foils with no Doppler broadening.

X- NoQma*

FIG. IV. 7. Resonance self-shielding

factors for thick foils with ft,l no Doppler broadening. 110 : CHAPTER IV1

FIG. IV. 8. Resonance self-shielding factors for thick foils with Doppler broadening included.

Detailed self-shielding studies

An article by Roe [18] gives an extensive mathematical treatment of absorption of neutrons in Doppler broadened resonances. Selander in his report [40] presents a semi-analytical method for computing the spectrum of the average flux in thin slabs (i. e. foils) and the corresponding self- shielding factors for cases in which scattering in the resonance is significant. Brose [41] has performed an experimental and theoretical investigation of the resonance absorption in gold, uranium and thorium foils. He calculates the resonance integrals with the narrow-resonance approximation. By comparison with the resonance integral at infinite dilution, theoretical self-shielding factors are obtained. For gold and uranium there is a good agreement between theoretical and experimental results, while for thorium there are discrepancies which are possibly due to the inaccuracies in the values of the resonance parameters. INTERMEDIATE NEUTRONS 111

Summarizing the theoretical work on the self-shielding corrections for intermediate neutron flux densities [42], one has the following cases (using T=Et) (a) Beam flux density, without Doppler broadening

GH = e"17'2 (I^HI^) )

(b) Isotropic flux density without Doppler broadening

G(T) = ÇdoM + Iiis) )ds r/2 or ' G(T> =7/l/T « S W** 0 E/2 (c) Isotropic flux density, with Doppler broadening; see the graphs given by Stewart and Zweifel [43], and Baumann [22, 42], based upon the original work of Roe [18] (d) Isotropic flux density, with large resonance scattering. No general theory treating the case of s elf-shielding effect for nuclides with large resonance scattering (e. g. 55Mn, 59Co and 186W) has been found in the literature. For particular cases, numerical calculations have been performed, e.g. for 59Co [40, 44] and for 55Mn [40].

Comparison between theory and experiment

As mentioned above, the calculated self-shielding factors G(T) for the case where Doppler broadening and scattering in the resonance can be neglected are in good agreement with experimental values for In, Au and W. Calculations performed by Selander [40] for cobalt and manganese have been checked by Eastwood [45] for the case of cobalt. Many experimental data on self-shielding factors or correlated data such as effective resonance integrals can be found in the literature. Some recent references are as follows: indium [22, 46, 47]; gold [41, 47-51]; tungsten [47]; cobalt [40, 44, 45]; copper [22, 52]; manganese [40]; and molybdenum [22].

IV. 4. FLUENCE MEASUREMENTS

IV. 4. 1. Experimental details

Resonance reactions of interest

Resonance reactions which have been applied for the determination of the intermediate fluence are mentioned in Table IV. IV. For recent data on resonance parameters, thermal cross-sections, differential cross-sections and resonance integral cross-sections, one should refer to the compilation by Hughes et al. (BNL 325) and to the Nuclear Data Sheets. The second edition of Reactor Physics Constants (ANL 5800) gives a table with calculated and measured values for resonance integrals. When looking at 112 : CHAPTER IV1 resonance reactions which might be applied, one has to consider the following quantities: The resonance energy Er of the main resonance (see, e. g., Table IV. IV) The value of the resonance integral cross-sections (see Tables IV. V and VI) The value of the parameter s0, which is proportional to the ratio of the pure resonance cross-section and the thermal cross-section. In Table IV. VII one sees clearly that s0 decreases with increasing values of Er The ratio Гу/Г, i. e. the fraction of resonance reactions leading to resonance activation TABLE IV. IV. DATA ON RESONANCE REACTIONS

Reaction Abundance Resonance energy Half-life (eV)

191Ir(n, y)192Ir 37.3 0. 654, 74.2 d 5. 36 '

193Ir(n, y)194Ir 62.7 1.303 19.0 h

103Rh(n, y)104Rh 100 1.257 4.4 min

115In(n, y)nbIn 95.72 1.457 54 min

I81Ta(n, y)182Ta 99. 99 4.28 115 d

197Au(n,y)198Au 100 4. 906 2. 70 d

109Ag(n, y)»°Ag 48.65 5.20 253 d

152Sm(n, y)153Sm 26. 63 8. 01 46. 2 h

107Ag(n,y)I08Ag 51.53 16.5 2.3 min

186W (n, y)187W 28.4 18. 8 24 h

127I (n, y)128I 100 20.5 25 min

74Se(n, y)75Se 0. 87 27.0 120 d

,5As(n, y),6As 100 47 26. 5 h

139La(n, y)140La 99. 91 73.5 •4-0.2 h

»°Cd(n, у)шса 12.4 89 48.6 min

i9»Pt(n, y)i"Pt 7.21 95 30 min

59Co(n, y)60Co 100 132 5.26 lyr

«Cu(n, y)«Cu 30. 91 227 5.1 min

203Т1(П, y)204Tl 29.50 238 3.9 yr

T1Ga(n, y)72Ga 39.6 295 14.1 h

55Mn(n, у)яМп 100 337 2.56 h

»Mo(n, y)i°>Mo 9. 36 367 14.6 min

98Mo(n, y)99Mo 23.78 480 67 h

"Cu(n, y)"Cu 69. 09 580 12.8 h

80Se(n, y)81Se 49.8 2000 56. 8 min

MZn(n, y^Zn 48. 99 2750 245 d

23Na(n, y)MNa 100 2850 15. 0 h INTERMEDIATE NEUTRONS 113

TABLE IV. V. COMPILATION VALUES FOR RESONANCE INTEGRAL CROSS-SECTIONS The symbols с and m denote calculated and measured values respectively; So=-RrI'/> CTo.

Г I1/v I Cut-off (bam) (barn) (barn) ' (eV)

"Ma с 0.12 0.25 [53] 0.4

с 0.067 [22] 0.622

m ~0.21 ~0.21 ~0.21 [54] 0.6

ш ~0.24. [53] 0.4

m 0.22 i 0.10 [55]

m 0.18 [56]

m 0.075 i 0.01 0.30 i 0.01 [57] 0.5

m 0.075 1 0.01 0.28 [22] 0.622

с 12 5.9 [53] 0.4

с 17 [58] 0.4

с 0.,7 0 i 0.06 10.0 [13]

с 11,4 [59]

m ~5 ~4.8 -9.4 [54] 0.6

m 5 11.8 [53] 0.4

m 11.7 i 1.5 [60]

ш 4.512.5 [61]

m 15.6 1 0.6 [62] 0.55

m 0.,615i0.01 5 7.8 i 0.8 [63]

m 9.5 1 5.0 15.4 [64]

m 9.0 (5.0) 14.0 1 0.3 [17] 0.6

m 8.15 i 0.60 (6.05) 14.2 1 0.6 [57] 0.5

m 14.0 t65]

m 12.7 4 1.2 5.4 18.1 [66]

m (9.4 i 1.4) 5.6 1 0.07 15.0 i 1.4 [67] 0.56

с 51.0 i 3.5 [68]

с 67.0 1 5.5 [69]

с 63.2 [59]

m 27.5 13.7 41.2 [54] 0.6

m 16.3 49.3 [53] 0.4

ш 38.3 1 4.0 [60]

m 81 1 4 [70]

m 75 1 5 [71] 0.5

m -70 [64]

m 55.2 i 4.5 17.1 72.3 1 5.0 [57] 0.5

m 74.6 [65]

m 1..6 0 1 0.08 53.1 13.5 16.8 69.9 1 3.5 [72]

с 2.153 [13]

с 4.28 [73]

m 4.4 [53] 0.4

m 3.72 i 0.13 [74] 0.64 114 : CHAPTER IV1

TABLE IV. V (cont,)

I' •l/v I Cut-of Nuclide с or m s, Réf. (bam) (bam) (bam) (eV)

"Cu m 3.09 ± 0.15 '2.0 5.1 i 0.15 [57] 0.5 (cont. ) m 3.17 10.18 5.0 [22] 0.622

m 5.1 "'"[73]

"Cu m 0.9 2.2 [53] 0.4

m 1.38 i 0.23 0.9 2.3 i 0.23 .[54] 0.5

- m 1.82 i 0.21 [74] 0. 64

"A! с 38.5 [59]

m 29.9 1.6 31.5 [54] 0.6

m (34.6) (2.2) 36.8 [53] 0.4

"Mo с . 3.9 [22] 0.622 3.4 ± 0.9

m 6.69 i 0.3 [75] 0.5

m [57] 0.5

m i 9.9 i 1.1 [22] 0.622

100 Mo с .3.1 [22] 0.622 1 6.2 i 1.6

m 20. 5 1 1 3.73 i 0.20 [76] 0.5

m 4.06 i 0.23 [22] 0.622

'"Ag с 1206 37 [53] 0.4 с 1440 : [64]

с 17.10 1400 i 65 [77]

m 1174 39 1213 '[54] 0.6

m 1160 [53] 0.4

m 1910 [64]

m 18.74 (1465) [78]

"sIn с 1752 87 [53] 0.4

с 3050 i 150 [63]

с 3213 179]

(54 min) с 2603 [80]

с 3200 [81] 0.55

(54 min i 13 s) с 3190 [22] 0.622

с 3038 [13]

с 3370 [58]

с 3188 [59]

m 2294 78 2372 [54] 0.6

(54 min) m 2640 [53] 0.4

m 2340 1 200 [82]

(54 min 13 s) m 3500 i 250 [61]

(54 min + 13 s) m 3650 i 350 [62]

(54 min + 13 s) m 20.2 3530 i 100 [63]

(54 min) m 2630 i 133 [17] 0.6

• m 2595 — [59] 0.5 INTERMEDIATE NEUTRONS 115

TABLE IV. V (cont.)

Г 'l/v I Cut-oi Nuclide с or m Ref. So (barn) (bam) (barn) (eV)

llSIn (cont.)

(54 min + 13 s) m 3300 i 850 [83]

(54 min) m 2550 i 80 " . [22] 0.622 (13 s) m ¡ i 650 ± 30 [22] 0.622 (54 min + 13 s) m 3200 i 100 [22] 0.622

(54 min) m 2500 ±85 [81] 1.30

(13 s) m 690 i 45 ! [81] 1.30

(54 min + 13 s) m 3190 1 120 • [81] 1.30

(54 min + 13 s) m ... „• 3480 ± 120, . [81] 0.55

1KW с 180 [53] 0.4

с 10.7.4 1.0 456 i [13] с 481 [58]

с 276 [59]

m 306 14 320 [54] 0.6

m 340 15 i i 355 [53] 0.4

m 170 i 20 .. [84]

m 290 1 35 [60] m 379 396 i 59 [47] 0.6 m 486 ± 54 14 500 i 54 [22] 0.622

"'Au с 1296 [54]

с • • ' . -1390 [53] 0.4

с 1591 [64]

с 1529 i 70 [23]

с t 1665 ... [71]

с 1487 [41,50] 0.68 - ' • с 1566 [13]

с 1584 - [58] 0.4

с 1437 [41,50]

m ... . ,1296 .. . .. 37 1337 [54] 0.6

m 1326 [85]

m 1513 45 1558 [53] 0.4

m 1513 [64]

m 1461.8 1553 j • [52] 0.64

m 1490 i 40 ! 1535 i íp [23] m 1491 , 43 1534 i 40 [71] 0.5 m 1567 [41,50] 116 : CHAPTER IV1

-г в

Iи iо> INTERMEDIATE NEUTRONS 117

JT1!

9 н о

Й H о <й оg ю и й й о fe H тз а . С О h ni S о 0 9. о 01 ф Л 01 о" S,2 Й œ to H Ио ш1шшЗ H и £ •я о E-i и H с Ы> ш Mо ф о m -гЧ +фJ м ¡S о С ф й я) ф •н +-> СЮ i" > 01 С < z о ф 2 ел ф 1 о 1н > 01 С < •н 01 ni oí oa j +-te> о С Ю lO О h о и о 01 й •а ш hф й с о "à" 01 01 и с o o <—I •гч о ф с <ч t- +->Й- о о ^ ^ oo (N x o o , i с <м ф ф, t= о +J DO 00 tt > О ф ф .g «tí S ^ I—I и M & I .Q II II и m й £ H ni о •J ь НИ и < Л у У 118 : CHAPTER IV1

TABLE IV. VIII.! SOME CHARACTERISTIC RESONANCE PARAMETERS 4E kT The value tD = дрй determines the Doppler correction to the cross-section.

Nucllde (eV) (tan) (eV) (eV) (iv) ^/Г ^ 'd

ll5In 1.457 29000 72 X 10"s 3.04 X 30"3 75 '/ 10"J 0.96 0.34 0.23 3.86 81 0.319 9.12 80 1.73 12.1 140 0.112 23.0 1.18 39.9 •. 140 3.6 ; 46.3 0.43 48.6 0.65

197 Au 49.06 37 000 124 X 10"' 15.6 X 10"1 139.6 X 10"J 0.89 0.36 0.13 46.5 21 (125) 0.13 58.9 800 (125) ~ 10 61.2 170 109 79.4 (170) 16

109Ag 5.20 34 000 140 X 10"' 12.5 x 10"' 152.5X10"1 0.93 0.63 0.21 30.5 125 6.8 40.2 137 5.0 55.7 144 36.1

1,SW 18.8 1 400 45 X10"> 266 X 10"' 311 X10"J 0.14 0.65 0.11

"As 47 3 100 296 X 10"J 39 X 10"' 335 X 10J ; 0.88 P 3.74 0.58 ; 92 653 269* 16 . 0.94 8.61 1.56 253 1 160 '..33 0 .- 70 ' 0.58 16.87 2.18 319 " 300 327 340

я Co 132 9 700 0.416 4.78 5.2 0.08 ' ,P-86 0.0085 4220 225 110 "Ga 95. 2 700 0.300 0.07 0.37 0.81 . 7.2'.' 1,01 290 4 300 0.35 7.4 7.7 0.05 1.05 0.007Ó 380 " 3 100 0.35 3.3 '... 3.6 0.10 2.93 0.03 770 700 0. 35 0.85 0. 52 0.29 ' 16.60 0.72

"Mn 337 2 830 0.6 21 21.6 0.02 0.56 • Q.013 1080 420 0.6 14 14.6 0.04 2.71 0;009T 2360 702 0.6 340 340 <0.01 0|25 з:'82 x 10"s "Cu 577 ; 1 550 0.4 1.26 1.30 0.31 14.10 . o! 56 . 2040 626 . 0.4 .. 32.0 - 32.4 . 0.01 2.00 3.19ХД0"' 5340 , 240 0.4 75.6 ч •:' 76.0 <0.01 -2.23 1.52 X 10"1

"Na 2850 370 0.34 404.6 405 <0.001 0.61 7.8 X'10"' INTERMEDIATE NEUTRONS 119

The parameter 2Ef/AF, which gives an indication of the probability that scattering in the resonance region leads to activation (see Table IV. VIII) The value tD= 4Ег'кТ/(А + 1)Г2, which determines the Doppler correction to the cross-section (see Table-IV. VIII). "The'correction for seíf-shielding can'suitably be calculated in the casé T>> I^and -tpfc 0; When tDdeviates from zero, the graphs given by Roe [18] are helpful.- When resonance scattering cannot be neglected, the' self-shièld:ihg 'correction has to be determined'by complicated computer calculations or by elaborate experimental techniques. Some-remarks1 on particular resonance detectors are given in the following: '

The reactions 19*Ir (n, y)192[r and'193Ir (ri, т)1941г

The nuclide 191Ir has large resonance at 0. 654 and 5. 36 eV, while the other stable isotope 193ir has a resonance at 1. 303 eV. The resonance activations- at'the energies of 0. 654 and 1. 303 eV have been applied by Jakeman for the measurement of the flux densities at these energies,' which are situated around the resonance at 1. 05 eV of 240Pu, in order to determine the 240Pu resonance escape probability in plutonium-fuelled reactor assemblies. Since, in general, fluence measurements in this low-energy region are performed with other detectors, such.as indium "and. gold, the special application of the reactions with the iridium isotopes will not be further discussed here. -•-• '<'---

The reaction 115In (n,-y)116min '

The differentiàl Cross-section curve foi" indium shows many resonances up to 130 eV. ' The first and most dominant resonance is situated at an energy of "1. 457 eV. For this résonance we have -the'difficulty that now thfe wing of the résonance perturbs the 1/v cross-section párt also below the Cadmium cut-Off energy. The decay scheme of the l16In produced is complex (80% HSi^n with 54-min half-life and 20% 116In with 13-sec! half-life). Baumann [22] determined the ratio for the thermal and resonance activation cross-sections for the 13-sec and 54-min activities of 116In. His "result (based upon 5 measurements with different thickness) is

• Л - 1-027 ± 0.008 - o(13s)/a(54m)th - > •••

The reaction 197AU (n, Y)198AU ,,.

Gold is widely used as activation detector, because of its suitable half- life, its available purity and the convenient decay scheme of 198Àu which permits absolute activity measurements with the j3-y coincidence technique. The resonance energy is 4. 9 eV. The 220'0-m/sec cross-section and the résonánce integral are now well known and therefore gold serves as a reference material for determinations of resonance integral cross-sections. From' thé values for the quantities Гу/Г and 2Ег/АГ it follows that the 4. 9-eV resonance is nèarly a pure activation-resonance. Calculated and experimental values for the self-shielding factors agree very well [41, 50]. 120 : CHAPTER IV1

The reaction i°9Ag (n, 7)110mAg

The reaction 109Ag (n, 7)110nA.g, which has a resonance at 5. 20 eV, is very interesting because of the long half-life (253 d) of the activity produced and of the relatively large resonance integral. The long half-life makes silver attractive for application as integrator for intermediate neutron flux density, i. e. as a detector for the intermediate neutron fluence during long irradiations. For this purpose one can use the cadmium ratio method or the two-foil method, in the latter case in combination with cobalt. The cross-section for the burn-up of n°mAg by the reaction UOmAg(n, Y)luAghasbeenreportedtobe 82 ± llbarns [78]. The complex decay of 110nAg, with its many gamma transitions and the corresponding complex gamma-ray spectrum, makes the standardization of these sources rather difficult so that applications up until now involved only relative measurements, which means that only comparative measurements have been made. This reaction, originally proposed by Hart et al. [35], has been studied further by Cabell [36] who applied a Ag-Al alloy (with 1% Ag) to reduce the self-shielding effect. When international standards of 110mAg sources are made available, this reaction might find more applications on a routine scale.

1S7 The reaction 186w (n, y) W shows a dominant resonance at 18. 8 eV and gives a suitable half-life (24 h). Baumann [22] compared the calculated self- shielding factors with his experimental results. The agreement was good, except for some intermediate foil thicknesses. Although this stated dis- • crepancy may be due to an error in the experiments, Baumann thinks it more likely due to the neglect of resonance scattering in the calculations. Nevertheless, an investigation of possible sources of experimental errors disclosed that shielding by cadmium, due to its resonance energy at 18. 5 eV, could give an erroneously low value for the measured integral. He, therefore, also applied boron filters.

The reaction 75As (n, y)76As

The reaction 75As (n, y)76As shows many resonances. The values of Гу/Г and 2Ег/АГ for the three lowest resonances (at 47, 92 and 253 eV) show that these resonances are mainly activation resonances, as Гу> >Гп and 2ЕГ/АГ>1. The self-shielding corrections can, therefore, be calculated with the analytical formula given in section IV. 3. 3. No experimental self-shielding data have been found in the literature.

TiL_e.r?action_7_1G_a_ (n,jy)_7fGa_

From the values for Гг/Г for the four resonances of 71Ga, situated at 95, 290, 380 and 770 eV, it follows that only for the 95-eV resonance is the probability for activation greater than the probability for scattering. For all resonances we have the condition that scattered neutrons do not contribute to activation. No experimental self-shielding correction data have been found in the literature. This reaction has not found wide applications. INTERMEDIATE NEUTRONS 121

Th_e_ r eaction_5_9C_о ( n,_ т)6_°Сo_

The reaction 59Co (n, 7)60 Co has two dominant resonances at 132 and 42'20 eV, of which the first is the most interesting. Resonance data have been determined by several authors. The recent data published by East- wood [72] seem to be the most accurate. The activation at the 132-eV resonance is complicated as the chance for scattering of the neutrons at this energy is large (Гу/Г = 0. 08), while moreover the scattered neutrons have a large chance to contribute to the activation (2Ег/АГ = 0. 11). There- fore the self-shielding correction cannot be simply calculated with an analytical formula. The experimental s elf-shielding data of Eastwood are in good agreement with the computer calculations performed by Selander [40]. The long half-life of 60Co (5. 24 yr) makes Co a suitable detector of the neutron fluence during long irradiations.

The reaction 55Mn (n, y)56Mn

Manganese has a first resonance at 337 eV. The l/v part of the resonance integral is nearly one half of the total resonance integral. The scattering contribution to the resonance activation cannot be neglected. The magnitudes of the quantities rr/Tand 2Er/Arfor the 337-eV resonance show that the resonance has the same character as the 132-eV resonance of 59Co. The second resonance at 1080 eV scatters the neutrons away from the resonance, as 2Ег/АГ = 2. 71. Self-shielding corrections have been calculated by Selander [40].

The reactions 63Qu (n, t)64Cu and 65Cu (_n, t)66Cu

With copper asvdetector, one has mainly the reactions 63Cu (n, 7) and 65Cu (n, 7) producing 64Cu with a half-life of 12. 8 h and 66Cu with a half-life of 5. 15 min, respectively. The first reaction has resonances at 580 and 2000 eV and the second at 227 and 2800 eV.

The reactions 98Mo (п,т)ИМо and 100Mo (п,т)101Мо

Only the molybdenum isotopes 98Mo and 100Mo show detectable resonance activation. Both isotopes have single tabulated resonances at an energy of several hundred electron volts which in fact make them very interesting. Baumann [22] has recently performed experiments with these two reactions. The counting procedure is somewhat complex because of the two-step decay schemes of the product nuclides. Baumann pointed out that the measured resonance integral of 98Mo greatly exceeded the value calculated from the parameters of the single tabulated resonance. He performed measurements with successive thickness of boron shielding to try to locate the difficulty. These experiments showed that a major portion of the 98Mo resonance integral originated at an energy in the vicinity of 100 eV, well below the tabulated resonance of 480 eV. There were also indications that 100Mo has significant resonance integral contributions at energies much higher than the tabulated resonance at 367 eV- He concludes that until the unknown resonances are identified neither of these molybdenum isotopes appears useful as a resonance detector. 122 : CHAPTER IV1

The reaction 23Na (п,7)24Ма_

Baumann [22] remarked that sodium has two interesting applications for reactor spectrum measurements. First, it approximates .the ideal l/v detector better than any other activation material since the absorption from the.single strong resonance is weak compared to the 1/v absorption,, and since that resonance is at the relatively high energy of 2. 85 keV. Secondly, the presencë of a single resonance at this high energy makes it a useful detector for spectrum measurements in the kilov.olt region. Un- fortunately, the features that make it useful for one of the.two applications are undesirable for the other. Cox [88] discusses the available data on the resonance parameters and points out that from the discrepancy between the experimental and computed cross-section curves, assuming a single isolated resonance at 2. 85 keV, it may be concluded that there is a negative contribution to the resonance. This complication makes the application of the sodium resonance for flux density measurements rather doubtful. .

Choice of detectors

The choice of a resonance detector is determined by. requirements concerning the material, the reaction data, and the product nuclide. Some general requirements can be written down as follows: The material should contain only one or predominantly one stable isotope so that possibilities for competing reactions with isotopes,in the material are minimized , The material should be available with as high a purity as possible in order to minimize perturbing reactions with impurities The material should be commercially available and not too expensive The material should be simple to handle and safe to use, preferably in metallic form ' ••• The material should hâve possibilities for preparing very thin detectors or for making dilute alloys or mixtures with a suitable carrier'material, in order to minimize self-shielding corrections 1 • The reaction should have a well-known cf(E) curve, consisting of a l/v part and preferably only one and narrow resonance peak, while the resonance scattering contribution to the total cross-section should be negligible • , The reaction should have a well-known resonance activation integral cross-section ' The reaction should have a convenient resonance energy and a convenient magnitude of the integral cross-section ' The product nuclide should have a well-known — and preferably simple — disintegration scheme- The product nuclide should have a suitable half-life • The product nuclide should be the only remaining longer-lived radioactive-one produced by irradiation so that counting should be possible without chemical separation ' • - The'product nuclide must have possibilities for rather simple'absolute ' activity determination, preferably with gamma-counting or coincidence- • • counting techniques, so that corrections for absorption and self- • absorption of /З-particles are not required. INTERMEDIATE- NEUTRONS 123

Because of-this long list of requirements, the number of resonance reactions in use or in development is much smaller, than the number Of reactions listed in Table IV. IV. In practice only a rather limited number of resonance reactions is applied because of the requirements on .the resonance parameters and the availability of accurate resonance activation integrals. ... For the determination of intermediate fluence during irradiations of several days or longer only the reactions 109Ag (n, y) 110nAg (half-life 253 days, resonance energy 5. 20) and 59Co (n, -y)60Co (half-life 5. 26 years,

EURATOM BOX FOR 3 mm 0 FOILS

FIG. IV. 9. Recommendations for cadmium boxes. 124 : CHAPTER IV1 resonance energy 132 eV) are suitable, as the reaction 64Zn (n, 7)65Zn (half- life 245 days, resonance energy 2 750 eV) has too low a value for the parameter s0. If the two-foil method is applied for fluence determinations in long irradiations, the detector couple Ag and Co seems most suitable. However, the absolute activity determination for 110mAg sources remains a problem when no calibrated sources are available.

Detector geometry and disposition

In practice, mainly very thin plane circular metallic foils or pressed pellets are used. The thickness is chosen in such a way that corrections for self-shielding are not required or can be applied with sufficient accuracy. The diameter of the disc-shaped detector is not a very important parameter. It is chosen in such a way that it meets the requirements of the space available and that the resulting induced activity is in a range suitable for easy counting with the counting equipment available. Sometimes one is interested in relative distributions along an axial or radial line. In these cases long but very thin wires are applied as activation detector. When cadmium covers are used as enclosures of resonance foils, special attention should be given to the dimensions as the cadmium cut-off energy is dependent among other things on the dimensions of the cadmium cover. The countries of the Euratom Community follow the recommendations of the Euratom Working Group on Reactor Dosimetry for the cadmium covers. Recently the American Society for Testing and Materials has also made a proposal concerning the cadmium cover (see Fig. IV. 9). As cadmium depresses the thermal flux density strongly, no thermal activation detectors should be positioned in the immediate surroundings of a cadmium cover. To avoid positioning errors in cadmium ratio measurements, the bare foil should be placed in an aluminium box, similar to the cadmium cover. When a series of detectors is irradiated, one should give proper attention to the position at which each detector is irradiated. To avoid interchange of foils, the detectors themselves and also the detector- holders should, if possible, be marked in a suitable way to identify them, not only before but also after irradiation. When the detectors are irradiated for the determination of the fluence incident on a sample or experimental assembly during a long irradiation with possible variations in the flux density, the detectors should be positioned in or around the sample (or assembly).

Irradiation conditions

For every irradiation of activation detectors one should record: The location of the detectors in the reactor The irradiation interval, specifying start and end of this interval The power level history of the reactor during irradiation, specifying all rises to power and all changes of power level, including the shut- down periods. In some cases the following data are also important: The core configuration of the reactor The position of the control members Location and type of experimental samples or assemblies in the neighbourhood of the detectors. INTERMEDIATE- NEUTRONS 125

When a relatively short irradiation is performed at constant flux density between two reactor shut-down intervals, one should also record carefully the history of the rise to power, in order to be able to apply a correction for activation during start-up. When the rise to power occurs exponentially, it is sufficient to note the time constant or the doubling time. For cadmium ratio measurements the bare foil and the covered foil should be irradiated in separate runs at the same position under the same conditions.

Counting techniques

Absolute fluence measurements are always based on the determination of the absolute activity of the product nuclide in the activation detector. As radiation detectors ionization chambers, proportional counters, Geiger- Mïiller counters,, scintillation counters and semiconductor counters can be used. The choice of the radiation detector depends among others on the disintegration scheme of the radionuclide of interest, the radiation purity of the source and the availability of calibrated reference sources of the same radionuclide. The nuclides 198 Au and 60Co can be calibrated by the (3-7 coincidence counting technique and by the 7-7 coincidence counting technique respectively. In general, gamma-counting procedures are preferred to beta-counting procedures. Often a scintillation detector coupled to a multichannel analysing assembly is used for relative and absolute counting of gamma-emitting nuclides. With this instrumentation the so-called photopeak method for standardization of sources is used. The detector efficiency and the peak- to-total ratio can be determined with sources calibrated elsewhere or with another procedure, or are taken from the literature [89, 90]. For details on absolute activity determinations the reader is referred to special publications on this topic [91-93].

IV. 4. 2. Data reduction and treatment

Correction for activation during reactor start-up

For the derivation of the correction formula for the activation during start-up of the reactor, the following case for the neutron flux density as function of the time may be considered.

During the time interval Tt = the flux density has a stationary value cps. Just before time ^ the flux density rapidly rises (exponentially or nearly exponentially) until the value cps is reached at time tx. The activation induced during the shut-down of the reactor can be neglected as at normal shut-down the flux density decreases very fast. If activation during start-up can be neglected, the following relations hold:

dN ... N = N0 a cps-XN

N(t2) =¿ N0a9s(l-e"XTi) and

A(t2) = N0 (jcps (l-e*XTi ) (IV. 50) 126 : CHAPTER IV1 where No is the total number of target nuclei under consideration; N is the total number of product nuclei; cr is the cross-section value for the reaction under consideration; X is the decay constant of the product nuclide; T¡ is the irradiation time; and A is the activity of the-product nuclei (A =. NX). For the case where the activation during start-up must be taken into account. Eq. (IV. 50) may be written as

A(t2) = N0 CTs (l-e"XTi) (1 + F) (IV. 51) where the quantity F denotes the fractional additional activation. The factor (1+F) is called the correction factor for activation during rise to power. For evaluation of the correction one must know the power history before time t}.' In some cases the.reactor is brought to power with a constant doubling time and then the correction factor is easily determined.1-- As can be expected, the correction becomes negligible when the irradiation time is much shorter than the half-life. •

Self-shielding correction data

S elf-shielding data for indium foils, shown in Table IV. IX, are taken from the work of Baumann [22]. This table lists computed resonance and thermal self-shielding factors for 115In activations in an isotropic flux density in a cavity.

TABLE IV. IX. SELF-SHIELDING DATA FOR-INDIUM- FOILS [22].

Natural indium

foil thickness Gres Gth Gres/Gth 1 . (mg/cmz)

0. 05 0. 988 1,000- 0. 988

0.1 0.977 1.000 0. 977

0. 2 0. 959 0. 999 0.960

0.5 0.920 0. 998 0. 922

1. 0 0.868 0. 997 0. 870

2. 0 0.796. 0. 993 0.801

5. 0 0. 649 0. 987 0. 658

10 0.519 0. 976 0.531

20 0.400 ,.. 0. 956 0.417 • -

30 . ' 0.334 'T ' 0.939 1 0. 357 ' "fe

40 0.294 0. 924 0.319

60 0.243 0. 897 0.271

100 0.192 0. 850 0.226

150 • 0,156 0. 800 0.195

200 0,134 0. 759 0.177

250 0.120 0. 720 0.167 INTERMEDIATE- NEUTRONS 127

Table IV. X lists self-shièlding data for gold detectors as reported by Brose [41, 50]. In the theoretical calculations the contributions from 56 resolved resonances and also from the unresolved resonances have been taken into account. The following approximation has been given by Brose:

О a - b JS/M

with ,

a = 0. 0273 ± 0. 0012

b = 0. 0541 ± 0. 0004 (gi cm"1) valid for the range

1. 5 s \ S/U 's 5 (cm g-i)

•Recently some experimental self-shielding data for gold wires have been reported [51]. Table IV. XI is based on this communication. The Work of Jacks [47] oh tungsten s elf-shielding is listed in Tablé IV. XII. Data for cobalt are listed in Tables IV. XIII-XV and that for manganese foils in Table IV. XVI. Copper foil self-shielding data as repor-ted by Baumann [22] are listed in Table IV. XVII.

TABLE IV. X. SELF-SHIELDING DATA FOR GOLD FOILS [41, 50]

Foil I • G thickness Gtheo exp_ ^theo"Gexp^^exp (barn) (cm) (%)

2 X 10"6 1556.83 0. 9936

4 x 10"6 1550. 04 0. 9893

8 X И"6 1577. 91 0. 9815

2 x ю-5 1507.41 0. 9621 0. 9644 - 0.24

4 x 10"5 1465. 83 0. 9355 0. 9340 + 0.16

8 x 10"5 1398. 77 0. 8927 0. 8852 + 0.85

2 x ю-" • 1252.38 0. 7993 0. 7852 + 1.80

4 x 10"4 1088. 91 0. 6950 0.6836 + 1.66

8 x ю-4 890.482 0. 5683 0.5612 + 1.27

2 x ю-3 628.570 0.4012 0. 3952 + 1.51

4 tf 10"3 ' 468.493 0.2990 0. 3020 - 0.99'

8 x 10"3 347. 671 0.221?,. 0.2219 . - 0. 0036

2 x 10"2 ' 234. 983 0.1450 0.1505 - 0.'35

4 x 10"2 ; -- 175.237 0.1118

8 x ю-2' 130.996 0. 0836 128 : CHAPTER IV1

TABLE IV. XI. SELF-SHIELDING DATA FOR GOLD WIRES [51]

Wire diameter

Nominal Average Average G( (10~3 in. ) (10~3 in. ) (cm)

0.5 0. 505 0. 00128 0. 703

1.0 0. 98 0. 00249 0.552

2. 0 1.98 0. 00503 0.410

4. 0 4. 05 0.01029 0. 302

6.0 6. 02 0.01529 0. 258

8. 0 7.98 0.02027 0. 228

10. 0 10. 01 0. 02542 0.208

TABLE IV. XII. SELF-SHIELDING DATA FOR TUNGSTEN FOILS [47]

Thickness (rag/cm2) G

0 1.000

5 0. 957

10 0.912

20 0.824

30 .. 0.764

40 _ 0.711

60 0. 630

80 0.574

100 0.538

140 0.478

180 0.439

22 0 0.413

250 0.403

IV. 4. 3. Data reporting

With a view to facilitating a re-analysis of presented data on flux density or fluence values in the future by an independent investigator, sufficient details should be mentioned in the report of the measurements. The following list of items, to be included in a report, is based on the recommendations of the IAEA Panels on In-Pile Dosimetry [94] and on In-Pile Fluence Measurements and on the proposals of the American Society for Testing Materials [95]. INTERMEDIATE- NEUTRONS 129

Detector data: mass, dimensions, chemical and isotropic composition Positioning data: description or drawing of irradiation geometry and capsules, type and dimensions of detector holders, covers and shields Irradiation data: duration of irradiation, time from the end of irradiation until counting, counting time, reactor power variations during irradiation, temperature conditions of detector materials and the moderator in the vicinity of the test Counting data: equipment, counting efficiency, calibration methods, activity at the time of counting, the calculated saturation activity per gram of target material (specifying the decay constant used in the calculation), measured or estimated amounts of impurity-induced activities Correction data: corrections applied (e. g. for the effects of burn-up and self-shielding) Final data: values of cadmium ratio, values for flux density or fluence (specifying the integral cross-section values used and their origin), error values associated with the various measured and calculated quantities.

TABLE IV. XIII. SELF-SHIELDING DATA FOR COBALT FOILS [40]

Foil thickness G (132 eV) (in.) (cm)

0. 0004 0. 001016 0. 8264

0. 0010 0. 00254 0. 7000

0. 0025 0. 00635 0. 5470

0. 0050 0.0127 0.4395

0. 0075 0. 01905 0. 3831

0. 010 0. 0254 0. 3476

0.015 0. 0381 0.3028

0. 020 0. 0508 0.2744

TABLE IV. XIV. SELF-SHIELDING DATA FOR COBALT FOILS [45]

Foil thickness Gexp(132 eV) (in.) - (cm) (excluding 1/v contribution)

0. 0004 0. 001016 0. 84

0. 0009 0. 002286 0. 70

0.0010 ' •0.00254 0. 63

0. 0020 ' 0.00508 0. 59

.0, 0036 0. 00914 0.46

0. 0040 0.01016 0. 45 130 : CHAPTER IV1

TABLE IV. XV. SELF-SHIELDING DATA FOR COBALT WIRES [45]

Wire diameter Cobalt content Gexp(132 eV) Gexp (in.) (cm) (mass °]o)

0. 050 0.127 0.104 1. 00 1.00

0. 050 0.127 0. 976 0. 95 i 0. 04 0. 99 ± 0. 01

0. 001 0.00254 100 0. 81 ± 0. 03 0.99 ± 0. 02

0. 005 0.01270 ' 100 0. 52 ± 0 02 0. 97 ± 0. 01

0. 010 0.0254 100 0.42 i 0 02 0. 94 ± 0. 01

0. 015 0. 0381 100 0. 38 ± 0 01 0. 92 ± 0. 02

0. 020 0. 0508 100 0. 34 ± 0 01 0. 90 ± 0. 02

0. 025 0. 0635 100 0. 32 i 0. 01 0. 88 ± 0. 03

TABLE IV. XVI. SELF-SHIELDING DATA FOR MANGANESE FOILS [40]

Foil thickness G(337 eV) G(1080 eV) (in.) (cm)

0. 004 0.01016 0.778 0. 703

0. 005 0. 0127 0. 745 0. 667

0. 006 0. 01524 0.716 0. 638

0. 007 0.01778 0. 692 0. 613

0. 008 0. 02032 0.671 0.591

IV. 5. SPECTRUM MEASUREMENTS

As shown previously, the response of a resonance detector in a 1/E spectrum is governed by the expression

J cr(E) Ç = I = Ii/V + Г (IV. 52)

ECd where I denotes the total resonance activation integral; Ij/y denotes the 1/v contribution; and I' denotes the peak activation integral.

Only the part I' is related to(the response from a narrow energy region, determined by the slowing-down density at the resonance energy, while the l\/y part tends to camouflage this narrow response. Therefore good spectrum measurements require resonance detectors with a high V/l1/v ratio. But in saying this one must keep in mind that Ij/y and I' have thus far only been defined sharply for a 1/E spectrum. The relation I = I' + Ij/y , based on the assumption of a pure 1/E spectrum, does not refer exactly to the response of a resonance detector for the case of a non-l/E spectrum, as the activation due to the l/v part of the cross-section cannot be written in terms of the quantity Ijyv used up till now. The activation due to the peak part of the cross-section can still be written in terms of I1, as the flux density might be considered as constant in the narrow peak region. INTERMEDIATE- NEUTRONS 131

TABLE IV. XVII. SELF-SHIELDING DATA FOR COPPER FOILS [22]

Foil thickness Nuclide !¿ff G (in.)' (cm) (barn)

63Cu 0 0 3.17 1. 000

0. 00186 0. 00472 2.72 ± 0. 08 0. 858

. 0.00187 0. 00475, 2. 68 ± 0. 08 0. 846

0.00189 0. 00480 2. 82 ± 0. 08 0. 890

0.00191 0. 00485 2.87 ± 0. 08 0. 905

' 0. 00423 0.01074 2. 51 i 0. 07 0..792

0. 00444 • 0.01128 2. 42 ± 0. 07 0. 764

0. 0052 0. 01321 2.36 i 0. 06 0. 745

0. 0052 0. 01321 2.32 ± 0. 06 0. 732

0. 00870 0. 02210 2. 07 ± 0. 04 0. 653 .

0. 00885 0.02248 2. 08 ± 0. 04 0. 656

0. 0205 0. 05207 1. 63 ± 0. 04 0. 514

0. 0205 0. 05207 1.60 ± 0.04 0. 505

65Cu 0 0 1.39 1.000

0.00186 0. 00472 1. 32 ± 0. 05 0. 950

0. 00423 0. 01074 1,23 i 0.05 0.885

0. 01013 0. 02573 1.18 ± 0,05 0.849

0. 01981 0. 05032 1.06 ± 0.05 0.763

Assuming a relation of the type cp(E) = 0(E)/E where в is now a slowly varying function of energy, One can write: CO

a - J o(E)tp(E)dE = 0(Er) I1 + J 0(E) a(E) ^ (IV. 53)

ECd ®Cd To determine the quantity 0(E) at a resonance energy for an unknown spectrum, one has to adopt a procedure for the evaluation of the 1/v part of the response: either by introducing approximations, or by making some assumptions on the shape of the spectrum, or by applying special techniques. Several techniques which have been used are listed,as follows: (a) The one-parameter spectrum method [96] ^ (b) The method of the semi-differential epithermal index [5] (c) The triple-foil technique [97] (d) The polygonal spectrum method [15]. The first method, in which the spectrum representation cp(E) = 0/E1+m is applied, is based on a numerical iterative procedure for the evaluation of the l/v contribution to the response integral. The second method uses the Westcott formalism, but redefines the quantity r and the dependent quantity s (which is related to the resonance integral cross-section), making these 132 : CHAPTER IV1 quantities velocity (i. e. energy) dependent. The third method also called "sandwich method" eliminates experimentally the 1/v contribution to the activation. The fourth method calculates the 1/v contribution to the activation and uses thereby a special model for the variation of the flux density per unit lethargy.

IV. 5. 1. Experimental details

The ideal resonance detector for intermediate spectrum measurements has a well-defined single and narrow resonance peak with a very small 1/v contribution. With several of these detectors it is possible to determine 6(E)%or r(E) at several energy bands. The resonance energy regions should be as narrow as possible and should be distributed with adequate intervals for the purpose in question. However at higher energies the ratio I'/11/v and the related quantity s0 = (2/^)1-/ s0 becomes smaller and smaller with increasing energy. Recently spectrum measurements with several detectors have been published [5, 11, 12, 15, 96-99]. The most common detectors for spectrum measurements are:

Nuclide Resonance energy

115In __ 1. 47 eV

197AU 4. 90 eV

186W 18. 8 eV

59Co 132 eV

55Mn 337 eV

The experimental procedure for spectrum measurements are in general the same as for fluence measurements. In planning experiments one should give special attention to the s elf-shielding corrections, trying to make the. detectors as thin as possible.

IV. 6. OTHER METHODS

IV. 6. 1. Recent developments with resonance detectors

Application of other shields than cadmium

In the reactor laboratories cadmium filters are used widely. Effective cadmium cut-off energies for point 1/v absorbers inside of spherical and cylindrical cadmium filters have been, calculated by Stoughton and Halperin [21]. They also published a compilation of cut-off energies for l/v detectors covered with other materials than cadmium: boron, gadolinium and samarium [100]. Farinelli [101] calculated values, of cadmium and gadolinium ratios for 1/v detectors (plus resonances above 1 eV) and for detectors having a non-l/v behaviour in the thermal or in the joining region, as a function of INTERMEDIATE- NEUTRONS 133 thickness of shield, geometry of irradiation, neutron temperature, epithermal index and shape of the joining functions. The cut-off of Gd is less sharp than that of Cd and is placed at lower energies due to the lower energy (0.03 eV) of the main resonance of Gd. The range of interest of Gd thicknesses is lower than for Cd because of the much higher microscopic cross-section at the resonance peak. The cut-off of gadolinium filters increases steadily with thickness, but in a way that depends very much on the values for the neutron temperature and epithermal index.

Application of boron shields

The resonance detectors normally used have their resonance energies below about 750 eV. At higher energies the 1/v contribution to the resonance integral cross-section I becomes in general too large. Measure- ments on the intermediate neutron flux density in the keV region are therefore at this time not a routine matter. Hyver [102] suggested the use of a normal resonance detector covered with a layer of borium containing material with a 1/v absorption law. He performed theoretical work on this line calculating the optimum shield thickness, and suggested the use of a sodium-detector :(resonance energy at 1710 èV) under a layer of borium. An interesting point is that-it can be imagined that other materials than sodium can also be used by suppressing the first resonance of Co, Mn, or Cu due to the 1/v absorption law of borium. For practical application the differential cross-section must be known rather well. Konijn [103] described a 'selective neutron detector in the keV region utilizing the 19F(n, y)20F reaction. The half-life of20F is, however, very • short (11. 56 sec). He used a pneumatic shuttle to bring the sample, a piece of Teflon, in and out the reactor. At the activation position he applied outside an aluminium tube a 1 mm cadmium cover, and an enriched 10B ' shield to a density of 1. 22 g10B/cm2. From cross-section data an effective activation cross-section was calculated (16 mb). The resonance energy is 30 keV. He compared the experimental results with those obtained from the better known resonance reactions 63CU(n, Y)64CU and 27Al(n, Y)28Al and obtained good agreement.

IV. 6. 2. Other fluence measurement methods

Although foil techniques are commonly used for fluence measurements because of the simplicity of the method, in special cases other methods are sometimes applied, such as nuclear emulsions or proportional counters. But as these special methods are more often used for spectrum measurements than for fluence measurements, they will be briefly mentioned in the following section.

IV. 6. 3. Other spectrum measurement methods

In addition to the activation detector method we have the following other neutron spectrum measurement methods: Proton recoil method Charged particle method Time-of-flight method. 134 : CHAPTER IV1

proton recoil method

nuclear emulsion

proportional counter

6Li (n,n)Vt reaction •

parameter (Ел+Е1 )

parameter Et

parameter y

i 10 Ло5 ïo1 " 10' energy tin' keV)

FIG. IV. 10. ' Comparison of measurable energy ranges for some spectrometric methods (from Ref. [106]).

These methods require special equipment and are only applicable in the high-energy region (first two methods) or in the low-energy region (third method). As these methods fall outside the scope of .this chapter, no details will be given here. An outline of these special techniques can be found in other publications, e.g. [104,105], while recent developments are described in the proceedings of the IAEA symposium on neutron dosimetry [93]. Only a few. remarks concerning recent developments are made below.2

The proton recoil method

The classical nuclear emulsion technique gives good results in the region between about 350 keV and 10 MeV, .but the gamma sensitivity restricts a wide application inside reactors. The exposure rate must be less than 0. 5 to 1 R/h, so that measurements inside reactors can only be performed in cold clean cores. From the work of Beets and co-workers [106, 107] who investigated a new emulsion type it can be concluded that with this new emulsion the detection limit is lowered to about 150 to 200 keV with a resolution of about 30 keV and that the gamma sensitivity is a factor 10 less. This is a promising progress,, but the application of nuclear emulsions remains limited to. cases where the gamma background is not too large. . In principle the proton recoil method with a gas-filled proportional counter is suitable to measure neutron spectra. But because of the physical dimensions of the counter tube, its application inside reactor core is difficult and often impossible. The energy range which has been measured in a zero-energy fast reactor was 30 to 500 keV [108].

2 A comparison of measurable energy ranges for some spectrometric. methods is shown in Fig. IV. 10, where the double line indicates the useful region under optimal conditions and the dashed line the more difficult region. INTERMEDIATE- NEUTRONS 135

The charged-particle detection method

Neutron spectrometry can be performed using the reaction 6Li(n, e)3H. The law of conservation of energy gives

En=Ea+.ET-Q with Q = 4. 787 MeV

The energy En can be determined when the sum Ea+ET is determined experimentally. For the data treatment the cross-section curve ст(Е) has to be known. , The neutron energy distribution can also be derived from the energy distribution of the tritons, provided that the differential cross-section distribution is known. The spectral distribution can also be derived when the distribution of the angle ф between alpha and triton particle is determined. Nuclear emulsionsToaded with lithium can be applied under optimal conditions in the range:

500 keV - 10 MeV if ET+Ea is measured

80 keV - 5 MeV if Ex is measured 1 keV - 7 MeV if ф is measured. These specialized techniques are still under investigation and are not routinely applied. The disadvantage of the emulsions is that they have a large gamma sensitivity. A recent development is the 6Li semiconductor spectrometer, consisting of two surface barrier semiconductors opposite each other, one of them being covered by a thin layer (about 1 цm) of 6LiF. The alpha and the triton particles are collected simultaneously in each of the detectors [109]. The advantage of this method is that the detector is small and has a high energy resolution. Moreover the gamma sensitivity is much less than for nuclear emulsions. With this type of spectrometer it is possible to measure neutron spectra above about 150 keV.

The time-of-flight method

The usefulness Of time-of-flight methods for obtaining information on the neutron spectrum inside a reactor is discussed in a book by Spaepen [110]. . Accurate measurements on beams extracted from a reactor can be performed in the energy range up to 100 eV.

IV. 7. CONCLUDING REMARKS

When using or presenting resonance integral cross-sections it is necessary to state the lower energy limit, the 1/v contribution, the correction factors applied and the nuclear data used. The recommendations of the European American Nuclear Data Committee (EANDC) should be followed. In cadmium ratio measurements the,type and the dimensions of the cadmium cover used should be specified. These procedures should be internationally standardized. At present there is a preference for the Westcott convention for flux densities and cross-sections. The unified formalism as proposed by Nisle, but not yet discussed in literature, requires full attention. There is a general need for more accurate values for resonance,parameters and for activation resonance integral cross-sections. The .following 136 : CHAPTER IV1 resonance detectors have been used in several laboratories: indium, gold, manganese, cobalt, copper, tungsten and sodium. S elf-shielding correction functions are important when thin detectors cannot be used. More experimental data for several resonance reactions are very welcome. They have to be compared with theoretical relations and with computer results for those reactions in which scattering cannot be neglected. Recommendations for standard procedures for measuring intermediate neutron flux density are not yet justified, because for many resonance reactions the required resonance data are not sufficiently well known. With respect to radiation damage studies, methods have to be elabo- rated for measurement of intermediate flux densities and fluences in the keV region. Because of space and temperature limitations inside a reactor, the use of covers or shields for the activation detectors is not always acceptable. Here we have severe experimental difficulties and much development work has still to be performed. From comparison of the experimental data and techniques for activation detectors for measuring flux densities and fluences of fast neutrons with those of intermediate neutrons it is obvious that the whole field of intermediate neutron measurements needs development.

REFERENCES TO CHAPTER IV

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CHAPTER V

FAST NEUTRONS

V.l. INTRODUCTION.

This chapter includes a general description of procedures used in fast neutron fluence measurements. After establishing the theoretical basis, the chapter will describe the experimental techniques routinely used in the irradiation and counting of threshold monitors. Finally the treatment of the count data - its reduction, interpretation and application - will be covered. Since the emphasis throughout the book is on routine, more established methods descriptions of the use of sulphur, nickel, iron and aluminium are given in some detail. Although not considered a fluence monitor, aluminium has been included because of its frequent use to determine flux densities. Other monitors and methods will be descussed only briefly. However, a bibliography is provided for the reader requiring additional information. He is also referred to Zijp's [1] review of fast fluence determinations. Fast neutron fluence is most commonly measured by use of so-called threshold monitors which require neutrons above a minimum energy for activation. Most materials used to monitor fast neutrons have thresholds above 2 MeV with the exception of a few fission monitors whose thresholds extend down to approximately 0. 7 MeV. As a consequence, there is a significant gap between the energy range monitored by resonance detectors (E < 0.01 MeV) and that monitored by threshold detectors. Unfortunately, neutrons in this unmonitored range cause most of the atomic displacements in materials. Therefore, only an extrapolation of measured data will provide fluence information which meet the requirements of materials studies. Determination or assumption of a neutron energy spectrum is necessary to extrapolate data and to interpret results of threshold monitors. Neutron spectra for fast neutron fluence measurements are generally calculated by reactor physics methods, as developed in Chapter II. However, the initial analysis in this chapter assumes a fission spectrum. Methods being developed of experimentally determining neutron spectra will also be described briefly. Fast-neutron fluence measurements are necessary in several areas of research. Perhaps the need is most urgent in studies of radiation damage to materials. Fast neutron fluence measurement is also important in fast reactor technology. Experimental determination of fluence and spectra is important to many of the design and operating parameters of the reactor such as speed of control, reactivity and fuel utilization.

141 142 : CHAPTER IV1

V. 2. NEUTRON SPECTRUM

For any irradiation a neutron spectrum must either be determined or assumed to obtain a spectral averaged cross-section and finally to calculate fluence. Spectra can be calculated using reactor physics methods, determined experimentally, or as a first approximation a fission spectrum can be assumed. A fission spectrum is the energy distribution of prompt neutrons emitted from the fission process. Although rarely encountered in reactor experiments, the fission spectrum is a convenient reference and has been used extensively in developing conventions for reporting data. An understanding of this spectrum and its application is helpful. Several semi-empirical representations of the fission spectrum have been proposed and are in common usage. Three of these, the formulae of Watt [2], Cranberg et al. [3] and Grundl and Usner [4], will be discussed for illustrative purposes. The Watt fission spectrum is represented by the equation:

In the method of Grundl and Usner the fission spectrum is represented by the equation:

These representations of the fission spectrum are illustrated in Fig. V. la and normalized data are presented in Table V.I. Exicept for minor departures in the high-energy range (E > 5 MeV), the Watt and Cranberg forms are very nearly the same, while the Grundl form differs significantly in the lower energies (Fig. V. lb). Throughout the chapter the Watt fission spectrum will be used. This choice is arbitrary and based primarily on its more widespread usage. The illustrated differences in fission spectrum forms may lead to differences in cross-section data averaged over a fission spectrum. For this reason, it is important to specify the form being used. An unperturbed fission spectrum is rarely, if ever, encountered in practice, thus the validity ôf the fission spectrum approximation in most applications depends upon the accuracy required. Reactor spectra are usually similar to the fission spectrum at energies above 3 MeV. However, spectra below this energy are very dependent upon reactor materials and geometry. Spectra typical of the core position from several types of reactors and a fission spectrum are presented in Fig. V. 2 for illustration and comparison. The relative flux densities for the four spectra are listed in Table V. II. It is evident that large differences in spectra occur. FAST NEUTRONS 143

ENERGY (MeV)

(a)

ENERGY (MeV)

(b)

FIG. V. 1. Watt, Cranberg and Grandi fission spectrum: (a) d(E) versus E; (b) Ratio of Cranberg/Watt and Grundl/Watt versus E. 144 : CHAPTER IV1

TABLE V. I. NORMALIZED FISSION SPECTRA

(MeV) E 0 (E) Watt ф (E) Cranberg 0 (E) Grundl

0.05 0.2717 0. 2709 0. 2945

0.1 0.3716 0. 3709 0.4006

0.2 0.4914 0.4909 0. 5242

0.3 0.5624 0.5623 0. 5941

0.4 0.6066 0. 6074 0.6348

0.5 0.6333 0. 6342 0.6567

0.6 0.6475 0.6487 0.6657

0.7 0.6526 0. 6534 0. 6654

0.8 0.6508 0.6523 0.6582

0.9 0.6437 0.6452 0.6460

1.0 0.6325 0.6345 0.6301

1.2 0.6015 0.6037 0.5910

1.4 0.5633 0. 5648 0.5466

1.6 0.5216 0.5230 0. 5003

1.8 0.4787 0.4796 0.4544

2.0 0.4361 0.4366 0.4101

2.5 0. 3373 0.3367 0.3111

3.0 0.2546 0.2530 0.2312

3. 5 0.1881 0.1862 0.1694

4.0 0.1373 0. 1348 0. 1229

4.5 0.09887 0.09668 0. 08840

5.0 0.07064 0.06840 0.06322

5.5 0.04995 0. 04820 0. 04498

6.0 0.03514 0. 03353 0.03187

6.5 0.02453 0. 02332 0.02251

7.0 0.01707 0. 01604 0.01584

7.5 0.01181 0.01101 0.01113

8.0 0.008131 0.007505 0.00779

9.0 0.003815 0. 003450 0.00380

10. 0 0.001765 0. 001565 0. 00185

11.0 0.000807 0.000702 0.000891

12.0 0.000366 0.000311 0. 000428 FAST NEUTRONS 145

1— Watt Fission Spectrum

Fast Reactor Spectrum (FERMI)

• Graphite Moderated Spectrum

Light-Water Moderated Spectrum (ETRt

(Spectra Normalized to Equal Flux Greater than 0.0674 MeV)

FIG. V.2. Reactor spectra compared to the Watt fission spectrum.

V. 3. DETECTOR RESPONSE

The response of a detector, or its activation by neutrons, is proportional to the product of the differential activation cross-section, a(E), and the differential flux spectrum

A = к J" cr (E)

The energy ranges of those neutrons causing activation may be seen in Fig. V. 3, which shows the calculated response curves for a fission spectrum. Curves are calculated from Eq. (V.4) with experimentally derived values for a (E) and the Watt spectrum for ç (E). All curves are normalized to unit area. As illustrated in the figure, neutrons of nearly the same energy initiate all the (n, p) reactions which are considered here, while neutrons of a significantly higher energy range initiate the 21A1 (n, ct) reaction. 1.46 CHAPTER V

TABLE V.II. : RELATIVE DIFFERENTIAL FLUX DENSITIES AT THE CORE REGION OF SEVERAL REACTORS

Sodium-cooled Water Graphite

ua (MeV) El Fission spectrum fast reactor moderated moderated (Watt) EFFBRb ETR [ 5] Hanford[6]

0.25 7.79 0. 60 Ó.15 0.60 0.10

0.50 6. 07 1.80 0.45 1.75 0.45

0.75 4.72 3. 90 0.75 3.10 1.10

1.00 3,68 6.60 1.20 4.40 1.65

1.25 2.87 9. 50 1.75 5.70 2.15

1.50 2.23 •11.30 2.40 9.30 3.65

1.75 1.74 11.70 , 3.00 8.00 4.25

2.00 1.35 11.35 3.80 8.15 6.25

2.25 1.05 9.90 4.60 7.50 6. 50

2.50 0.821 8.10 5.70 7.20 6.60

2.75 0.639 6. 50 7.50 7.-50 6.85

3.00 0.498 4. 90 8.90 6.10 6.85

3.25 0.388 3.70 9.50 4.50 6.80

3.50 0.302 2.70 ., 9.70 5.10 6.65

3.75 0.235 2.00 8.80 4.30 . ' 6.60

4.00 0.183 1.40 8.00 3.90 6.50

4,25 0.143 0.95 "' 7.20 3.40 6.40

4.50 0.111 0.65 , 6.15 2.90 6.40

4.75 0.0865 0.45 5.00 2.60 6.35

5.00 0.0674 0.30 3.75 2.30 6.20

a u = In ( with E0 = 10 MeV. Eo ^ Power Reactor Development Co., Atomic Power Development Associates, Inc., brochure Enrico Fermi Atomic Power Plant: Irradiation Test Facility, (1965).'

The response of the detectors may also be characterized by the energy range causing the bulk of the activation. Assuming a Watt fission spectrum, calculations show that 90% of the activity induced in the monitors generally used for fast neutron measurements is caused by neutrons with the energy ranges listed in Table V. III. Most of these monitors show a response which falls in essentially the same energy range. Unfortunately, none of the fast neutrón monitors are sensitive below 0.9 MeV. Thus, a large neutron energy gap exists between fast and intermediate neutron measurements. This gap is of extreme importance in irradiation damage studies and is the source of much ambiguity and uncertainty iri fast fluence measurements. Differences in the response function of detectors are the basis for methods developed' to determine spectra from activation data. Such methods will be described later'in section V. 6. FAST NEUTRONS 147

FIG. V. 3. Response curves for fast fluence monitors.

TABLE V. III. 90% ENERGY LIMITS OF RESPONSE

Lower limit Upper limit : ' ' Reactions ' (MeV) (MeV) '

27 Al (n, a) 6.8 H. 1

32S(n,p) r . 2.7 , 6.,6

54 Fe (n, p) 2.4 . 6.7

58 Ni (n, p) ' 2.5 '• 6.6 '

238 U (n, f) 1. 7 • .5.5 ,

' 237 Np (n, f) • 0.86 . , .• >4.4 -

232 Th (n, fi 1.7 6.1

V. 4. .FLUENCE MEASUREMENTS

V.4.1. Experimental details

Choice of detectors ' ' • •

The choice of detectors is often difficult; however, it may determine the success or failure of the monitoring effort. Several factors which dictate the choice of fast fluence monitors are: The purpose of the study The. duration of irradiation The accessibility of monitors after discharge from the reactor The mechanical design of the test apparatus The environment to which the monitor may be exposed. Each of these will be discussed. 148 : CHAPTER IV1

The purpose of an irradiation experiment is a primary consideration in selection of monitors. Often the energy range of importance to the experiment dictates the choice of monitors. An example is high- temperature embrittlement of metals by hydrogen or helium gas produced through the (n, p) and (n, a) reactions. These reactions generally have very high threshold energies (ET= 3 to 7 MeV) and therefore a monitor which responds to this energy range would be desirable. Many of the threshold monitors commonly used in materials studies have similar response curves, i.e. are all sensitive to neutrons above 3 MeV, and other experimental considerations may dictate the final monitor selection. The duration of an irradiation and the accessibility of monitors influence monitor selection because of the half-life of product isotopes. Many monitors are precluded from use in materials or fuel irradiation testing because these experiments often involve lengthy irradiation. Further- more, the monitors may be unavailable for counting for several days or weeks after discharge from the reactor because a 'cooling' period and hot cell disassembly of the experiment are required due to the activity induced in the experimental assembly. If the half-life of the product isotope is too short, radioactive decay during the period between discharge and counting could reduce the specific activity below an acceptable level. The minimum activity level for accurate measurement depends primarily upon the radiochemical facilities and methods employed. In general, if the delay between discharge and analysis exceeds 5 half-lives, the results are questionable. • Monitors with short half-lives are sometimes used to map flux during low-power tests. In such tests the monitors can be counted soon after withdrawal from the reactor. Results are extrapolated on the basis of power measurements to estimate flux during full-power operation. Specimens or structural members can sometimes be used as self- monitors to obtain a fluence estimate. Metals or alloys which contain an appreciable concentration of an isotope normally used as a monitor can be analysed for the product nuclide activity. Fluence estimates are generally approximate because of analytical difficulties in separating the activity of the desired isotope from the myriad of others and because of inhomogeneity of composition in alloys. However, when no monitors, per se, accompany the specimen, this method can be used to obtain fluence estimates. Such estimates are certainly more accurate than any obtained by other means, particularly if the specimen was exposed during extended irradiation periods in which the intensity variations were complex. A very important factor to be considered when choosing detectors is the time required to cause saturated activity. Saturated activity, which occurs when the generation rate of the.product isotope equals the rate of decay, is related to flux, cross-sections and the decay constants. The rate of production of an isotope in a nuclear reaction is

(V.5) where Nq is the concentration of the parent (target) isotope; ст is the reaction cross-section of the target isotope; and

- = (X + VCT (h)N = Л N (V.6) FAST NEUTRONS 149

where X is the decay constant of the product isotope;

N0a

Saturated activity is proportional to the equilibrium concentration of the product isotope and depends only on the flux and concentration of the parent for a given detector. The time constant that, determines the approach to saturated activity is the effective in-reactor half-life derived from the effective decay constant by:

T+ = 0.693 Л"1 ( V.8)

The dependence of the effective half-life on thermal flux is extremely important when measuring fluence with nickel. Data presented in Fig. V.4 show that the effective half-life diminishes rapidly for thermal fluxes greater than I X 1013n-cm"2' sec"1. The effective half-life is reduced to 29 days by a thermal flux of 1 X 1014 n-cm"2: sec'1 and to only about 8. 5 days by a flux of 5 X 1014 n- cm-2- see"! What this does to the build-up of 58Co activity for flux densities of 1 X 1013, 1 X 1014 and 5 X 1014 n- cm"2- sec"1 is illustrated in Fig. V. 5. Build-ups were calculated by considering the fission spectrum equivalent flux to be equal to thermal flux, using a fission spectrum averaged cross-section of 113 X 10"27cm2. Calculations for the curves in Figs V.4 and 5 were made with a thermal cross-section value of 1650X 10"24 cm2. A nearly linear relation between time and build-up is seen when 150 : CHAPTER IV1

FIG. V. 5. 58 Co activity build-up and saturation in several thermal flux densities.

the thermal and fast flux densities equal 1013 n-cm"2-secNickel would be a good fluence monitor in this case since saturated activity is not approached. Saturated activity is approached within a relatively short time however, at the higher flux densities. The concentration of 58Co is essentially independent of time after 25 days in aflux of 5 X1014 n-cm"2-sec"1. The approach to saturated activity makes fluenoe measurements (per se) impossible. In this case, the activity level depends upon the flux density and fluence can be estimated only by assuming the flux density deduced from saturated activity was typical of the entire irradiation. Similar calculations were made for the build-up of activity in the 27 Al,(n, a), 32S(n, p) and 54Fe (n, p) reactions for an irradiation in which the thermal and the fission spectrum equivalent flux were 1 X 1014 n-crn'~2-sec"1. These data, presented in Fig. V. 6, clearly demonstrate the limitation of the 27Al(n, a) and 32S(n, p) reactions as fluence monitors for irradiations extending beyond a few days (because build-up is not proportional to the time t). Build-up of 54Mn in iron is seen to be nearly linear for the 140 days considered in these calculations, thereby making accurate fluence measurements possible.' '"' Calculation of the time to reach saturated activity for monitors prior to extended irradiations is a very helpful exercise to determine the suit- ability of det.ectors and to aid in interpreting activation data. FAST NEUTRONS 151

time, (d)

FIG. V. 6. Activity build-up of fast fluence monitors. '•••".

Availability and costs of monitors often influence their selection. Pure isotopes such as 54Fe, which are often necessary if analysis is made with gross gamma measurements,• are expensive; ' Natural iron, nickel, and • aluminium can be obtained with purities exceeding 99.9% at relatively low cost. Alloyed flux monitors are also'available at reasonable prices, but any inhom'ogeneity of the alloy could cause error. .

Detectors in use • '••'

Monitor selections must also'be made with due consideration to past.' practice in -the area of' study. Dosimetry for materials studies almost invariably includes 54Fe(n, p) and 58Ni (n, p) monitors.- Inclusion of these monitors would ensure better correlation with data from other investigators than inclusion of sulphur or aluminium. Fission monitors are used by a number of experimenters. Monitors with very long half-life products, 46Ti(n, p) 46Sc and'63Cu(n,

Detector geometry'

•The geometry or physical form of detectors is often quite important, particularly when the detectors are analysed directly without chemical 152 : CHAPTER IV1 dissolution. Calibration of counting facilities are generally made on the basis of a standard size and shape of detector. The counting geometry, which determines the fraction of activity reaching the analyser, can be determined precisely for a standard detector and thereby facilitate subsequent analysis of large numbers of samples for that detector geometry. Counting comparisons between laboratories are similarly improved through the use of standard detector geometries. Experimental conditions unfortunately serve to frustrate attempts to standardize detector geometry. Irradiation experiments or surveillance devices may not provide sufficient room, or protection for bulkier geometries that might be the preferred geometries. The three types of geometry most commonly used for fast neutron fluence detectors are wires, foils and pellets. For most experiments short lengths of thin wire (approximately 6 mm long by 0. 5 mm diameter) are very convenient. Whenever possible, wires are bent to some pattern for identification; then nickel wires could be identified from iron wires, for example, even during remote operations. Long wires are often used to map irradiation facilities. Where possible, wires can be extended the length of a facility to determine flux gradients. Foils are commonly used in dosimetry experiments where a large variety of monitors are irradiated simultaneously. Fission monitors such as uranium, neptunium and thorium are irradiated as foils to facilitate control of the concentration of the fissionable isotope, and to minimize effect of self-shielding and self-absorption. Fissionable isotopes are deposited on the surface of an inert material such as aluminium. Sulphur is genérally irradiated in pellet form. According to the ASTM [7] method, pellets are normally made 1/8 inch thick or greater to obtain maximum counting sensitivity independent of small differences in weight. Count-rate data are cited [ 7] to show that a 3/16 inch pellet would be infinitely thick for the most energetic beta-ray particle from 32P. Very uniform sized pellets of high-purity flowers of sulphur can be obtained commercially. Many experimenters make their own pellets with a pharma- ceutical press. Uniformity of pellet size is particularly important for accurate flux measurements with sulphur. A type of pellet being developed for use in very high temperatures consists of various metals (e. g. Fe, Ni, Al) sintered into a ceramic matrix. This type of monitor would permit fluence measurements at temperatures over 1000°C. However, it has been necessary to precalibrate this type of monitor to determine the monitor content within the ceramic. This precalibration involves an irradiation to a very low fluence in a calibrated facility and a determination of the induced activity levels.

Detector placement

Ideally, neutron dosimeters would be positioned in an experimental assembly to measure directly the fluence incident on the sample. Careful monitor placement is important since local perturbations in flux and spectrum should be detected by the monitors. Although perturbations of fast neutron flux and spectrum are generally of much smaller magnitude than those of thermal flux, they may be of sufficient magnitude to introduce substantial errors into fluence measurements. Perturbations and flux gradients are caused by moderating materials that modify the fast neutron spectra FAST NEUTRONS 153 or the neutron flux. Therefore, monitors should be located so that the fluence incident on specimens can be measured directly, or determined accurately by interpolation of data from monitors on several sides of specimens studied. The position of the specimen and monitor should coincide but this is seldom possible because of mechanical restrictions, or simply because the size and number of specimens preclude placement of dosimeters adjacent to each specimen. In such cases monitors must be located within the irradiation facility in sufficient numbers to permit accurate estimation of the fluence at unmonitored points. Fluence at an unmonitored point is obtained by determining flux density contours and interpolating. This type of fluence determination is.required in most material studies. Irradiations are often performed in test reactors where flux gradients are particularly steep or in which environment at a specimen position would be incompatible with the monitors. Under these conditions it is impractical to couple a monitor with the specimens and it is necessary to estimate fluence at the specimen from results at the monitored positions. Fast neutron spectra are calculated for a specific position in an experimental assembly such as the intersection of the reactor midplane with the experiment. Some monitors-should be located near this position for normalization. Additional monitors should also be located along the length of an experiment so variations caused by leakage near the boundaries of the fueled regions and other local perturbations would be detected. The need for multiple detector locations within an experiment is particularly acute in experiments conducted in reactors with relatively small cores. Because spectral shifts and flux gradients may be severe, detailed mapping and monitoring of these facilities are required. Recording and reporting of monitor locations is an absolutely essential requirement. The costs of dosimetry are nearly always negligible compared to the costs of irradiation tests, yet careful dosimetry practices are essential to secure valid data. Proper selection and careful placement of monitors are necessary; however, results can easily .become confused unless there is complete and accurate reporting of monitor locations. In- formation required includes the location of the irradiation facility in the reactor and the exact location of the sample in the irradiation facility.

Effect of environment

Environmental factors can have profound effects on dosimetry and must be given due consideration during the design of experiments. The most important factors will be the temperature of the monitors during irradiation and the chemical compatibility of the monitors with their surroundings. These factors determine the choice of monitors as well as their placement and handling. Temperatures well above the melting point of cadmium and sulphur are frequently encountered in high-flux facilities. Irradiation of these materials becomes impossible unless they can be thermally grounded or encapsulated. If cadmium is not used, this imposes restrictions on the use of nickel dosimeters unless thermal fluxes can be accurately determined. Thermal burn-out of the metastable isomer of 58mCo (ath ~ 175 000 barns) and 58Co (crth ~ 1650 barns) must be eliminated by shielding or accurately estimated if nickel activation data are to be interpreted correctly. 154 : CHAPTER IV1

Cadmium shielding of nifckel monitors is not possible in high-temperature tests unless cadmium oxide or another cadmium ceramic is used. Utili- zation of this material or even of cadmium metal at lower temperatures has had limited success. Oxides of cadmium often deposit on the surface of the monitor and the activity of the cadmium isotopes mask the 58Co activity. Pure sulphur, which melts at 113°C, is eliminated from many tests, but the useful temperature range may be extended by use of sulphur compounds. The compounds most commonly used and their melting temperatures are: aluminium sulphate Al^SO^g (770°C); ammonium sulphate

(NH4)2S04 ( 280°C); lithium sulphate Li^SO^ (860°C); and magnesium sulphate

MgS04 (1124°C). A note of' caution must be injected on the use-of sulphur- bearing compounds in dosimetry. Determination of the sulphur content of these salts can be very difficult because of uncertainties in the amount of hydrated water present during analysis. Substantial errors can he introduced by incorrect assumption of an anhydrous state and incorrect values of molecular weight. ,••••.. The melting point of aluminium, 660°C, may limit its usefulness as a monitor. However, aluminium, like sulphur, is used much more frequently in low-power short-term experiments to determine flux rather than fluence because of the half-life of the activated species. Thus aluminium is not • often used at high temperatures. It is necessary to protect monitors from reactor coolant and from other materials which might contaminate or react with the .monitors. Im- purities, introduced either through contact or chemical reaction, may have competing activities which make determinations of the monitor activity either impossible or unacceptably inaccurate.' Usually, it is advisable to encapsulatë dosimeters in an inert or non-reactive container made of material such as quartz or aluminium. '

Post-irradiation handling

Improper handling of monitors between discharge and counting can cause considerable error. Ideally, monitors should be retrieved during operation, or immediately after discharge. However, if monitors are in- corporated in experiments, gamma activity levels often make retrieval impossible for days or weeks after discharge, and only then in a hot cell. If hot cell disassembly is foreseen, retrieval of monitors must be considered early in the experimental designs. Careful planning at this stage will be particularly valuable in avoiding difficulties during actual disassembly. ' • - Hot cell operations must be preplanned-so monitors can be retrieved, identified and separated while remaining uncontaminated. Since many monitors of different types are irradiated in most experiments, there is an obvious need for a system of remotely identifying and separating monitors. Contamination, which makes accurate analysis difficult or impossible, can happen very easily by contact. Therefore, operations-must be conducted with care and forethought. • It is generally instructive to run through the disassembly procedure on a test basis before attempting the disassembly of an actual experiment. : • • FAST NEUTRONS 155

Reactor operation

Ideally, a dosimetry experiment would involve insertion into a reactor, irradiation, and discharge of the monitors during a period of constant reactor power. Time expended in charge and discharge should be very small compared to the time of irradiation. This type of experimentation facilitates accurate data reduction and permits the use of monitors .which might otherwise be excluded because of their short half-lives. This type of testing, typically conducted in 'rabbit' facilities, finds a variety of applications. Measurements of flux densities, determination of relative cross-sections and preparation of radioactive standards for reference in gamma spectroscopy are some of the applications. Idealized reactor operation is certainly not typical of materials studies. Experiment's are inserted in a reactor for several cycles during which time power variations may be large. Monitors may not be available for analysis for days or several weeks after discharge because of activity in án assembly. The need for accurate fluence measurements under these 'less-than-ideal1 conditions is critical and the appropriate monitors and methods of analysis must be provided to satisfy normal experimentation practices. However, to provide accurate fluence measurements under these circumstances, data are needed on reactor operations. The time of reactor shut-downs and start-ups are essential. Records of power variations are of less interest •unless extended operations were conducted where power varied by a factor of ten or more. . '

V. 4.2. Data reduction .

Data reduction as considered here is the calculational procedure used to obtain fluence or flux densities from activation data. The methods will be presented for simplified cases and equations will be derived for more complex applications. Calculations in this section are made with the assumption that the neutron spectrum is a fission spectrum and cross- sections are fission-spectrum-averaged cross-sections. Treatment of realistic spectra can be done by substituting appropriate cross-section values into these general equations. The general equations to calculate fluence and flux density from activation of monitors will be derived, assuming that the flux density was constant during the irradiation and that there was no formation of a metastable isomer. The rate of increase in product isotope concentration is

- N0a? - AN .

where N is the concentration of the product isotope; N0 is the concentration of the parent isotope; and A, equal to X +

:N0 a

If Eq.(V. 15) is inverted, solved for (p¡ t (e.g. $f) and N2 replaced by (A/X) exp (Xgtg), it becomes

exp(x2V Л1Л2* Ф, = NX г л •^lAJl-e-V 1 + стЛ,(1 + л л 1 - е" > ' AgAj/ / m 2V Л9. -Л- (V.16)

Calculation of accurate fluence values from activation of 5SNi is very dependent upon accurate determination of thermal flux. Martin and Clare [9] have demonstrated that corrections of 50% can be required at thermal flux densities of 1 X 1014n- cm"2-sec_1because of the extremely high burn- up cross-sections of 58Co and 58mCo. Reported values of the branching ratio, the ratio of 58Co atoms produced directly per 58Ni(n, p) reaction, are 0. 61 by Mellish et al. [10], 0. 66 by Hogg et al. [11], and the two values 0. 68 and 0. 70 by Passell and Heath [ 12]. Martin [13] reported that the mean branching ratio can be considered independent of spectra variations on the basis of experiments in five diverse reactor spectra which yielded mean branching ratios of 0. 75 and 0. 76. Computation of fluence from fission monitor activations is accomplished in essentially the same manner as in the reduction of other flux monitor data except that a fission yield (Y) term must be included. The reader should be warned of the possibility of error introduced by transmutation of the fissionable isotope to another fissionable isotope with a larger fission cross-section. Fluence is determined from fission monitors by analysis of a fission product which has a relatively high fission yield and a long half-life. The 158 : CHAPTER IV1

140La daughter of 140Ba and "Mo are probably the two most common iso- ' topes which are used. 137Cs is another fission product which is attractive since it has a long half-life, an easily distinguished gamma peak and a high fission yield (5.8%).

Reactor down time

Flüence measurements are typically made during periods of intermittent reactor operation and varied power levels. Multicycle irradiations are particularly common for materials studies and surveillance programs. Reduction of activation data for extended multicycle irradiations can be quite uncertain and difficult; however, assumptions which simplify calculations are sometimes valid. Problems are encountered in proper consideration of decay during reactor down times and of approach to saturation of the detector activities. Proceeding through a hypothetical example, which is typical of irradiation studies, will expose the problems and suggest possible simplifications. Consider an irradiation experiment conducted in a test reactor during four 42-day cycles, with the reactor operated 50% of the time. Let the fission- spectrum flux density and the thermal-flux density each be 1X 1014n • cm'2' sec"1 and consider that 32S, 54Fe and 58Ni detectors are included in the experiment. The build-up of activity in these detectors and the fluence from the activity of each detector will be calculated. • The build-up of 32P, 54Mn and 58Co activities in the detectors is illustrated in Figs V. 7, 8 and 9 respectively. Decay and burn-out of 58mCo have not been considered. The activity build-up of 32P and58Co show a significant decay during reactor down times and the increase in activity from the end of one operating period to the end of the next diminishes as saturated activity is approached. The half-life for 32P is 14. 3 days and the effective half-life for 58Co would have been 28. 9 days in this experiment.

SHADED AREAS INDICATE REACTOR OPERATING TIME

0 R^SNNNi ! ISSSSNSM t^S^^M 0 21 42 63 84 105 126 147 TIME (D) , FIG. V. 7. 32 P activity during a multi-cycle irradiation. FAST NEUTRONS 159

TIME (D)

FIG. V. 8. яМп activity during a multi-cycle irradiation.

FIG.V. 9 58 Co activity during a multi-cycle irradiation.

The activity of 54Mn builds up nearly linearly with only a small fraction (4.5%) decaying during the 21-day periods when the reactor was not operating. -Treatment and interpretation of activation data from these detectors obviously must differ. . • . The daughter-to-parent ratios obtained by stepwise calculations can > be used to calculate fluences to obtain an estimate, of the effect of assumptions which must be made. Equations (V. 10) and (V. 15) can approximate down time by adjusting the parameters t and t0 (the time of irradiation and 160 : CHAPTER IV1 the time between discharge and counting). Simplifying assumptions which are often made are:

(a) Let t be the time which the reactor operated (operating time) (b) Let t be the time from the beginning of the first start-up to the last shut-down (total time) (c) Let t be the operating time and let to be the time elapsed from discharge to counting plus the intermediate down time.

Flux densities and fluences were calculated for each of these assumptions

and compared with the true values (f = 1 X 1014 n. cm"2, sec"1 and Ф = 7. 26 X 1020n- cm"2). The ratios of calculated-to-true values in Table V. 4 demonstrate the accuracy of each approximation. Calculation based on operating time gave the most accurate determination of flux density with both iron and sulphur, although a 20% error is incurred with sulphur. Use of the second assumption, t equal to total time, gave a very accurate estimate of fluence with the iron monitor but a very poor estimate of flux density (43% error). This approximation gave about the same magnitude of error for sulphur as the first assumption. The third method is seen to be totally unacceptable for this hypothetical irradiation.

TABLE V. IV. INFLUENCE OF REACTOR OPERATING HISTORY ON FLUENCE AND FLUX DENSITY

Ratios: calculated to actual values

54 Conditions Fe (n,p) 32 S (n, p) Flux density Fluence Flux density Fluence

t' = 84 d 0.933 0. 933 0.805 0.805

t„ =0 -

t = 147 d 0. 570 0.999 0.794 1. 390

4 =° t = 147 d 0.660 1.150 1. 680 29.480

t„ = 63 d

t' = operating time; t = total time; t0 = shut-down time.

This exercise was certainly not intended to establish criteria but is presented to illustrate the problems and magnitude of errors which can occur. Similar calculations, which can be done quickly for particular experiments, are often very helpful in evaluating and interpreting activity data. A more exact solution for fluence can be obtained by iterative methods in which a trial fluence, calculated from the observed activity, is used to calculate monitor activity for the known reactor operating history. The solution is completed when agreement between calculated and observed activities is obtained. This type of approach is easily programmed for computer solution. FAST NEUTRONS 161

Relation of fluence and corrected activities /

Discussions to this point have treated the reduction of activation data to flux or fluence but have not considered the meaning or accuracy of cross- section data used in these calculations. Fluence data depend heavily upon the cross-section values and it is essential that monitor cross-sections be understood and correctly applied. A point which cannot be overemphasized is the necessity for reporting cross-section data used to calculate fluence in irradiation experiments. Fluence data can be refined or renormalized using improved values of cross-section or of neutron spectra if the original cross-section data are reported. However, all too often potentially valuable data must be discounted because it is not possible to determine how fluence estimates were made or how accurate they might be. Interpretation-of data is similarly difficult if the fluence is undefined. Examples are irradiation experiments in which a fluence in units of 'fast neutrons' is reported. The actual meaning of this unit can be determined if cross-section data are reported. Otherwise, data are of limited utility. Very substantial errors may result from cross-section uncertainties. Uncertainties in the measured value or in the assumed spectra-averaged cross-sections calculated from the data can cause errors of 50%. Errors easily as large can be caused by uncertainties in the normalization of data. Measurement of monitor cross-sections used to determine fast- neutron fluence is very difficult. Generally cross-sections have been calculated by the ratio of activity induced in the isotope of interest to that in a reference isotope (e. g. 32S or 21 Al). If an incorrect value for the cross- section of the standard, or an incorrect half-life assumption were made for either, errors would result. Thus a major task in obtaining differential cross-section data has been a determination and evaluation of uncertainties in each measured point. Once obtained, differential cross-sections are used to determine spectral-averaged cross-sections using group-averaged cross-sections consistent with group structures of the spectral calculation. Differential cross-sections are also necessary to determine the range of neutron énergies to which the monitor responds in a particular spectrum. The response curve, that is the curve of the function cr(E) tp (E), is used in the selection of monitors and in the determination of spectra from monitor activation data. The differential cross-section curves for the 32S(n, p), 58Ni(n, p) and 27Al(n, a ) reactions are relatively well established because many measurements have been made for each. The differential cross-section for the 54Fe(n, p) reaction, however, is uncertain because insufficient data lead to energy gaps where no measurements have been made. An excellent reference for cross-section data of reactions useful for neutron dosimetry is the work of Barrall and McElroy [14] which presents and discusses all of the data through 1964.

58Ш_(пгр}^Со The measurements that have been made of the 58Ni(.n, p) 58Co reaction are presented in Fig. V. 10. The data of Meadows and Whalen [15] were used to determine the curve in the region from 1 to 2.7 MeV. At higher energies there is substantial scatter in the data and uncertainties in the individual measurements of about 20% in the cross-section and 10% 162 : CHAPTER IV1

RUJlîfl FAST NEUTRONS 163 in energies. .There are.no data between about 8. 5 and 12. 5 MeV and extreme scatter at the very high energies. The differential cross-section curve of Barrall and McElroy [14] yields an average cross-section of 113 mb when averaged over a Watt fission spectrum.

_2¿L(.P.i PÎ^Ç There have been many measurements of the 32S(n, p) 32P cross-section that make it possible to establish the differential cross- section in rather fine detail, see Fig. V. 11. However, some uncertainty exists below 4 MeV because of conflicting data. According to Barrall [ 14], the average cross-section in a fission spectrum (Watt) [2] is 0.067 barns if microscopic cross-section data are used as originally reported byHurliman and Huber [ 16], Leeser et al. [17] and Ricamo [ 18 ]. A value of 0. 061 b is obtained by Barrall when the data of these authors are renormalized to account for discrepancies believed to have been introduced by S02-filled ionization chambers. Many measurements have been made in the energy ranges between 5 and 8. 5 MeV and between 12. 5 and 16 MeV, while in between there are data which disagree by 50% (see Fig. V. 12). The differential cross-section curve synthesized by Barrall yields an average value of 0. 705 mb when averaged over a Watt fission spectrum.

^íle.Cl.jJ-^M.1! The differential cross-section of the 54Fe(n, p)54Mn reaction has not been well established. Data reported to date are presented in Fig. V. 13 (courtesy R.C. Barrall) [14] along with the differential cross- section postulated by Barrall. The absence of data between 6 and 12 MeV and below 2. 5 MeV has caused significant differences in fission-spectrum averaged cross-section values calculated by various investigators. Barrall [14] calculates a value of 97 mb, Carroll and Smith [19] 68 mb and Helm [20] 81 mb for a Watt fission spectrum. It is to be hoped that further

REFERENCES TO FIG.V.10

[1] ALLAN, D. L., Nucl. Phys. 24 (1961) 274. [2] ALLAN, D.L., Proc. Phys. Soc.. Lond. 70 (1957) 195. [3] BARRY, J.F., J. nucl. Energy A/B 16 (1962) 467. [4] CROSS, W.G. et al., Bull. Am. phys. Sor. 7 (1962) 335. [5] GLOVER, R. N., PURSER, K. H., Nucl. Phys. 24 (1961) 431. [6] GLOVER, R. N., WEIGOLD, E., Nucl. Phys. 29 (1962) 309. [7] GONZALEZ, L., RAPAPORT, ]., VANLOEF, J. J. , Phys. Rev. 120 (1960) 1319. [8] HASSLER, F.L., PECK, R. A. Jr., Phys. Rev. 3 (1962) 1011. [9] JERONYMO, J.M.F. et al., Nucl. Phys. 47 (1963) 157. [10] KUMABE I,, FINK, R. W., Nucl. Phys. 15 (1960) 316. [11] MEADOWS, J. W., WHALEN, J.F., Phys. Rev. 130 (1963) 2022. [12] NAKAI, K., GOTOH, H., AMANO, H., J. Phys. Soc. Japan 17 (1962) 1215. [13] NEUERT, H., POLLEHN, H., Table of Cross Sections of Nuclear Reactions With Neutrons on the 14-15 Energy Range, Rep. EUR-122e (1963). [14] PREISS, I. L: , FINK, R.W., Nucl. Phys. 15 (1960) 326. [ 15] PREISS, I. L., FINK, R. W., Nucl. Phys. 15 (1960) 326 (for the effect of s8mCo). [16] HYDER, M., private communication cited in BEAUGE, R., Rep. DEP/SEPP-148 (1963). [17] PURSER, K.H., TITTERTON, E. W., Aust.J.Phys. 12 (1959) 103. [18] STOREY, R.S., JACK, W., WARD, A., Proc. Phys. Soc., Lond. 75 (1960) 526. 164 : CHAPTER IV1

sujEg FAST NEUTRONS 165 measurements will define the differential cross-section of this very important fast-fluence monitor more clearly. Until such data are available iron monitors should be supplemented with another monitor whose differential cross-section is more certain if absolute fluence measurements are required. Other reactions The fission cross-section of 238U is the most thoroughly measured of any potential fast-neutron monitor. The cross- section averaged over the Watt fission spectrum for the 238U(n, f) reaction is 299 mb. The value for the Cranberg [3] and Grundl [4] forms of the fission spectrum are 310 and 312 mb respectively. Cross-section data for the 46Ti(n, p)46Sc and 63Cu(n, а-)60Со reactions are extremely limited and no differential cross-section curves are known. These reactions are well suited for relative fluence measurement for extended irradiations but cannot be used for absolute measurements until more data become available.

Fission-spectrum-averaged cross-sections

The spectrum of neutrons emitted from fissioning 235U provide a very useful reference for determining and reporting activation cross-sections. This spectrum,which has been represented mathematically and experimentally reproduced with varying degrees of success, is used to obtain spectral- averaged cross-sections that can be adjusted to any known spectrum. Fluence measurements reported for an irradiation in which the spectrum is unknown are often calculated and reported using a fission-spectrum- averaged cross-section.

Fission-spectrum-averaged cross-sections (5f) reported in the literature have been calculated from the differential cross-section using the relationship

, (E)dE 0 1 where

REFERENCES TO FIG. V. 11

[ 1] ALLAN, D. L., Nucl. Phys. 24 (1961) 274. [2] ALLEN,'L. efal., Phys.Rev. 107 (1957) 1363. [3] BLEULER, E., Helv. phys. Acta 20 (1947) 519. [4] COLLI, L. et al., Nuovo Cim. 10 1711 (1960) 634. [5] DEMENTI, V.S., TIMOSHUK, D.V., C.r. (Doklady) Acad. Sci. URSS 2^(1940) 926. [6] EUBANK, H.P., PECK, R.A.Ir., HASSLER, F. L., Nucl. Phys. 9 (1958) 273. [7] FELD, B.T. et al., Final Report on the Fast Neutron Project, Rep. NYO-636 (1951). [8] HURLIMAN, T., HUBER, P., Helv. phys. Acta 28 (1955) 33. [9] KLEMA, E.D., HANSON, A. O., Phys. Rev. 73 (1948) 106. [10] LEVKOVSKIY, V.N., Soviet Phys. IEPT 18 (1964) 213; I. exptl theoret. Phys. (Russian) 45 (1963) 305. [11] LUSCHER, E. et al., Helv.phys. Acta 23 (1950) 561. [12] PAUL, E.B., CLARKE, R. L., Can.I.Phys. 31 (1953) 267. [13] RICAMO, R., Nuovo Cim. 9 81 (1951) 383. [14] SANTRY, D. C., BUTLER, J. P., Can. I. Chem. 41 (1963) 123. 166 : CHAPTER IV1 FAST NEUTRONS 167 irradiation in converter facilities whose spectra approximated a fission spectrum. Foils of. several isotopes to be studied are irradiated with an isotope whose cross-section is known. Cross-sections can then be calculated from the relative activations. The accuracy of these determinations depends upon uncertainties in the reference monitor and upon the validity of approximating the test spectrum by a fission spectrum. Departures from a fission spectrum are less at higher energies so this technique is more apt for determining cross-sections of the high-energy reactions. Zijp [1] presents an excellent tabulation of fission-spectrum-averaged cross-sections for threshold reactions. Variations in the values reported reflect the difficulties inherent in obtaining cross-section data.. As an example, fission-averaged cross-sections ranging from 0.44 to 0.85 mb have been reported for the 21Al(n, о ) reaction. However, many of the experimental measurements agree within 5% of the calculated value. Similar ranges are found in values reported for most of the other reactions including the 54Fe(n, p) and 58Ni(n, p) reactions. Particular emphasis must be placed on determining the validity of the cross-section reported in the literature.

REFERENCES TO FIG.V.12

BAYHURST, B. P., PRESTWOOD, R. J., J. inorg. nucl. Chem. 23 (1961) 173. BONAZZOLA, G.C. et al., Nucl. Phys. 51 (1964) 337. •BORMANN, M. et al., J. Phys. et Rad. 22 (1961) 602. BUTLER, J. P., SANTRY, D. C., Can. J. Phys. 41 (1963) 372. CINDRO, N., KULIS1C, P., STROHAL, P., Phys. Lett. 6 (1963) 205. CSIKAI, J., GYARMATÍ, В., HUNYADI, I., Nucl. Phys. 46 (1963) 141. DEPRAZ, M. J., LEGROS, G., SALIN, M. R., J. Phys. et Rad. .21 (1960) 377. FORBES, S. G.,. Phys. Rev. 88 (1952) 1309. GABBARD, F., KERN, B.D., Phys.Rev. 128 (1962) 1276. GLAGOLEV, V. N. et al., JETP (Soviet Phys.) 9 (1959) 742j JETP 36 (1959) 1046. GRUNDL, J. A., HENKEL, R. L.,. PERKINS, B. L., Phys. Rev. 109 (1958) 425. ARON, P.M. et al., Atomic Energy 16 (1964) 370. IMHOF, W. L., private communication from Lockheed, cited by LISKIEN, H., PAULSEN, A., Rep. EUR-10E (1961). . ... JERONYMO, J.M.F. et al., Nucl.Phys. « (1963) 157. KERN, B. D., THOMPSON, W. E., FERGUSON, J. M., Nucl. Phys. 10 (1959) 226. KHURANA, C.S., HANS, H. S., Nucl. Phys. 13 (1959) 88. KHURANA, C.S., HANS, H. S., Proc. Symp. Low Energy Nuc. Phys. Bombay, 1961, 21-9. KUMABE, I., J. phys. Soc. Japan 13 (1958) 325. : KUMABE, I, et al. , Phys. Rev. 106 (1957) 155. MANI, G. S., McCALLUM, G. J., FERGUSON, A. T. G., Nucl.Phys. 19 (1960) 535. MUKHERJEE, S.K. et al., Proc. Symp. Low Energy Nucl. Phys., Bombay, 1961. PAUL, E. B., CLARKE, R. L., Can. J. Phys. 31 (1953) 367. POULARIKAS, A., FINK, R. W., Phys. Rev. 115 (1959) 984. SANTRY, D. C., BUTLER, J. P., Can. J. Chem. 41 (1963) 123. SCHMITT, H..W., HALPERIN, J., Phys.Rev. 121 (1961) 827. STROHAL, P., CINDRO, N., EMAN, В., Nucl.Phys. 30 (1962) 49. TEWES et al., Neutron Excitation Function, Rep. WASH-1028 (1960) 67. YASUMI, S., J. Phys. Soc. Japan 12^(1957) 443. GRUNDL, J., Study of Fission Neutron Spectra With High-Energy Activation Detectors, Rep. LAMS-2883 (1963).• [30] LISKIEN, H., PAULSEN, A., J. nucl. Energy 19 (1965) 73.

FAST NEUTRONS 169

Effective cross-sections

The method of effective cross-sections is widely used to determine flux density of neutrons with energies above a given threshold energy. An effective cross-section is calculated by the expression

/ с (E).

where F is the fraction of neutrons in the spectrum above energy Ej. This relationship is used primarily in calculation of fluences or flux densities

when a fission spectrum is assumed. As an example, for 5f = 68 mb and F = 0. 692 the effective cross-section determined for E¿ = 1 MeV in a fission spectrum is

а, со

CT(Ei) = 0^92" = 98 mb

Because the effective cross-section depends upon both the threshold energy and the spectral shape, it is essential that spectra as well as effective cross- sections and threshold energies be reported so that fluence data can be properly interpreted.

REFERENCES TO FIG. V. 13

[1] ALLAN, D.L., Nucl. Phys. 10 (1959) 348. [2] ALLAN, D. L., Nucl. Phys. 24 (1961) 274. [3] ALLAN, D.L., Proc. phys. Soc., Lond. 70 (1957)195. [4] CROSS, W.G. et al.. Bull. Am. phys. Soc. 7 (1962) 335. [5] JOHNSON, R.G., SALISBURY, S., VAUGHN, F.S., Rep. LMSC-4-50-62-1 (1962). [6] MARCH, P. V., MORTON, W.T., Phil.Mag. 3 (1958) 143. [7] NEUERT, H., POLLEHN, H., Tables of Cross Section of Nuclear Reactions with Neutrons in the 14-15 Energy Range, Rep. EUR-122e (1963). [8] POLLEHN, H., NEUERT, H., Z. Naturf. T1 a 16 (1961) 227; [9] STOREY, R.S., JACK, W., WARD, A., Proc. phys. Soc., Lond. 75 (1960) 526. [10] VANLOEF, J.J. , Nucl. Phys. 24 (196,1) 340. [11] CARROL, E.E., Jr., SMITH, G.G., Rep. WAPD-T-1726 (1964). [12] MALMSKOG, LAUBER, Rep. EANDC(OR)23. [13] SALISBURY, S. R., CHALMERS, R. A., unpublished data. Lock-Research" Laboratories, Palo Alto, Càlif. (1965). ' [ 14] LISKIEN, 'H., PAULSEN, A., Compilation of Cross-Sections for Some Neutron Induced Threshold Reactions, Rep. EUR 119. e, (1963). 170 : CHAPTER IV1

Although activations are induced by a very limited range of neutron energies, effective cross-sections provide an estimate of fluence over a wider energy range provided the spectrum employed is accurate. This is actually the common practice of determining fluences of neutrons of a monitor such as Ni that does not respond to neutrons with energies less than 3 MeV. Effective cross-sections become highly dependent upon accurate and detailed spectral determination particularly for threshold energies below 1 MeV since spectra diverge markedly below that energy. Examples of spectra in irradiation facilities in a graphite-moderated reactor, a light- water-moderated reactor, a fast (Fermi) reactor and a fission spectrum are presented in Fig. V. 2. Effective cross-section values for E¡> 1. 0 MeV, Ej >0.5 MeV and Ei > 0. 18 MeV in each spectrum for the 58Ni (n, p)60Co reaction are given in Table V. V to illustrate the variations which are encountered.

TABLE V. V. SPECTRUM-WEIGHTED 58Ni (n, p) 60Co ACTIVATION CROSS-SECTIONS

Fluence energy Reactor type limit Light-water Heavy-water Watt fission Graphite moderator moderator moderator spectrum

0 (Ej > 1 MeV) 101.6 180.5 161.2 157.9

® (Ej >0.5 MeV) 61.1 113.0 111.3 125.3

Ф (Ei > 0.18 MeV) 39.2 79.3 76.1 112.2

Fluence data calculated for an assumed fission spectrum or using a differential cross-section can be corrected whenever more accurate spectra or cross-sections become available, providèd the original assumptions are known. The correction is made by multiplying the original fluence by the ratio of the effective cross-sections, i.e.

Ф 5E = kA = Ф1 5¿ ' (V. 20)

and

'*'=*(%) . (V-21) where ¿ and are the corrected values of .fluence and effective cross- section respectively. In Eqs (V. 20) and (V. 21) E¿ may be different from E^. The calculation of accurate fluence data from activation of monitors requires more than proper reduction of activation data to specific activities. ' A real obligation is incumbent on the experimenter to obtain, use and report the best possible spectral and cross-section data. If a fission spectrum must be assumed, this should be stated so that the data can be adjusted if more accurate information becomes available. FAST NEUTRONS 171

Conventions for reporting neutron fluence

Conventions used to report fast-neutron fluence are more ambiguous than those used for thermal and intermediate neutrons. Ambiguities and lack of precision in fast-neutron conventions stem from the extreme variations in fast-neutron spectra and from the need to extrapolate foil-activation .data over wide energy ranges. The objective in providing a reporting convention for radiation damage studies is basically to correlate effects caused in diverse neutron spectra. If that objective were accomplished, the same radiation effect would be observed at a particular reported value regardless of the neutron spectra, provided all other experimental parameters were the same. Ideally, proper normalization would provide a basis for correlating data from various experiments, would permit the accurate use of test data in the design of reactor components and would permit more definitive study of the effects of variables other than neutron spectra on damage. The ambiguities in fast neutron fluence measurements make each system much less than ideal and obligate investigators to provide the information necessary to recon- struct fluences. Type of monitors, cross-sections and neutron spectra are essential and activation rates are certainly desirable. The conventions which have been most commonly used to report fast- neutron fluence are: fluence above 1 MeV, equivalent fission fluence and megawatt days per adjacent ton of fuel (MWd/adjacent ton). A brief discussion of the implied assumptions and application of each should be useful. The practice of reporting fluence greater than 1 MeV is probably the most common. This would seem to imply that all neutrons above 1 MeV have been counted, which is usually not the case. A fission spectrum is often assumed in these measurements and fluences calculated by an effective fission-spectrum cross-section for energies greater than 1 MeV (see Eq. (V.18)). When data are reported in this convention the author has a definite obligation to report the spectrum used. When the calculations are made with accurate cross-sections and spectra, this does provide a valid means of reporting fluence data. When fluence is reported for E > 1 MeV in materials studies, the assumption is sometimes made that damage is proportional to the flux above 1 MeV. According to analyses of the displacement process, neutrons with energy greater than 1 MeV cause less than half the displacements in iron and even less in lighter elements. As a result, this reporting convention has not been adequate to correlate irradiation damage in diverse neutron spectra. The equivalent fission flux, another commonly used reporting convention, is defined as the flux of fission neutrons which would produce the same reaction rate in the detector as the actual flux. Activation data are reduced with fission-spectrum-averaged cross-sections, but the use of a fission spectrum is definitely not implied. Advantages are that the assumed spectrum is known and the same cross-section values are used for all- irradiations, thus reducing errors in calculating or reconstructing'fluences. When this convention is used in materials studies, a damage index is calculated for various facilities to correct for difference in damage caused by given 'fission fluences' by spectral variations. Proper use of this convention requires publication of cross-section data for reconstruction of 172 : CHAPTER IV1 fluence and of damage indices and spectra for interpretation of material, damage. The advantage is that fluence data can be clearly interpreted provided the cross-sections are given. Unfortunately the inference is often drawn that a fission spectrum actually is present. The practice of reporting fluence in units of megawatt days, or megawatt days per ton of adjacent fuel has been used extensively in the past for irradiations conducted in large reactors. Irradiations were monitored with reactor operating history and fluences were normalized lo integrated power. The ratio of fast neutron flux to fission rates is certainly not constant from reactor to reactor since fission rates depend upon thermal neutron flux which is much more sensitive to fuel burn-up and control-rod movement than to fast flux. This system provided self-consistent data for each reactor but no basis for correlation among different reactors. Proper treatment of neutron spectra and damage processes may change the relation between damage and fluence by factor of two for extreme spectra. Such a change can be critical in many very important applications. Examples are: Prediction of safe lifetimes for reactor pressure vessels from accelera- ted tests conducted at in-core positions Estimation of clad rupture from test data Prediction of dimensional changes in graphite moderators from test data generated in.light-water-moderated or fast reactors Design of fast-reactor components from data generated in thermal reactors. Each of the above cases represents an instance where the need for proper correlation is critical and where the refinement afforded by such correlation is essential. If proper correlation is not accomplished, the variations introduced by different reactor spectra contribute to the spread of experimental data. With a larger spread in data experiments are less efficient in distin- guishing the effects of variables. Therefore proper correlation through a suitable reporting convention will enhance the efficiency of radiation tests. Accurate neutron spectra are required to correlate fluence properly for radiation-damage studies. Spectra are determined through reactor physics calculations as described in Chapter II-. Although experimental techniques have not been developed which give accurate, detailed spectral resolution in the energy range above 0. 1 MeV, partial verification of calculated spectra is sometimes possible by comparing calculated values of monitor activations or reactor parameters with experimental observations. Complete reliance upon calculated spectra is necessary when attempting to extend data to a new reactor design. Spectra definition must be made in relatively fine energy groups to calculate activation rates and damage processes adequately. A group structure having lethargy intervals of Д/л = 0. 25 and lethargy1 values from ц = 0 to д = 4 (energy from 10 to 0. 18 MeV) is recommended. Spectral definition is much more important than calculation of absolute neutron fluxes since fluxes can be determined by monitors if spectra have been calculated. Moreover, calculations of damage indices and effective cross- sections involve only the spectral shape.

1 Lethargy u = In (E0 /Е) with E„ = 10 MeV. FAST NEUTRONS 173

Correlation of irradiation damage in materials also requires a model describing the damage mechanism. Most of the work to date has assumed that radiation damage is proportional.to displacement production. Although complex defect nucleation and annealing processes occur, it. has been postulated that the effects of these processes would be the same if all experimental variables (i. e. temperature, test methods, material composition, stress) were the same ex'cept for neutron spectra. To date this general assumption has been adequate to correlate data on graphite and metals; however, research may demonstrate that more refined models will be needed to describe the phenomenon under study. In calculating displacement production the neutron spectra, rather than absolute intensity, are used since absolute displacement rates are not generally necessary to correlate radiation effects. A model which describes the dependence of displacement production on neutron energy is required. The displacement cross-section is equal to the product of the microscopic transport cross-section of the material and the number of displacements caused by the collision of a neutron of a particular energy with a lattice atom, or

.£d(E) = N(E) ES(E) ( V. 22)

where ED is the differential displacement cross-section; N(E) is the number of displacements per primary knock-on as a function of neutron energy;

and Es is the elastic scattering cross-section of the material. The term N(E) is based on a displacement-theory model such as those of Kinchin and Pease [ 21], Snyder and Neuf eld [22], Thompson and Wright [23], de Halas [24] or others. While the model is the most uncertain part of an analysis, uncertainties in the absolute number of displacements do not influence the accuracy of a method for correlating radiation damage if the functional form is correct. The displacement cross-section term should be modified if processes other'than displacements are significant sources of damage. Examples are coefficients or cross-sections for reactions which may produce damage through transmutations,- recoil energy or gas generation. However, in . most techniques utilized to correlate damage, the assumption is made that damage is proportional to displacements. Several techniques have been proposed to correlate damage through proper normalization of fluences. Methods have been developed by Dahl and Yoshikawa [6], Thompson and Wright [23], Rossin [25], Zijp [1] and Grounes [26] that utilize energy-dependent damage models and detailed reactor spectra to correlate radiation effects. Each of these treatments relates displacements to ultimate damage, yet flexibility exists to account for damage caused by other mechanisms. Most of these methods produce very similar damage indices, that is ratios between the damage effectiveness of various spectra, but differ in the proposed manner of reporting fluences. Some of these methods incorporate additional refinements in the models utilized to calculate displacements but end results are similar. A brief description of these methods will be given followed by a discussion of other methods that have a different basis. The method of correlating radiation damage developed by Dahl and Yoshikawa [ 6] proposes reporting neutron fluences above an energy, E^, that is calculated to give a constant ratio of displacements to integral flux in diverse spectra. The objective of this treatment was to report fluence 174 : CHAPTER IV1 in terms that are familiar to materials scientists, yet have a sound technical basis. The inherent errors in the Ф (E > 1 MeV) system were corrected without a radical departure from accepted terminology. A value of was calculated for each material (i.e. Fe, C, Zr) such that an 'effective damage cross-section' for a material would have a constant value for all spectra. The 'effective damage cross-section' К is calculated according to

7 E (E)

Ф (E > El) sometimes leads to the misconception that only neutrons with energies greater than ELare considered in the analysis. Actually all neutron energies and mechanisms are considered and fluences are determined by reduction of activation data of monitors (i.e. 54Fe and 58Ni) with effective cross-sections calculated with the actual spectrum. This method has been utilized in the analysis of graphite irradiations and of metal irradiation studies, particularly in predicting transition temperatures in pressure vessel steels. The damage functions applied to equivalent fission-spectrum fluènces by UK investigators are calculated in much the same manner. Displace- ments and relative damage functions are calculated for each irradiation facility and have been tested where possible by experimental means. The damage functions which have been used agree very well with measurements of electrical resistivity changes in graphite in a variety of irradiation facilities. Damage functions are now calculated using the model developed by Thompson and Wright [23] and (extremely) detailed neutron spectra calculated with the Monte Carlo [27] method. The treatment of Thompson and Wright [23] is scientifically well founded and can incorporate modifications suggested by more detailed analyses of the displacement process. The principle disadvantage is that two quantities must be reported and authors sometimes fail to mention damage functions after reporting equivalent fission fluences. Methods proposed by Rossin [25], Shure [28], Pawlicki [29] and others are somewhat similar. Each cites the necessity for considering the actual spectrum, utilizes an energy-dependent damage model and the fission spectrum for a reference. The methods developed by these investigators report damaging exposure in terms other than conventional fluence notation. These are techniques which are as exacting as the preceding two and can also be revised as fundamental advances are made in understanding the mechanisms of radiation damage in materials. FAST NEUTRONS 175

Rossin's method [25]., described briefly as an example, involves a Radiation Damage Unit (RDÜ) which is the damage caused by a unit flux having a fission-spectrum distribution. Mathematically the unit is denoted by the following integral

fo¿ (E) Of (E) dE . a¿=°—^ ( V. 24) .i (E) 0 1 where cr¿ (E) is a differential damage cross-section. In his initial formulation Rossin assumed the displacement rate proportional to neutron energy; however, any other model can be used instead. From the definition one sees that 1000 neutrons having a fission spectrum • distribution would deliver 1000 RDU by definition. To find an exposure rate, the spectrum for the location of interest, multiplied by appropriate RDU cross-sections, expressed in the same manner as activation cross- sections, is integrated over energy. Rossin recommends that where possible, differential flux be normalized to reactor power. Thus RDU/MW-sec are calculated and can be integrated over the power history of an irradiation to give an exposure in RDU. This damage unit can also be normalized to the activation of a monitor such as 54Fe. The RDU/54Fe ratio derived in this way could effectively .make a specimen containing iron its own monitor providing the activation data are properly reduced. • Rossin has applied his method to correlate data on mechanical properties of copper and iron irradiated in fast and thermal reactor core positions. Shure [ 28] and Pawlicki [29] used a quite similar treatment in studies of pressure-vessel life. The principle advantage of Rossin's method is a relatively precise and unambiguous unit. The method utilizes detailed data on neutron spectra and damage theory and is flexible enough to incorporate refinements. This method requires detailed reporting of RDU cross-sections, normalization bases and spectra for interpretation and reconstruction. The main disadvantage is that the RDU is markedly different from terminology utilized by people conducting radiation-damage experiments. Many of the same advantages and disadvantages characterize the methods of Shure [28], Zijp [ 1] and Pawlicki [29] which are developed in much the same manner. With each of the models damage ratios calculated between actual spectra and differential damage modes agree quite well. The most significant points are the necessity for realistic spectra represented in relatively fine energy detail and the use of damage mechanisms to weigh the relative effectiveness of neutrons. Beeler [30] analyses damage mechanisms in specific materials and specimen geometries by calculating the motion of atoms in a lattice struck by incident energetic neutrons. Formation of clusters are calculated by Monte Carlo [27] methods in BeO and bcc iron for sample geometries used in test and service irradiations. Transmission through the sample is considered as well as focussing and channelling within the lattice, in calculating the nucleation and destruction of clusters. Beeler [30] concludes that irradiation induced change in mechanical properties of a-iron can be correlated more exactly by assuming damage to be proportional to the number of displacement spikes caused by primary knock-on atoms (PKA) 176 : CHAPTER IV1 with energies greater than 2. 7 keV rather than to the gross number of displacement spikes. This detailed and sophisticated type of treatment may prove to be necessary in predicting radiation damage where specimen geometries are very different. A damage model based on Beeler's analysis could be applied in any of the preceding methods where necessary. Several methods assume that damage is proportional to the average energy of neutron spectra. For example, Kantz [31] estimates average energy by a threshold-reaction technique using the two reactions 32S(n, p) and 239Pu(n, f) with effective thresholds of 3. 0 MeV and 0. 001 MeV respectively. By this technique the average energy isE = 7.42 R + 0.3 MeV where R is the ratio of the flux above the sulphur threshold to that of above the plutonium. This technique has been applied to studies of carrier removal rate in n-type silicon.

V. 4. 3. Summary of fluence measurements

In summary then, the simplest method for correlating data, and the most commonly used, reports Ф (E > 1 MeV) on the basis of an assumed fission spectrum:. This maybe adequate where spectral variations are" ; small but otherwise can lead to errors of 100 to 200%. Methods based on the assumption that damage is proportional to average 'fast' neutron energies are somewhat more precise, but are more appropriate for irradiations involving heavy elements in which damage is more nearly proportional to neutron energy. Methods which consider energy-dependent damage models and actual neutron spectra have refined structural materials data to a degree where uncertainties resulting from neutron spectra are equal to or less than those introduced by other experimental variables. Additional refinement will be made as the damage mechanisms are better understood. For all methods of reporting exposures, the need for complete reporting of cross-sections, spectra and damage ratios is critical.

V. 5. SPECTRAL DETERMINATION FROM MONITOR ACTIVATIONS

Although a number of methods for spectral determinations from monitor activations have been developed, none is widely accepted or applied. However, they should prove useful for many applications when developed further and tested more fully. Ideally, a number of monitors, each sensitive to a unique and narrow energy range, should be available. Relative activations'would then define spectra in the energy ranges covered. Unfortunately, fast-neutron spectra are very difficult to determine in accurate detail by monitor activations because ideal materials do not exist. For example, no monitors are adequate to-cover the energy range from about 1 MeV down to less than 0.01 MeV. Moreover, cross-section uncertainties are a definite limitation. Also, the choice of monitors is even more limited by experimental conditions that may preclude the use-'of monitors with short half-lives'than in normal fluence or flux density measurements. All methods for synthesizing spectra from activation data are quite complex mathematically and require computer solution. A few of these methods will be briefly described here, with the recommendation that the more interested reader refer to Zijp's [1] review. FAST NEUTRONS 177

The activation of detectors may be expressed in the form

oo A¡ = JTj (E) q> (E) dE (V. 25) 0 where the subscript i refers to the ith detector and the integral indicates the reaction rate at saturation per target nucleus of the ith detector in the differential neutron flux

Mathematical methods:

(1) . Flux integral method in which the actual cross-section curve is replaced by a step function so integral flux densities for definite energy ranges are obtained (Hughes [32], Hurst [33]) (2) Step curve method: the spectrum is represented as a step curve (histogram) with suitably chosen energy intervals (Delattre [34], Brownell [35]) (3) Polygonal method in which the spectrum is represented by a polygon (Uthe [ 36]) (4) Simple polynomial method in which the spectrum is represented by a polynomial in E (Uthe [36]) (5) Expansion in orthonomial combination, combinations of cross- section curves (Hartman [37], Trice [38]) (6) Expansion in orthonomial combinations of simple polynomials, (banning and Brown [39]) (7) Method with successive exponentials (Dierckx [40]). Perturbation methods include: (8) Semi-empirical deviation method (Dietrich and Thomas [41]) (9) Polynomial deviation method [1] - (10) Orthonomial deviation method [1] (11) Method of spectral indices (Grundl and Usner [4], Uthe [36]) (12) Iterative method with successive exponentials (Bresesti et al. [42]) (13) Iterative perturbation method (McElroy et al. [43]). Weighing methods: (14) Series expansion method with weighing function (DiCola and Rota [44]) (15) Relative deviation minimization method (DiCola and Rota [44]).

Few of these methods have been applied to synthesize spectra or calculate flux from activation data. Perhaps the most extensively tested is the method of McElroy et al. which involves selection of an initial spectral approximation and subsequent perturbation to obtain a best-fit simultaneous solution for a system of ten or more activation integral equations. The 178 : CHAPTER IV1 recently developed computer code SAND II will yield integral and differential flux in a 621-group histogram over a region of 10"10 to 18 MeV. Effects of cross-section uncertainties are reduced with a best fit rather than an exact solution over the overlapping sensitivity ranges of various monitors. Spectra calculated with SAND II compared with spectra obtained by other experimental and analytical methods including time-of-flight neutron spectrometers, diffusion, transport and Monte Carlo calculations, show agreement within about 30% when ten or more monitor types could be used. Fast-neutron spectral synthesis from activation data is very complex and at present somewhat limited and uncertain because of restricted monitor selection and cross-section inaccuracies. Application to calculate detailed spectra for irradiation damage studies may be limited because of the very few monitors which are sensitive to neutrons with energies between 0.01 and 1 MeV. However, many of the limitations are being overcome as methods are being refined and tested.

V. 6. OTHER METHODS OF FLUENCE MEASUREMENT

Threshold activation monitors are not the only means of measuring fast neutron fluence. In general, other methods depend upon fluence measurement by activations for calibration. These methods may then be used for applications where activation monitors would be unsuitable. The most common technique has been the use of materials as self- monitors. For example, property changes in electronic devices, graphite and metals have been used to determine fluence. Resistivity changes in semi-conductor devices have been used in non-recoverable space experiments and for many space-oriented studies. Similar experimental techniques have been used to determine fluence in structural materials irradiations. Electrical resistivity changes in graphite were the basis for the work reported by Bell et al. [45] in formulating the Calder Equivalent Dose. Mechanical and physical property changes in-metals have been used occasionally to determine fluence. Caution is necessary in employing such property changes to monitor exposures. The response of the property in question and the effect of other variables must be known accurately. The dependence of the property change upon fluence must not change during the irradiation. Unfortunately, the relation between fluence and a property change can be masked by the effects of other variables. As an example, annealing effects caused by temperature changes are often very large and could produce substantial error. Several groups are developing the technique of monitoring fast-neutron fluence by observing changes in quartz density. This method appears promising but no reported usage or direct comparison with other methods could be found. Measurement of fission rates by analysis of fission products or fission tracks is being developed for a range of applications. In one such technique fission fragment tracks are counted in mica, glass or even a plastic. Price and Walker [46] reported that disordered regions caused by fission frag- ments are selectively etched and rendered visible by HF in a few minutes. A low-power optical microscope is used to count tracks quickly. Utiliza- tion of this technique provides a fluence monitor which does not saturate FAST NEUTRONS 179 and does not decay; hence the monitor can function as a true time integrator [47]. Martin [48] is evaluating a fast-fluence monitor technique based upon detecting helium produced in beryllium by (n, (?) reactions. A mass spectrometer analyses the helium released by melting the beryllium in vacuo. Reported accuracies are very good. This technique may be very useful for extended intermittent irradiations. The preceding methods are generally suited to measuring fluence in high flux densities. Several other techniques, largely limited to low flux densities, could be very helpful in validating or correcting spectra obtained from computations. Although measurements have to be made at very low flux densities - fast neutron spectra appear to be nearly independent of flux density. Thus spectral measurements made at low reactor power or in critical facilities might be applicable to normal reactor operations. Semiconductor detectors are being developed which rely on measurement of the energies of the products from the 3He (n, p) and 6Li (n, a) reactions. The energy of the charged reaction products is equal to the incident neutron energy and the Q value of the reaction. A neutron spectrometer consists of helium gas or a thin layer of 6LiF between two closely spaced semiconductor detectors (sandwich geometry) each of which then records one of the reaction products which are emitted at about 180° from one another. An extensive review of neutron spectrometers has been prepared by Dearnaley and Northrop [49]. The semiconductor spectrometer systems are compact and relatively rugged, their response is linear and independent of the type of incident particle and the Q value provides a built-in bias against low-energy background events. Some of the disadvantages are low efficiency, significant count background from (n, a) and (n, p) reactions in a silicon semiconductor, susceptibility .to radiation damage at relatively low fluences 1012 n-cm"2) and interference of reactor gammas in 3He counters due to the low Q value (0. 76 MeV compared to 4. 78 MeV of the 6Li reaction). Shielding is required in thermal reactors for the 6Li spectrometer because of the large thermal cross-section for the 6Li (n, a) reaction. The spectrometer systems have not been used routinely to date since electronic development is generally required. However, they should provide a means in the future of mapping neutron spectra, particularly in epithermal and fast reactor critical assemblies.

REFERENCES TO CHAPTER V

[ 1] ZIJP, W. L., Review of Activation Methods for the Determination of Fast Neutron Spectra, Rep. RCN-37 (1965). [2] WATT, B.E., Energy spectrum of neutrons from thermal fissions of 235 U, Phys.Rev. 87 (1952) 1037. [3] CRANBERG, L., FRYE, G., NERESON, N.. ROSEN, L., Fission neutron spectrum of 235 U, Phys. Rev. 103 (1956) 662. [4] GRUNDL, J.A., USNER, A., Spectral comparisons with high energy activation detectors, Nucl. Sci. Engng 8 (1960) 598. [ 50 BEMENT, A. L., DAHL, R. E., IRVIN, I. E., Fast Neutron Flux Characteristics of the ETR, G7 Hot Water Loop, Rep. BNWL-89 (1965). [6] DAHL, R. E., YOSHIKAWA, H. H., Neutron spectra calculations for radiation damage studies, Nucl. Sci. Engng 17 (1963) 389-403. 180 : CHAPTER IV1

[7] Tentative Method for Measuring Fast-Neutron Flux by Radioactivation of Sulfur, ASTM Designation E 265-65T (1965). [8] MORGAN, W. C., Foils: A Program for Computing Neutron Exposures from Foil-Activation Data, Rep. HW-81367 (1964). [9] MARTIN, W. H., CLARE, D.M., Determination of fast-neutron dose by nickel activation, Nucl. Sci. Engng 18 (1964) 468-73. [ 10] MELLISH, С. E., PAYNE, ]. A., ОТ LET, R. L., Flux and Cross-section Measurements with Fast Fission Neutrons in BEPO and DIDO, Rep. AERE-I/R-2630 (1958). [11] HOGG, C.H., WEBBER, L. D., YATES, E. C., Thermal Neutron Cross-sections of the 58 Co Isomers and the Effect on Fast Flux Measurements Using Nickel; Rep. IDO 16744 (1962). [ 12] PASSELL, Т. O., HEATH, R. L., Cross-sections of threshold reactions for fission neutrons: Nickel as a fast flux monitor, Nucl. Sci. Engng 4 (1961) 308-15. [ 13] MARTIN, W. H., Effect of neutron spectrum on the branching ratio of the 58 Ni(n, p) 58 Co reaction, Nucl. Sci Engng 18 (1964) 531-32. [ 14] BARRALL, R. C., McELROY, W. N., Neutron Flux Spectra Determination by Foil Activation, Rep. AFWL-TR-65-34 Vol. 2 (1965). [ 15] MEADOWS, J. W., WHALEN, J. F., Ni58 (n, p) Co58m- g cross-section and isomer ratio from 1.04 to 2.67 MeV, Phys. Rev. 130 (1963) 2022-25. [16] HURLIMAN, T., HUBER, P., The reaction cross-section of S (na) Si and S (n, p) P for neutron energy of 2.2 to 4.0 MeV, Helv. phys. Acta 28, (1955) 34-48.'

[17] LEESER, D.O., Radiation effects on reactor metals, Nucleonics, 18 9 (1960) 68. [18] RICAMO, R., Risonanze (n, n)ed (n, p) ncl p31 eS32, Nuovo Cim. i (1951) 383-402. [19] CARROLL, E.E., Jr., SMITH, G.G., Iron-54(n, p) cross-section measurement, Nucl. Sci. Engng 22 (1965) 411-15. [20] HELM, J. W., High temperature graphite irradiations, Carbon 13 (1966) 493-501. [21] KINCHIN, G.H., PEASE, R. S., The displacement of atoms in solids by radiation, Rpts Prog. Phys. 1£ (1955) 1-51. [22] SNYDER, W.S., NEUFELD, J., Disordering of solids by neutron irradiation, Phys. Rev. 97 (1955) 1636; 99 (1955) 1326; 103 (1956) 862. [23] THOMPSON, M. W., W.RIGHT, S. B., A New Damage Function for Predicting the Effect of Reactor Irradiation on Graphite in Different Neutron Spectra, Rep. AERE-R 4701 (1964). [24] deHALAS, D. R., "Theory of radiation effects in graphite, " in Nuclear Graphite (NIGHTINGALE, R.E., Ed.), Academic Press, New York (1962) 195-238. [25] ROSSIN, A. D., Degradation of impact energy of steel as a function of neutron exposure, Trans. Am. nucl. Soc. 6 (1963) 389-90. [26] GROUNES, M., "Review of Swedish work on irradiation effects in pressure vessel steels and on the significance of the data obtained, " in Proc. Symp. Effects of Radiation on Structural Metals, 1966, Atlantic City, N. J. [27] WRIGHT, S. B., Calculation of High Energy Neutron Spectra in Heterogeneous Reactor Systems, Rep. AERE-R-4080 (1962). [28] SHURE, K., Radiation damage exposure and embrittlement of reactor pressure vessels, Nucl. Appli- cations 2 (1966) 106-15. [29] PAWLICKI, S.S., Neutron-exposure criteria for reactor vessels, Trans. Am. nucl. Soc. 6 (1963) 149-50. [30] BEELER, J.R. Jr., Primary damage state in neutron-irradiated iron, J. apple. Phys. 35 (1964) 2226-36. [31] KANTZ, A. D., Average neutron energy of reactor spectra and its influence on displacement damage, J. appl. Phys. 34 7 (1963) 1944-52. [32] HUGHES, D.J., "Pile Neutron Research, " p. 93 Addison-Wesley, Cambridge Mass., 1953. [33] HURST, G.S. et al.; "Techniques of Measuring Neutron Spectra With Threshold Detectors, " Rev. Sci. Instruments 27 153 (1956). [34] DELATTRE, P. "Les Méthodes de Détermination des Spectres de Neutrons Rapides i Г aide de Détecteurs à sevil, " CEA-1979 (1961). [35] BROWNELL, G. L. et al., "Neutron Spectroscopy and Dosimetry at the Medical Therapy Facility of the MIT Reactor, " Neutron Dosimetry, Vol. I, p. 51, IAEA, Vienna, 1963. [36] UTHE, P.M., "Attainment of Neutron Flux Spectra Foil Activations, " UCRL-5403 (March 1957) (Also WADC-TR-57-3). [37] HARTMAN, S.R., A Method for Determining Neutron Flux Spectra from Activation Measurements, WADC Tech. Rep. 57-375 (Oct. 1957). [38] TRICE, J. B., Measuring reactor spectra with thresholds and resonances, Nucleonics 16 7 (1958) 81. FAST NEUTRONS 181

[39] LANNING, W. D., BROWN, K. W., Calculated and Measured Neutron Energy Spectral Distributions Using the Threshold Detector Techniques, Rep. WAPD-T-1380 (Sept. 1961). [40] DIERCKX, R., Threshold detectors for measuring fast neutron spectra, Nucl. Instrum. Meth. 15 (1962) 355-56. [41] DIETRICH, O. W., THOMAS, J., "Fast neutron spectra by threshold activation, " Physics of Fast and Intermediate Reactors (Proc. Seminar Vienna, 1961) 1, IAEA, Vienna (1962) 377. [42] BRESESTI, M. et al., Fast Neutron Spectrometry in Pile by Threshold Detectors, Rep. EUR-289.e (1963). [43] McELROY, W. N., BERG, S., GIG AS, G., Neutron flux spectral determination by foil activation, Nucl. Sci. and Engng 27 (1967) 533. [44] DiCOLA, G., ROTA, A., Analysis and Development of the Series Expansion Methods in Threshold Detectors Activation Data Handling, Rep. EUR-588. с (1964). [45] BELL, J. C. et al., Stored energy in graphite of power-producing reactors, Phil. Trans. R. Soc. Ser. A 254 (1962) 361-95. [46] PRICE, P. В., WALKER, R. M., Observations of charged-particle tracks in solids, I. appl. Phys. 33 (1962) 3400-12. [47] PREVO, P.R., DAHL, R. E., YOSHIKAWA, H. H., Thermal and fast neutron detection by fission- track production in mica, I. appl. Phys. 3£(1964) 2636-38. [48] MARTIN, W. H., "Measurement of neutron fluence in irradiation experiments performed in the UK, " Presented at IAEA Panel on In-Pile Fluence Measurements, 11-15 Oct, 1965. [49] DEARNALEY, G., NORTHUP, D. C., Semiconductor counters for nuclear radiation, John Wiley, New York, 2nd ed. (1966).

INDEX

Absolute calibration: 62 Fission monitors, neptunium: 152 Activation cross-section: 145 Fission monitors, thorium: 152 Aluminium cobalt: 63 Fission monitors, uranium: 152 Aluminium, cross-section: 163, 166 Fission neutron spectrum: 87 Aluminium monitors : 154 Fission products: 75 Argon: 71 Flux depression: 55, 57, 58, 61, 106 Atomic displacements: 16 Flux perturbation: 54 B>—coincidence counting: 59 Fuel burn-up: 58 BFg-counters: 69 Gadolinium: 132 Boron: 75, 132 Gadolinium ratios: 132 Boron ionization chambers: 69 Gas loops-. 71 Boron shields: 133 Gold: 59, 66, 68, 74 Buckling measurement: 58 Graphite: 58 Cadmium covers: 47, 53, 124 Grundl and Usner spectrum: 142 ' ' Cadmium cut-off: 19, 50," 99 Half-width of resonance: 88 Cadmium ratio: 47, 50, 53, 66, 67, 98, 99 Heavy water: 58 Chargedrparticle detection method: 135 High-temperature embrittlement: 148 Cobalt: 53, 58, 59, 64, 65 Hydrogenous: 58 Collectrons: 72 Indium: 59 Conventional thermal flux density: 46, 53, 82 Ionization chamber: 65 Copper: 59 Iron, cross-sections: 163, 168 Copper cobalt: 63 Joining energy Ej: 87 Counting techniques: 125 Lethargy: 12 Cranberg spectrum: 142 Light water: 58 Cross-sections, effective: 169 175Lu: 66,69 Cross-sections, fission-spectrum-averaged: 165 176Lu: 66,67 Crystalline moderator: 23 Lutetium: 65, 66, 68 Cut-off function: 83 Manganese: 59 Detector geometry: 151 Maxwell-Boltzmann neutron distribution: 45, 65 Diffusion theory: 27, 57 Maxwellian component: 48 Displacement cross-section: 173 Maxwellian spectrum: 48, 51 Displacement spikes: 16, 176 Mean flux density: 54 Doppler broadening: 111 Mean velocity: 46 Doppler broadening resonances: 110 Megawatt days per adjacent ton: 171, 172 Doppler correction: 89, 118 M ELU SINE reactor: 68 Dysprosium: 59 Method of semi-differential epithermal index: 131 Effective cadmium cut-off: 47, 48 Moderator temperature: 67 Effective cross-section: 49, 52, 83 Moments method: 27 Effective cross-section factors: 84 Monitoring of irradiation experiments: 18 Effective decay constant: 149 Monte Carlo: 23, 27, 57 Effective half-life: 149 Most probable velocity: 45 Effective resonance integral: 92 Multiple scattering: 80, 81 Embrittlement, elevated temperature: 1 148 Nd: 75 Embrittlement of austenitic steels: 17 Neutron capture: 14 Epicadmium neutrons: 47 Neutron temperature: 13, 19, 23, 52, 83 Epithermal index r: 83 Nuclear heating: 16 Epithermal neutrons:. 47, 66 1/E spectrum: 48, 81 Filled proportional counters: 39 One-parameter spectrum method: 131 Fission monitors: 157, 165 1/v activation detector: 47, 49

183 184 ÍNDEX

1/v subtraction technique: 94 Self-shielding correction data: 126 Orthonormal methods: 177 Semi-empirical spectrum representation: 86 Parasitic activities: 59 Spectral indice method: 177 Perfect 1/v absorber: 19 Spectrum for an infinite hydrogen-moderated Photographic plates: 39 reactor: 12 Photomultiplier: 71 Standard pile: 58 Placzek function: 55 Standard source: 65 Plutonium: 75 Successive exponential methods: 177 Polygonal spectrum method: 131, 177 Sulphur-compound monitors: 154 Polynomial methods: 177 Sulphur, cross-sections: 163, 164 Primary knock-on atoms: 175 Sulphur, monitor: 152 Proton recoil techniques: 39, 134 Threshold activation detectors: 37 Radiation damage: 16 Threshold reactions: 17 Reaction rate: 46, 52 Time-of-flight method: 135 . .

Removal cross-section method: 27 /( Transport theory: 27, 57 Resonance capture: 14 Triple-foil method: 106, 131 Resonance integral cross-section: 88 Two-foil method: 105 Resonance integrals: 47 2200 m/sec flux density: 82 . .

Resonant detector: 59 Types of resonances: 80 4 Response function: 14, 94, 145 238U: 75 Rupture of chemical bonds: 15 U depletion: 75 Samarium: 132 U fission chambers: 69 Scattering by a monatomic gas: 23 Unified formulation of Nisle: 83 Scintillation counter: 65, 71 Watt fission spectrum: 142 Scintillator crystal: 71 Westcott's formalism: 48, 51, .82 . Se If-monitors: 148 Zirconium cobalt: 63 Self-shielding: 50, 55, 107

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