Atlas of Neutron Resonances
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Atlas of Neutron Resonances Resonance Parameters and Thermal Cross Sections Z=l-100 5 th Edition
S.F. Mughabghab National Nuclear Data Center Brookhaven National Laboratory Upton, USA
ELSEVIER Amsterdam Boston Heidelberg London New York Oxford Paris San Diego San Francisco Singapore Sydney Tokyo
Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands The Boulevard, Langford Lane, Kidlington, Oxford 0X5 1GB, UK First, second and third editions published 1952, 1955 and 1973 by Brookhaven National Laboratory Fourth edition published 1981 by Academic Press Fifth edition published 2006 by Elsevier Copyright © 2006 Elsevier BV. All rights reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without prior written permission of the publisher Permissions may be sought directly from Elsevier's Science & Technology Rights Department in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333; e-mail:
[email protected]. Alternatively you can submit your request online by visiting the Elsevier web site at http://elsevier.com/locate/permissions, and selecting Obtaining permission to use Elsevier material Notice No responsibility is assumed by the Publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library. Library of Congress Cataloging in Publication Data A catalog record for this book is available from the Library of Congress. ISBN-13: ISBN-10:
978-0-444-52035-7 0-444-5203 5-X
For information on all book publications visit our website at books.elsevier.com Printed and bound in The Netherlands 06 07 08 09 10 10 9 8 7 6 5 4 3 2 1
To Pavel, Charles and Suzanna for their support
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Preface This book is the fifth edition of what was previously known as BNL-325, Neutron Cross Sections, Volume 1, Resonance Parameters (Third Edition, 1973) by S. F. Mughabghab and D. I. Garber. The first edition of the BNL-325 reports which appeared in 1955 was prepared by Donald J. Hughes and John A. Harvey. The fourth edition was published by Academic Press in two parts in 1981 and 1984. As with the last two editions, only recommended parameters are presented. For more detailed information relating to experimental data, the user is encouraged to consult the web site of the National Nuclear Data Center at Brookhaven National Laboratory, www.nndc.bnl.gov. In addition to the extensive list of detailed individual resonance parameters for each nucleus, this book contains thermal cross sections and average resonance parameters as well as a short survey of the physics of thermal and resonance neutrons with emphasis on evaluation methods. The introduction has been expanded to include commonly used nuclear physics formulas and topics of interest such as direct or valence neutron capture, sub-threshold fission, the nuclear level density formula, and the treatment of electric dipole radiation in terms of the Fermi Liquid Model. As in the last edition, additional features have been included to appeal to a wider spectrum of users. These include (1) spin-dependent scattering lengths that are of interest to solid state as well as nuclear physicists and neutron evaluators, (2) Maxwellian average 30-keV capture cross sections that are of importance to astro-physicists, (3) s-, p-, and d-wave average radiative widths, gamma strength functions for s- and p-wave neutrons, and (4) nuclear level density parameters. The various neutron strength functions are compared with optical model calculations and the radiative widths are calculated within the approach of the generalized Fermi Liquid Model and then compared with experimental data. Extensive application of the Porter-Thomas distribution, coupled with Bayesian analysis in the resonance region, was made in order to determine the parity of neutron resonances. The objective of achieving consistency between the thermal cross sections on one hand and the resonance parameters on the other is met by postulating negative energy resonances. Immediately preceding the resonance parameter tables, one can find the contributions to the thermal capture and fission cross sections from positive energy resonances for each spin state (for odd target nuclei) as well as the direct capture component calculated within the framework of the the Lane-Lynn approach. This information is required in nuclear structure investigations carried out with thermal neutrons and in the design of a thermal neutron polarizers. The previous editions of BNL-325 have been widely used and extensively cited. vii
viii
Preface
We hope that this new edition will continue to be a prime resource which will satisfy the needs of the casual and serious users of neutron cross sections as well investigators interested in this rich field.
S. F. Mughabghab National Nuclear Data Center Brookhaven National Laboratory September 2005
Acknowledgments A project of this magnitude would not be possible without support from many individuals. The list of people whose help must be acknowledged is necessarily long. Three people must be specially thanked. These are Dr. Pavel Oblozinsky, Dr. Charles Dunford, and Dr. Alejandro Sonzogni. Without their support and encouragement, the production of this fifth edition would not be possible. The scope of the project consisted of four parts: (1) preparation of the pertinent up-todate body of data; (2) evaluation and recommendation and (3) computer coding, and (4) checking and production. (1) Production of this book demanded an accurate and complete body of the world's resonance parameter and thermal cross section data. The computerized files of the CSISRS data library of the National Nuclear Data Center (NNDC) were used. In addition, the other three international data centers Data Bank (NEA), NDS (IAEA) and CJD (Russia) transmitted to us data collected from their areas of responsibility. (2) To produce resonance parameter listings in a readily usable form for checking, computer coding was ably carried out by Dr. R. Kinsey. (3) As an aid in the evaluation procedure, computer codes for physics checking and calculation of quantities from the voluminous amount of resonance parameters were required. The author wishes to express his gratitude to Dr. Charles Dunford, Dr. Jongwa Chang (KAERI),and Dr. Soo-Youl Oh (KAERI). (4) This volume was produced from computer-generated postscript files. The program to compose pages, including upper and lower symbols and Greek letters was written by Dr. R. Kinsey; subsequent modifications were carried out by Dr. T. Burrows. Their contributions were invaluable. The conversion of these files to pdf format, the checking of the final results, and the production of the figures was carried out by Dr. B. Pritychenko. Also, Dr. D. Rochman contributed in the production of the figures. The aid of Dr. D. Winchell in producing the numerous equations in Latex format is greatly appreciated. The author wishes to thank Mrs. J. Totans and Mrs. M. Blennau for their aid in the production task and Mr. R. Arcilia for his tireless effort in providing the necessary PC services. In addition, the author wishes to thank Dr. P. Oblozinsky, Dr. J. Tuli, Dr. M. Herman and Dr. A. Sonzogni for their suggestions and criticisms regarding the manuscript. Discussions of thermal capture cross sections with Dr. N. Holden are acknowledged. Many physicists outside the NNDC have aided in this publication. The author is grateful to the many experimentalists throughout the world who made special ix
x
Acknowledgments
efforts to provide their recent resonance parameter and thermal data prior to publication. In particular, thanks are due to Dr. P. Koehler, Dr. L. Leal, Dr. K. H. Guber, and Dr. R. Sayer of Oak Ridge National Laboratory, Professor R. Winters of Dennison University, Dr. P. Mutti of the Institute Lau-Langevin, Dr. P. Schillebeecks, Dr. J. Wagemans, Dr. A. Borella, and Dr. F. Corvi, and of EC-JRC-IRMM, Belgium, Dr.Ohkubo of Japan Atomic Research Institute, Dr. Yu. Popov of the Joint Institute of Nuclear Research, Dr. G. V. Muradian of the Russian Research Center Kurchatov Institute, Dr. L. De Smet of the University of Gent, Dr. Wisshak of the Institute of Kernphysik, Professor R. Block of Rensselaer Polytechnic Institute, and Dr. R. Firestone of Lawrence Berkeley National Laboratory. Last but not least this project could not have been completed without the support and encouragement of my wife, Suzanna. This research was carried out under the auspices of the United States Department of Energy under Prime Contract No. DE-AC02-98CH10866. Partial financial support from Ohio University and the Korean Atomic Research Institute (KAERI) is gratefully acknowledged.
Contents Preface
vii
Acknowledgments
ix
Contents
xi
List of Figures
xiii
List of Tables
xv
1 Thermal Cross Sections 1.1 Introduction 1.2 Scattering Cross Sections 1.3 Capture Cross Sections 1.4 Thermal Fission Cross Sections 1.5 Paramagnetic Scattering 1.6 Potential Scattering Length or Radius R'
1 1 1 6 13 16 16
2 Resonance Properties 2.1 S-, P- and D-Wave Neutron Strength Functions 2.2 Average Level Spacings and Level Density 2.3 Radiative Widths and 7-Ray Strength Functions of S- and P- Wave Resonances 2.4 Resonance Integrals
23 23 38 44 58
3
Individual Resonance Parameters 71 3.1 Determination of Spins of Neutron Resonances 71 3.2 Determination of the Parity of Neutron Resonances 75 3.3 Scattering Widths: Relationship Between Sdp and T°n 79 3.4 S- and P- Wave Radiative Widths 80 3.5 Alpha Widths of Neutron Resonances and the (11,7a) Reaction . . . 85 3.6 Neutron-Induced Fission 86
4
Notation and Nomenclature 4.1 Thermal Cross Sections 4.2 Resonance Properties
107 107 108 xi
xii
CONTENTS 4.3 Resonance Parameters 109 4.4 Weighted and Unweighted Averages, Internal and External Errors . I l l 4.5 Reference Code Mnemonics 112
Bibliography
121
List of Figures
1.1 Comparison of the paramagnetic scattering cross sections measured at a neutron energy of 0.0253 eV with the calculations of Mattos [49]. The solid curve is an eye-guide to the theoretical values
17
1.2 Variation of R' with mass number A. The solid curve is based on the deformed optical model with parameters Vo=43.5 MeV, ro=1.35 fm, Vso=8 MeV and surface absorption W/j=5.4 MeV. The dotted curve describing the trend at low mass numbers is based on spherical optical 20 model calculations using the same parameters 2.1 Comparison of the theoretical with experimental values of the s-wave neutron strength function. The solid curve represents deformed optical model calculations of Ref. [87] while the dotted curve is based on spherical optical model calculations of the present work
26
2.2 Comparison of the theoretical with experimental values of the p-wave neutron strength function. The solid curve represents deformed optical model calculations of Ref. [87] while the dotted curve is based on spherical optical model calculations of the present work
27
2.3 Comparison of the theoretical with experimental values of the d-wave neutron strength function. The solid curve represents spherical optical model calculations of Ref. [88]
28
2.4 Variation of the s-wave neutron strength function with mass number for the Cd, Sn and Te isotopes compared with the theoretical prediction (solid line) of a doorway state model [94] and optical model results (dashed line) which include an isospin term [119]. The dotted curves represent optical model calculations without an isospin term [87]
36
2.5 Level density parameters aexp (solid triangles) derived from average resonance spacings are compared with those calculated on the basis of Eq. 2.40 Ogiobai (open circles); see text for details 2.6 The average s-wave radiative widths plotted versus mass number. Note the decrease of < F7o > with A
43 56
2.7 The average p-wave radiative widths plotted versus mass number. . . . 57 xiii
xiv
List of Figures
3.1 Potential energy as a function of the deformation parameter (3 according to Strutinsky's [283] calculation. The predicted double-humped fission barrier offers a description of the clustering of subthreshold fission strengths which was first observed in 240 Pu. The symbols are explained in the text 94 3.2 The measured subthreshold fission widths of 240 Pu plotted versus neutron energy. The clustering of fission strengths at certain neutron energies was interpreted by Lynn [4] and Weigmann [276] in terms of Strutinsky's double-humped fission barrier. The solid curve is a Breit-Wigner shape fit of the data, the parameters of which are reported by Auchampaugh and Weston [288] 97 3.3 The measured subthreshold fission widths of 238 Pu plotted versus neutron energy. The solid curve which is a differential fit of the data describes the intermediate structure observed at a neutron energy of 285 eV. For details see the text 98 3.4 The left-hand side of the figure represents the distribution of subthreshold fission widths of 238 Pu below a neutron energy of 500 eV and the right-hand side section describes the distribution of reduced neutron widths. The solid curves are integral chi-square distributions with v=\ and 3 degrees of freedom 99 3.5 The subthreshold fission widths of 234U showing two intermediate structures located at neutron energies of 580 and 1227 eV. For details, see James et al. [296] 102
List of Tables 1.1 1.2 1.3 1.4 1.5
Common Standards for Scattering Lengths Cross Section Standards The Westcott gw Factor for Non-l/v Capture Cross Sections Direct Capture Coefficients Sff Bound Scattering Cross Section
6 8 10 12 14
2.1 2.2
Penetrability Factors, Pi = kRVg, for a Square Well Potential S-Wave Strength Function Values of Isobars. All values from the present evaluation S-wave level spacing (Do), nuclear level density parameters (a,exp and ^global) derived using Eqs. 2.34 and 2.40, spin disperssion parameter (CM), predictions of capture widths (F* and F*) and experimental value (F«)
24
2.3
3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9
37
60
Angular Distribution of Dipole 7-Rays Due to p-Wave Capture in Target Nuclei With 1=0 77 Angular Momentum Spin Factors for s —> p, p —> s and p —> d Transitions for Nuclei with Zero Target Spin 82 Thermal (11,70!) and (n,a) Cross Sections 86 235 U Average Fission Widths of the Various Channels 88 Derived Information on the K*" Channel Contributions in Neutron Induced Fission of 235 U 89 Comparison of Measured and Calculated F 7 / Widths 90 238 Pu Parameters of the Intermediate Resonance at 285 eV 100 238 U Class II Parameters for the Intermediate Clusters at 721.6 and 1211 eV 104 Heights of the Bottom of the Second Well for Some U and Pu Isotopes Obtained from Level Spacings of Class I and Class II Resonances. . . . 105
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Chapter 1
Thermal Cross Sections 1.1
Introduction
The aim of the following survey is to present (1) the various considerations employed in the assessment of the experimental data, (2) the relationships, methods and principles that have been used in arriving at the recommended parameters, (3) the systematics of s- and p-wave neutron strength functions, radiative widths, and potential scattering radii and (4) the standards for capture, scattering cross sections, and neutron, as well as 7-ray energies. The cited relationships can serve as an easy reference guide. In addition, should the need arise, because of absence of experimental data, the user may appeal to the systematics of data as displayed in the form of large scale figures. This survey is not intended to be an exhaustive review of the field of neutron physics. For comprehensive and detailed reviews, the user may consult any of the excellent articles or books on the subject [1]-[12]. A discussion of recent results of parity non-conservation (PNC) in transmission measurements can be found in the review article of [13].
1.2
Scattering Cross Sections
The coherent neutron scattering lengths are basic nuclear quantities which represent the link between measured cross sections on the one hand and wave functions and nuclear potential on the other. Thus, scattering lengths can shed light on theoretical models dealing with the neutron-nucleus interaction. For example, accurate knowledge of the singlet and triplet scattering lengths of H is basic for theoretical models dealing with the n-p interaction. Scattering lengths of the four nucleon systems, n-3H and n-3He, can elucidate the nucleon nucleon interaction. Experimental values of the thermal neutron scattering lengths of 3H and3He favor the Yukawa potential over the exponential form as indicated by the calculations of Kharchenko and Levashev [14]. Also, spin-dependent scattering lengths of light and medium weight nuclei can test the accuracy of shell model calculations [15]. From the applied point of view, scattering lengths are used as "tools" in various studies of crystals and solids. 1
2
1. Thermal Cross Sections
What has just been stated can be recognized from the following discussion which is restricted to low energy neutron scattering. At large distances from the scattering center, the wave function describing the incident and elastically scattered neutron can be written in the form [16]
^-
(1.1)
f{9) describes the scattered wave amplitude in the direction 9 relative to the incident beam. For s-wave neutrons, f{9) is given by
fo =
ik{e2iSo-1)
= e
- -]rsinSo
(L2)
where k is the wave number and So is the s-wave phase shift which is related to the scattering length, a, by the definition /sin S \ - hm —-—o = a k^o \ k )
, . (1.3)
thus, = -ha
(1.4)
More generally, the phase shift S is related to the logarithmic derivative of the wave function at the nuclear surface and the scattering length by the following relationship which is applicable to second order terms in k 2 1 1 =kcotd=- -+ reSk
i.£|f) ip dr J
r=R
a
(1.5) 2
where reff is the effective range of the nuclear potential. Note that for k -> 0, the negative inverse of the scattering length is equivalent to the logarithmic derivative of the wave function at the nuclear surface. The scattered wave for I = 0 neutrons then has the form
c
kr and in the limit of h —> 0 -e r As a result, the scattering cross section for slow s-wave neutrons is
1.2. Scattering Cross Sections
Aqr
as = — sin2 So = 4TTO2
(1.6)
It can be shown [17] that the spin-dependent scattering lengths a+ and a_ associated with spins states / + 1/2 and / — 1/2 (where / is the spin of a target nucleus) can be written in terms of the Breit-Wigner formalism as
(L7)
where the summation is carried out over resonances with the same spin. The total coherent scattering length is then the sum of the partial scattering lengths weighted by their spin statistical factors g+ and
where
27Ti
and
9
(L8
- = 2iTi
The coherent scattering length for each spin state Eq. 1.7 can be separated into real and imaginary parts
r
( T?
T? \
\
1lj\Ilj — Hi'j 1 Tp \2 _i_ "P2
31
j
T
"P
. ^—T /\jL njL / j AfTp Tp .\2
j _i_ "P2
31
3
j
<>
(1-9)
= ar + iai where ar and a, are respectively the real and imaginary components of aCOh It is interesting to note that the imaginary part can be related to the absorption cross section, aa by ai
( L1 °)
=17
where k (in cm" 1 ), the wave number, is given by
k = ^ = 2.1968 x 109 (-AT)
^E
(1.11)
A \A + 1/ E is expressed in the laboratory system in eV and A is the nucleus-neutron mass ratio. Substituting Eq. 1.11 in Eq. 1.10, one then obtains for the imaginary part of the coherent scattering length in fm (10~13 cm)
1. Thermal Cross Sections
at = 1.748 x 1Q-3 ( J ^ - J ; ) V ^ o
(1-12)
The relation between wave length in Angstrom units and the neutron energy (1.11) is
Note that, for a 1/v absorption cross section, a, is energy independent. However, when a neutron resonance is located near thermal energies, as in the case of 113Gd, 149 Sm and 155Gd, the neutron cross section no longer obeys the 1/v law and hence a< is strongly energy dependent. For details on the measurement of the imaginary part of the coherent scattering lengths, we refer the reader to the original work of Peterson and Smith [18]. With the exception of a few nuclei, neutron resonances are located far away from thermal energies. Under these conditions and for Ej > Tj the coherent scattering length (in fm.) takes the simple form
= R'- 2.277 x 103
) £ ^
(1.14)
In this expression it is assumed that the imaginary part, at, is negligible. Another important relation which is required in the analysis of experimental data is the one between the free nuclear, a, and bound, b, coherent scattering lengths
b= f l i - p j a + Zx&ne
(1.15)
where bne is the neutron-electron interaction length. Its most accurate value was determined by Koester et a!. [19] to be: bae = -(1.38
0.03) x 10~3 fm
For an informative discussion of the free and bound coherent scattering lengths, the reader is referred to the review article of Sears [20]. The coherent, incoherent, and total free scattering cross sections can be described in terms of the coherent scattering lengths by _) 2
o-ilic (spin) = kirg+g- (a+ - a_) 2
(1.16)
(1-17)
1.2. Scattering Cross Sections
Os =CTcoh+ o-inc(spin) = 4n(g+a+ + g-aJ)
(1.18)
For an element with several isotopes, having different coherent scattering lengths, an additional incoherent scattering cross section arises and is expressed by ffinc(isotopic) = 47r^2^2 -fjfk(aj
- ak f
(1.19)
where fj and a,j are respectively the fractional abundance and the scattering length of the j-th isotope. The coherent scattering amplitude of an element is given in terms of the isotopic values by
Note that from Eq. 1.19 the spin incoherent cross section is identically zero if a+ = a_ and in this case, as well as for nuclei with / = 0, the total scattering cross section is equivalent to the coherent scattering cross section; i.e. O"s =
ffcoh
(1.21)
Very interesting specialized techniques based on the wave properties of the neutron (interference, refraction, reflection and diffraction) have been developed which resulted in highly improved determinations of the coherent scattering lengths of the elements and their isotopes. These as well as other methods are described in detail in the excellent review article by Koester [11]. It is noteworthy to mention here that the absolute and highly precise values of the coherent scattering lengths are determined by the neutron gravity refractometer introduced by Maier-Leibnitz [21]. In this kind of measurement only the height of a freely falling neutron is required for the determination of the coherent scattering length. Values of the coherent scattering lengths are listed in Table 1.1, most of which are measured by this technique, and which can be adopted as standards. In addition, a technique [22] called neutron interferometry has been developed for obtaining interference effects between two spatially separated coherent thermal neutron beams. By means of this technique, it is possible to determine the phase of the neutron wave function, which is related to the coherent scattering length. The reader is referred to the informative article of Werner [23] describing the various fundamental experiments which can now be realized via neutron interferometry. Another significant development dealing with coherent scattering lengths is the determination [24] of the magnitude and the sign of the incoherent scattering lengths, b+ — &_. This is achieved by measuring [25, 26] the dependence of the precession angle of the neutron spin on the nuclear polarization of the target nucleus (pseudo magnetism method). When the incoherent scattering lengths, derived from these measurements, are combined with precise values of the coherent scattering
1. Thermal Cross Sections Table 1.1: Common Standards for Scattering Lengths. Target Element
b(fm) 1
H D C
7 3
0
4
Al
5
Si
0
Cl
8
Ge
6
Bi
4
lengths, accurate values of the spin-dependent scattering lengths, a+ and a_, are established. The spin-dependent scattering lengths play a major role in shell model [15], meson-exchange [27] and valence-capture calculations [28, 29]. The simplicity of the basic information which can be derived from the latter is illustrated in the next section.
1.3 1.3.1
Capture Cross Sections Breit-Wigner Formalism
Generally, and in most cases, a description of the thermal capture cross sections can be conveniently provided by the Breit-Wigner formalism. However, in some cases which will be discussed subsequently, the capture cross sections can be adequately accounted for in terms of a non-resonance contribution, i.e. direct capture mechanism. This capture mechanism plays a dominant role in light nuclei as well as in the vicinity of nuclei with magic neutron or proton numbers. For an isolated, neutron resonance, one can write the Breit-Wigner formula for capture reaction in the form
(1.22)
1.3. Capture Cross Sections where
2.608 xlO 6 (A + lV gTn ^ ) —
(1.23)
It is important to realize that the total width in the expression for y is energy dependent via the penetrabilities in the widths, F n , F a , Tp and F^, and is to be evaluated at the neutron energy where the cross section is calculated. For the majority of nuclei, the resonances are located far from the thermal region and satisfy the condition that EQ 3> F. As a result, the contribution from known resonances to the capture cross section at a neutron energy of 0.0253 eV (corresponding to a velocity of 2200 m/sec) can be approximated by the incoherent sum
= 4.099 x l O « ( ^ ) f : ^ ^
(1.24)
It is implicitly assumed here that the number of channels for 7 decay is large. The superscript on the capture cross section in Eq. 1.24 designates a 2200 m/sec value which is expressed in barns when the widths F^-, F7J- and the resonance energy are in eV. The summation extends over N resonances. Expressions similar to Eq. 1.22 can be written for aa, av and ad by substituting F o , Tp or Td for F 7 . 1.3.2
Westcott Factor, gw
Neutron capture reaction cross sections are usually measured by techniques (such as activation or pile oscillator method) on a relative basis, i.e. a ratio measurement to some standard value. The most common standards for thermal neutron capture cross sections are gold, cobalt and manganese, in that order of importance. Early cross section measurements often used the absorption cross sections of natural boron and lithium as standards. These cross sections depend on the (n,a) reaction in10B and 6Li respectively. However, because of the variable abundance of 6Li and 10 B in nature, a large uncertainty in the corresponding natural element cross sections is introduced, even though the isotopic cross sections are well determined. The values in Table 1.2 refer to cross-sections for a neutron velocity of 2200 m/sec. Generally, cross sections are measured in a non-thermal neutron spectrum. Westcott [29] treated the problem of converting the effective capture cross section, a, to a 2200 m/sec value by describing the neutron spectrum as consisting of a Maxwellian distribution characterized by temperature, T, and an epithermal component, proportional to dE/E. For an isotope whose cross section does not vary inversely with the neutron velocity, the effective cross section is
1. Thermal Cross Sections
Reaction 197
Au(n,7)
9
Co(n,7)
a
Mn(n,7)
5
59 55
Table 1.2: Cross Section Standards. Thermal Cross Section Resonance Integral (barns) (barns)
35
Cl(n,7)
10
43.6
8 2 5
6 2 to
2 3
Li(n,a)
746
e
f
4 4
9
4
h
2
a
c
d
Li(abs)
^(n/y)
6
773
3 6
Total Capture
0.4
B(n,7)
B(abs)
Comments
0.3326
i
0.0007
: Due to an independent beta decay branch in the 10.5 minute isomer, the activation cross section is 0.05 barns smaller than the value quoted in the table. 6 : Boron from Searles Lake Borax (California) used at Argonne, Brookhaven and CBMN Geel. c : Boron from unknown source (est. Turkish) used at Harwell and Fontenay aux Roses. d : Boron from California Howlite. e : Boron from USSR Inyoite. f: Boron from Argentine Ulexite. 9 : Lithium from Searles Lake Li2NaPC>4. h : Lithium from Searles Lake Li2CO3. *: Lithium from Union Chemique Beige (LiCl from an unknown source).
1.3. Capture Cross Sections
a = a°(gw+rs)
(1.25)
where gw is the Westcott factor (gw is introduced here to avoid confusion with the spin statistical factor, g), the epithermal index, r, is approximately the fraction of the total neutron density in the epithermal component, and s is a temperature dependent quantity given by
(1.26) where / ' is the reduced resonance integral. The gw factor is the ratio of the Maxwellian averaged cross section, a, to the 2200 m/sec cross section (Eq. 1.25, with r = 0) 1
f°° 4 v3
where VT is the most probable velocity for a Maxwellian spectrum specified by temperature T and is given by En = kT = - m u |
(1.28)
where k is the Boltzmann constant and m is the mass of the neutron. At T = 293° K (room temperature), E n = 0.0253 eV and vT = 2200 m/sec. Note that the Westcott factors are dependent on the temperature T via (Eq. 1.28). It can be easily shown that if the cross section varies as 1/v, then the Maxwellian capture cross section is equivalent to the 2200 m/sec value and then gw=l. The gw factors can be calculated from Eq. 1.27 and are close to unity in most cases. Some nuclei with significant departures of gw from unity (called non-l/u nuclei) are listed in Table 1.3 where the values are from ENDF/B- IV [30]. The accuracy of the gw factors depends on the accuracy of the slope of the cross section. The only significant study of the accuracy of the gw factors has been made by Westcott [29] for the fissile nuclei. The standard deviations were of the order of 0.1-0.3%. 1.3.3
The Lane-Lynn Formalism
As early as 1956, it was realized by Groshev and collaborators [31] that gamma-ray spectra resulting from the capture of thermal neutrons exhibited features which are not in complete accord with the statistical theory of the decay of complicated highly excited compound states. These observations were the basis for the radiative capture theory of Lane and Lynn [32] in terms of a direct capture mechanism treated in the framework of R-matrix formalism. Phase space is divided into external and internal regions separated by a hard sphere or interaction radius, R. The direct capture process takes place predominantly outside the nuclear region while the
10
1. Thermal Cross Sections Table 1.3: The Westcott gw Factor for Non-l/u Capture Cross Sections.
Nuclide
factor
iisCd
1.34
135
Xe
1.16
Sm
1.64
149
151
Eu
0.89
176
Lu
1.75
182
Ta
1.64
239p u
1.13
243
0.90
Am
compound internal resonance capture mechanism is confined inside the nucleus. Lane and Lynn [32] derived an algebraic expression for the direct component of the capture cross section (Eq. 4 of Ref.[32]) which was subsequently simplified [33] under the assumption that the imaginary part of the coherent amplitude is negligible when compared to the real part. The approach of the study of direct capture via the Lane-Lynn [32] formalism is clearly described and outlined in detail by Mughabghab [33]. The radiative capture cross section due to the capture of an s-wave neutron from an initial state, i, in channel spin s into a final p-orbit, / , (i.e. electric dipole transitions) can be written in the form [33]
<7j/ (direct) = er7/ (hard sphere) 1 + where
,,
ff7/(hard
,
, , 0.062 fZ\2 sphere) = (J
2J/
(1.30)
1.3. Capture Cross Sections
11
where interaction radius = 1.35 x A1/3 fm, coherent scattering length for channel spin * corresponding to spins Jj = / 1/2, Jf = total spin of the final state, / = spin of the target nucleus, Sdp = (d,p) spectroscopic factor, Ej = energy of 7-ray due to capture of an s-wave neutron feeding a final p-state, En = the energy of the incident neutron = 0.0253 eV for 2200 m/sec neutrons. If two channel spins are involved, as in the case of target nuclei with non-zero spin, then the expression for direct capture is generalized to R asi
= =
(1.32) The summation is extended over the two possible initial channel spin states, i, corresponding to I + 1/2 and / - 1/2, the curly brackets are Wigner's 6j symbols, and the various quantities in the brackets are: Ji = total spin of the initial state "Si/2> $3/2 = (d,p) striping amplitudes for the P\/2 and P%/2 components of the final state, i.e., Sdp = S^,2 + S\,2 Note that since the following relationship holds
E
Sff = (2Jf + l)Sdp
(1.33)
then for the special case where the coherent scattering amplitudes are equal, i.e. a+ = a_, Eq. 1.31 simplifies to Eq. 1.29. As a convenience, Sff values for target nuclei with spins ranging from / = 1/2 to 7/2 are calculated according to Eq. 1.32 and listed in Table 1.4. The total direct capture cross section, cr7(D), is then determined by summing the contributions due to the electric dipole transitions over final states o-7 (D) = E °7/ (direct) /
(1.34)
Although the signatures of direct capture have been known for a long time, it was not until the work of [33] that a quantitative verification of Eq. 1.29 was realized in the reactions 13eXe(n,7)137Xe and 138Ba(n,7)139Ba. The impressive
12
1. Thermal Cross Sections Table 1.4: Direct Capture Coefficients Sff. O2 *«/
(2J> + l)Sdp
1
(Ji = i*+1/2) "1.2
0
#1/2
2 1/2 3/2 5/2 0
C(?2 Ol3 3/2
1/2)
-
w
f1*y/o C nr JL ^03/2
4fC2 , O2 1 ^v"l/2 ^ "3/2J KO2
3/2
= /
0
1/2
1
(^
4/oc* /KQ ^2 f \^^3/2 ~~ VO"l/2j
2S 1 / 2 ) 2
_i_
0
0 2 W3/2
2
4 ^ XH^C
^" \^r 3t.'3 /2 "T"
°"3/2 l/ri
qf
0
/K Q \2 ~ V0iJl/2)
2(^3/2 ~ ^1/2)
2 1 / , / F o _T o^^V " ^ 3 / 2 ~i -5 1 / 2 )
2"v^3/2 ~r *
7
'"3/2
5/2
nip;:! fif
3
3
1052/21/2
7/2
9(S 2 /2 + 5 2 /2 )
1 2 3 4 3/2 5/2 7/2 9/2 3 4
,/nq /Eq
\2 \2
0
0 |(4S 3 / 2 + v 8/nn
/*$ C! \2
7^^3/2 — V^1/2J
—f"\/3iS / -1-
0
105 2 /2 0
T(*-53/2 — v " " 1 / 2 / 3 / /^Q /KG ^2
2S 1 / 2 ) 2
2 7 _ /*> c
_L
2"tvt^'^3/2 ~
1/7*7
T\ * ' " 3 / 2 — V " " 1 / 2 /
I2
0 success of the Lane-Lynn [32] theory led [28] to the extension of the calculations to the light isotopes, 9Be,10 B, 12 C,13 C, 19 F, 27A1 [34]-citemughabghab87. In view of these findings, it was warranted to calculate the direct capture component at a neutron energy of 0.0253 eV via Eq. 1.34. These values are included at the top of the resonance parameter tables. In addition, cr7(+),
(1.35)
It is assumed here that interference terms between the direct and the compound components are absent or negligible. Because of the absence of nearby s-wave reso-
1.4- Thermal Fission Cross Sections
13
nances in some nuclei, the internal compound resonance contribution is negligible, resulting in a7(B) = <77(_D). Another approach to valence capture in the thermal region can be found in [37]. Prestwich and Kennett followed the Lane-Lynn approach to determine the direct E2 contribution to the capture process. A simplified closed form expression is derived (Eq. 22 of ref [38]) which was applied to the reaction 20Ne(n,7)21Ne produced by thermal neutrons.
1.4
Thermal Fission Cross Sections
Accurate knowledge of the thermal neutron fission cross sections and neutron yield parameters are of paramount importance in the design of nuclear reactors and for use as basic standards in similar or different cross section measurements of other nuclei. Because of this circumstance, considerable experimental as well as evaluation effort was expended in obtaining very precise and consistent constants at a neutron energy of 2200 m/sec for the fissile isotopes 233 U, 235 U, 239 Pu, and 241 Pu. The parameters under consideration are the absorption (cra,) fission (erf,) and capture (
r, = 9^ = -— an 1+ a
(1.36)
a=-^-
(1.37)
c a = erf + c 7
(1.38)
where 9= total number of average neutrons emitted per neutron induced fission = Vp + i/4, 9P= average number of prompt neutrons per neutron induced fission, and 9d= average number of delayed neutrons per neutron induced fission. In addition, when the scattering cross section is known, the absorption cross section can be determined absolutely to a high degree of accuracy from a measurement of the total cross section by the following expression:
(1.39)
The total scattering cross section can be measured relative to a vanadium or lead scatterer, or can be derived from the sum of the coherent and incoherent scattering cross sections:
14
1. Thermal Cross Sections Table 1.5: Bound Scattering Cross Section.
Isotope
233
U
235
U
2
239
Pu
6
241
Pu
3
0
*2 = *«h + <4c
(1-40)
The superscript b refers to the bound atom value. Because of the impact of the scattering cross sections on the accuracy of the absorption cross sections (Eq. 1.39), an evaluation of the scattering cross sections of 233 U, 235U, 239 Pu, was achieved in the present work by considering all the scattering data as well as carrying out a fit of the thermal capture and scattering cross sections in terms of a Breit-Wigner formalism. The results of the present evaluation are summarized in Table 1.5. The fission cross section can be described as a sum over positive and negative energy resonance contributions in the framework of the Breit-Wigner formalism: 2.608xl0YA + l \ ^
glV/j
Note that, within the summation sign, En (at 0.0253 eV) cannot be neglected for the fissile isotopes because of the presence of negative and/or positive energy resonances near the thermal region. The Westcott factors relate cross sections which are measured in a Maxwellian reactor spectrum to the corresponding values at a neutron energy of 0.0253 eV through the following equations: (1.42)
1.4- Thermal Fission Cross Sections
15
af = gfaof
(1.43)
0"a = 9a^a
(1-44)
where the cross sections on the left-hand and right-hand sides are the Maxwellian and the 2200 m/sec values respectively. For convenience, from now on, we shall drop the hat symbol on the Maxwellian values. Therefore it follows that the Westcott factors gw(w = 7, / , a) can be calculated from Eq. 1.27. From Eq. 1.42-1.44 and the definition of the absorption cross section, it readily follows that the Westcott factors are interrelated by:
(1-45) For a cross section which varies as 1/v in the thermal range i.e. a(v) = Vo/va°, Eq. 1.27 shows readily that gw = 1. Therefore, cross sections of isotopes, such as 235 U, 239 Pu, and 241 Pu, which depart from the 1/v law have Westcott factors differing from unity. Such departures are due to the presence of positive and/or negative energy resonances near the thermal region. In the past, several detailed evaluations [39]-[41] of the thermal constants have been carried out by various groups using one form or another of a least-squares analysis [42] of the total body of thermal data. In 1975 and 1977, Lemmel [40],[43] summarized the results of the International Atomic Energy Agency evaluation. His main finding is that there seems to be apparent and unresolved inconsistencies between the evaluated values obtained from the 2200 m/sec data and those derived from the Maxwellian data. Since that time, several data sets have appeared in the literature and become available through private communications. In addition, our knowledge of related fundamental data has improved. These include the following: The capture cross sections of the standards, 59 Co, 55Mn, and (n,a) cross sections of 6Li and 10B were updated. The half lives of 233 U, 234 U, and racy [44, 45].
239
197
Au and the
Pu are presently known to a better accu-
Inconsistencies of the various v values of 252Cf were resolved. These developments warranted carrying out two evaluations: one at the National Nuclear Data Center by Divadeenam and Stehn [46], the other at the Central Bureau of Nuclear Measurements at Geel by Axton [47]. The latter evaluation was devoted exclusively to the analysis and treatment of the 2200 m/sec data while the former considered three sets: 1) the combined Maxwellian and the 2200 m/sec data, 2) the Maxwellian data separately, and 3) the 2200 m/sec data separately. The results of the evaluation of Divadeenam and Stehn [46] show that the discrepancies
16
1. Thermal Cross Sections
between the Maxwellian and the 2200 m/sec evaluations indicated by Lemmel [40] are essentially resolved. Since the evaluation of Divadeenam and Stehn [46] is based on a wider spectrum of data sets, their recommended values for 233U, 235U, 239 Pu and 241 Pu are adopted in the present work.
1.5
Paramagnetic Scattering
In the rare earth region and for elements in the ionic state such as oxide samples, the interaction between the magnetic moment of the neutron and that of the ion must be taken into account. Such an interaction gives rise to a paramagnetic cross section, apm, which has the form: [48]
where: = the classical electron radius, e2/mec2 7 = the magnetic moment of the neutron, H = the effective magnetic moment of the ion in Bohr magnetons, F2 = is the squared magnetic form factor averaged over all angles. Calculations of the energy dependent paramagnetic cross sections have been carried out, for example, by Mattos [49] for the rare earth ions Ce, Pr, Nd, Gd, Tb, Dy, Ho, Er, Tm, and Yb. The paramagnetic cross sections (apm) can be determined from analysis of total cross sections or total scattering cross sections when the capture cross sections and nuclear scattering cross sections are known. Recommended values for apm are given at 2200 m/sec and they range from 0 b for Er to 3 b for Nd. To obtain an idea of the variation of the paramagnetic scattering cross section with mass number, calculated values of apm at 2200 m/sec obtained from the work of Mattos [49] are plotted in Fig. 1.1 and are compared with the experimental values [50]-[53]. Such a plot serves as a means of obtaining an estimate of the paramagnetic scattering cross sections of elements when experimental values are not readily available, as was done in the 150Sm investigation of Eiland, et al.[54].
1.6
Potential Scattering Length or Radius R'
The potential scattering length or radius, R', is an important parameter which is required in the calculation of scattering and total cross sections. In addition, it is interesting to recall here that the variation of R! as well as the neutron strength function (see next section) with mass number A, provided a dramatic confirmation [55, 56] of the optical model at low neutron energies, thus demonstrating the inadequacy of the strong coupling model [57]. In analogy with the coherent scattering length, R' is defined by the relation R, =
_l i m
1.6. Potential Scattering Length or Radius R'
17
40
35
. Paramagnetic Scattering on Rare Earth Ions (En = 0.0253 eV) A
30
Experimantal Values — Calculated Values
25
20
E Q.
15
10
0 55
60
65
70
75
Atomic Number Z Figure 1.1: Comparison of the paramagnetic scattering cross sections measured at a neutron energy of 0.0253 eV with the calculations of Mattos [49]. The solid curve is an eye-guide to the theoretical values.
18
1. Thermal Cross Sections
where Re means the real part of and 5'QP is the optical model s-wave phase shift. In the optical model, the interaction between the incident particle and the target nucleus is replaced by a complex potential V(r) = Vof(r)+iWg(r)
(1.48)
where VQ and W are the magnitudes of the real and imaginary parts of the optical potential respectively. The radial dependence of the potentials is specified by the functions /(r) and g(r). For simplicity, Feshbach, Porter and Weisskopf [55] assumed a spherical square-well potential /(r) = g(r) = 0 for r > R f(r) = g(r) = - 1 for r < R In the weak coupling approximation, (W < VQ) the following expression is derived [55] for R'
Kw^)
(L49)
where 1m \ f 22 22 22 X = Xi + iX2 = (k R + ^-(V0+ iW)R JJ
(1.50)
Note that R' = R when X\ = mr/2 where n is an integer In addition, R' has extreme values whenever X\ = mr. Thus, R' modulates around the nuclear radius R. In contrast, the strong coupling model (W > Vo) predicts that R' = R. The second term in Eq. 1.49 is related to the distant s-wave level contribution, R°°, and in It-matrix formalism Eq. 1.49 can be written in the simple form for s-waves R> = R(l - i?°°)
(1.51)
where
The summation is carried out over the distant resonances. Other studies of the variation of R' with mass number A, which included deformation and potential parameters effects, (See section II-A) can be found in references [58]-[59]. Experimentally, five methods were exploited in the extraction of the potential scattering radius:
1.6. Potential Scattering Length or Radius R'
19
In the resolved resonance region, the ability to fit the shape of the total and scattering cross sections, (particularly in the wings of, as well as in between, resonances) enables one to extract R'. For example, the scattering cross section can be expressed in the form [55]
a* = wX
e
2ikR'
(1.53)
The interference between potential and resonance terms of an s-wave resonance, which produces a minimum on the low energy side of a resonance (for small kR values), provides a means of obtaining an R' value. In the absence of resonances, the scattering cross section at low neutron energies Eq. 1.53 reduces to the potential scattering cross section a. = o-pot = 47r(i?')2
(1.54)
As pointed out previously, in R-matrix analysis R' is deduced with the aid of Eq. 1.51 when R°° is determined from the shape fit of the cross section. In the unresolved energy region, by averaging the Breit-Wigner total cross section over an energy region (<100 keV) containing many s-wave resonances, one obtains
(at) = 4w(R')2 + 2?r2A2v/I?So + O(S0)2
(1.55)
where So is the s-wave strength function. In the keV energy region, the contribution of higher order terms, O(So)2, is small and as a result can be ignored. By carrying out a least-squares fit to the average total cross section, one can derive R' as well as So. A variation on the second method is the measurement of the change of the transmission function at one neutron energy with respect to sample thickness [60]. Values of R' can also be obtained with the aid of Eq. 1.14 when the measured coherent scattering lengths are combined with known information of positive and negative energy neutron resonances. Extensive use of this procedure was made previously [61, 62] by several investigators. Differential neutron elastic measurements can provide information on the sand p-wave strength function as well as the potential scattering radii RQ and R[ [63]. This method was applied for the first time by Nikolenko and collaborators [64] to measure these parameters for the even-even Tin isotopes.
1. Thermal Cross Sections
20
12
\
'
i
r
10
Deformed Optical Model Spherical Optical Model i
0
20
40
60
80
.
i
i
.
i
100 120 140 160 180 200 220 240
A Figure 1.2: Variation of R' with mass number A. The solid curve is based on the deformed optical model with parameters Vo=43.5 MeV, ro=1.35 fm, Vso=8 MeV and surface absorption W B = 5 . 4 MeV. The dotted curve describing the trend at low mass numbers is based on spherical optical model calculations using the same parameters.
Our present knowledge of the variation of R' with mass number is summarized in Fig. 1.2, where the recommended R' values are plotted versus mass number and are compared with optical model calculations. The solid curve is based on deformed optical model calculations with parameters Vb=43.5 MeV, ro = 1.35 fm, Vgo=8 MeV and surface peaked absorption potential, VFD=5.4 MeV. The dotted curve describing the trend at low atomic mass numbers is based on spherical optical model calculations using the same parameters. As shown, the optical model calculation adequately describes the general trend of the measurements. In particular, the deep minima at mass numbers, A = 10, 48
1.6. Potential Scattering Length or Radius R'
21
and 142, are well reproduced. However, above mass number A = 220 the predictions of the optical model fall short of the experimental data. The general agreement between theory and experiment yields evidence of the semi-transparency of the nucleus, i.e., the weak coupling model gives a better description of the scattering radii of the nuclei than the strong coupling model. It may be recalled that the latter model predicts that R' = 1.35XA1/3. Another corroboration of the weak coupling model comes from studies of the s- and p- wave neutron strength functions as discussed in detail previously and illustrated in Figs. 2.1 and 2.2. One of the interesting problems which can be investigated is the spin dependence of the scattering radius. Early studies by Firk et al. [65] of 51V revealed a possible spin dependence of R'. Since that time, a few additional studies [66]-[70] carried out on 7Li, 59Co, 61Ni, 63Cu, 187Os reported similar results for the spin dependence of R'. Another interesting feature, which is exhibited here and previously predicted by the optical model, is the behavior of R' below mass number 40. A peak in R1 appears at about A = 12. At smaller A values, R' decreases, approaching a value of about 3.0 fm at mass number A = 3.
This Page is Intentionally Left Blank
Chapter 2
Resonance Properties 2.1
S-, P- and D-Wave Neutron Strength Functions
2.1.1 Introduction As early as 1937, the importance of the neutron strength function was realized by Bethe [71]. According to the extreme compound, or black, nucleus model [57] the strength function is constant for all nuclei, and for s-wave neutrons is given by
Do
= 1 x 10"4
(2.1)
-KK
where (F°) is the average s-wave reduced neutron width, Do is the average swave level spacing, ko is the wave number for a 1 eV neutron while K is the wave number inside the nucleus. For a potential well depth of 42 MeV, the black nucleus value of the strength function is lxlO~ 4 (Note that the strength function is a dimensionless quantity). Subsequently, measurements carried out by Carter et al. [72] indicated that strength functions deviate from a constant value, particularly around the mass region A = 160. These results along with the variation of the average cross sections with mass number [73] formed the basis of the optical model calculations of Feshbach, Porter and Weisskopf [55]. Before presenting the results of these calculations, the definition of the strength functions for various partial waves, £, is presented.
1
(2£+l)Dt
(2£+l)AE^yj
nj
y
''
where the summation is carried over N resonances in an energy interval AE, gj is the spin statistical weight factor for angular momentum J = I +1+ | , and F^ is the reduced neutron width and is related to the neutron width Tnj by
(2 3)
vt
"
23
24
2. Resonance Properties Table 2.1: Penetrability Factors, Pi = kRVe, for a Square Well Potential. I
Vt
0
1
1
K2R2/(1 + K2R2)
2
KiRi/(9
3
K0R0/(225 + 45K2R2 + 6K4R4
+ 3K2R2 + K4R4)
Values of the penetrability Vt for £ = 1 — 3 derived for a square well potential are shown in Table 2.1 and can be determined with the aid of the following relation k2R2 = 8.7953 x lQ-BEnA2's (^-j)
(2-4)
For convenience, the relationship between the p-wave reduced neutron width and F n for R = 1.35 x A1/3 fm is given here. 11369. The error on the strength function is estimated from the the relation (2/N)l/2So (which is an asymptotic approximation for large number of resonances) based on the Porter-Thomas [74] distribution of widths. For a limited number of resonances, Muradyan and Adamchuk [75] and subsequently Slavinskas and Kennett [76] applied the likelihood method to find the most probable value of the strength function and the associated asymmetrical errors. Later, Liou and Rainwater [77] applied the Monte Carlo method to determine So and Si and their errors. 2.1.2
Optical Model
As discussed previously in the section pertaining to potential scattering length, Feshbach, Porter and Weisskopf [55] introduced the optical model to predict R' and So, as well as average cross sections as a function of mass number and neutron energy. Gross features of the experimental data were reproduced. The s-wave strength function is related to the imaginary part of the optical model phase shift by the expression
So = (2-^)
lim « * )
\ w J *»o \
k J
(2.6)
2.1. S-, P- and D-Wave Neutron Strength Functions
25
More generally, the strength function for the various partial waves is given by
^Tt
forl«l
T( = 1 — (1 — irViSi)2
(2.7)
(low energy approximation)
In the weak coupling approximation, Feshbach, Porter and Weisskopf [55] derived (for a square well potential) the following relationship for the s-wave strength function
D
2kRX2 fl-Sin2X1/2X1\ TTX1 \ Xi + cos2 Xx
[Z.t
It is clear from Eq. 2.8 and Eq. 1.50 that the maxima of the s-wave strength function can be found from the relation 2m,
N
1/2
(2.9)
For Vo= 50 MeV and R = 1.35 x A 1 / 3 fm, Eq. 2.9 reduces to 2.1 x A1/3 ~ (ra+l/2)7r
(2.10)
and hence the maxima occur at about mass numbers A ~ 11, 52, 144 and 305. It is remarkable that the predicted positions of the maxima are in reasonable agreement with measurements which exhibited two broad resonances at mass numbers A = 55 and 150. These broad resonances are referred to as the 3s and 4s giant resonances and are associated with the nodes of the continuum single particle wave function corresponding to n = 3 (A~52) and 4 (A~144) respectively. Improved measurements [78] of s-wave strength functions around A~160 revealed that the peak was split into two components located at A~150 and ~180. Margolis and Troubetzkoy [58] interpreted the splitting of the 4s peak in terms of a spheroidal model with a square well potential. Subsequently, Chase, Wilets and Edmonds [80] used a Saxon-Woods potential with a surface-peaked imaginary potential and rotational degrees of freedom (deformation) for target nuclei to calculate s-wave strength functions and R' in the deformed mass region. Agreement with experiment was significantly improved as compared to previous calculations. In addition, by adopting a particular rounded potential well, Fiedeldey and Frahn [81]-[83] derived closed form expressions for the s- and p-wave strength functions as well as R'. One of the interesting questions discussed is the splitting of the p-wave strength function into pi/2 and p 3 / 2 components [81] due to the spin-orbit coupling term in the potential. However, it is to be emphasized that such a splitting requires about double the normal value of the spin-orbit potential (i.e. Vso ~ 16 MeV).
— Deformed Optical Model — Spherical Optical Model 10
4s
2s
CO X
0.1
20
40
60
80
100
120 140 160 180 200 220 240 260 280
A Figure 2.1: Comparison of the theoretical with experimental values of the s-wave neutron strength function. The solid curve represents deformed optical model calculations of Ref. [87] while the dotted curve is based on spherical optical model calculations of the present work.
o 3 a 3
to
— Deformed Optical Model — Spherical Optical Model 4p 10
s C-f-
S3-
0.1 20
40
60
80
100
120 140 A
160
180
200
220
240 260
Figure 2.2: Comparison of the theoretical with experimental values of the p-wave neutron strength function. The solid curve represents deformed optical model calculations of Ref. [87] while the dotted curve is based on spherical optical model calculations of the present work.
to OO
1
20
,
1
40
,
1
60
,
1
,
80
1
,
100
1
,
120
1
,
140
1
,
160
1
180
200
220
240
260 280
A
Figure 2.3: Comparison of the theoretical with experimental values of the d-wave neutron strength function. The solid curve represents spherical optical model calculations of Ref. [88].
o 3 a 3
2.1. S-, P- and D-Wave Neutron Strength Functions
29
In 1962, Buck and Perey [84, 85] developed techniques to study neutron scattering over a wide energy and mass range employing non-local potentials. They calculated s- and p-wave strength functions using an axially symmetric potential for rotational nuclei and pure quadrupole vibration model for dynamically deformable nuclei. Subsequently, Jain [86] and Mughabghab [87] used Buck and Perey's computer code to study the effect of parameter variation on strength functions. Making use of the corresponding S-matrix elements evaluated by Buck and Perey, Divadeenam [88] computed d-wave strength functions. It should be pointed out that in most of these calculations, the optical model strength functions are calculated for neutron energies 10-100 keV. A comparison of theoretical and experimental values for both s- and p-wave strength functions indicated a lack of detailed agreement in certain mass regions: The region between the 3s and 4s giant resonances, Fig. 2.1. The region between the 3p and 4p giant resonances, Fig. 2.2. Variation of the s-wave strength function with mass number for the Sn, Te isotopes [89, 90]. The 2p giant resonance is not very well denned due to ambiguity of the application of the optical model for light nuclei. Various investigators have attempted to resolve the above mentioned problems. These are briefly outlined. Moldauer [91] accounted for the deep minimum at A = 100, (Fig. 2.1), by concentrating the imaginary part of the optical potential outside the nuclear surface. The justification for this approach can be based on the Pauli exclusion principle which inhibits the interaction of slow neutrons with nucleons in the interior of the nucleus. Following the suggestion that the s- ad p-wave neutrons see different absorptive optical potential, Turinsky and Sierra [59] adopted an angular momentumdependent imaginary surface potential in their calculations and achieved good agreement between measurement and theory, not only in the deep s- and p-wave mimina but also throughout the mass range. Lane et al. [92] and Sugie [93] attributed the departure of the experimental s-wave strength function from the average trend of the optical model predictions to nuclear shell structure effects. The first observation of a general decrease of So with mass number A was reported by Fuketa and Harvey [89] for the tin isotopes. This behavior was accounted for by Shakin [94] in terms of 2 particle-1 hole, or doorway, state model (See Fig. 2.4 and below). Furthermore, in extensive investigations of the Te isotopes, Tellier et al. [90] showed that these isotopes manifested the same trend as the tin isotopes. A possible explanation for this behavior can be made by including an additional term, the isospin-dependent term (Lane potential [95]), in the imaginary potential
W(isotopic) = t
T—Wl
= (N — Z)—A
for neutrons,
30
2. Resonance Properties
where t and T describe the isobaric spin vectors of the incident particle and the target nucleus respectively. Resorting to coupled channel calculations of Tamura [96] (JUPITOR Code), Newstead et al. [97] succeeded in fitting the Sn and the Te data with reasonable values of Wi (Wi = 45 MeV for Sn and Wi = 63 MeV for Te). For light nuclei (A<50), appreciable energy dependence in the strength functions are caused by violations of low energy approximations and fluctuations due to large spacings of resonances. The latter situation would make it difficult to obtain appropriate averages. Moldauer [98] attempted to resolve these problems by redefining the reduced neutron width. This is carried out by removing the energy dependence from the scattering width.
2.1.3
Lane-Thomas-Wigner Model
The giant resonance model of Lane, Thomas and Wigner [99] attempts to link the R-matrix formalism to the optical model giant resonance. In particular, the spreading of the single particle strength into compound nuclear resonances due to the nuclear two body interaction leads to the sum rule between compound nuclear reduced widths, j ^ n , and the single particle reduced width, -y^p for states with the same quantum numbers, i.e. ?n = 72p
(2-12)
In R-matrix formalism, the R-function for single particle states, p, (for brevity j'p is used instead of j^p) is given by
The strength function is then described by
b
Imfl(fi).
l
W-j
T
^
It can be seen from Eq. 2.14 that the strength function has a resonance shape as a function of neutron energy. Furthermore it is instructive to point out the dependence of S on W under two limiting conditions S~W ~ 1/W
(in between single particle resonances) (near a single particle resonance)
(2 15)
Another interesting relation between So and W for s-wave neutrons is derived by Porter [100] in the optical model context.
2.1. S-, P- and D-Wave Neutron Strength Functions
31
77? S
° = V^^ zirzn K Jo
W(r)\U0(r)2\dr
(2.16)
where Uo(r)2 is the s-wave neutron wave function at zero energy. The relationship between the strength function and the imaginary potential has important implications. Early theoretical attempts [101]-[104] to derive the imaginary potential from basic principles led to the following expression W(r) = ir\(0\V\
cUP(r))\pEL
(2.17)
where $o is the ground state, $ c are states of the target nucleus, Up are states of incident particle, V is the two-body interaction, and PEL is the density of compound states. The bar over the expression indicates an average over the product states $cC/p. It is obvious that nuclear structure effects are introduced through the level density which is known to depend on shell closure, pairing and excitation energy. 2.1.4
Doorway States
The giant resonance model of Lane, Thomas and Wigner [99] attempts to link the two extreme kinds of states, viz. the narrow compound (F n ~ few eV) nuclear resonances to the wide single particle resonance (Tsp ~ few MeV). Resonance states whose widths are Yn < Yd < F s p are referred to as doorway states by Feshbach and his collaborators [105]. The doorway states have 2 particle 1 hole (2p-lh) configuration or correspond to a particle-vibration configuration if vibrational states are excitable [106]. Analogous to the Lane, Thomas and Wigner model [99] the strength function in the vicinity of an isolated doorway is represented by r t r 4-
[E - Edf + (Fd/2)2
(2.18)
where
F^ is the escape width to the entrance channel and F^ is the spreading width for decay into more complicated states. As mentioned previously, Shakin [94] was the first to calculate the doorway state widths and energies for the tin isotopes. Overall agreement was achieved when doorway state resonances spanned an energy range A= 3 MeV. This is both due to the variation of doorway state widths and doorway spacings D
32
2. Resonance Properties
In Fig. 2.4, the strength functions of the Sn and Te, as well as Cd, isotopes are shown. For comparison, theoretical values of Shakin [94] are included. One striking feature is that the strength functions of the Sn isotopes are in general much lower in magnitude than either the Cd or Te isotopes. This may be attributed to the proton shell closure in Sn(Z = 50) and the availability of 2 holes (particles) in Cd (Te), thus implying a larger level spacings for the Sn isotopes as compared to the Cd and Te isotopes. For a detailed understanding of the strength function variation, the filling of neutron shells and the effect of excitation energies have to be considered. Nevertheless, the systematics of strength functions described in Fig. 2.4 indicate that an interesting picture emerges if one recalls that Lynn [4] interpreted Shakin's [94] averaging interval A as some spreading width, Wap-i/u which averages over several 2p-lh state resonances. Therefore it is tempting to interpret the Sn, Te, Cd data in terms of a sub-giant resonance model for the doorway state (In Eq. 2.14, W —> W2P-ih)- According to this model, So oc l/W2P-ih m the vicinity of a doorway state, thus describing the data. To explain the Sn and Te data, Nikolenko [107] followed a similar approach whereby he employed the Lane-Thomas-Wigner giant resonance model [99] and additionally introduced superconducting-type pair correlations in the wave functions. Sirotkin and Adarnchuk [108] studied the effect of doorway states on So in the mass region 90 < A < 140. Their conclusions are similar to those of Block and Fechbach [109] in that So is proportional to the doorway state density. Since the density of states depends on excitation energy of the compound nucleus, these authors attribute the local fluctuations in So to the variation of neutron separation energy in the compound nucleus. As part of an extensive program of study of deformed nuclei, Malov and Soloviev [110] investigated the fragmentation of the single particle states into the quasiparticle-phonon (particle-vibration) states for vibrational nuclei. A formalism was developed to calculate s- and p-wave strength functions in terms of the distribution of the single particle states. It should be pointed out that no effort was made to calculate the doorway energies or widths of the states. Reasonable agreement was found between experimental and theoretical s- and p-wave strength functions for nuclei ranging from Sm to Fm. On the other hand, Beres and Divadeenarn [111] applied the particle-vibration doorway model for the interpretation of intermediate structure in 208 Pb+n. Another significant application of the doorway model was attempted by Divadeenam, Beres and Newson [112]. A basis of shell model 2p-lh states in the compound system of 49 Ca and 89Sr were diagonalized via an effective two-body interaction to predict doorway state energies, elastic escape widths and s-, p-, and d-wave strength functions. The study of 88 Sr+n and 48 Ca+n 2p-lh doorway states [112] resulted in some interesting conclusions. As expected, the doorway state spacing is 1.5 - 2.5 MeV in the case of 48 Ca. Furthermore, theory predicts the absence of strong S1/2 resonance below neutron energy of about 1 MeV as evidenced by measurements [113] (refer to resonance parameter table of 48 Ca). Theory also indicates that near A = 90 the shell model allows many more p-wave than s-wave doorways; due to the proximity of the p-wave giant resonance, p-wave doorway widths are larger in magnitude than the s-wave doorway widths.
2.1. S-, P- and D-Wave Neutron Strength Functions 2.1.5
33
Experimental Techniques
Strength functions can be extracted from different types of measurements: a) high resolution total cross section, b) average total cross section, c) average capture cross section, d) neutron transfer reaction to the continuum states, e) differential scattering measurements. High Resolution Total Cross Section The Breit-Wigner or R-matrix formalism is employed to extract individual sand p-wave resonance parameters. With the aid of Eq. 2.2, strength function quantities are derived. Alternatively, strength functions can be determined from the slope of a plot of reduced neutron widths (divided by 2^+1) versus neutron energy. Average Total Cross Sections, < at > As pointed out previously, average total cross section in the unresolved resonance region, where only s- and p-wave contributions are significant, is related to the s- and p-wave strength functions as well as the potential scattering radius (Eq. 1.55). By carrying out a least-squares fit to < at >, strength function and R' values can be determined. Average Capture Cross Sections, < a1 > Analogous to the average total cross section in the keV energy region, the average capture cross section can be written in the form [114, 115]
All symbols have their usual meaning and are defined in Ref. [114, 115]. It is important to recall here that certain assumptions were made in the derivation of this equation. They are: — The strength functions Si are independent of J and are £, —
Dj=Dobs/gj,
— F 7 is independent of J, I and En, — F = Tn + F 7 , i.e. the inelastic channel, as well as other exit particle channels, is considered closed. Transfer Reactions, (d,p) The (d,p) experimental results enable one to study strength functions both in the continuum and bound regions. For example, the relation between the (d,p) spectroscopic factor, SdP, and reduced neutron width, which is given in a succeeding section, enables one to obtain strength function values from SdP-
34
2. Resonance Properties Differential Scattering Measurements It is of interest to draw attention to a method of extracting the s-wave as well as the pi/2 and p 3 / 2 components of the p-wave strength functions from scattering measurements in the keV energy region. The differential elastic scattering cross section is expanded in terms of Legendre polynomials with coefficients Bo, Bi, and B2, which are related to the strength function (Sj/ 2 , Sj , 2 , S3 ,2 ) and the distant level contribution R°°. This technique, which was suggested by Popov [63], was successfully applied by Nikolenko et al. [64] to the even-even tin isotopes, 116Sn. 118Sn, 120Sn, 122Sn and 124Sn. For example for 124Sn these authors obtained S° / 2 = , S} / 2 = , S*/2 = , and R'o= 3 fm. The former value is in agreement with the recommended value, S° = . It is interesting to note that for 124Sn the p]y2 component of the strength function is significantly different from the p3/2 component, suggesting that the p 3 / 2 single particle state is tightly bound.
2.1.6
S- P- and D- Wave Neutron Strength Functions
Introduction The total available body of data on s-, p-, and d-wave strength functions is summarized in Figs. 2.1-2.3 and comparisons between measured strength functions and optical model calculations are made. Inspection of the figures shows that generally good agreement between experimental and theoretical values is achieved. However, there are mass regions where significant discrepancies still exist. This circumstance provides strong incentive for further experimental and theoretical studies of neutron strength functions, particularly from the point of view of obtaining refinements in the optical model parameters. Because of the increased relative improvement in the accuracies of the derived s-wave strength functions for numerous isotopes with the same mass number and the determination of spins of many resonances, the isospin and spin dependence of the strength functions will be examined. Strength functions for various partial waves will be discussed separately.
S- Wave Strength Functions a. Isospin Interaction An inspection of Fig. 2.1 shows large departures of the measured s-wave strength functions from optical model calculations occur, particularly in the mass region A = 110-140. To some extent, this is due to the uncertainty in experimental values derived from a limited number of observed and identified s-wave resonances. However, a closer examination of the data indicates that in several cases So values for nuclei with the same mass number are significantly different. To shed additional light on this aspect of the data, the available cases where this holds true are assembled in Table 2.2. In spite of the limited accuracy of the data, a trend of increasing So, with a decreasing N-Z for the same mass number readily emerges. In particular, attention is drawn to the large and significant difference in the strength function values for the two isobars, 5Q2Sn and 522Te, 5o4Sn and 524Te, 52°Te and sfBa. A similar effect was earlier noted by Tellier and Newstead [116]
2.1. S-, P- and D-Wave Neutron Strength Functions
35
in their investigation of the strength functions of the Te isotopes. This behavior is a manifestation of the influence of the isospin interaction. The importance of this interaction, which arises because of Heisenberg forces and exchange effects, had been previously stressed by Lane [95, 117]. Such an interaction produces an additional term in the optical potential of the form: V (isospin) = t —I I ;— V\ I V\ 4\ A J
=
I
Vi
for forincident incident neutrons neutrons
/ „ nn% {2.20)
for incident protons
where t and T describe the isobaric spin vectors of the incident particle and the nucleus of mass A respectively. Estimates of V\ in the 50-120 MeV range were made [95] Evidence for an isospin term in the optical potential was reported by Perey [118] which was derived from measurements of proton elastic scattering in the range of 9 to 22 MeV. An isospin term was required in the real part of the optical model potential, which is described by
F<SO = 27(—^— J MeV
(2.21)
Note that the magnitude of the isospin term is about double of that deduced by Lane [95]. The decreasing values of the s-wave strength functions of the Te isotopes with increasing mass number have been successfully described by Tellier and Newstead [116] in terms of an isospin interaction term, which was included in the imaginary potential. Their result is as follows:
W = 12.8 - 62.8 —j—
MeV
(2.22)
Furthermore, the s-wave strength functions of the Cd isotopes follow the same systematic trend as that of the Te isotopes and are well reproduced [119] by the Tellier and Newstead potential. One can recall that in the mass region corresponding to the minima of the s-wave strength function, the s-wave strength function is approximately directly proportional to the imaginary part of the optical potential. Because of this dependence of So on W, a decrease in the s-wave strength function with increasing neutron excess can then be readily accounted for in terms of an isospin potential (Eq. 2.15). In conclusion, it seems that the behavior of the s-wave strength function in the mass region A = 100-140 can be described either in terms of a doorway state formalism or an isospin interaction. At this stage, it is not clear whether there is any connection between these two descriptions. b. Spin-Spin Interaction It is possible that a spin-spin interaction term may exist. If such a term is present and is incorporated into the real part of the potential,
2. Resonance Properties
36
50Sn
52Te
A = 3 MeV
A = 2 MeV
108 112 116 112 116 120 124
124 128
A Figure 2.4: Variation of the s-wave neutron strength function with mass number for the Cd, Sn and Te isotopes compared with the theoretical prediction (solid line) of a doorway state model [94] and optical model results (dashed line) which include an isospin term [119]. The dotted curves represent optical model calculations without an isospin term [87].
its influence will be to displace the peak of the s-wave strength function of one spin state relative to the other spin state. As a result, the effect will be most pronounced in the vicinity of the 3-s and 4-s single particle states, i.e. at A = 55 and 155 and will be negligible in the minima of the s-wave strength function. However, as it is well established, deformation effects in the mass region around A = 155 have a large influence on the strength functions. Recent measurements of total cross sections with a polarized neutron beam and polarized targets of 141 Pr, 159 Tb, lfl5 Ho, 167Er and 169Tm by Alfimenkov and collaborators [120] indicated that the spin-spin interaction term in the deformed region is quite small. Analysis with a spin-spin interaction term included in the imaginary part of the potential yielded Wgg = 6 MeV for the absorptive spin-spin potential. In order to search for
2.1. S-, P- and D-Wave Neutron Strength Functions
37
Table 2.2: S-Wave Strength Function Values of Isobars. All values from the present evaluation. Target Nuclide 106
Pd
r a 106 CM 46 48 ^ a
108pj 46 r u
48 Cd iioPd
Target Nuclide
So
9 7 7 7
48°Cd
7 7
112/"^ 48 ^ u 112c,, 50 a u
1 4
122c,, 50 Dn 122 T 52 ^
2 9
124c,, 50 Dn
5l
8
Te
128v p 54 ^ e 130 rTU 130TD „
i 8 Ba 8
5? La i|2 42 6
C e
Nd
So
5 4 5 1 5 5 8 7 7 9
a spin-spin interaction, it will be of great interest to carry out similar measurements and extend the energy region to higher energies for the following isotope: 133 Cs, 135 Ba, 137 Ba, 141 Pr, 143Nd and 145Nd, where the effect of the spin-spin interaction is expected to be strong.
P- Wave Strength Functions As in the case of the s-wave strength function, the p-wave counterpart can be determined from the slope of the cumulative sum of reduced neutron widths of pwave resonances plotted against neutron energy or from an analysis of the average capture, total or differential elastic scattering cross-sections in the keV range. In the former method, due to the broad distribution of reduced neutron widths, the accuracy of the derived value is essentially limited by the number of assigned p-wave resonances. On the other hand, in the other methods, an accurate knowledge of the average s-wave parameters as well as the scattering radius (for total cross section measurements) is required in order to unfold the p-wave contribution. In some cases, the onset of inelastic neutron scattering due to the presence of excited levels very close to the ground state of the target nucleus can also limit the usefulness of this method. In addition, because of the availability of mono-energetic neutron beams produced by reactors with the aid of Sc and Fe filters, Sb-Be photo-neutron sources and Van de Graaff generators, accurate cross section measurements carried out at one specific energy are possible. Subsequently, one can follow the method of Bilpuch et al. [121] to extract p-wave strength functions. The p-wave strength functions thus
38
2. Resonance Properties
obtained by these various methods are derived in different energy regions. Because of this situation, one has to recognize the possibility of an energy dependence in the strength function due to the presence of doorway states or fluctuations in the measured cross section. However, in general, the p-wave strength functions derived from average cross-sections have significantly better accuracies than those obtained from individual resonances. The evaluated p-wave strength function values are graphically shown in Fig. 2.2 and are compared with calculations based on the optical model-computer program of Buck and Perey [84], which takes into account the permanent deformation as well as pure quadrupole vibrations of the nucleus. Also shown is a spherical optical model calculation for nuclei with mass numbers less than 40. Inspection of Fig. 2.2 reveals the following features: a) As is readily evident, the observed 3-p peak is well reproduced by the optical model calculation. Contrary to previous claims [122], the 3-p peak is not split into its p!/ 2 and p 3 / 2 components with a minimum at A = 100 having a value Si = 1.5 xlO~ 4 . Such a result would require that the spin-orbit potential is at least twice the normally accepted value [123]. b) There seems to be an indication of another peak in the mass region 20-40 corresponding to the 2-p giant resonance. c) Of particular interest is the hint of a small peak around mass number 160. This is due to improved data in this mass region. At this point, it is noteworthy to remember that the spherical optical model predicts a minimum at A = 160 and a maximum at A = 240 with a value Si ss 6xlO~ 4 . In contrast, as is predicted by Buck and Perey [84] and as observed here, the strong rotational motion in this mass region produces a splitting of the 4-p giant resonance with peaks located at about mass numbers 160 and 230.
D- Wave Strength Functions Detailed information on the d-wave strength functions is scarce because of difficulties in unfolding their contribution from those of the s-wave strength functions. This is due to the occurrence of the s- and d- wave giant resonances at approximately the same mass number and their tendency to cancel each other in the energy region 100-1000 keV. The known values of the d-wave strength functions are all derived mainly from average capture and total cross section measurements. Our present knowledge of the d-wave strength functions is summarized in Fig. 2.3. A comparison of the measured values with a spherical optical model calculation, carried out by Divadeenam [88] who adopted the potential parameters of Buck and Perey [84], is also displayed. As shown, the data seem to indicate a maximum in the rare earth region with a possible splitting into two peaks at A = 150 and 172, which can be attributed to deformation effects, and another possible maximum below mass number 75. This behavior is similar to that of the s-wave neutron strength functions.
2.2
Average Level Spacings and Level Density
Various investigators have used different methods to evaluate the average level spacings. These are outlined in the informative review article by Liou [124] as
2.2. Average Level Spacings and Level Density
39
well as in the Proceedings of the Specialists Meetings on Neutron Gross Sections of Fission Product Nuclei [125]. A description of the decay of a compound nucleus via particle and gamma-ray emission can be carried out in the framework of the statistical theory of nuclear reactions. To be able to achieve this objective, the level density parameter (see below) of the Fermi gas model is required. One of the simple and most direct ways of determining this important parameter can be obtained from measurements of neutron resonances with the same Jw in an energy interval AE containing many resonances, JV.
^
(2-23)
where (Dtj) is the average level spacing of resonances specified by orbital angular momentum, I, and spin j (for brevity, the subscripts Ij will be dropped out in the following discussion). However, because of instrumental energy resolution and Doppler broadening of resonances, weak resonances may escape detection. Furthermore, in certain mass regions particularly in and near the 3p giant resonance, p-wave resonances are readily detected especially in capture measurements. Because of this situation, corrections for missing weak resonances or differentiation between s- and p-wave resonances have to be carried out. The latter problem can be partially resolved experimentally since there are various methods of distinguishing between strong s-wave and p-wave resonances. However, the experimental problem for the weak resonances still remains. A partial solution to this problem can be obtained via the statistical distribution laws of level spacings and PT distribution of neutron reduced widths along with applications of Bayes' theorem. A graphical method of evaluating the average level spacings for the case of s- and p-wave resonances is carried out by plotting the total number of observed resonances as a function of energy then a least-squares fit in the linear region is made; the inverse slope, which is equal to the average spacing, is then determined. A departure from linearity may indicate that resonances are missed or in some mass regions p- or d-wave levels are included. Under these conditions a cutoff energy is stipulated and a least squares fit to the data is determined. However, this method is not always reliable. In 1956, Wigner [126] remarkably surmised the distribution of level spacings on the basis of the behavior of the eigenvalues of a symmetric matrix with random Gaussian distributed elements. This is presently known as the Gaussian orthogonal ensemble (GOE). The Wigner distribution of level spacings of resonances with the same spin and parity takes the simple form
(2.24)
where (D) is the average level spacings and D is the nearest neighbor spacings. Note that the Wigner [126] distribution describes the level repulsion effect, i.e.
40
2. Resonance Properties
small spacings are unlikely to occur. Also note that the maximum value D m of the Wigner [126] distribution occurs at Dm = (D) V ^
(2.25)
An estimate of the error on (D) is derived from the distribution and is given by /4-TT\1/2
A {£>>=( —j£-
{D)
^ ' _ 0.52 (D)
(2.26)
Subsequently Gaudin [127] derived the level spacing distribution function as a rapidly converging infinite product in the limit of large dimensions and numerically compared his distribution with Wigner's surmise [126]. The difference between the two distributions is found to be less than about 5% for spacings less than two times the average value, which fact makes it difficult to distinguish experimentally between the two distributions (See Table lof Ref. [127]). Because of its simple form, and since it represents a good approximation to the exact distribution, the Wigner distribution is widely adopted by experimentalists in the analysis of resonance data. It is interesting to note that the Gaussian orthogonal ensemble predicts [128] a value of -0.25 for the short range correlation coefficient between adjacent levels. The negative sign can be understood in terms of level repulsion. In investigations of random symmetrical matrices, Garrison [129] determined a statistical uncertainty of 3 to the value of this correlation coefficient. Dyson and Mehta [130] introduced a statistics (A3) which describes long range correlations and is denned as the mean square deviation of a staircase plot (cumulative number of resonances versus energy) from a best fit straight line. That is, AE
A3 = min -^J
(N(E) - AE - BfdE
(2.27)
A,B
where y = AE + B represents the fit corresponding to the minimum value of A3. Dyson and Mehta [130] derived the average value and standard deviation (S.D.) of A 3 . The results are
(A3) =
0.0687)
S.D. = 0.11 The A3 statistics was used extensively by the Columbia group [131] as a sensitive test for the detection of spurious levels (resonances due to impurities or due to p-wave neutrons) or non-observation of weak resonances. Departure of the experimental A3 value as determined by Eq. 2.27 from the theoretical value gives an indication of these problems.
2.2. Average Level Spacings and Level Density
41
Because of the extreme complexity and random character of the initial state [132], \/fn" is normally distributed with zero mean and as a result F^ obeys the Porter-Thomas (PT) distribution. [74].
P{x) = ^ = e - i /2ix
(2.29)
where
This is a special case of a more general distribution called Chi-squared distribution which is represented by
(2.30)
where v is a parameter known in statistics as the number of degrees of freedom and T{v/2) is the gamma function. One feature of the Chi-squared distributions is that as v increases, the distribution becomes peaked at x = 1. When these combined methods are applied in the analysis of neutron resonances, reliable values of average level spacings are determined. As pointed out previously, the neutron level spacings can give valuable information on the level density parameter a as well as fundamental quantities such as the effective nucleon mass. For a review of the effective nucleon mass and the Migdal constants refer to [133]. On the basis of the Fermi gas model, the level density formula [134] is
p{U,
J) = C^+1K-^J+l^2e2^U
(2.31)
where U is the effective excitation energy and a is the spin cutoff or dispersion parameter. For slow neutrons,
U = Sn - A
(2.32)
where Sn is the neutron separation energy of the compound nucleus and A is the pairing energy which for compound nuclei A + l is specified by the following A = 8n + Sp
even-even
= 8P = Sn
even-odd odd-even
= 0
odd-odd
(2.33)
42
2. Resonance Properties
Sn and Sp are the neutron and proton pairing energies and can be obtained from Ref. [135]. Other modified forms of the level density formula (Eq. 2.31) were reported by Newton [136], Lang and Le Couter [137], Ericson [138], Gilbert and Cameron [139] and Dilg et al. [143]. Additionally, level density parameters can be obtained on line from www-nds.iaea.org/RIPL-2. Because of its widespread application in various nuclear model calculations, the Gilbert-Cameron level density was adopted here to determine the level density parameters a from the presently recommended average level spacings for s- and p-wave resonances
In this relation, U is the effective excitation energy which is determined by U = Sn + 0.5 x Emax — A {Emax is the upper energy of observed resonances with known scattering widths). In the present analysis, the spin dispersion parameter O~M is specified by a modified form of the Lang-Lecouter expression [137, 140] ry
a2M =< m2 > {—aXcotX + n) x SC
(2.35)
where n=0, 1, 2 for even-even nuclei, odd mass nuclei, and odd-odd nuclei respectively, and <
2 m
> = 0.146 x A2/3
(2.36)
where < m 2 > is the mean-square magnetic quantum number for single particle states [141], and cos\ = e~& SC = 1 +
(2.37) sheU
(l-
e ^ WJ
(2.38)
The latter equation describes the shell correction to the level density parameter, as developed by Ignatyuk et al. [142]. In this expression, EsheU is the shell energy, obtained from the work of [140] and 70 is a constant which is determined experimentally; its value is 4 MeV^ 1 [140]. According to the Thomas-Fermi model for a finite nucleus, an expansion of the level density in powers of A~x/Z yields the following relation:
where aidm is the level density parameter at high excitation energies where shell effects are washed out, and av, as and ac are the volume, surface and curvature components of the level density parameter respectively. The contribution of shell effects is introduced into this equation such that L
, Eaheu (
= aidm 1 + —JT- U -
^T^Al e A
'
/o A(\\
( 2 - 40 )
I
re
40
Nuclear Level Density Parameter, a
35
I
30
s
25
A A
20 15 10 5 0 20
40
I
I
I
I
60
80
100
120
I
I
I
I
I
I
140
160
180
200
220
240
260
A Figure 2.5: Level density parameters aexp (solid triangles) derived from average resonance spacings are compared with those calculated on the basis of Eq. 2.40 agiobai (open circles); see text for details.
44
2. Resonance Properties
A least-squares fitting procedure for the data yielded the following results 9 MeV"1, as 7 MeV"1 [140]. Because of the coupling of a andCTM,an iterative procedure was carried out with the aid of Eqs. 2.31, 2.35 to determine a andCTM-These are tabulated in columns 5 and 7 of Table 2.3. For comparison with aexp, the level density parameters as determined by shell effects (Eq.. 2.40), called agiobai, are listed in column 6. Note that agioi,ai is basically specified by nuclear size and shell effects as well as the constants av, as and ac. Other level density parameters can be found in Refs. [139]-[148] and their systematics have been discussed by Rohr et.al. [149].
2.3
Radiative Widths and 7-Ray Strength Functions of Sand P- Wave Resonances
The description and determination of neutron radiative widths has been the subject of extensive experimental and theoretical investigations. Of particular interest from a theoretical and evaluation point of view is the determination of the dependence of radiative widths on neutron resonance spacing, spin and parity of the initial state, excitation energy, nuclear size (i.e. A) and nuclear structure effects. These aspects will be discussed briefly. At present, it is recognized [150]-[152] that the partial radiative width from an initial state i to a final state / is composed of three components so that one can write
{Tliff2
= (I^cn)) 1 / 2 + (r 7i/ (sp)) 1/2 + (r 7i/ (ds)) 1/2
(2.41)
where the successive terms on the right hand side designate contributions due to compound nucleus, single particle, and doorway-state respectively. These represent the statistical, direct and semi-direct components respectively. 2.3.1
Statistical Aspects of Partial Radiative Widths
At this point let us turn our attention to the statistical aspect of the radiative process. Because of the extreme complexity and random character of the initial state [132], \fT-yif {en) is normally distributed with zero mean and as a result the partial radiative widths populating a fixed final state, / , from various initial states i (resonances) obey the Porter-Thomas distribution [74].
(2.42)
This is a special case of a more general distribution called Chi-squared distribution which is represented by
2.3. Radiative Widths and j-Ray Strength Functions of S- and P- Wave Resonances
45
(2
"43)
where v is a parameter known in statistics as the number of degrees of freedom and T(v/2) is the gamma function. One interesting feature of the Chi-squared distributions is that as v increases, the distribution becomes peaked at x = 1. The distributions of the total radiative widths is determined by the total number or partial radiative widths i.e. the number of final states or exit channels which is identified with v in Eq 2.30. When the number of exit channels is small, one then expects fluctuations in the total radiative widths of resonances. By contrast, the capture widths cluster close to a constant value when the number of exit channels becomes large as in the case of large mass number nuclei. The first theoretical estimate of average radiative widths were carried out by Blatt and Weisskopf [16] on the basis of the single particle model in conjunction with the assumption of uniform distribution of gamma-ray strength over the resonances in accordance with the relation
£-
(2.44)
In this expression F 7 /(sp) is the single particle estimate of the radiative transition populating a final state / , D is the average level spacing of resonances with the same spin and parity and DQ is the spacing of single particle states which is about 0.5-1.0 MeV. Rough estimates of the El, Ml, E2 and M2 partial radiative widths are given by [16]
= 8.9 x l O "
2
^ /
3
^ - ^
(2.45)
(2.46)
= 7.9 x 1 0 " 8 A 4 / 3 £ ; 7 / ^ ^
(2.47)
(r 7/ (Af2)) = 1.9 x 10~8A2/3E°f—— (2.48) Do In these equations, the radiative widths are expressed in eV when the 7-ray energies are denoted in MeV. D and Do are to be expressed in the same units. Several important points are to be emphasized with regard to the derivation of Eqs. 2.45-2.48:
46
2. Resonance Properties
a) The spin statistical factors which include Clebsch-Gordon and Racah coefficients are not included in these relations (see footnote on page 626 of Ref. [16]; also refer to table 3.2 and Eq. 3.23). In any thorough analysis, these spin factors must be included particularly in p-wave investigations. As an example, some transitions such as p3/2 —> d3/2 are hindered by a factor of five relative to p 3 / 2 —> S1/2 transitions because of these spin statistical factors [153]. b) An approximate expression for the radial overlap integral is derived [16]
<2-49'
where L is the multipolarity of the 7-ray. Comparison with radial integrals for El transitions derived on the basis of a Saxon-Woods potential indicates [4] that Eq. 2.49 is overestimated by about a factor of two. c) The neutron effective charge, e, for electric dipole transition which is denoted by the following expression e= - e |
(2.50)
must be applied in Eqs. 2.45-2.48. d) The numerical constants in Eqs. 2.45-2.48 depend on the assumed value of the nuclear radius. The constants quoted above are based on R = 1.35 x A1/3 fm. In analogy with neutron strength functions, reduced photon strength functions for El and Ml transitions are defined [154] by <**i> = w1:^'
(2-5i)
(2.52)
According to Eqs. 2.45-2.48, the theoretical estimates of (feei) and (AMI) are 8.9 x 1(T 2 /A) and 2.1xl(T2/-Do respectively. A detailed examination of available thermal and resonance neutron capture data was carried out by Bartholomew [154] who showed that the (fc^i) and (kMi) values for the various isotopes clustered around mean values of 3xlO~9 MeV~3 and 4xlO~9 MeV~3 for El and Ml radiations respectively. Agreement with the theoretical {UEI) and (&MI) was achieved on the basis of a 15 MeV for Do, which is much larger than the single particle spacing. This large Do can be considered as a reflection of the fact that the Weisskopf single particle values are overestimated because of the various points discussed previously. Another interpretation may be made in terms of the distribution of gamma strength over the giant dipole resonance region. Because of these considerations it is felt that no physical significance should be attached to the Do value; it should be viewed as an adjustable parameter. Subsequent surveys of photon strength functions were made by Bollinger [155],
2.3. Radiative Widths and 'y-Ray Strength Functions of S- and P- Wave Resonances
47
Bartholomew et al. [156], Jackson [157], Kopecky [158], and Lone [159] to which the interested reader is referred. In addition, McCullagh et al. [160] examined the 7-ray spectra due to resonance capture in nuclei ranging from 19 F to 238 U with the conclusion that (kE1) = (2.9 and (kMi) = (30
0.3) x 10~9 ) x 10~9
MeV"3
well described the data. Although the El photon [157] strength function value is in good agreement with Bartholomew's determination, there seems to be a large discrepancy with regard to the Ml strength. A possible explanation may be due to experimental problems associated with the measurements of Ml transitions which are weaker than El transitions. This results in difficulties in obtaining meaningful averages because of the experimental sensitivity of detecting weak 7-rays. Since the level spacing of the studied nuclei vary by about three orders of magnitude, the results of these investigations can be considered as providing a verification of the linear dependence of the radiative widths on the average level spacings. This dependence was first demonstrated by Carpenter [161] in his studies of the El radiative strength functions of even-odd target nuclei ranging from 143Nd to 201 Hg. At this stage, we would like to draw attention to the El and Ml estimates of partial radiative widths as derived by Lynn [4]. These values are based on the valence neutron model in combination with the strong coupling model. The results are
< r 7 / ( £ l ) ) L = 0.16 x l O " 9 ^ (MeV) A5/6D (eV)
(2.54)
(2.55)
where the partial radiative widths are expressed in eV. We designate these widths by the subscript L (Lynn) for the purpose of differentiation. When compared with the Weisskopf estimates, some interesting features emerge: a) The nuclear size dependences of the El and Ml partial radiative widths represented by Eqs. 2.54-2.55 are different from those of Eqs. 2.45-2.48. In particular, {kE1)L = 0.16 x 10-gA1/6
(kMi)L
= 0.66 x 10~9A1/3
(2.56)
(2.57)
b) Eqs. 2.56-2.57 indicate that the photon strength functions as defined by Eqs. 2.51-2.52 are slowly increasing functions of the mass number. c) The derivation of Eqs. 2.45-2.48 bypasses the use of a Do value and consequently avoids the ambiguity resulting in its physical interpretation.
48
2. Resonance Properties
2.3.2
Giant Dipole Resonance Considerations
Brink-Axel Model An alternative approach to the study of electric dipole radiative strength function is based on the giant dipole resonance (GDR) observed in photonuclear reactions. The large body of experimental photonuclear data showed that the Lorentz shape gives a reasonably good approximation to the GDR
where ag, Fg and Eg are respectively the peak cross section, full width at half maximum and peak position of the giant dipole resonance. Deformed nuclei are represented by two such Lorentzian shapes corresponding to oscillations along the major and minor axes. A compilation of the GDR parameters can be found in [162]. Axel [163] advanced the idea that the reaction mechanism which produces the GDR is also responsible for the (n,7) process near the neutron separation energy. By equating the gamma absorption cross section to the giant dipole resonance, using the principle of detailed balance, and invoking the Brink [164] hypothesis, that each excited state has built on it a giant dipole resonance of the same shape as the ground state except for a displacement, Axel [163] obtained for the 7-ray strength function the following relation
D
( E E )
(2m)
+ ET
It is assumed that the giant dipole resonance behavior can be extrapolated to low energy 7-rays. By numerical approximations, Axel [163] showed that Eq. 2.59 (for r g = 5 MeV, A = 100 and Eg = 80xA^1/?3) can be written in the following simple forms in the two energy regions: For E 7 fa 7 MeV,
— = 2.2x10
^?(MeV)y
y1QQj
(2J0)
VlOO/
(2.61)
= 6.1 x 10"15B^ For E 7 < 3 MeV
D
= 1.6x " " " "
\7(MeV)y 15
= 4.4 x 10~ i It is interesting to note that near E1 = 7 MeV, the GDR prescription predicts an .Ej! energy dependence (which is faster than the single particle behavior) and an
2.3. Radiative Widths and j-Ray Strength Functions of S- and P- Wave Resonances
49
8 3
A / dependence. A supporting evidence for the E^ dependence of partial radiative widths can be found in the pioneering investigations of Bollinger and Thomas [165] on target nuclei 105 Pd, 155 ' 157 Gd and 185 Ho. Their results demonstrated that the average 7-ray intensities are a smooth function of energy and do not depend (except for spin factors) on any specific nuclear structure properties of the final states. In addition, the data for both El and Ml transitions seem to follow an E^ dependence giving a supporting evidence for the Brink-Axel treatment. A survey [160] of El strengths in terms of the Axel treatment indicated that the constant in Eq. 2.60 is (4.2 0.4) xlO~ 15 MeV~3 which is smaller than the predicted value 6.1 xlO~ 15 MeV" 3 . The Fermi Liquid Model According to the Fermi Liquid Model (FLM) for a nucleus which is characterized by strongly interacting quasi-particles, the El photon strength function for spherical nuclei is expressed by [166]
where Fo and Fx are the Landau-Migdal Fermi liquid force constants describing the interaction of the quasi particles [167]. In this model, the damping width of the GDR, Ffl , is energy-s as well as temperature-dependent, and is given by [166]: Tg(E7,T) = ^(E?f+4n2T2)
(2.63)
9
where F° = Tg(Ey = Eg,T = 0) is the damping width of the GDR at its peak energy, and T is the nuclear temperature of the final state. As shown in Eq. 2.63, the spreading width is composed of two components: one which is due to the decay of particle-hole states to more complicated configurations, the other is produced by thermal effects arising from the collisions of quasi particles in the nuclear volume. Generalized Fermi Liquid Model None of the above models describes adequately the gamma-ray strength functions for both spherical and deformed nuclei. A unified model which can account for the El transition strengths for both was proposed in [168]. On phenomenological grounds, and based on Tourneux's results [169], as well as the fact that (Eq. 2.63) does not take into account surface contributions, this equation was generalized by including the dipole-quadrupole contribution in Eq. 2.62 [166], such that
- 8 674 x - 8.674x where F™ is a modified form of Eq. 2.63 and is represented by the relation
50
2. Resonance Properties
t4ir2T2)+Tdq((32,E7)
(2.65)
In this expression, the second term represents the damping width due to the interaction between the dipole and quadrupole vibrations as calculated in [169]; C is a constant determined as in [166], by the condition that F™ (Eg, 0) = F°. In this relation it is assumed that the interference between the two terms of the right hand side of Eq. 2.64 is absent or negligible. With the aid of the relation between the restoring force and the B(E2) value [170], it can be easily shown that Tourneux's [169] reported relation for the width of the giant dipole resonance due to the dipole-quadrupole interaction can be simplified to
(2-66)
where 2.35 = 2(2/n2)1/2 is a conversion factor from standard deviation to full-width at half maximum for a Gaussian distribution, /?2 is the deformation parameter of the nucleus as derived from the B(E2) values, and E2 is the energy of the vibrational state [169]. For spherical nuclei, Ei is the energy of the first excited 2 + state; for deformed nuclei, an average value of the energies of the /? and 7 vibrational states can be adopted.The evaluated deformation parameters can be found, for example, in [171]. For deformed nuclei, the sum of two incoherent terms (2.66), with the appropriate Lorentzian GDR parameters, are required. It is emphasized that the dipolequadrupole term, Eq. 2.66, was derived in [169] for spherical nuclei. In the investigations reported in [168], it was shown empirically that this interaction term applies equally to deformed nuclei. Other approaches to the description of the gamma-ray strength functions in terms of a modified GDR model can be found in the works of Kopecky et al. [172] and Plujko and Kavastshyuk [173].
2.3.3
Non-statistical Component of Radiative Widths
As indicated previously and to be discussed subsequently in some detail, nonstatistical effects in the radiative capture of resonance neutrons are at present well established [153] in certain mass regions and particularly in the neighborhood of the 2p-, 3s- and 3p-giant resonances. Examples [174] are the 85 keV p-wave resonance of 24Mg, the 429, 612 and 818 eV resonances of 98Mo, the strong p-wave resonances in 90Zr [237]. These non-statistical effects are well accounted for in terms of the valence neutron model which will be described in the next chapter. In obtaining average values of radiative widths, the non-statistical average component has to be subtracted out. Similar investigations for light nuclei 24Mg, 28Si, 32 S can be found inRef. [175].
2.3. Radiative Widths and j-Ray Strength Functions of S- and P- Wave Resonances 2.3.4
51
Total Radiative Width: Theory
Here we present briefly some interesting results derived from theoretical investigations of total radiative widths. Cameron's Treatment One of the first attempts to calculate total radiative widths in terms of Blatt and Weisskopf's estimate of El partial radiative width was made by Cameron [176]. The total radiative width for electric dipole radiation can be written in the form e2 (R\2D{U)
)
fu
E*dE
L
E)
(2 67)
"
where e is the effective charge, U is the effective excitation energy in MeV and D(U) is the spacing of levels with the same spin and parity at excitation U. With the aid of this expression and Newton's level spacing formula [136], Cameron [176] derived the following expression for the total radiative width: (F7(meV)) = 5.2,4
where o=
5.97 (JN + JZ +
I)1/2 A1/3
For slow neutrons, U = Sn — A. J^ and Jz are the effective average total angular momenta of single-particle states near the Fermi surface (which are tabulated in Ref [176]). Sn is the neutron separation energy and A is the pairing energy which is defined previously, Eq. 2.33. Implicit in this derivation is the adjustable constant Do which is varied to a value of 45 MeV in order to fit the experimental data. Eq. 2.68 describes the decrease of F 7 with A and can take account of shell structure effects via the variables JN and JzGiant-Dipole Resonance Treatment By integrating the dipole integral (Eq. 36 of Ref. [163]) over a constant-temperature level density formula D oc e~E~//T, Axel [163] derived the following approximate expression
= 1.03 x 10" 5 yl 7/3 T 5 Malecky et al. [177] derived a more general relation corresponding to Eq. 2.69.
52
2. Resonance Properties
In addition, Johnson [178] carried out calculations of total radiative widths on the basis of the Brink-Axel formalism for nuclei in the 3p giant resonance. The result is
(2.70)
where ag, Tg, and Eg are the parameters of the giant dipole resonance. Johnson [178] parameterized the combination agT2g/Eg by relating it to the atomic weight by
E
5
9
where C is a constant. According to Johnson's conclusion [178], the radiative width varies as U5/2A~4/3. Additionally, Moore [179] and Lynn [180] carried out calculations of total radiative widths of the trans-actinium isotopes on the basis of the Brink-Axel treatment.
Thermo dynamical Approach Treating the nucleus as a heated system Kuklin [181] applied thermodynamical considerations to arrive at the following expression (r 7 (eV)} = 0.77 x W-6a'A7/3T7
(MeV)
(2.72)
It is interesting to note that in the Fermi liquid theory, the constant a1 is related to the imaginary part of the optical potential by the relation,
where Ep is the Fermi energy and for the case of neutrons is given by
-69" The then existing data of total radiative widths were reasonably described [181] by a' = 12 MeV^1 which leads (for Ep = 40 MeV) to an imaginary value of the optical model of W = 0.03E2. For E = 8 MeV, W = 1.9 MeV which is in reasonable agreement with values obtained from scattering experiments. In addition, by treating the nucleus as a radiating black body, Benzi and his collaborators [182, 183] arrived at the following simple expression for the total
2.3. Radiative Widths and j-Ray Strength Functions of S- and P- Wave Resonances
53
radiative width, 2
T3 A
(2.75)
3
7.32T (MeV) The radius parameter was adjusted to the reasonable value ro = 1.2 fm to obtain general agreement between theory and experiment for nuclei ranging from 151Sm to 244 Cm. The temperatures T are related [184] to the excitation energy U by the BCS equations and are of the order 0.5 - 1 MeV. The experimental values are reproduced by Eq. 2.75 to an accuracy of 30%. Applying the theory of finite Fermi systems, Zaretskii and Sirotkin [185] obtained an expression for the total radiative width within the framework of the shell model approach of the theory of nuclear reactions. The result is, (r7(eV)> = 3 x 10" 6 a ( ^
2 / 3
)
(2-76)
where U, or1 and a" 1 are expressed in MeV. Reasonable agreement with experimental data was achieved [185] for a RS 0.1, which corresponds to an imaginary part of the optical potential of 5 MeV. Note that this is more than double the value obtained by Kuklin [181] in his analysis of radiative widths. It is interesting to point out that for a oc A Eq. 2.76 closely reproduces the A dependence found experimentally by Malecky et al [177].
Generalized Fermi Liquid Model Within the generalized Fermi Liquid model approach, [168] , total radiative widths are computed on the basis of the gamma-ray strength functions specified by Eqs. 2.64-2.66, as well as a level density with a constant nuclear temperature. Furthermore, the dependence of deformation on thermal fluctuations as described by Mughabghab and Sonzogni [186] is included in these calculations through the following equations
yf(^
!^)
(2.77)
with Vo = 0.80 x as x A 2 / 3 (l - x) ^7826(^^) ^ ^ Jeff
2
(2.78) ]-
1
(2.79) (2.80)
54
2. Resonance Properties
where as is the surface energy, og=17.94 MeV, J e // is the effective moment of inertia, and Irig%A is the moment of inertia of a rigid rotor. The last term in Eq. 2.77 is negligible in resonance neutron capture. The nuclear temperature T for a Gilbert-Cameron level density formula is obtained on the basis of the thermodynamic relation
h = wl09p{u) = ^ ~ w
{2M)
It should be noted that this relation converges to the familiar expression T = y/U/a at high excitation energies. The total radiative widths estimated on the basis of the level density parameters, a&%v and dgiobaii are summarized in columns 8 and 9 of Table 2.3. In the last column of this table are included the measured average radiative widths as evaluated in this study. On close inspection of this table, the following general features stand out: For Z <40, a large discrepancy exists between calculated and measured capture widths. For the majority of the nuclei, the estimated width is much smaller than the measured one. For 41
These findings can be interpreted in terms of two possible reasons: Peak position of the GDR. With increasing A, the peak position of the GDR is shifted closer to the neutron separation energy than nuclei with smaller A (note that for spherical nuclei J s =77xA~ 1 / 3 ). As a consequence, its influence dominates for nuclei with large mass numbers, as in the actinide region. By contrast, at low mass numbers its role in the radiative process is diminished. As a result, other reaction mechanisms come into play. Energy dependence of the 7-strength function. First, it should be noted that the major contribution to the El dipole integral comes from an energy region about 1-2 MeV below the neutron separation energy. Second, the calculated capture widths for nuclei with low mass numbers and small separation energy are more susceptible to the energy dependence of the GDR model or KMF model at low energy than those at high mass
2.3. Radiative Widths and j-Ray Strength Functions of S- and P- Wave Resonances
55
numbers. Third, the energy dependence of the 7-strength function is well established for gamma-ray energies above 4 MeV for the GFL model [168]. However, very little experimental evidence is available from capture gammaray spectroscopy on this subject at low energies. Results obtained from other particle reactions [187, 188] show an enhancement over the predictions of the KFM model [166] at low energies. There is some evidence for the energy dependence of the gamma-ray strength function investigations of Schiller et al. [189] and Voinov et al. [188] where pronounced pygmy resonances are observed at about gamma-ray energies of 2-3 MeV for compound nuclei 161Dy, 162 Dy, m Y b , 172 Yb. As an explanation, these authors propose a pygmy Ml resonance. However, the nature of its multipolarity is not known. This indicates that the energy extrapolation of the GDR to energies below ~ 3 MeV does not hold out. In conclusion, detailed examinations of Table 2.3 indicates that the GFL model gives good description for those nuclei when Eg — Sn < 8.5 MeV. 2.3.5
Systematics of Average Total Radiative Widths: Experimental
The first experimental attempt to study systematically average radiative widths and interpret them in terms of Blatt and Weisskopf's [16] estimates of partial radiative widths was carried out by Levin and Hughes [190]. Subsequently, by carrying out a similar study, Stolovy and Harvey [191] arrived at the following empirical expression for
(2.82)
Malecky et al. [177] noted that the previously derived relations of total radiative widths can be expressed in term of a combination of the parameters A, a, U, and T. A detailed least-squares analysis was carried out for a wide range of nuclei ranging from 59 Co to 246 Cm. The resulting total radiative width has the following form
(2.83)
where / is the spin of the target nucleus and U and a" 1 are expressed in MeV. As pointed out by Malecky et al. [177], there is general agreement between the experimental values and those calculated on the basis of Eq. 2.83 with few exceptions, which include target nuclei 77Se, 91Zr, 197Au and 232 Th. Our present knowledge of the variation of the average s-wave radiative widths with mass number A is illustrated in Fig. 2.6. Two interesting features emerge from this plot: (a) the general monotonic decrease of F 7 with A in the mass region 60 - 190 and (b) the maxima at about the mass numbers A = 50 and 208, which are related to the magic numbers Z = 28 and 82. It should be noted that large fluctuations are observed from resonance to resonance for s-wave radiative widths around A = 50 which are attributed to valence capture.
en
10
S-wave Average Capture Widths
\—i
v Even A Target Odd A Target
A
0.01
J
20
I
.
40
60
80
100
.
I
120
A
140
160
180
200
220
240
260
Figure 2.6: The average s-wave radiative widths plotted versus mass number. Note the decrease of < F 7 0 > with A.
S3
a
10 a-
P-wave Average Capture Widths
S
s S5-
5
CD
S
1
i
v
1
TSfr-r-ab *
*
Even A Target Odd A Target
Co
i
0.01 20
40
60
80
100
120
140
160
180
200
220
A Figure 2.7: The average p-wave radiative widths plotted versus mass number.
240
260
58
2. Resonance Properties
The variation of p-wave radiative widths with mass number is shown in Fig. 2.7, which demonstrates the same qualitative features as for s-wave radiative widths. A detailed comparison of s- and p-wave radiative widths shows generally the following results:
s > p at A « 5 0
(2.84)
These differences can be understood in terms of channel or valence neutron capture which will be discussed in some detail subsequently.
2.4
Resonance Integrals
The epicadmium dilute resonance integral for a particular reaction <JR{E) (
1=
VR(E)^JEC
(2.85)
&
where Ec is determined by the cadmium cutoff energy, which depends on the cadmium thickness. We have assumed that the cutoff energy, Ec, equals 0.5 eV. It is assumed that the dE/E neutron spectrum is not perturbed by the absorbing materials. Explicit expressions have been derived by Story [192] for this integral for the case where the cross section is represented by a Breit-Wigner shape. However, if the resonances are not located close to Ec and the cross section is described by a sum of single-level Breit-Wigner contributions, the resonance capture integral (in barns) can be approximated by the very simple expression
J7 = f 2.608 x where the summation is extended over resolved resonances and all units are expressed in eV. Similar expressions for the fission integral could be written. The contribution to the integral from the unresolved energy region has been treated by Dresner [193]. The 1/v contribution to the resonance integral for a cadmium cutoff energy of 0.5 eV is given by
(2.87)
2.4- Resonance Integrals
59
For EQ = 0.0253 eV and _Ec=0.5 eV, the 1/v contribution to the resonance integral simplifies to I(l/v)
= 0.45cr°
(2.88)
The capture and fission resonance integrals have been computed for each isotope from the resonance parameters and have been compared with measurements. If large discrepancies exist between measured and calculated values, these cases have been brought to the attention of the user. The resonance integrals are compiled by Gryntakis and Kim [194, 195] and Westcott gw factors are computed [195] from the resonance parameters in the temperature range 0° to 200° C for the following nuclei: 109 Ag, 113 Cd, 115 In, 149 Sm, 151 Eu, 152 Eu, 155 Gd, 157 Gd, 176 Lu, 182 Ta, 191 Ir, 193 Ir, 197 Au, 226 Ra, 2 2 9 Th, 2 3 1 Pa, 232 U, 233 U, 234 U, 235 U, 236 U, 238 U, 2 3 7 Np, 2 3 9 Pu, 241 Pu, 241 Am and 242 Am.
Table 2.3: S-wave level spacing (Do), nuclear level density parameters (aexp and agiobai) derived using Eqs. 2.34 and 2.40, spin disperssion parameter (<7M), predictions of capture widths (F* and Fjj,) and experimental value (F^)
Nuclide (Target)
Spin sn (Target) (keV)
23
3/2+
24
0+
Na Mg 28 Si 29 Si 30 Si 31p 32g 33g
34g 35
Cl C1 40 Ar 37
39K 41K
40
Ca Ca 43 Ca 44 Ca 45 Sc 46 Ti 47 Ti 48 Ti 49 Ti 42
50Ti
0+ 1/2+ 0+ 1/2+ 0+ 3/2+ 0+ 3/2+ 3/2 0+ 3/2+ 3/2+ 0+ 0+ 7/20+ 7/2-
0+ 5/2
0+ 7/2-
0+
6960 7331 8474 10609 6587 7937 8642 11417 6986 8580 6108 6100 7800 7534 8363 7933 11131 7415 8761 8880 11627 8142 10939 6372
Do (eV)
aexp (MeV"1)
108 ± 14d 474 ±72 d 329±22.0d 117±ll d 212+24d 40.3±4.5d 149 ±412 d 21 ± 2d 135 ± 12 d 22.3±2.5d 13.2±1.4d 51.9±2.3d 11.2±1.0d 14.8±1.4d 47±3d 14.9±1.2d 1.5±0.2d 25.7±2.8d 1.03±0.05d 20.1 ±1.5 d 1.64+ 0.13d 20.8±2.5d 4.5±0.3d 24.9+ 1.5d
4.01 3.40 3.05 2.80 3.46 4.72 3.78 3.82 5.17 4.72 7.27 7.67 6.05 6.19 5.57 7.35 7.19 7.41 7.60 6.34 6.21 6.74 6.20 8.13
^global
r
(MeV"1) 3.97 3.40 3.24
4.39 5.09 5.53 5.20 6.77 8.74 6.60 8.04 6.53 8.13 8.25 8.42 8.07 7.79 7.85 7.70 7.36 7.46
2.95 1.70 1.72 1.21 1.63 3.00 2.15 1.50 2.32 3.29 3.79 2.84 3.73 4.08 2.55 2.82 1.95 2.79 4.25 2.85 1.94 2.77 1.88 2.52
7
r
7
(meV)
(meV)
140 145 184 333 78 460 1350 1623 214 106 14 90 84 77 33 105 609 45 170 309 474 186 397 40
118 88 153 183 116 660 825 656 160 78 17 59 63 35 20
77 392 31 142 167 237 121 222 53
o
Fc 7 (meV) 121 ± 94 1610 ± 360
2150 ± 1170 1900 1600 ± 580 446 ± 32
1000 ± 540 940 ± 180 1500 ± 900 1100 ± 200 750 ± 40 1300 ± 400 870 ± 430 1380 1190 ± 480 2300 ± 1400 740 ± 330 270 ± 90
S s S
CO
Table 2.3, continuation. Nuclide Spin s« (Target) (Target) (keV)
Cr 52 Cr 53 Cr 54 Cr 55 Mn 54 Fe 56 Fe 57 Fe 58 Fe 59 Co 58 Ni 60 Ni 61 Ni 62 Ni 64 Ni 63 Cu 65 Cu 64 Zn 66 Zn 67 Zn 68 Zn 70 Zn 89 Ga 71 Ga 70 Ge
7/2-
0+ 0+ 3/2-
0+ 5/2-
0+ 0+ 1/2+ 0+ 7/20+ 0+ 3/20+ 0+ 3/23/20+ 0+ 5/20+ 0+ 3/23/20+
7311 9261 7939 9719 6246 7270 9298 7646 10044 6581 7492 8999 7820 10596 6838 6098 7916 7067 7979 7052 10198 6482 5834 7654 6520 7416
Do (eV)
8-exp
d d d d d d d d
22.0+1.7 6.4+0.5 d
d d d d d d
21.1+ 2.0d 722+ 47 510+ 60 2940+ 130 4700+ 400 367+ 19 3785+ 190 3510+ 179 316+ 41 326+ 41 1170+ 230
(MeV- 1 )
(MeV- 1 )
7.30 6.27 5.95 5.64 7.28 7.91 5.83 6.69 5.82 8.32 8.28 6.07 7.35 6.42 8.15 8.96 9.16 10.76 9.13 9.78 8.47 11.12 8.96 10.68 12.78 11.99
7.46 7.10 7.19 7.89 8.49 9.39 6.23 7.44 8.17 8.82 8.34 6.53 7.93 8.46 9.18 10.27 9.04 10.11 9.13 11.03 11.13 11.96 12.79 11.67 12.78 12.68
7
3.92 2.89 2.82 2.01 2.90 4.08 2.60 2.74 1.98 2.86 4.21 2.76 2.97 2.13 3.12 3.22 4.45 4.73 3.45 3.58 2.45 3.67 3.66 4.71 5.36 3.66
r
F
7
(meV)
(meV) 7
(meV)
114 287 218 341 69 111 324 147 346 55 157 347 141 355 68 37 120 55 94 53 149 30 46 184 74 83
108 200 123 121 43 67 267 107 120 45 157 281 113 157 48 25 126 67 78 38 67 24 15 140 72 69
1340 1100 2000 1400 1500 750 150 1600 900 470 1830 1130 110 540 40 2030 1150 1160 360 2000 300 910 250
onam
51y 50
!NS
b
&1
§r
5000 395 726 60 400 20 440 60 320 40 266 222 185
19 10 30
Oi 1—I
Table 2.3, continuation. Nuclide Spin sn (Target) (Target) (keV) 72
0+
73
9/2+
Ge Ge 74 Ge 76 Ge 75 As 74 Se 76 Se 77 Se 78 Se 80 Se
0+ 0+ 3/2-
0+ 0+ 1/2-
0+ 0+
79Br siBr
3/23/2-
84Rr
0+ 0+
86Rr 85
Rb Rb 84 Sr 86 Sr 87 Sr 88 Sr
87
89y
90Zr 91
Zr 92 Zr 93 Zr 94 Zr
5/23/2-
0+ 0+ 9/2+
0+ 1/2-
0+ 5/2+
0+ 5/2+
0+
6782 10196 6505 6072 7328 8028 7419 10498 6963 6701 7892 7593 7121 5515 8650 6082 8530 8424 11112 6359 6857 7195 8635 6735 8221 6462
Ci bO
Do (eV)
(MeV- 1 )
(MeV- 1 )
2070+ 290 99 10 3000+ 1500 4820+ 760 6 284+ 50 505+ 65 121+ 11 1480+ 200 3500+ 1500 53.8+ 2.8 145+8 4040+ 450 25170+ 1200 185+6 2030+ 210 3830+ 130 3000+1000 312+ 22 26100+ 2000 3290+ 130 6890+530 502+ 28 4310+ 510 298+ 30 3684+ 222
12.32 12.35 12.23 12.13 13.36 13.29 13.51 10.36 12.62 11.52 13.27 12.19 9.86 8.94 10.00 10.40 9.03 9.12 8.31 7.99 9.41 9.14 9.14 10.85 10.70 11.79
13.46 12.91 13.37 12.86 13.52 13.04 13.90 13.32 13.69 12.72 13.71 12.82 14.46 14.92 11.46 11.09 13.04 11.51 10.39 10.63 10.58 10.67 10.67 12.51 11.36 14.59
^global
3.66 2.02 3.59 3.43 5.60 3.81 3.75 2.42 3.63 3.44 5.67 5.7 4.47 4.27 4.99 4.69 3.90 3.58 2.59 3.21 4.58 3.24 2.63 3.54 2.82 3.68
po 7 (meV)
p6 7 (meV)
75 150 70 57 121 76 57 320 81 89 117 106 181 120 206 77 267 200 489 132 121 106 178 69 98 45
57 133 54 48 117 82 53 158 63 65 110 91 59 25 136 63 92 100 254 55 83 66 83 44 80 23
pc 7 (meV) 150 197 195 115 340 250 230 390 230
35 +6 40 25 +9 20 18 54 20
313 243 160 153 250 192 235
5 +5 30 18 +8 16 20 to
150 120 134 170 20 134 16 141 40 157 20 80 20
190
CB Co
3 s>
8 is
1re re CO
Table 2.3, continuation. Nuclide Spin Sn (Target) (Target) (keV)
Do
S-exp
ra
^global 1
1
(eV)
(MeV- )
(MeV- )
13500+ 3000 84.6+ 4.8 2340+ 201 1524+ 141 75.4+ 8.6 972+ 75 46.5+ 5.8 858+ 42 663+ 63 12.8+ 0.5 21.7+ 2.3 345 21 18.5 2.1 375 70 373 36 29.3 1.5 219 21 10.9 0.5 183 12 14.8 0.8 135 9 318 27 14.9 0.6 15.1 0.6 156 18 96 + 9
11.47 12.56 9.35 11.45 11.59 13.38 13.33 15.96 18.50 16.82 12.72 15.07 13.43 16.91 17.90 17.10 15.53 13.98 17.42 14.08 19.27 18.98 17.61 18.84 14.18 16.38
15.88 10.17 10.37 12.30 13.41 14.20 15.03 15.96 18.37 15.79 14.09 15.21 15.62 16.81 17.87 16.18 15.44 15.75 16.90 16.82 17.84 18.41 16.14 17.25 13.84 15.35
F
7
rc
(meV) 7
(meV)
(meV) 7
74 92 125 128 185 74 106 42 26 92 215 74 198 50 40 83 78 184 49 179 33 29 66 47 135 69
28 84 92 102 119 61 73 39 28 108 156 70 127 48 39 93 77 128 52 106 41 31 81 49 146 80
66 + 8 173 + 4 155 + 9 128 + 9 162 + 7 96 + 9 130 20 72 + 6 64 + 4 137 + 8 193 11 113 15 186 15 104 24 99 4 163 + 7 148 10 151 5 89.5 3.5 169 39 77 + 5 56 + 3 143 + 6 141 7 157 + 9 102 + 8
Co
O 96
Zr 93 Nb 92 Mo 94 Mo 95 Mo 96 Mo 97 Mo 98 Mo 100 Mo 99Tc 99
Ru 100 Ru 101 Ru 102 Ru 104 Ru 103 Rh 104p d io5pd 106p d 107p d 108JM
nopd 107 A 109 A 106
g g
Cd
108 C d
0+
9/2+ 0+ 0+
5/2+ 0+
5/2+ 0+ 0+
9/2+ 5/2+ 0+
5/2+ 0+ 0+
1/20+
5/2+ 0+
5/2+ 0+ 0+ 1/2
1/20+ 0+
5575 7228 8070 7369 9154 6821 8643 5925 5398 6764 9673 6802 9219 6332 5910 6999 7094 9561 6536 9228 6154 5726 7271 6809 7924 7327
3.69 5.05 3.08 3.55 2.48 3.73 2.34 3.81 3.86 6.05 2.46 3.83 2.81 3.94 3.97 6.15 3.85 2.53 3.91 2.63 3.91 3.86 6.19 6.39 3.75 3.85
s
1 Co""
s
Table 2.3, continuation. Nuclide Spin (Target) (Target) (keV) 110
Cd Cd 112 Cd 113 Cd 114 Cd 118 Cd
0+ 1/2+ 0+
ii3In ii5In
9/2+ 9/2+
m
112
Sn Sn 117 Sn 118 Sn 119 Sn 120 Sn 122 Sn 124 Sn 121 Sb 123 Sb 122 Te 123 Te 116
i24Te
1/2+ 0+ 0+ 0+ 0+ 1/2+
0+ 1/2+
0+ 0+ 0+ 5/2+ 7/2+
0+ 1/2+
0+
125
1/2+
128
0+ 0+ 0+
Te Te 128 Te 130 T e 127j
5/2+
6976 9394 6540 9043 6141 5777 7274 6785 7743 6943 9327 6484 9108 6170 5946 5733 6807 6468 6929 9424 6569 9113 6287 6082 5929 6826
Do (eV)
(MeV- 1 )
(MeV- 1 )
240 14 26.9 2.1 178 6 24.8 0.6 247 24 645 60 10.7 0.8 9.04 0.41 158 47 465 18 61 + 7 870 120 90 20 1134 39 3090 90 4893 210 10.0 1.5 24.0 1.7 190 10 25.1 2.8 220 30 42.7 8 730 66 1700 240 3390 540 9.7 + 0.8
15.78 13.69 17.60 14.42 18.02 17.21 16.05 17.44 14.49 14.42 12.50 14.36 12.25 14.40 12.73 12.01 17.65 16.46 16.08 13.70 16.49 13.13 14.75 13.39 11.90 17.53
16.39 16.42 17.50 16.97 17.61 17.55 16.46 17.00 14.82 17.55 15.65 16.04 15.44 15.52 14.42 12.68 16.54 15.60 17.07 16.32 16.29 15.34 14.96 13.05 10.56 15.85
^global
F
3.92 2.63 3.91 2.41 3.84 3.76 6.11 6.22 4.01 4.03 2.66 3.93 2.57 3.80 3.61 3.31 6.12 5.81 4.20 2.53 4.01 2.69 3.76 3.43 2.99 5.95
F
7
7
(meV)
(meV)
(meV)
67 173 50 139 38 37 99 66 100 68 186 58 201 34 65 70 59 55 80 198 65 226 75 88 103 84
59 103 52 88 40 35 89 69 90 40 96 40 101 26 44 57 68 60 66 118 65 142 69 91 114 111
71 + 6 106 6 77 + 5 98 + 10 54 + 4 50 + 5 75 + 5 78 + 5 53 + 3 117+ 20 45 + 7 100 16 36 + 3
90 + 8 9 7 + 13 74 + 4 121 10 59 10 120 11 53 15 35 + 5
So Co
S s o
'-a
3 CO
110
10
co' Co
Table 2.3, continuation. Spin Nuclide (Target) (Target) (keV) 129j 129
Xe Xe 131 Xe 132 Xe
130
133 C g 134
Cs 130 Ba 134 Ba 135 Ba 136 Ba 137 Ba 138 Ba 139 La 140 Ce 141pr 142
Nd
143 N d 144
Nd 145 Nd 146 Nd 148 Nd 150 N d 147p m 144
Sm 147 Sm
7/2+ 1/2+
0+ 3/2+
0+ 7/2+
4+ 0+ 0+ 3/2+
0+ 3/2+
0+
7/2+ 0+
5/2+ 0+
7/20+
7/20+ 0+ 0+
7/2+ 0+ 7/2-
6463 9256 6605 8935 6440 6891 8828 7494 6973 9108 6906 8611 4723 5161 5428 5843 6123 7817 5755 7565 5292 5039 5335 5895 6757 8142
Do (eV)
(MeV- 1 )
(MeV- 1 )
19.0 1.4 30.7 2.8 174 28 42.0 6.4 630 230 20.4 0.9 32 11 54.8+5.5 360+48 40+30 1326+87 290+40 19490+2000 208+10 3730+470 112+7 1125+100 37.6+2.1 450+50 17.8+0.7 235+29 179+14 169+11 3.6+0.5 770+45 5.7+0.5
16.67 13.42 16.56 12.63 14.29 15.62 12.10 17.00 14.52 12.15 12.16 9.95 11.67 15.48 13.58 15.48 14.48 14.74 17.59 16.97 20.95 22.96 21.78 22.20 15.10 17.63
15.53 15.48 15.01 14.02 13.06 14.50 13.64 17.49 13.25 14.12 12.85 11.96 12.43 13.99 14.43 15.63 15.68 16.82 17.80 18.65 19.83 21.41 21.31 20.41 17.85 19.40
^global
F
5.55 2.63 3.84 2.68 3.50 5.55 4.27 4.33 3.85 2.86 3.49 2.41 3.05 5.17 3.44 5.57 3.72 2.77 3.90 2.63 3.98 4.04 4.13 6.77 3.16 2.77
7
r
7
(meV)
(meV)
85 234 68 253 92 92 295 127 105 290 130 340 49 39 54 59 67 112 46 81 26 15 24 39 106 77
103 152 89 182 114 109 200 115 132 183 107 194 39 50 44 55 52 75 42 60 30 18 26 49 63 57
rc (meV) 7 87 + 6 122 9 113
8
120 3 169 6 100 20 79 + 5 121 9 82 + 5 77 + 6 55 20 50 + 2 43 11 83.5 2.5 67 + 8 73 + 4 51+4 58 + 5 54 + 6 39 + 5 67 25 68 + 6 74 + 5 73 + 2
Co O
S a
Table 2.3, continuation. Nuclide Spin (Target) (Target) (keV) 148
Sm
0+
i49Sm
7/2-
i50Sm
0+
151
Sm 152 Sm 154 Sm 151 Eu 153 E u 154
Eu Gd 154 Gd 155 Gd 158 Gd 157 Gd 158 Gd 160 Gd 152
5/2-
0+ 0+ 5/2+ 5/2+ 3-
0+ 0+ 3/2-
0+ 3/2-
0+ 0+
i59Tb 160 Dy
3/2+
1 6 1 T~)-y-
5/2+
162 Y)-\r
0+
163 D y
5/2-
164
Dy
0+
0+
165Ho
7/2-
162Er 164 E -.
0+ 0+ 0+
166E~.
5871 7987 5597 8258 5868 5807 6307 6442 8151 6243 6435 8536 6360 7937 5943 5634 6475 6454 8197 6271 7658 5716 6243 6903 6650 6436
Do (eV)
(MeV- 1 )
(MeV- 1 )
65 11 2.2 0.2 55 + 9 1.27 0.09 45.9 2.8 131 14 0.71 0.07 1.14 0.08 0.92 0.17 16.1 1.2 13.8 0.8 1.8 0.2 30.5 1.7 4.47 0.33 87 + 3 189 19 3.82 0.24 27.3 1.7 2.1 0.15 62.9 3.6 7.28 0.37 144.1 9.6 4.20 0.12 6.9 1.2 20 + 3 38.0 1.8
17.63 19.98 23.57 20.49 22.81 20.57 24.90 23.36 18.75 23.34 22.72 18.65 21.40 18.23 20.52 19.84 21.71 21.23 18.46 19.97 17.71 20.00 20.51 21.97 20.79 20.02
19.53 21.13 22.33 21.74 21.98 21.03 22.58 21.96 21.52 22.52 22.25 21.62 21.45 21.14 21.14 20.49 21.02 21.45 21.04 20.68 20.51 20.37 20.29 22.09 21.54 20.38
^global
F
2.75 2.70 4.23 2.65 4.28 4.17 7.33 7.16 7.09 4.86 4.71 3.32 4.74 3.39 4.57 4.42 6.87 4.83 3.54 4.67 3.13 4.49 6.38 5.30 5.35 5.16
F
7
7
(meV)
(meV)
(meV)
43 67 24 61 29 39 44 61 124 33 42 114 50 108 50 47 76 55 115 53 99 50 80 71 77 79
31 56 27 53 33 37 59 73 86 36 45 79 51 73 47 43 79 55 81 49 66 49 85 71 71 77
44 + 4 62 + 2 60 + 5 92 + 4 60 + 5 79 + 9 91 + 9 93 + 3 94 + 2 55 + 3 74 + 3 110 3 88 12 99 + 6 90 + 6 111 15 97 + 7 106 6 107 3 112 20 105 8 114 11 84 + 5
?®
So Co
S
s
1 '-a
3 CO
89 + 4
co' Co
Table 2.3, continuation. Nuclide Spin sn (Target) (Target) (keV) 167 Er
7/2+
i68Er
0+ 0+
170 Er 169Tm
1/2+
169 Y b
3/2
170
0+
Yb 171 Yb
1/2-
172 Y b
0+
173
Yb Yb 176 Yb 175 Lu 178 Lu 174
174 R f 176 H f
5/20+ 0+
7/2+ 70+ 0+ 0+ 0+
177 H f 178 H f 179 R f
9/2+
180 H f
0+
180
Ta 181 Ta 182 Ta
9-
7/2+ 3-
iso w 182 W 183 W
1/2-
184 W
0+
0+ 0+
7771 6003 5682 6592 8470 6614 8020 6367 7464 5822 5566 6289 7073 6709 6383 7626 6099 7388 5695 7577 6063 6934 6681 6191 7412 5734
Do (eV) 4.0 94 125 7.28 1.47 34.9 6.08 75.7 8.06 170 240 3.45 1.61 21 29 2.4 44.1 4.09 94 1.02 4.17 2.9 22.5 63.4 12 81
0.2 +8 25 0.43 0.26 5.6 0.41 4.9 0.36 12 18 0.15 0.16 +5 +3 0.3 2.9 0.3 11 0.08 0.04 1.6 6.5 2.7 +1 +5
aexp (MeV- 1 )
(MeV- 1 )
17.78 19.47 19.85 20.95 18.20 19.80 18.17 18.90 17.35 18.87 18.99 20.94 20.46 20.52 20.80 18.79 20.80 19.07 20.53 20.67 21.26 20.39 20.48 19.81 18.29 20.80
20.66 20.40 20.23 22.47 21.33 20.24 20.89 20.39 20.53 20.28 20.43 20.61 21.18 21.60 21.28 21.44 21.11 21.12 20.85 20.67 20.70 21.47 21.14 20.71 20.96 20.88
aM
3.92 5.00 4.90 6.70 4.03 5.13 3.79 4.96 3.81 4.81 4.77 6.66 5.68 5.29 5.17 4.22 5.07 3.63 4.93 5.24 6.62 5.34 5.07 4.88 3.72 4.79
(meV)
(meV)
(meV)
135 68 56 85 176 56 124 80 123 63 54 73 91 71 69 107 62 94 52 106 79 86 80 72 100 48
90 61 54 71 115 54 86 66 77 52 44 78 84 62 66 75 60 71 51 108 83 78 75 62 78 49
88 + 2 86 + 6
Co
o a
s
re ^+
86 + 7
re to
g Co
76 + 4 74 + 5
77 + 5 6 3 + 14 45 65 50 62 50 51 61 67
+5 +5 +4 +6 +5 +1 +2 +2
51+4 73 + 6 52 + 4
c
Table 2.3, continuation. Nuclide Spin Sn (Target) (Target) (keV) 186 W 185
Re 187 Re
186 O 187
s
Os
188 Qg 189
Os 191j r 192j r 193 Ir 192pt 194pt 195pt 196pt 197 Au 198Hg 199 H g 200 H g 201Hg 202Hg 204 H g 203
Tl 205 Tl 204p b 206p b 207p b
0+ 5/2+ 5/2+
0+ 1/2-
0+ 3/23/2+ 4+ 3/2+
0+ 0+ 1/2-
0+ 3/2+
0+ 0+ 0+ 3/2-
0+ 0+ 1/2+ 1/2+
0+ 0+ 1/2
5467 6179 5872 6290 7990 5920 7792 6198 7772 6067 6256 6105 7922 5846 6512 6664 8028 6230 7754 5994 5668 6656 6503 6732 6738 7368
Do (eV)
(MeV- )
(MeV- )
93 + 7 2.87 0.14 3.69 0.15 24.90 0.4 4.56 0.20 40 + 2 3.2 0.2 1.68 0.13 0.64 0.30 3.98 0.36 31.4 1.3 82.0 9.4 19.2 1.6 214 38 15.69 0.68 69 + 3 100 30 519 69 233 20
21.46 22.04 22.53 20.96 18.84 21.20 19.01 23.59 20.54 22.00 20.71 18.92 15.81 17.29 17.53 17.12 12.50 14.04 10.61
20.83 20.68 20.55 19.57 20.55 19.74 19.90 19.59 19.82 18.10 19.13 17.36 16.55 16.26 14.77 13.57 12.81 11.49 10.82 8.84 6.40 8.89 6.11 8.28 5.02 4.52
584 3000 2129 14340 12000
54 240 96 210 900
8-exp
^global 1
11.16 8.31 9.46 5.98 4.15
F
1
4.73 6.66 6.60 4.60 2.95 4.52 2.90 6.52 5.13 6.15 4.67 4.35 3.07 3.94 5.35 3.65 2.46 3.18 2.21 2.63 2.25 3.74 2.85 2.67 1.75 0.90
7
rb
(meV)
(meV) 7
38 69 56 56 102 45 84 65 132 72 68 82 204 96 150 126 400 172 600
42 83 73 68 90 55 74 104 141 120 83 102 173 111 233 232 355 292 435 486 1660 526 1380 492 2070 3770
301 610 432 1290 5070
F
7
(meV) 51 56 57 60 76 82 87 80
+ + + + + + + +
5 3 3 4 4 4 4 8
93 + 8 60 1 72 13 122 20 8 7 + 15 128 + 6 120 11 289 17 333 27 386 92 Co
1980 630 1100 734 2140
360 70 200 65 11
s
s>
o CO
1 CO CO
Table 2.3, continuation. Nuclide Spin Sn (Target) (Target) (keV) Ra 228 Th 229Th 230Th 232
Th 231 Pa 232 Pa 233p a 232
U 233 U 234 u 235
U U 237 U 238 U 238U 236
238
U 238U
238
U
237Np
238p u 239p u 240p u 241p u 242p u
9/20+ 0+ 5/2+ 0+ 0+ 3/22-
3/20+ 5/2+ 0+ 7/20+ 1/2+ 0+ 0+ 0+ 0+ 0+ 5/2+ 0+ 1/2+ 0+ 5/2+ 0+
4604 4561 5249 6794 5118 4786 5549 6529 5220 5762 6845 5298 6546 5126 6153 4806 4806 4806 4806 4806 5488 5646 6534 5242 6310 5034
pa
a-exp
(eV)
1
1
(MeV- )
(MeV- ) 7.34 27.83 27.15 27.15 27.27 27.70 26.86 26.10 27.10
3730 30.3
300 5.4
10.40 30.30
0.53 8.7 15.82 0.51 0.47 0.59 4.6 0.45 10.92 0.49 14.74 3.5 20.26 20.26 20.26 20.26 20.26 0.52 9.0 2.07 12.8 0.83 17.0
0.15 1.4 0.55 0.03 0.04 0.09 0.7 0.05 0.47 0.02 0.44 0.8 0.72 0.72 0.72 0.72 0.72 0.04 0.7 0.07 0.6 0.08 1.0
25.33 29.93 30.31 29.93 26.61 31.31 27.81 25.20 27.96 25.88 28.05 29.12 29.12 29.12 29.12 29.12 29.12 29.13 26.40 24.53 27.52 25.63 27.94
26.30 26.36 26.63 26.56 27.08 27.14 27.14 27.14 27.14 27.14 25.96 27.14 25.89 25.87 26.40 26.34
7 (meV)
3.12 4.74 3.78 3.78 5.14 5.05 7.68 7.69 5.41 3.95 5.28 4.03 5.20 5.45 5.19 5.19 5.19 5.19 5.19 7.46 5.41 4.23 5.36 4.05 5.34
159 14 61 23 19 35 54 28 39 59 29 50 29 37 23.5 7.4e 10.0/ 13.0s 17.3'1 43 43 64 32 40 28
Tb 7 (meV) 421 17 31 50 30 24 48 52 41 47 53 35 46 34 46 28 9.2e 12.5^ 16.3 s 21.8^ 60 49 56 39 58 33
pc 7 (meV)
43 8 26 2 36 2 43 4 26 2 24.7 0.7 39.7 4 47
2
34
40 25 38.1 23.4
5 1 1.7 0.8
23.4
0.3
40.7 34 43 31 36 22
0.5 3 4 2 6 1
Co
onant
209 B i 226
Do
Co""
Table 2.3, continuation. Nuclide Spin s« (Target) (Target) (keV) 244p u 241
Am 242 Am 243 Am 242 Cm 243 Cm 244 Cm 24B Cm 246 Cm 247 Cm 248 Cm 249 Bk 248 Qf 252 C f
0+ 5/255/2-
0+
5/2+ 0+
7/2+ 0+
9/20+
7/2+ 9/20+
4771 5538 6364 5367 5693 6801 5520 6458 5156 6213 4713 4970 6625 4804
Do (eV)
(MeV" 1 )
(MeV- 1 )
21 + 4 0.55 0.05 0.29 0.02 0.60 0.06 15.4 1.9 1.11 0.07 12.1 0.9 1.21 0.05 34.0 5.0 1.79 0.15 33.6 3.2 1.22 0.09 0.72 0.03 27 + 3
28.86 28.60 26.64 29.32 25.04 23.22 26.41 23.96 25.66 24.20 28.16 28.98 24.59 27.63
26.50 25.23 26.00 25.72 24.71 25.12 24.86 25.43 25.38 25.96 26.29 25.68 24.45 24.72
&exp
°: calculated on basis of <xexp. b : calculated on basis of aglobalc : measurement.
(meV)
22 40 56 35 50 77 39 65 37 57 23 31 47 34
29 58 61 52 52 63 46 56 38 48 28 44 54 48
7
pc 7 (meV) 20 + 2 45 + 2 39 + 1 34 + 6 37 + 2 28.1
1.8
25.5 1.4 36 + 2
^ S
,
. e : f: 3 : h :
7.32 5.96 7.38 5.10 3.89 5.07 4.02 5.01 3.97 4.96 7.23 3.86 4.62
pa 7 (meV)
re
energy in keV. isomeric-shape isomeric-shape isomeric-shape isomeric-shape
capture capture capture capture
width width width width
F7/j F7// F7JJ F7//
for for for for
/?2—0.24. /32=0.36. ^2=0.50. /32=0.70.
§ a g
a §
Ȥ
Chapter 3
Individual Resonance Parameters Level widths are generally determined from a shape or area analysis of the reaction cross section, with use of certain formalisms, described in Refs. [l]-[6] and [196,197]. For fissile nuclei, the various resonance reaction formalisms were described by de Saussure [198]. The general considerations that were taken into account in the recommendation of the widths of the resonances are the following: (a) energy resolution, (b) sample thickness, (c) type of measurement, (d) isotopic enrichment, and (e) type of analysis. In addition, charged particle reaction data such as (d,p) and inverse reactions such as (7,n) and (a,n) reactions were considered. In the recommendations of spins and parities, clear-cut decisions have been made, depending on the merit of the methods. The different techniques of spin and parity determinations have been reviewed and assessments have been made.
3.1 Determination of Spins of Neutron Resonances A knowledge of resonance spins is required for studies of (a) the distribution of level spacings or of level densities, (b) spin dependence of the strength function, average radiation widths, fission widths, and average reduced neutron widths, (c) Porter-Thomas distribution of partial radiation widths, (d) valence neutron capture predictions, (e) channel theory of fission. Because of these and other considerations we outline briefly the various methods for determining the spins of neutron resonances. 3.1.1
Total Cross Section Measurements
The peak cross sections of a resonance is expressed by
(3.1) 71
72
3. Individual Resonance Parameters
for resonances with F n 3> F 7 , <7o = 4TTA2<7. Hence, an experimental determination of the peak cross sectionCTOand the resonance energy EQ establishes the spin of the resonance. This method is used extensively in the mass region A < 60. In the heavy mass region and for non-fissile nuclei where the condition Yn 3> F 7 is no longer valid, one could carry out a shape analysis of good resolution transmission data to extract gTn and F values. With the aid of the relationship
F7 = F - ^
>0
(3.2)
one could determine the resonance spin. 3.1.2
Scattering Measurements
In a scattering measurement, the scattering area, As, is expressed by
^
(3.3)
When this information is combined with gTn and F, it yields the spin of the resonance. 3.1.3
Capture Measurements
Similarly, in a total capture measurement, the capture area, A1 is
(3.4) With the aid of available gTn and F 7 values, the spin of a resonance can be deduced. This method could be applied to the determination of the spins of p-wave resonances of 5 2 Cr by plotting the gTnT7/T values of resonances versus neutron energy. Assuming that F n 3> F 7 and that F 7 is approximately constant one expects the experimental points to cluster roughly around two lines,
Primary and Low-Energy 7-Rays Detected by Ge-Li Detectors
In s-wave neutron capture 7-ray measurements, the observation of reasonably strong primary 7-rays populating low-lying states with spins I + 3/2 or I - 3/2 establishes the spin of the capturing state as I + 1/2 or I - 1/2, respectively. Underlying this conclusion is the assumption of primary dipole transitions. Because of the PorterThomas distribution of partial radiative widths or the unavailability of low-lying
3.1. Determination of Spins of Neutron Resonances
73
states with these spins, the selected high-energy 7-rays may be either weak or absent in the spectrum of a resonance. One could then alternatively rely on measurements of low-lying 7-rays. As shown by Huizenga and Vandenbosch [200], using a simple statistical model, the 7-ray population of a low-lying state is sensitive to the spin of the capturing state. By considering two suitable low-energy 7-rays and forming intensity ratios for the various resonances, one finds that these ratios separate into two distinct groups. In 1953, Sailor [201] observed a large difference in the isomeric ratios, which are related to the population ratios, of the first two levels of 115In. Subsequently Draper et al. [202] studied the low-energy 7-rays of 115In(n,7)116In and showed that the various intensity ratios form two groups. Later, Wetzel and Thomas [203], using Ge-Li detectors, established that the intensity ratio is related to the spin of the capturing state. The BNL fast chopper group [204] has extended this technique to a variety of nuclei in the rare-earth region. The combination of primary and low-energy 7-rays proved to be a powerful tool for the determination of spins of neutron resonances. Corvi et al. [205] successfully made use of this technique to determine the spins of 235U resonances below 41 eV. 3.1.5
The 2-Step Cascade Method
Bollinger et al. [206] introduced a novel approach for the determination of neutron resonance spins, known as the 2-step cascade method. The intensity of 7-rays for such a cascade populating the ground state is measured by recording the spectrum of the pulse heights for coincident counts in two Nal crystals. Under certain simplifying assumptions (such as a constant F 7 and a (2J + 1) dependence of level density) it can be easily shown that the intensity for such a cascade is proportional to (2J+1)" 1 . As a result, it would be expected that the 2-step cascade intensities for the resonances would separate cleanly into two groups. This method was applied by Bollinger et al. [206] to even-odd target nuclei m ' 1 1 3 Cd, 135Ba, 105Pd and m Hf, for which the intensities behaved according to expectation. Since then, spin assignments of resonances of these target nuclei have been made with the aid of other techniques. To assess the experimental validity of the 2-step cascade method, spin assignments derived by this and other methods have been compared by us and the excellent agreement with other techniques warrants the extensive use of the 2-step cascade method in spin determinations. 3.1.6
The 7-Ray Multiplicity Method
Another method, developed by Coceva et al. [207], relies on the idea that the multiplicity, or average number of steps per cascade, depends on the spin difference between the initial and final states. By using two Nal crystals viewing a sample, they measured the intensity ratio R = Is/Ic where Is is the intensity of counts measured with a single pulse-height detector and Ic is the intensity of coincidence counts measured with two Nal detectors. For s-wave capture, expectations are that R will separate into two distinct classes. The applicability of these ideas has been verified experimentally by determining the resonance spins of 105Pd, 99>101Ru, 95 97 ' Mo and m ' 179 Hf. On the other hand, nuclei such as 121-123Sb and m Y b did not exhibit the expected spin effect in the ratio Is/Ic [208].
74
3. Individual Resonance Parameters
3.1.7
Polarization Measurements
One of the direct measurements of resonance spins is the use of polarized neutrons and targets. Under these conditions, the interaction cross section can be written in the simple form a = Co + fnfNVP
(3.5)
where / „ and /jv are the neutron and nuclear polarization and (TQ and ap are the polarization-independent and polarization-dependent cross sections and are expressed by
,
g-ax
(3-6)
ap = g-{cr+ — o--) These quantities are related through the transmission effect r by the relationship
r- J ^
« -\{l + MndNNcrot)^-
(3.7)
where Tp and TA are the transmissions through the target when the neutron beam is polarized parallel or antiparallel to the applied magnetic field, respectively, <j> is the efficiency for reversing the beam polarization, t is the sample thickness, and N is the number of target nuclei per unit volume. Since one can measure the quantities f (1 + 4>)fn and fNNaot, a polarization measurement determines ap/ag- Note that for a well-isolated resonance — = -1
1 +1
for J = I - 1/2 for J = J + 1/2
Reddingius et al. [209] and Keyworth et al. [210] achieved spin determination of resonances of 235U and 237 Np.
3.1.8
Level-Level Interference in Scattering or Fission Channel
It can be shown that the resonance-resonance interference term due to scattering can be written in the form
(3.9) where
Xi =
2(E-E1)
and X2 =
2{E - Eh)
3.2. Determination of the Parity of Neutron Resonances
75
A shape analysis between strong resonances can give an indication of the spins of the resonances. Cote et al. [211] applied this method to deduce the spins of resonances of 55Mn. This method is used also in partial capture cross section measurements for spin determinations. Similarly, shape analysis of the fission cross section using multilevel formalisms can separate the resonances into two classes of spins. For example, Moore and collaborators [212] were successful in deducing the two sets of spin states of 241Pu. 3.1.9
Degree of Symmetry of Fission: Ratio of Symmetric to Asymmetric Fission
One of the indirect methods for the determination of spins of fissionable nuclei is a measurement of the degree of symmetry in the fission process. Cowan and collaborators [213] developed a method in which a sample is placed on a rotating wheel and the ratio of the fission products 115Cd and "Mo is determined radiochemically. This ratio is related to the ratio of symmetric and asymmetric fission. A comparison between the 235U spin assignments of Cowan et al. [213] and Keyworth et al. [210] below 33.5 eV shows reasonably good agreement, suggesting that there is a correlation between spin and symmetry of fission.
3.2
Determination of the Parity of Neutron Resonances
The parity of neutron resonance states is determined by the orbital angular momentum (I) of the incoming neutron and the parity of the target nucleus. Some of the methods described previously for spin determination may shed light, in some cases, on the I value of the incident neutron. For example, the observation of a strong primary 7-ray transition to the ground state of 94Nb (I = 9/2+) with spin and parity 6 + indicates that the capturing state is 5~ under the reasonable assumption that the radiation is dipole in character. Hence, the capturing state is formed by p-wave interaction of neutrons. Higher orbital angular momenta may be excluded on the basis of the strength and Wigner limit of a resonance. Similarly, in the 7-ray multiplicity method, the determination of intensity ratios different from those expected for s-wave capture may indicate interaction of p-wave neutrons with the target nucleus. However, these techniques are indirect in nature. Some of the direct methods are outlined below. 3.2.1
Interference Effects
Presence of an interference minimum on the low-energy side of a low energy resonance gives evidence that the resonance is formed by s-wave neutrons. The resonance strength must be reasonable to allow experimental observation of the dip. As an illustration, the 45-eV resonance of 95Mo exhibits the interference minimum to allow a definite £=0 assignment for it. 3.2.2
Angular Distributions of Elastically Scattered Neutrons
The differential cross section can be written in the form
76
3. Individual Resonance Parameters
da(a's';as)
X2 °° = —*— V BL(a's'; as)PL(cos0)dn 2s + l
(3.10)
L=O
where the channel index a describes the type of incoming particle and state of the target nucleus, and s defines the channel spin. PL(cos9) are the usual Legendre polynomials, and BL(a's';as) coefficients are related to the Z coefficients and collision matrix. For details, the reader is referred to the work of Biedenharn et al. [214]. For elastic scattering of neutrons, a = a'. In general, for an unpolarized neutron beam on an unpolarized target, summation of s and s' must be carried out for the entrance and exit channels. However, for zero-spin nuclei the channel spin is unique, and for the case of an incoming neutron, s = s' = 1/2. From the shape and interference patterns of the differential cross sections, the £ value of the incoming neutron can be inferred. Asami et al. [215], applied this method to determine the parity of the 1.15 keV resonance of 56 Fe. Also, Kirouac and Nebe [216] exploited the same technique for the assignments of spins and parities of 4 0 Ca resonances. 3.2.3
Angular Distribution of Capture 7-Rays
It is possible to utilize high-energy capture 7-rays by measuring their angular distributions for the purpose of investigating the parity of the initial, as well as final, states. The theory of angular correlation for the case of a particle in and a 7-ray out gives cos6)
(3.11)
which would enable one to calculate the angular distributions of 7-rays due to neutron capture. For the definitions of the symbols, the reader is referred to Sharp et a l . [217]. Applied to the case of p-wave capture and target nuclei with 1=0, one obtains the angular distribution patterns shown in Table 3.1. Also shown in the last column is the 7-ray intensity ratio at angles of 90° and 135°, specified by the neutron beam and 7-ray directions. By observing an anisotropy in the 7-ray capture spectra recorded at two angles, one can readily deduce the spins and parities of the initial and final states. It should be noted, however, that the observation of a symmetrical angular distribution does not permit a distinction between S1/2 and pi/2 resonances. Using Nal crystals, McNeil et al. [218] applied these methods to investigate the 7-rays of the 62-eV resonance of 124 Sn. Subsequently, Slaughter et al. [219] reported similar measurements for the 45.8-eV resonance of 118 Sn and the 62.0-eV resonance of 124 Sn. With the advent of high-resolution GeLi detectors, Bhat et al. [220], Mughabghab et al. [51, 221], and Chrien et al. [222, 223] exploited these methods for the determination of low energy p-wave resonances in 116,118,120,122^ 98 M O ; i o 8 p d a n d 56 Fe a s w e l l as spin assignments of low-lying compound states. Angular distributions can also determine channel spin admixtures for £ = 0 resonances in target nuclei where I > 0 as demonstrated by Chrien et al. [222] for 93 Nb.
3.2. Determination of the Parity of Neutron Resonances
77
Table 3.1: Angular Distribution of Dipole 7-Rays Due to p-Wave Capture in Target Nuclei With 1=0. Transition
W(0)
14 13
3/2 -> 5/2
3.2.4
1^(90 deg) W(135deg)
3/2 -> 3/2
2^r(,7 - 3 s « n 0)
8 11
3/2 - > l / 2
ifeP + Wfl)
10 7
1/2 - > l / 2
symmetrical
1
Method of Low Energy 7-Rays
Because of Porter-Thomas fluctuations of partial radiative widths or unavailability, in certain mass regions, of low-lying final spins with J/ = / 3/2, a lack of observation of primary high energy 7-rays feeding such states does not provide any information on the spin and parity of the capturing state. For such cases, low energy 7-rays can be fruitfully utilized for the determination of the parities, as well as spins, of resonances. This technique was successfully applied [153, 224, 225] for assigning p 3 / 2 resonances in 98Mo at 12, 402, 818, 1122, 1920, 2615, 3264 and 3794 eV and in 120Sn at 365, 920, 1287, 1424, 3853 and 4332 eV. However, other methods, discussed previously, have to be used to assign P1/2 resonances. More recently Zanini et al. [226] succeeded in determining the spin as well as the parity of neutron resonances of i°7>i°9Ag. Other useful information can be derived from such investigations. For example, the spin of the final state of 121Sn at 869 keV was found to be 5/2 and consequently an E2 transition feeding this state from the 950 eV s-wave resonance was observed [225].
3.2.5
Angular Distribution of Photoneutrons
Jackson and Strait [199] and Berman et al. [227] made use of the threshold photonuclear reaction (introduced by Bertozzi et al. [228]) to study the properties of the neutron resonance states. By measuring the angular distributions of the ejected photo-neutrons and by assuming dipole transitions, information on the spin and parity of compound resonance states can be deduced. Spins and parities for the compound nuclei 51Cr, 61Ni and 57Fe have been reported by Jackson and Strait [199].
78
3. Individual Resonance Parameters
3.2.6
The Wigner Limit
In the atomic mass region of light nuclei, one could apply the Wigner limit Tw to place an upper limit on the £-value of the incoming neutron. tr
rw = —skVt (3.12) mR where Vi is the probability of a neutron passing through a centrifugal barrier. Values of Ve for t < 3 are derived for a square potential well. These are included in Table 2.1. The experimentally determined width of a resonance must not exceed the Wigner limit corresponding to a particular ^-value. 3.2.7
Statistical Considerations
The application of statistical arguments such as the Porter-Thomas distribution of reduced neutron widths, Dyson and Mehta's A3 and F statistics, and Bayes' theorem may give some indication of the p-wave assignment of resonances. The principal use of these methods is the ability to estimate average quantities such as p-wave strength functions and p-wave mean level spacings. The first investigators to apply Bayes' conditional probability for the determination of parities of 238 U resonances were Bollinger and Thomas [229]. Subsequently, Perkins and Gyullassy [230] and Oh et al. [231] extensively applied this procedure in the evaluation of resonance parameters. The probability that a resonance with a given gYn value is a p-wave resonance is expressed by
P(p-wave) = |1 + ^ X |
(3.13)
where
X=
So [9.863 x
and values of A M are respectively, 1, 2/3, and 1/2 for target nuclei with spins 1=0, 1/2 and > 1 respectively. Mizumoto et al. [232] applied Bayes' theorem to estimate d-wave contribution in target nucleus 206 Pb by using
3.3. Scattering Widths: Relationship Between Sd
79
1/2
P(d-wave) =
(3.14)
5 {SiPj
where
X=
Pi Si
where Pi, P2 are the p-wave and d- wave penetrabilities, Si and 52 are the p-wave and d-wave strength functions, and (Di) is the average p-wave level spacing. It is emphasized that these probabilistic methods do not give unique £ assignments since there is an overlap between the weak s-wave and strong p-wave resonances. Such a problem was clearly pointed out by Garg et al. [233] in the case of 64Zn resonances. For example, application of Bayes' theorem to the 39.44 keV resonance yields a probability of 99.9 % that it is an s-wave resonance, which is at variance with the experimental observation that it is a p-wave resonance based on its symmetrical shape. This illustrates the great danger of assigning £ values on the basis of Bayes' theorem.
3.3
Scattering Widths: Relationship Between Sdp and F
Scattering widths of positive energy neutron resonances above the neutron separation energy are determined by the usual conventional techniques of applying shape or area analysis in transmission, scattering, and capture measurements. As an example, for the case of transmission measurement carried out on a thin sample, the area of a resonance is proportional to gTn while for a thick sample measurement it is proportional to gTnT. Combination of those two types of measurements can yield values of Tn, F 7 and J. However, for negative energy resonances below the neutron separation energy, it is difficult to determine unambiguously the position, the spin, and the reduced neutron width of a resonance from a shape analysis. This difficulty can be circumvented for light and medium weight isotopes by resorting to information derived from charged particle, particularly (d,p), data. Specifically, there exist a relationship between the s-wave reduced neutron width and the (d,p) spectroscopic factor, Sdp- Since extensive use is made of this relationship, we present it here and determine the normalizing factor. In particular, the single particle dimensionless bound reduced neutron width for s-wave resonances is determined by a comparison of the experimental (d,p) and (n,n) data. The (d,p) spectroscopic factor is defined as the square of a ratio of dimensionless reduced neutron widths,
(3.15)
80
3. Individual Resonance Parameters
where 02N is expressed by:
in
Q2
(3.16)
and the reduced neutron width 7^ for s-wave neutrons is defined as
ln
2=
rn
<"7>
2kR
Substituting Eqs. 3.16-3.17 into Eq. 3.15, and using an interaction radius R=1.35xA 1 / 3 fm, one obtains
^f
(3.18)
where F° is expressed in eV. It is to be noted that 62p is model dependent. For example the use of harmonic oscillator wave functions [234] gives 02 = 0.036. Instead of relying upon model calculations, we apply Eq. 3.15 to the C + n system to derive 62p. Meadows and Whalen [235] carried out a precise measurement of the total cross section of natural carbon (98.89% 12C) in the energy region 100 - 1500 keV. Within the framework of single level R-matrix analysis, the authors obtained an excellent fit of the total cross section throughout the whole energy region. The derived reduced neutron width of the bound level at -2020 keV is 7^ = 540 keV for an interaction radius of 4.80 fm. This bound level was studied by Darden et al. [236] by (d,p) stripping reaction, for which a spectroscopic factor Sdp = 1.1 was obtained. Substituting these values in Eq. 3.15, one derives
9ip = 0.175 and hence Sdp = 4.21 x l O " 4 ^ / 3 ^
(3.19)
This relation is used extensively for light nuclei in converting the (d,p) spectroscopic factors to estimated reduced neutron widths.
3.4
S- and P- Wave Radiative Widths
The investigation of radiative widths of individual resonances can yield interesting information on the neutron reaction mechanism. As was pointed out previously, radiative widths of light nuclei as well as those in the 2p-, 3p- and 3s- giant resonances may fluctuate with resonance energy. This phenomenon can be attributed
3.4- S- and P- Wave Radiative Widths
81
to either direct processes (valence or channel capture) or limited number of exit channels in the distribution function (Eqs. refpx, 2.43). In the channel or valence capture model, the incoming s-(or p-)wave neutron is pictured as being captured into a bound orbit which is characterized by a large single particle p (s or d) orbit without the formation of a complicated compound nuclear state. In this process, enhanced electric dipole radiation is emitted. The channel contribution to the s-wave radiative width from the external region of the nucleus is given by [32]
where k7 is the photon wave number, ©?(= 7n/^2/m-^-2) a n d ® / ( = Sdp) are dimensionless reduced neutron widths of the initial and final states respectively, Yf = KfR, and the reduced matrix element for the operator Y^ can be written as [237]
(2J> +
1 ) W i + m j f
o)
(3.21)
The subscripts i and / refer to the initial and final states respectively, W is the Wigner 6-J symbol, and Cm is the Clebsch-Gordon coefficient. In Table 3.2, the angular momentum coupling coefficients for the transitions s—>p, p—^s, and p—^d are presented for nuclei with zero target spin. In Eq. 3.20, the dipole radial integral for slow s-wave neutrons was calculated by integration in the external region (R to oo) and by considering the outgoing wave function for a p-orbital in a square well potential to have the form (l + l/kfr)e~kfr.
(Radial Integral) z = 4 (j^r)
(3-22)
yi
It is important to emphasize that the core of the nucleus is assumed to be inert, i.e., core transitions are neglected in the derivation of Eq. 3.20. Of particular interest is the 7-ray energy dependence of the direct capture component of the partial radiative width as illustrated in Eq. 3.20 which demonstrates an E^ instead of an E^ dependence. On the other hand, the direct capture cross section at thermal energy Eq. 1.29, exhibits an E1 energy dependence. By considering the motion of a single neutron in a potential well and by adding the contribution from the internal region of the nucleus, Lynn [4] represented the valence component of the partial El radiative width by -*
f-i
Tjif (valence) = —
i
H-\^\
/>o
;
/
Jo
X,
I
.
- » - - F - | | - v ^ / l * l l *
~r
-r
.
\
^
(3.23)
82
3. Individual Resonance Parameters
Table 3.2: Angular Momentum Spin Factors for s —> p, p —> s and p —> d Transitions for Nuclei with Zero Target Spin. Transition Type
f(AM)2 I
Sl/2
""> Pl/2
3
Sl/2
""> P3/2
2 3
Sl/2
1 3
Pl/2 " > d 3 / 2
2 3
P3/2 ""> Sl/2
1 3
P3/2 ~ > d 3 / 2
1 15
P3/2 " > d 5 / 2
3 5
Pl/2 "">
The radial overlap integrals are evaluated by Lynn [4] by adopting the wave functions of a Saxon-Woods potential with a spin orbit coupling term which is normalized throughout space. The values of these integrals are tabulated by Lynn [4]. It is of interest to compare these radial integrals with those described in Eq. 3.22 for the case of an s—>p transition A = 70. For the transition 3si/2 —>-2p3/2 Lynn presented a value I2 (barn) = 0.088. This is to be contrasted with the larger value of 0.16 barn calculated according to Eq. 3.22 for E 7 = 7.7 MeV and R = 1.35XA1/3 fm. As will be discussed shortly, the source of the discrepancy is due in large measure to the normalization procedure of the wave functions. Applying the optical model for the study of valence capture and working in the framework of K-matrix formalism, Lane and Mughabghab [238] derived a general relation for the valence component of the El partial radiative widths if (valence) = Fj n
Im
Im tan£(opt)
(3.24)
where D is the dipole operator, UE(opt) and 6(opt) are the optical model initial state wave function and the phase shift respectively. It is interesting to note that the valence component is expressed in terms of a ratio of optical model quantities. In the vicinity of a giant resonance, Eq. 3.24 is simplified to the following form r 7 i / (valence) = - ^ |< uf\D\u0 >f cos2 ka
(3.25)
1 On
where Fon is the natural decay width of the giant resonance (single particle state). It is important to note here that uo(r) is normalized inside the nuclear surface
3.4- S- and P- Wave Radiative Widths
83
\u0\2r2dr = l
(3.26)
This criterion on the normalization of the incident wave function would introduce a correction factor to the radial overlap integrals as calculated by Lynn [4]. In addition, the use of square well potential quantities for the penetrabilities in the calculation of reduced neutron widths 7^ (in the 0 | expression) results in another ambiguity in the application of Eq. 3.23. To obtain the penetrabilities of a diffuse edge well for s-wave neutrons, Vogt [208] has shown that the square well penetrabilities must be multiplied by a factor / . Empirically, Vogt [208] derived a relationship between / and the diffuseness d of the surface thickness of the potential and obtained / = 1 + 6.7d2
(3.27)
where d is expressed in fm. To illustrate the importance of the normalization procedure in the 3p and 3s giant resonances, the wave functions of the single particle initial states are considered as lightly bound by about 100 keV and are normalized in the interior region according to Eq. 3.26. For A = 120 and d = 0.69 fm, the correction factor due to the normalization of the wave function in the interior region is calculated as 4.2 which is identical in this case to the correction obtained from Eq. 3.27 due to diffuseness. Here it is assumed that Eq. 3.27 holds also for p-wave neutrons. Thus the two corrections cancel out and the optical model results of Lane and Mughabghab [238] are equivalent to the valence calculations of Lynn [4] for p-wave neutrons at low energies (i.e. cos2ka = 1). On the other hand, in the 3s region and at low neutron energies, the total correction factor is computed as 1.5 for A = 45 and 70. Thus the optical model results in the 3s region are greater than the valence results by this factor according to Eq. 3.23. These results are in full agreement with the calculations of Barrett and Teresawa [239] who applied the Lane-Mughabghab [3] optical model formula, Eq. 3.24, to the target isotopes 56Fe(3s region) and 90Zr (3p region) and carried out a detailed comparison between the optical model and the valence model at various neutron energies. In addition, Allen and Musgrove [246] applied the Lane-Mughabghab [238] optical model formula in the investigation of the systematics of valence neutron capture throughout the periodic table and tabulated in the form of figures the ratio of the imaginary parts of the optical model quantities represented by Eq. 3.24. Another approach to the study of valence capture was carried out in the framework of the shell model by Boridy, Mahaux and Cugnon [240]-[244]. The latter two authors generalized Eq. 3.24 to higher energies and in addition presented a readily calculable expression for the valence capture component. This can be written [245] in the following general form for s- and p- wave neutrons. p
o,_2
rf
IT Wf T^
O2
72
£7*/ _ » * _ _ £ _ E 3 Z kn)Q
f
(2/+1)
2;
. s , 2(AM)2 , ^ W"'')l 4 M
(3 28)
"
84
3. Individual Resonance Parameters
where 8TT2
e.2
9 (he)
= 1.64412 x 10" 6 MeV" 2 fm~2
(3.29)
where kn is the incident neutron wave number and (AM) are the angular momentum spin coupling coefficients described by Eq. 3.21 and displayed in Table 3.2. For s—^p and p—^s transitions, J2(ji,jf) values as a function of A and Era are graphically Cugnon and Mahaux presented by Cugnon [244]. In the calculation of J2(ji,jf) [243, 244] emphasized that the optical model potential parameters must be chosen so as to reproduce the experimental scattering lengths and strength functions. The valence component of the radiative width is then independent of the imaginary part of the optical model potential. On the other hand, it is of interest to recall here that, in contrast, the statistical component of the radiative width depends on the imaginary component W of the optical model potential as shown, for example, in Eqs. 2.72-2.73. The first quantitative verification of the valence neutron model was demonstrated for the case of p-wave resonances of 98 Mo (429, 612, 818 eV) and 96 Zr (302 eV) by Mughabghab et al.[152, 237]. In addition, the fluctuations of total radiative widths of strong p-wave resonances in the even-even target nuclei were interpreted quantitatively [152] in terms of the valence model. Subsequently, Toohey and Jackson [174], measured the ground state radiative widths for various p-wave resonances in 90 Zr using the inverse reaction (7,11) and found that the average valence component is about equal to the statistical component. In extensive capture measurements, above a neutron energy of 2.5 keV, the Australian-Oak Ridge collaboration (Allen, Boldeman, Macklin, Musgrove [246]) determined s- and p-wave total radiative widths of resonances for numerous isotopes. The results of the measurements were compared with optical model valence calculations. Good agreement was found particularly in the 3s region. Furthermore, Harouna and Cugnon [247] applied Eq. 3.28 in the investigation of s-wave total radiative widths of isotopes in the mass region spanning 4 0 Ca and 60 Ni. The comparison between theoretical and experimental values indicates that valence capture plays an important role in this mass region. In the 2p giant resonance, the partial radiative widths of the 85 keV p-wave resonance of 24 Mg were well accounted [248, 249] for in terms of the valence model. Beer, Spencer and Kappeler [250] measured high energy 7-ray transitions to lowlying states due to neutron capture in 56 Fe and 58>60Ni and carried out a comparison of experimental and theoretical values of partial radiative widths according to both Lynn's [4] and Cugnon-Mahaux's approaches. Subsequently, Beer [251] investigated the validity of the valence model in the mass region 40 < A < 70 and concluded that the model can account for the total radiative widths of the low-lying resonances of 50 Cr and 54 Fe. At this stage, we would like to note that the valence neutron model appears to fail to explain the 100 Mo(n,7) 101 Mo data [252]. The ground state transitions (P3/2 ~* s i/2) due to p-wave neutron capture in the p 3 / 2 resonances of 100 Mo at 1260, 1696 and 2418 eV are absent [252] although the valence neutron model predicts strong transition feeding this state. In contrast, the piy2 —> S1/2 transition
3.5. Alpha Widths of Neutron Resonances and the (n,ja) Reaction
85
of the 1069 eV resonance has a valence component comparable to the statistical component. The disappearance of valence effects in the p 3 / 2 resonances can be partially accounted for in terms of the fact that the p 3 / 2 single particle state is bound in 100Mo. This would result in a smaller value for the normalized radial integrals, due to destructive interference effects, with an accompanying decrease in the valence component [254, 238]. Additional experimental as well as theoretical investigations are required in order to shed additional light on this interesting and exciting field of neutron capture 7-ray spectroscopy.
3.5
Alpha Widths of Neutron Resonances and the ( Reaction
Measurement of the alpha decay of neutron resonances provides another possible means for studying the properties of highly excited states of nuclei. Such investigations have been carried out extensively in the resonance region covering the energy range up to a few hundred eV by Popov and collaborators [255]-[257]. The spins of individual neutron resonances can be determined in some cases by invoking the principle of parity conservation. As an illustration, the reaction 147 Sm(n,a) 144 Nd leading to the ground state of 144Nd (spin and parity 0 + ) is considered. S-wave neutron capture by 147Sm, whose ground state spin and parity are 7/2~, leads to the formation of two compound states with spins of 3~ and 4~. Because of parity conservation, the observation of alpha particles leaving the residual nucleus 144Nd in its ground state establishes that the resonance spin is 3 and the angular momentum of the alpha particle is 1=3. On the other hand, the alpha transition 4~ —> 0 + is parity forbidden. However, the non-observation of an alpha transition to the ground state does not give any positive information on the spin of a resonance due to the PT fluctuations of alpha particle widths. The distribution of the total alpha particle widths have been studied by Balabanov et al. [257] for 147Sm and 149Sm. The experimental data were compared with chi-squared distributions for each spin sequence. The number of degrees of freedom, veff, for spin 3~ and 4~ resonances of 147Sm were found to be 1.8 and 2.5 respectively. In addition, the alpha widths for the 3~ resonances of 149Sm obey a chi-squared distribution with i/eff=2.6. In contrast, the 4r resonances of 149Sm appear to follow a much narrower distribution {veff =6.6) than expected on theoretical grounds. Since the transition 4~ —> 0 + is forbidden by parity conservation, it is possible that soft gamma rays can be emitted and then followed by an alpha particle transition to the ground state of the residual nucleus. As a result, we have the two-step process (11,7a). The resulting alpha particle spectrum due to this process will then have a continuous shape superimposed between the discrete alpha groups corresponding to transitions to the ground and first excited states of the residual nucleus. This reaction was first observed by Oakey and MacFarlane [258] in 143Nd( n,7a) 140 Ce. A study of the shape of the continuous alpha particle spectrum can yield important information on the multipolarity of the soft gamma rays and the reaction mechanism. One of the interesting findings derived from such investigations is that magnetic dipole 7-ray transitions between highly excited states play an important role in the decay of compound states in this reaction. The interested reader is referred to the excel-
3. Individual Resonance Parameters
86
Table 3.3: Thermal (11,7a) and (n,a) Cross Sections. Target Nuclide
o-n>7a(mb)
33g
1.75
37
Ar
40K 59
67
0
[260]
< 5xlO 3
3
0
< 1 < 13
Zn
~ 0.04
2
Te
3
143
Nd
5 4
147
Sm
< 0.035
i49Sm
3
[260] [260]
3
Ni
123
Reference
ff«,a(mb)
[260, 261] [260]
6 2
[262] [260] [262]
6 1
[263] [263]
lent review article by Popov [259]. At this point a summary of thermal [258]-[263] (11,7a) and (n,a) cross sections is presented in Table 3.3.
3.6 3.6.1
Neutron-Induced Fission Channel Theory of Fission
Because of its simplicity, the body of available data for fissile material is described in terms of the single-level Breit-Wigner formalism. However, recent use of the SAMMY code [264], authors reported fission widths derived on the basis of the multilevel Reich-Moore formalism [265] with two fission channels (F/i and (F^)For an understanding of the energy and spin dependence of the fission widths, it is necessary to discuss briefly the channel theory of fission. To estimate the fission width, Bohr and Wheeler [266] applied statistical considerations to arrive at an expression which was subsequently reinterpreted by Aage Bohr [267]: D(J7T)N(J7r)
(3.30)
where (F/(J 7 r )) and D(Jn) are respectively the average total fission width and average level spacing of resonances of specified spin and parity, and N{Jn) is the
3,6. Neutron-Induced Fission
87
available number of transition states. In modern terminology, N(JW) is known as the number of exit fission channels. This simple relation plays a major role in accounting for some of the principal features of nuclear fission, such as (1) the steplike increase of the fission cross section with increasing excitation energy, (2) spin dependence of the fission widths, and (3) mass distribution of fission fragments, etc. Of particular importance in the development of the theory of fission was the introduction by Aage Bohr [267] of the concept that the spectrum of transition states at the saddle point is finite. The nucleus at the saddle point is considered thermodynamically cold since a large fraction of the available energy is in the form of potential energy of deformation. The number of fission channels for a particular 2W is small and can be related to the vibrational and rotational modes of the nucleus at the saddle point; in other words, the spectrum of transition states at the saddle point resembles the rotational and vibrational spectra of deformed nuclei near the ground state. Then the exit fission channels can be characterized by the parity, the J and K quantum numbers (the projection of the angular momentum quantum number on the symmetry axis of the nucleus). A description of the exit fission channels or transition states for even-even compound nuclei in terms of simple collective motions has been given by Lynn [4]. For excitation energies below the fission barrier (subthreshold fission for a single-humped potential), the above relation for the average fission width has to be modified to take into account barrier penetration or transmission factors:
P/ =
1 + exp(27r(F/ - E)/hu)
where Vf and E are the fission barrier height of a particular fission channel / and the excitation energy respectively, and hu is the circular frequency of the oscillator. The parameter hw gives a measure of the curvature of the potential barrier. Estimates of hu range from about 0.3 to 0.9 MeV. Note that Pf(E) is unity for E > Vf and is 0.5 at E = Vf. The relation for the barrier penetration factor was first derived by Hill and Wheeler [268] for a potential barrier of an inverted simple harmonic oscillator. Therefore the general expression for (Tf) becomes
} = -^- ££
1+exp(27r(F/_B)/M
1
where the summation is extended over / fission channels. 3.6.2 Distribution of Fission Widths Early in the history of fission, it was held that fission widths would behave much like radiation widths (numerous exit channels, and therefore widths with small dispersion), but experimental findings soon showed that the fission widths do not behave according to these expectations but fluctuate strongly from resonance to resonance of a particular nucleus. Analysis of the 235U fission widths carried out by Porter and Thomas [74] indicated that the number of degrees of freedom is
3. Individual Resonance Parameters U Average Fission Widths of the Various Channels.
^=3 J7r=4~
3 parity forbidden
AlA
235
AlA
Table 3.4:
1
0
2
0
0
4
0
5
. At this point it is important to note that the effective number of degrees of freedom, v e //(J 7r ), obtained from analysis of the fission widths on the basis of a chi-squared distribution, is generally different from N( J71"). This arises because the degrees of opening of the various contributing fission channels may be different, which results in unequal average partial fission widths for F/i, F/2, F/3. Under this condition, the distribution of the total fission widths(F/ = F/i + F/2 + F p ) no longer exactly obeys a chi-squared distribution. Thus, one could define an effective number of degrees of freedom by the relationship.
Variance Tf(J*)
(3
"33)
In terms of the penetration factors or "degree of opening" of the various fission channels, it can be shown that
(3-34)
Note then that veff is generally larger than iV/p71"). A study of the statistics of fission widths for partly open fission channels was carried out by Cook and Rose [269] for the cases of two and three channels. This was achieved by convoluting Porter-Thomas distributions with different mean values for the fission widths. The distributions for two and three partly open fission channels were explicitly given by Cook and Rose [269]. Subsequently, Musgrove et al. [270] applied the three channel distribution formula to the data of Moore et al. [271] who assigned the 3 and 4~ spins to numerous 235U resonances based on polarized neutrons and polarized target. The distributions of the J71" = 4~ and 3~ fission widths were successfully fitted [270]. Furthermore, contrary to previous views, it was demonstrated that three fission channels are required to describe the fission widths of the 4~ resonances. The results are summarized in Table 3.4. The numbers in parentheses nearest to the partial fission widths correspond to the K" quantum numbers of the transition states. The analysis of Musgrove et al.
3.6. Neutron-Induced Fission
89
Table 3.5: Derived Information on the K71" Channel Contributions in Neutron Induced Fission of 235 U. Calculated A2 values J 7 r =3"
(J:K) (3:0) (3:1) (3:2) (3:3)
J 7r =4"
(J:K) (4:0) (4:1) (4:2) (4:3) (4:4)
-2.92 -2.19 0 +3.65
Exp.
8
parity forbidden -2.48 -1.17 +1.02 +4.08
Exp.
4
[270] indicates that the main contribution to the fission process in the low energy 3~ and 4~ resonances is due to the K7r= 1~ and 2~ channels as well as the K=0~ channel for the J=3~ resonances. This interpretation is in accord with the results of Keyworth et al. [210] who combined the data of Pattenden and Postma [272] on the angular distribution of fission fragments from an aligned 235 U nucleus with their measured spin assignments. The expression for the angular distribution of the fission fragments is derived by Dabbs et al. [273] and is represented by W(6) = 1 + A2f2P2 (cos 6)
(3.35)
where f2 is the alignment parameter, P2 is the Legendre polynomial of order 2 and A2 is given by Pattenden and Postma [272],
The average A2 values extracted [210] from the measurement for the J 7r =3^ and 4~ resolved resonances are compared with calculated values in Table 3.5. As is readily evident, the derived < A2 > coefficients can be interpreted as due to admixtures of various (J:K) components. It is of particular interest to point out that with the aid of the admixture coefficients derived by Musgrove et al. [270] and the A2 values of Table 3.5 calculated from Eq. 3.36 one obtains < A2 > = 5 and -1.33+0.30 for the 3 and 4~ resonances respectively. Within the experimental errors these are in satisfactory agreement with the corresponding experimental values -1.26+0.08 and -1.80+0.04.
90
3. Individual Resonance Parameters
Table 3.6: Comparison of Measured and Calculated F 7 / Widths. Tlf (meV)6
Tlf (meV)° (II)
Tlf (meV)° (I)
8.3 3.2
4.6 1.6
0.15
2.0xl0 ~
238p u 239p u
0.02
0.003
16
12
<0.1 3.0 (J=l+) 4.7 (J=0+)
241 p
2
1.4
3
Target nucleus
237 N
241
p
u
Am
1.5 (J=4- ) 3 .0 (J=3" )
Tlf (meV)c exp.
~3 7 (4 3 (3
7
4 8 8 9
(1+) (0+) (3+) (2+)
5x10
242
° Ref. b Ref. c Ref. d Ref.
3.6.3
[259] [274] [275] [276]
(n,7f) Reaction
Since the fission thresholds of even-even compound nuclei in the actinide region are about 1 MeV below the neutron thresholds, it is possible for a compound nuclear state formed by slow neutron capture in this mass region to decay by radiating a low energy 7-ray (EJ<1 MeV) and subsequently undergoes a fission event. This is the (n,7f) reaction which is similar to (n,7a) reaction discussed previously. Rough estimates of the gamma-fission widths were first made by Stavisky and Shaker [277]. Subsequent calculations made by Lynn [274] for 233U, 235U, and239 Pu, indicated that the magnitude of F 7 / is roughly about 10% of the radiative width of the corresponding nuclide. More recently, Popov [259] extended these calculations to other nuclei and treated the problem in a manner similar to the (n,7a) reaction. In Table 3.6, a summary of the theoretical estimates as well as the measured F 7 / [278]-[275] values is presented. In the calculated F 7 / values of Popov [259], the index I refers to the prompt fission component while the index II takes into consideration the additional fission component which occurs through the isomer state. Since several variables influence the magnitude of the F 7 / widths, large uncertainties exist in the
3,6. Neutron-Induced Fission
91
calculated results. These include the following: (a) the fission barrier parameters, (b) the temperature value of the level density formula and (c) the shape of the primary gamma ray spectrum and the multipolarities of the emitted low energy gamma rays. Because of the small magnitude of the F 7 / and the large values of the fission widths, particularly for resonances of even-even compound nuclei, a direct experimental observation of this reaction is difficult to make. However, if the fission process is inhibited because of the unavailability of the proper Kw transition state or its location relative to the neutron binding energy (i.e. partly open channel), the likelihood of detecting the (n, 7f) reaction is enhanced. In the (n,7f) reaction, since a compound state can decay to various states by radiating several 7-rays, the distribution F 7 / is characterized by a large number of degrees of freedom. This can be considered as one of the possible signatures of the (n,7f) reaction. Other methods include an establishment of anti-correlations between the average number of neutrons per fission, v, and the average total kinetic energy, < E1 >, on one hand and the fission widths, F/, of resonances on the other hand. Bowman et al. [282] were the first to invoke the (n,7f) reaction to interpret the relatively narrow distribution of the fission widths of low-energy neutron resonances of 238 Pu. There are, however, difficulties in reconciling the experimental values of the observed fission widths with theoretical estimates of F 7 / (Table 3.6). Subsequent measurements of the fission neutron multiplicity, v, and average fission gamma ray energy < Ey > for 238 Pu resonances below 110 eV by Shackleton et al. [278] exhibited an anticorrelation between these quantities for the 1 + resonances which was presented as evidence of the (n,7f) reaction. In addition, Ryabov et al. [279] carried out measurements of the ratio of the fission 7-rays and fission events, R, for 239 Pu resonances in the energy range 0.26195.4 eV. Large fluctuations up to about 15% in the values of R were observed for the J*' = 1 + resonances (i.e. with small fission widths) and the R values are strongly correlated with the reported v and < E1 > values [278]. On combining the results from these two experiments, the following parameters were derived [279]: < F7/ > = 9 meV (for J* = 1+ resonances) and < E7f 9 MeV. The former value is in reasonable accord with the theoretical estimate of Lynn [274] (F 7 / ~ 3.0 meV) and the latter is in accord with expectation.
3.6.4
Subthreshold Fission
Shortly after Strutinsky [283] reported his prediction of a double-humped fission barrier for the actinides, Lynn [4] and Weigmann [276] independently gave an interpretation of the intermediate structure, observed by Migneco and Theobald [284] in the subthreshold fission cross section of 240 Pu, in terms of the states associated with the second minimum of the potential energy as a function of deformation. An exhaustive review of subthreshold fission was given by Bjornholm and Lynn [10]. In this section, due to space limitations, we can present only a brief description of this interesting phenomenon which is schematically represented in Fig. 3.1. The symbols are defined as follows:
92
3. Individual Resonance Parameters Dj D//
= =
average level spacing of class I compound resonance states. average level spacing of class II states which are associated with the second minimum of the potential energy as a function of deformation B. This can be determined from the number of observed fission clusters in a certain energy interval. h WJ4 = curvature parameter of the first fission barrier A. % UJB = curvature parameter of the second fission barrier B. U/ = Sn — Si = effective excitation of class I levels. The pairing energies can be found, for example, in the work of Nemirovsky and Adamchuk [135] or the systematics of Newton [136]. U/j = effective excitation energy for class II levels measured from the second minimum. E// = U/ - U// + &i - &n = excitation energy of the second minimum measured from the ground state of the compound nucleus. = inner barrier height. — outer barrier height. If the energies, spins, and parities of compound resonance states in a certain energy interval happen to match those of one of the intermediate states in the second potential well B, the probability of the nucleus to fission is enhanced. Note that the intermediate states in the second well then act as doorway states in the (n,f) process. This picture provides a natural interpretation for the clustering observed in the subthreshold fission cross sections of even-even target nuclei such as 238 U and 240 Pu and the odd-even nucleus 237Np. Each fission cluster is associated with one intermediate state in the second potential well. One of the resonances in the central region of each cluster carries a large fraction (~80%) of the fission width of the cluster. Furthermore, one of the interesting consequences of this interpretation is the existence of an anticorrelation between the fission widths and the reduced neutron widths i.e. the central resonances with large fission widths are characterized by small reduced neutron widths (see for example 238U and 240 Pu resonance parameter tables). The fission widths as a function of neutron energy appear on the average to follow a Breit-Wigner shape. A quantitative description of this observation was presented by Lynn [4] and Weigmann [276] in terms of Strutinsky's [283] double-humped fission barrier. The result is given by,
where
3.6. Neutron-Induced Fission
93
Ff\i F1"
= fission width of class I level at energy E\< = r(f, (/) = decay width (fission width) of class II intermediate level at Ev' The distribution of class II fission widths follow a PorterThomas distribution [10] < Hyy, > = average value of the squared coupling matrix element between class I levels, A', and a class II level, A" r4= T{i(C) = (2TT/DI) < Hl,x,, > = spreading width of the intermediate state A". At this stage, it is important to distinguish between the following coupling schemes: a) Very weak coupling to narrow class II states. In this case, < H\,y, >/Di < Di and the fission width of class II states is obtained from v/ 240
(3.39)
Pu and 238 U provide examples of this coupling scheme. b) Moderately weak coupling to narrow class II states where, Di K
{H^A
K Du
(3 40)
Example of this coupling can be found in 2 3 4 U. c) Moderately weak coupling to broad class II states. Note that in the uniform picket fence model of Lane et a l . [238] an additional term, W D / / 2 T T , is present in the denominator of Eq. 3.38, and a background term is added to Tyf. A reasonable estimate of the "non-resonant" background term can be obtained from an expression derived by Ignatyuk et al. [285] by application of the WKB approximations. The penetration probability through two barriers A and B is approximated by the relation, -l
PB\
P(E) = \
)
„ 2 , r,2 , sin A
(3.41)
where PA and PB are the penetration probabilities through barriers A and B respectively and $ is a phase connected with the second well. For example, when the neutron energy in the first well matches one of the states in second well, i.e. it resonates, $ = (n+ 1/2)TT (where n is an integer), then APAPB P
(3-42)
On the other hand, when $ = n-zr, which corresponds to direct penetration through the two barriers without formation of a resonant state, then
Pmin = \PAPB
(3.43)
94
3. Individual Resonance Parameters
9
The Double-Humped Fission Barrier 8 7
B
6 o o c LU
Class I
ro 4 c CD
Isomeric Fission
1 0
leous Fission 0.2
0.4
0.6
0.8
1.0
1.2
1.4
Deformation p Figure 3.1: Potential energy as a function of the deformation parameter j3 according to Strutinsky's [283] calculation. The predicted double-humped fission barrier offers a description of the clustering of subthreshold fission strengths which was first observed in 240 Pu. The symbols are explained in the text.
3,6. Neutron-Induced Fission
95
In terms of the escape and spreading widths, this can be written in the form 1 FT 4 Pmin = ^-F^-Dj
(3.44)
It is of interest to point out that the forms of Eqs. 3.37-3.38 are quite similar to the strength function expression in the vicinity of a doorway state (for example see Eq. 2.18). Currently, three methods are applied in the determination of F1", T^, and < H2 >. These are as follows: a) The differential method whereby a Breit-Wigner shape fit of the distribution of fission widths is made with application of various weighting techniques. b) The integral method of Lane et al. [286]. In this method, the cumulative sum of the fission widths E v F/x' is plotted as a function of energy E,v The resulting staircase function is fitted to: E
W
7T
\
W
c) The matrix element method of Lynn and Moses [287]. Information on the barrier heights of the inner and outer potential wells, and V / B , can be derived from the subthreshold fission parameters, < Fj^f,,, > and < Fy,c >, with the aid of the following relations:
Diin
DII\1 , __n_{ViB-m
l ( 3 4 7 )
Other methods include fitting the fission cross section in the subthreshold region using expressions similar to Eqs. 3.46-3.47. Estimates of the excitation energy of the second minimum can be obtained from a knowledge of the level spacings DI and DII with the help of the commonly used level density formula of Gilbert and Cameron [139].
£jL = {^j exp
2(aiIUn)1/2]
(3.48)
where a/ and a// are the level density parameters of class I and class II states respectively at the Fermi energy. The simplifying assumptions that a/ = a// and S\ = <52 are customarily made to be able to calculate Uu from which E/j is estimated.
96
3. Individual Resonance Parameters
If the radiative width of a pure class II state can be determined experimentally (except for the 721 eV resonance of 238U no success is reported for others because the central resonances have small scattering widths), in principle one can also determine the excitation energy E// of the minimum in the second well or vice versa. However, the dependence of the radiative width on excitation energy has to be known. One can apply the relations discussed in the "Total Radiative Width" section to arrive at the following:
14= where n can be 2, 3/2, or 5/2 depending on whether one adopts respectively the Weisskopf, the thermodynamical, or giant-dipole estimates. Alternatively, one can carry out detailed calculations within the framework of a particular model to arrive at better estimates, as was done for 238 U (refer to section 238U subthreshold fission). Finally, a brief discussion of a few interesting examples of subthreshold fission is presented here. 240
Pu Subthreshold Fission
Dramatic observations of intermediate structure in subthreshold fission were made first by Paya et al. [289] on 237 Np and two years later by Migneco and Theobald [284] on 240 Pu. Subsequently, the subthreshold fission cross section of 240 Pu was measured with high resolution in an extended energy range 0.50-10 keV by Auchampaugh and Weston [288]. Fig. 3.2 shows four fission clusters at 782, 1405, 1936, and 2695 eV out of the approximately 22 observed clusters below 10 keV. This information leads to an average level spacing of class II states of 0 eV. The solid line is a shape fit to the data using the parameters which were extracted by Auchampaugh and Weston [288] by the integral method. The decay and spreading widths determined by Bjornholm and Lynn [10] using the matrix element method are given in Ref. [10]. In both methods, the decay width is determined from the sum rule of fission widths. On the other hand, the two methods yield quite different values for the spreading widths and hence the coupling coefficients. Note that for 240 Pu F^ ~ F^ and < H2 >/Df
Pu Subthreshold Fission
The subthreshold fission cross section of 238 Pu was measured by Drake et al. [290] and Silbert and Berreth [291] using underground nuclear explosions (Persimon and Pommard) and later by Budtz-Jorgensen et al. [292], using both a Van de Graaff and an electron linear accelerator. The resonance parameters below 500 eV were derived by Silbert et al. [291, 293] by combining the capture and fission cross section measurements from Persimmon. Intermediate structure in the fission cross
97
3.6. Neutron-Induced Fission
240 Pu: Subthreshold Fission
D : = 13.6(7) eV D n = 450(50) eV 10000
782 eV
1405 eV 1936 eV
2695 eV
I
i
1000
1
100
I.
10
g
I*
(0
0.1
0.01
500
1000
1500
2000
2500
3000
Neutron Energy (eV) Figure 3.2: The measured subthreshold fission widths of 240 Pu plotted versus neutron energy. The clustering of fission strengths at certain neutron energies was interpreted by Lynn [4] and Weigmann [276] in terms of Strutinsky's double-humped fission barrier. The solid curve is a Breit-Wigner shape fit of the data, the parameters of which are reported by Auchampaugh and Weston
3. Individual Resonance Parameters
98
238
Pu: Subthreshold Fission
Dj=9.0(6) eV DTI=600(130) eV 10000
1000 CD
|
100 c
g 'to
10
0.1 100
200
300
400
500
600
Neutron Energy (eV) Figure 3.3: The measured subthreshold fission widths of 238 Pu plotted versus neutron energy. The solid curve which is a differential fit of the data describes the intermediate structure observed at a neutron energy of 285 eV. For details see the text.
CJi
238-
238
Pu
100
100 ,^-r
Pu
S ft-
r f Distribution =1 rf> = 45.6 (meV)
10
10
S
v=3 = 6.9 (meV)
0.1
i
2
.
i
4
.
i
6
.
i
8
.
i
.
i
.
i
10 12 14
0.1
i
i
i
i
i
0.5
1.0
1.5
2.0
2.5
Figure 3.4: The left-hand side of the figure represents the distribution of subthreshold fission widths of 238 Pu below a neutron energy of 500 eV and the right-hand side section describes the distribution of reduced neutron widths. The solid curves are integral chi-square distributions with v=l and 3 degrees of freedom.
CO CO
100
3. Individual Resonance Parameters Table 3.7:
238
Pu Parameters of the Intermediate Resonance at 285 eV. Lynn and Moses (1) (2)
Present Analysis (1) (2)
r^, c (eV)
27.9
4.287
22.1
4.18
T[if(eV)
4.287
27.9
4.18
22.1
44.05
6.72
33.4
6.3
4
2
section below 500 eV is clearly visible in Fig. 3.3 with the first cluster located at about 285 eV. An estimate of the average level spacing of class II states is D/j = 0 eV. Applying the method previously described, one obtains E/j = 3 MeV. The fission distribution of the 285 eV cluster represented in Fig. 3.3 shows three striking features: a) its broad width, b) its maximum value (the resonance at 285 eV has a fission width of 3.3 eV) and (c) the presence of a large "background" component. The solid curve is a shape fit obtained in Ref. [294] using the differential method, the parameters of which are included in Table 3.7. For comparison, the parameters of the class II states extracted by Lynn and Moses [287] applying the matrix element method are also included in Table 3.7. Two interpretations of the subthreshold fission of 238 Pu are possible: (1) the resonance compound states are strongly coupled to a narrow class II state or (2) the resonance compound states are weakly coupled to a broad class II state. The latter choice is favored by Lynn and Moses [287] on the basis of the comparatively large reduced neutron width of the 285 eV resonance (F° = 1.61 meV) and the narrow level hypothesis. The distribution of subthreshold fission widths of class I resonances was first examined by Silbert and Berreth [291] and later by Derrien [295]. The integral distribution is illustrated in Fig. 3.4 which shows that two different chi-squared distributions [295] with v = 3 and 1 and respective average fission widths 6.9 and 45.6 meV are required to fit the data. On the other hand, from a study of the differential distribution, Silbert and Berreth [291] concluded that v = 4 and < Tf > = 5.3 meV. The possibility that p-wave resonances may play a role here (i.e. additional fission channels are available) can be eliminated by inspection of Fig. 3.4 which represents the distribution of reduced neutron widths below a neutron energy of 500 eV. No evidence for an excess of small widths which can be attributed to p-wave interactions can be noted. Alternatively, the following is advanced as a plausible interpretation. Due to the coupling between class I and class II states, the observed fission widths of the compound resonance states are expected to follow a Breit-Wigner shape (with superimposed Porter-Thomas fluctuations) along with a background, which can be associated with the direct penetration through
3.6. Neutron-Induced Fission
101
the two barriers. For 238 Pu, those fission widths with < F^ > = 6 meV and v = 3-4 correspond to the background term, an estimate of which can be obtained from the formula Tbf = l/27r(r t r i /Df / )D. Applied to the 285 eV cluster, and with the aid of the parameters of Table 3.7, one readily obtains Fy = 5.5 meV which is in surprising agreement with the experimental value, 5.3-6.9 meV. Similarly, one can derive Fj = 0.63 or 0.22 meV for 240 Pu depending on whether the parameters of Bjornholm and Lynn [10] or Auchampaugh et al. [288] are used. These are to be compared with a value of < F/ > = 0.8 meV (y = 4) obtained from the distribution analysis of 240 Pu fission widths having values less than 16 meV. 234
U Subthreshold Fission
The subthreshold fission cross section of 234U was measured extensively by James and collaborators [297]-[296]. The intermediate cluster at 580 eV and another possible one at 1227 eV are shown in Fig. 3.5. The solid curve is a Breit-Wigner shape fit to the data generated using the weighted maximum likelihood method. To reproduce the solid line, it was necessary to assume that W, the actual widths of the two Breit-Wigner shapes, are double the values reported by the authors [296]. If the full width is associated with the spreading width, then for the 580 eV 0 eV and F f = T[if = 0.096 eV. Since the intermediate state, Tl = Y^,c = spreading width satisfies the condition Di < T1/ < Du, then 234U represents a good example of moderately weak coupling. The average level spacings for class I and class II states are 7 eV and 3 keV respectively. Finally with the known level density parameters, one derives E/j = 9 MeV. 236
U Subthreshold Fission: (n, 7f) Reaction?
All the subthreshold fission widths of resonances measured by Theobald et al. [299] in the energy interval 5.45-415 eV exhibit small fluctuations around an average value of 0.35 meV. The number of effective degrees of freedom veff is found to be 18. This large value of veff coupled with the small values of the individual fission widths suggests that the (n,7f) process may play a significant role here. Another possible interpretation is that the energy region 5.45-415 eV may constitute the tail of an intermediate structure with maximum strength located at a higher energy. An estimate of the average level spacing of class II states can be derived by assuming that the location of the second minimum (below the neutron separation energy) in 237 U is about the same as those in 235U and 239U (Table 3.9). With this assumption, one readily derives DH = 0 eV. With the aid of Eq. 3.44 and this value of D/j one obtains the estimate F^F1" = 51 eV2 which suggests that possibly we are dealing here with another case of moderately weak coupling. At this point, it is interesting to draw attention to the 236U fission cross section measurements (Pommard data) of Cramer and Bergen [300]. In addition to the subthreshold fission resonances below 450 eV, reported previously, the Pommard data exhibits another fission resonance at 1292 eV as well as unresolved peaks in the energy interval 1874-1983 eV. Since the observed fission resonances are characterized by large scattering widths, it is possible that due to insufficient time resolution the central fission resonances (with small F n ) have escaped detection in this measure-
102
3. Individual Resonance Parameters
234
U: Subthreshold Fission Widths
D.=10.6(5)eV
DM=2.1(3)keV
1227 eV
10
1E-3 0
200 400 600 800 1000 1200 1400 1600 1800 Neutron Energy
Figure 3.5: The subthreshold fission widths of 234U showing two intermediate structures located at neutron energies of 580 and 1227 eV. For details, see James et al. [296].
3.6. Neutron-Induced Fission
103
ment. Therefore, it will be of great interest to carry out high resolution fission cross section measurements of 236U in an extended energy range to distinguish between these two interpretations. Furthermore, the Pommard data indicate that the fission barrier height of 236U (V/A ° r V/s, whichever is the higher) is about 1 MeV above the neutron separation energy. 238
U Subthreshold Fission
The observation of prominent resolved clusters at 720 and 1210 eV and weaker groups at 2.5, 7.5, 11, 15, 27, and 35 keV was first made by Block et al. [301]. These authors noted that the fission strengths are anticorrelated with the neutron strengths in agreement with theoretical expectations. Extensive high resolution measurements of subthreshold fission cross section were later carried out by Difilippo et al. [302, 303] at the Oak Ridge electron linear accelerator. In these measurements, 38 fission clusters were observed below 68 keV yielding an average level spacing of class II intermediate states D/j = 5 keV below the first inelastic threshold at about 45 keV. The parameters of class II states for the clusters at 721.6 and 1211 eV, which were analyzed [303] in terms of the differential method, are presented in Table 3.8 where (1) and (2) refer to the two possible interpretations: very weak coupling and moderately weak coupling respectively. From these data alone, it is not possible to distinguish between these two interpretations. In order to shed additional light on this problem, attempts to measure 7-ray spectra due to neutron capture in the low energy resonances were made by Weigmann et al. [304] and Browne [305]. However, the two results were discrepant. The basic problem to be faced is that the central resonances which carry the bulk of the fission widths are characterized by small neutron widths. Consequently their capture rates are quite small while the background due to fission 7-rays and neighboring resonances is large. Later attempts by Auchampaugh et al. [306] and Oberstedt [307, 308] et al. were succesful in estimating the capture widths of the 721 eV resonance. In the former measurements the capture width of the 721 eV resonance was reported as 6 meV while in the latter investigation, it is estimated as 1 meV. Since this width is considerably smaller than those of other 238U resonances, this is attributed to the formation of a shape isomer in 238U. At this point it is instructive to calculate the average radiative width of the shape-isomeric resonances within the framework of the GFL model, discussed previously. On the assumption that a/ = an and Eu=1.52 MeV (Table 3.9), one readily obtains for the isomeric state F 7 //=7, 9, 10, 11, 13, 15 and 17 meV for /32=0.24, 0.32, 0.37, 0.40, 0.50, 0.60 and 0.70, respectively. It should be noted that /?2=0.24 correspond to the ground state deformation of 239U. On the other hand, if £77=1.9 MeV (Table 3.9), then the calculated radiative widths are reduced by about 20% from the above values. The effective radiative width of an isomeric state can be obtained from the following equation
r 7 = (1 - C7/)rV + CnT^u
(3.50)
where C// is the class II fraction. With F7/=23.4 meV, F 7 //=7 meV (minimum value), C//=0.8 [307] one obtains F7=10.3 meV, which is not in accord with the
3. Individual Resonance Parameters
104
Table 3.8: 238U Class II Parameters for the Intermediate Clusters at 721.6 and 1211 eV. 721.6 eV
1211 eV
(1)
(2)
(1)
(2)
x»/(eV)
1.8 x l t r 3
0.89
0.4xl0- 3
4.61
r{f,c(eV)
0.89
1.8xlO-3
4.61
0.4xl0- 3
3- 1
6.3xlO- 3
16
1.4xlO-3
r
result F7 6 meV. Then it appears that the present calculations support the result of Oberstedt et al. [307, 308], provided that the distribution of the radiative widths of isomeric states is narrow as in the case of the compound states in the first potential well. Height of the Second Minimum, E/j The height of the second minimum derived from the level spacings of class I and class II states on the basis of the Gilbert-Cameron [139] formula are compiled in Table 3.9. The EH value for 244 Pu is obtained with the aid of the subthreshold fission data of Moore et al. [309]. Two interesting features which emerge from the present study are the general decrease of En with A and the nearly constant value of Sn - En for those nuclei in table 3.9. Comparison of the present values with those compiled by Bjornholm and Lynn [10] indicates a systematically low trend in the values extracted from D/ and DJJ. If one accepts the functional form of the level density formula and the values of the level density parameter, a/, the source of the discrepancy can be traced to our lack of knowledge of D/j, a// and <S/j. The assumption made in the analysis that all the observed clusters are due to s-wave interaction may not be correct. This is due to the circumstance that the actinides lie in a mass region in the vicinity of the 4p giant resonance (see Fig. 2.2). As a result, p-wave neutron interaction plays an important role, particularly in the keV range. Since D/j is determined in an energy region extending up to a few keV, it is possible that some fraction of the observed clusters may be due to p-wave interaction which would then lead to underestimated D/j values. Another alternative solution to the problem is the possibility that the level density parameter for class II states is about 15% larger than that of class I states.
Table 3.9: Heights of the Bottom of the Second Well for Some U and Pu Isotopes Obtained from Level Spacings of Class I and Class II Resonances. Target Nuclide
Eau (MeV)
Ehu (MeV)
Dx (eV)
Bn — (eV)
(MeV- 2 )
(MeV)
Jo
e 2 ?
I
234
U
9
3
7
0
236
U
e
4
4
e
238
U
6
3
2
0
6
29.12
238
Pu
3
2
7
0
3
27.14
240
Pu
8
3
6
0
8
22.52
242
Pu
3
3
0
0
6
27.94
244
Pu
8
0
/
8
28.60
9
27.96 Gn
d
28.05
o
present study. Ref. [10]. present study (Table 2.3). assumed. derived from systematics. all observed clusters are assumed to be due to s-wave interaction. o
This Page is Intentionally Left Blank
Chapter 4
Notation and Nomenclature The following is a table of definitions of quantities used in the present edition of this book. A quantity with a superscript c indicates that it is calculated from resonance parameter information or related measurements. Other superscripts such as 0, 1, r, ', are used in connection with thermal capture cross sections and scattering widths as described below.
4.1
Thermal Cross Sections
a®
=
cr7
=
a^
=
a^
=
9w
=
aa
= = = = = = = = =
3^= neutron radiative capture cross section measured at 2200 m/sec. oni= neutron radiative capture cross section measured in a Maxwellian flux. &n7= neutron radiative capture cross section measured with reactor neutrons. The value of the activation cross sections leading to the formation of the ground and isomeric states (g or m) are followed by two brackets. The first set of brackets indicates the half life and type of state; the second specifies the spin and parity b[18.7 sec 4 6 Sc m ][-l]. of the state. For example capture cross section calculated from resonance parameters or derived from that of the natural element. Wescott factor which describes the shape of the cross section at low energies. ona= neutron cross for alpha particle emission. (Tnp= neutron cross section for proton emission. &nf= neutron fission cross section. neutron absorption cross section. paramagnetic cross section. total neutron cross section. (Jnn= "free" neutron nuclear scattering cross section. "free" nuclear coherent scattering cross section. "free" nuclear incoherent scattering cross section.
107
108
4- Notation and Nomenclature
bCoh b(+)
= =
b(—)
=
o-coh
=
a(+)
=
a(—)
=
R' R'(+) R'(-) v vp v& vsp tj a
= = = = = = = = =
<7po
=
dpi
=
<7p2
=
CTao
=
aa\
=
&a2
=
J7 If
= =
bound coherent scattering length, generally a complex quantity. b+= bound scattering length due to the interaction in the parallel spin direction i.e. 1+1/2. &_= bound scattering length due to the interaction in the antiparallel spin direction i.e. 1-1/2. "free" nuclear coherent scattering length, which can be derived from the value of &coj». a+= scattering length due to the interaction in the parallel spin direction i.e. 1+1/2. a_= scattering length due to the interaction in the antiparallel spin direction i.e. 1-1/2. s-wave potential scattering length or radius. s-wave potential scattering length or radius for spin 1+1/2. s-wave potential scattering length or radius for spin 1-1/2. vp + Ui, average number of neutrons emitted per fission. average number of prompt neutrons per fission. average number of delayed neutrons per fission. average number of prompt neutrons per spontaneous fission. v(af/aa) = number of neutrons emitted per neutron absorbed. o^/c/ = capture to fission cross section ratio. At a neutron resonance, this is equivalent to F 7 / F / . (n)P) cross section with the formation of a residual nucleus in the ground state (n,p) cross section with the formation of a residual nucleus in the 1st exited state (n,p) cross section with the formation of a residual nucleus in the 2nd excited state (n,a) cross section with the formation of a residual nucleus in the ground state (n,a) cross section with the formation of a residual nucleus in the 1st exited state (n,a) cross section with the formation of a residual nucleus in the 2nd excited state radiative capture resonance integral. fission resonance integral.
4.2 Resonance Properties Ia Ip D Do
= = = =
absorption resonance integral. For fissionable nuclei, Ia=Ly+If. proton emission resonance integral. mean observed level spacing. s-wave average level spacing.
4-3. Resonance
Parameters
109
.Di £>2 50
= p-wave average level spacing. = d-wave average level spacing. = (#r° ) / (.Do) = s-wave neutron strength function. For brevity, So is given in units of 10~4. = (#r^) / 3 (Di) = p-wave neutron strength function. For brevity, 51 51 is given in units of 10~4. S*2 = (ffF^) / 5 (D2) = d-wave neutron strength function. For brevity, 52 is given in units of 10~ 4 . S7 = (F 7 ) /D = 7-ray strength function. = (F7o) /Do = 7-ray strength function for s-wave resonances in units 570 of 10" 4 . 571 = (F 7 i) /Di = 7-ray strength function for p-wave resonances in units of 1(T 4 . (r 7 o) = average radiative width for s-wave resonances. = average radiative width for p-wave resonances. (F 7 i) (F72) = average radiative width for d-wave resonances.
4.3
Resonance Parameters
Immediately preceding t h e resonance parameter tables, t h e following information can b e found. /" % Abn
= =
Sn
=
<7 7 (+)
=
<77(-)
=
a7{B)
=
<77(.D)
=
spin of t h e target nucleus; superscript on t h e I indicates t h e parity. percentage abundance of isotope. These a r e taken from K.J.R. Rosman a n d P.D.P. Taylor, P u r e Appl. Chem. 70, 217 (1998). Neutron separation energy for t h e compound nucleus A + l . These are adopted from Audi et al., Nucl. Phys. A729, 337 (2003). <7° 7 (+) = 2200 m / s e c neutron radiative capture cross section in barns calculated from positive energy s-wave resonances with spin 1+1/2. c ° 7 ( - ) = 2200 m/sec neutron radiative capture cross section in barns calculated from positive energy s-wave resonances with spin 1-1/2. the contributions from negative energy resonances with spins 2 including the direct capture component. Then (jy(exp.) =
110
4- Notation and Nomenclature =
<7/(+)
txf{—) =
cr°^(+) = 2200 m/sec neutron fission cross section in barns calculated from positive energy s-wave resonances with spin 1+1/2. c°/(—) = 2200 m/sec neutron fission cross section in barns calculated from positive energy s-wave resonances with spin 1-1/2. the contributions from negative energy resonances with spins . Then af(exp.) =
In the resonance parameter table, the following quantities may be found. A quantity enclosed by parentheses () or square brackets [] indicates that it is assumed or preferred respectively. Square brackets enclosing the orbital angular momentum I shows that it was determined by Bayesian analysis. EQ
=
resonance energy.
o
I
=
F Fn F7 Fa Fd Tf
= = = = = =
Tfx r^2 rp
= = =
ra
= r 7 +r a +r p +r d +r / .
F°
=
Tl 7^
00 j 1 g
a
energy gap in the measurements. This signifies that the experimental measurements spanning the energy region specified by this symbol have not been carried out as yet. total width of resonance at Eo. neutron width at Eo. radiative width. a width at Eo. deuteron width at Eo. fission width. Positive and negative signs associated with this quantity are interpreted as follows: pairs of resonances with fission widths having the same sign interfere destructively while pairs with opposite signs interfere constructively. fission width in channel number 1. fission width in channel number 2. proton width at Eo.
Tni^/leV/Eo = reduced neutron width at 1 eV for s-wave neutrons. = Tny/leV/Eo[l + 91/k2R2)] = reduced neutron width at 1 eV for p-wave neutrons, k = 2w/X, i?=nuclear radius = 1.35XA1/3 (fin). = Tnj2Pi = reduced neutron width in R-matrix formalism. Pi = neutron penetrability for a specified orbital angular momentum of the incoming neutron. Values for a square well potential are presented in table 2.1. = peak cross section at Eo. = = 1 + 1 + 1 / 2 = spin of the resonance state. = orbital angular momentum of the incoming neutron. = (2J+l)/[(2s + l)(2/ + l)] = statistical wight factor. For neutrons the spin s = l / 2 so that g = (2J + 1)/[2(2I + 1)]. = ( as in 2agTn) fractional abundance of the isotope.
4-4- Weighted and Unweighted Averages, Internal and External Errors
111
Note that the recommended widths are R-matrix derived parameters for Z<30 and are single-level BreitWigner parameters for Z>31 unless otherwise indicated. The widths are generally expressed in the laboratory coordinate system except in the case of light nuclei where the CM. is used.
4.4
Weighted and Unweighted Averages, Internal and External Errors
The weighted average of n independent measurements X1+AX1, X2+AX2 AXn is given by
Xn +
(4.1)
X =
where W» = (A ^. )2 The internal and external errors are defined as -1/2
(4.2)
internal error =
external error =
(4.3)
n(n-
The error associated with X is the larger of the internal and external errors. The unweighted average of these same measurements is given by 1/2
n
n(n — 1)
4- Notation and Nomenclature
112
4.5
Reference Code Mnemonics
References were abbreviated following closely the CINDA (An Index to the Literature on Microscopic Neutron Data published annually by the International Atomic Energy Agency) notation. Preceding the reference can be found the letters J, R, C, B or T to indicate whether the quoted reference is a journal, report, conference, book or thesis. A positive or negative sign following the reference indicates respectively that the information is fortified by private communications or the values of another author are quoted in the present author's publication. Only the leading author is indicated. AAA AAEC/EAAEC/TMAC ACJ ACR ACR/A ACR/B ACS ACT.EL ADADP AE AEAEC-TRAECDAECLAECUAEEW-MAEREAF AJ AJ/L AJ/S AJN AKE AMA ANCRANE AND ANLANS
Astronomy and Astrophysics Australian AEC report series Australian AEC Technical Memos (Anal.Chem.) Analytical Chemistry (Acta Chem.Scand.) Acta Chemica Scandinavica (Acta Crystallogr.) Acta Crystallographica (Acta Crystallogr.,Part A) Acta Crystallographica, Part A (Acta Crystallogr.,Part A) Acta Crystallographica, Part A (J.Amer.Chem.Soc.) Journal of the American Chemical Society The Actinide Elements, National Nuclear Energy Series, Div IV, Vol 14A, McGraw-Hill (1954) United States Dept. of Defense reports (Annalen der Physik) (Atomnaya Energiya) see SJA for English translation Aktiebolaget Atomenergi reports USAEC, Div. of Technical Information translations USAEC, Div. of Technical Information reports Atomic Energy of Canada Ltd. reports USAEC, Div of Technical Information reports AEEW-Winfrith report series AERE-Harwell reports to Philosophical Magazine (Ark.Fys.) Arkiv foer Fysik (Astrophys.J.) Astrophysical Journal Astrophysical Journal Astrophysical Journal, Supplement (Arab J. of Nuclear Science and Application) (Atomkernenergie) (Ark.Mat.,Astron.Fys.) Arkiv Foer Matematik, Astronomi och Fysik Aerojet Nuclear Corp. reports (Ann.of Nucl.Energy) Annals of Nuclear Energy, formerly Annals of Nuclear Science and Engineering Atomic Data and Nuclear Data,USA Argonne National Laboratory reports (Trans.Amer.Nucl.Soc.) Transactions of the American Nuclear Society
4-5. Reference Code Mnemonics AP APA APP AREAEEARI ARN AUJ BAP BAS BKE BKN BMIBNLBOS BTI CCCEACFCHP CJC CJP CLCNAEMCNP COOCPCR CRCR/B CSCU(PNPL)CUCZJ/B DA DA/B DOK DPEANDC(E)-
113
(Ann.Phys. (New York)) Annals of Physics (New York) (Acta Phys.Austr.) Acta Physiea Austriaca (Acta Phys.Pol.) Acta Physiea Polonica Arab Republic of Egypt Atomic Energy Establ. reports. (Int.J.Appl.Radiat.Isotop.) International Journal of Applied Radiation and Isotopes (Annu.Rev.Nucl.Sci.) Annual Review of Nuclear Science (Aust.J.Phys.) Australian Journal of Physics (BuU.Amer.Phys.Soc.) Bulletin of the American Physical Society (Bull.Acad.Sci.USSR,Phys.Ser.) Bulletin of the Academy of Sciences of the USSR, Physical series, English translation of IZV (BuU.Boris Kidrich Inst.Nucl.Sci.,Electron.) Bulletin of the Boris Kidrich Institute of Nuclear Sciences, Electronics, Vol. 18 (BuU.Boris Kidrich Inst.Nucl.Sci.Eng.) Bulletin of the Boris Kidrich Institure of Nuclear Sciences, Nuclear Engineering Battelle Memorial Institute reports Brookhaven National Laboratory reports (Trans.Bose Res.Inst., Calcutta) Transactions of the Bose Research Institute, Calcutta (Bull.of the Tokyo Inst. of Technol.) English Edition Chicago University Metallurgical Laboratory report series Centre D'Etudes Nucleaires, Saclay reports Chicago University Metallurgical Laboratory report series (Chin.J.Phys.(Taipei)) Chinese Journal of Physics, Taipei (Can.J.Chem.) Canadian Journal of Chemistry (Can.J.Phys.) Canadian Journal of Physics Oak Ridge National Laboratory, Clinton Pile reports Cekmece Nuclear Research Centre reports (Chin.J.Nucl.Phys.(Beijing)) Chinese Journal of Nuclear Physics (Beijing) USAEC Chicago Operations Office contract Chicago University Metallurgical Laboratory report series (Comptes Rendus,Serie B) see CR/B USAEC, Div. of Technical Information reports (Comptes Rendus,Serie B) Comptes Rendus Hepdomadaires des Seances de L'Academie des Sciences, Serie B, Physique Chicago University Metallurgical Laboratory report series Columbia University progress reports Columbia University reports (Czech.J.Phys.,Part B) International Issue (Diss.Abstr.) Dissertation Abstracts (Diss.Abstr.B) Dissertation Abstracts International B (Sciences) (Dokl.Akad.Nauk SSSR) Doklady Akademii Nauk SSSR, see SPD for English translation Du Pont Savannah River Laboratory reports Reports from EURATOM and member countries to the EuropeanAmerican Nuclear Data Committee
114 EANDC(J)EANDCEEN EN EIREON ERDA-NDCEURFRNC-THGAHP HPA HWICDIDOUP ININDC(ARG)INDC(AUS)INDC(CAN)INDC(CCP)INDC(EGY)INDC(NOR)INDC(SEC)INDC(TUR)INDSWGIRE IZV JAERIJAP JCP JET JIN JINRJMM JMS JNE JNE/A JP/A JP/C JP/G
4- Notation and Nomenclature Reports from Japan to the European-American Nuclear Data Committee European-American Nuclear Data Committee reports (Ergeb.Exakt.Naturw.) Ergebnisse der Exakten Naturwissenschaften (Energ.Nucl.(Milan)) Energia Nucleare (Milan) Eidg.lnst.Reaktorforsch. Wuerenlingen reports (Euronuclear) Euronuclear US ERDA Nuclear Data Committee reports Euratom reports Reports from France other than CEA General Atomic reports (Health Phys.) Health Physics (Helv.Phys.Acta) Helvetica Physica Acta Hanford Laboratory reports Bull.Centr Po Jadernym Dannym, Obninsk Philips Petroleum Co., Idaho, reports (Indian J.Phys) Indian Journal of Physics Idaho Nuclear Corp. reports Argentine reports to the International Nuclear Data Comittee Austrian reports to the International Nuclear Data Committee Canadian reports to the International Nuclear Data Committee USSR reports to the International Nuclear Data Committee Egyptian reports to the International Nuclear Data Committee Norwegian reports to the International Nuclear Data Committee International Nuclear Data Committee report series Turkish reports to the International Nuclear Data Committee Reports of the International Nuclear Data Science Working Group (IEE.Trans.Nucl.Sci.) IEE Transactions on Nuclear Science (Izv.Akad.Nauk SSSR,Ser.Fiz.) Izvestiya Akademii Nauk SSSR, Seriya Fizicheskaya, see BAS for English translation JAERI, Tokai-mura reports (J.Appl.Phys.) Journal of Applied Physics (J.Chem.Phys.) Journal of Chemical Physics (Sov.Phys. Jetp) Soviet Physics, English translation of ZET USSR (J.lnorg.Nuel.Chem.) Journal of Inorganic and Nuclear Chemistry Joint Institute for Nuclear Research, Dubna, reports (J.Magn.Magn.Mater.) Journal of Magnetism and Magnetic Materials (J.Mass Spectrom.Ion Phys.) Journal of Mass Spectrometry and Ion Physics (J.Nucl.Energy) Journal of Nuclear Energy (J.Nucl.Energy,Part A) Journal of Nuclear Energy Part A, Reactor Science (J.of Physics,Part A) Mathematical and General (J.of Physics,Part C) Solid State Physics (J.of Physics,Part G) Nuclear Physics
4-5. Reference Code Mnemonics JPJ JPR
115
(J.Phys.Soc.Jap.) Journal of the Physical Society of Japan (J.Phys.(Paris)) Journal de Physique (Paris) formerly Journal de Physique et le Radium JPR/L (J.Phys.(Paris)jLett.) Journal de Physique (Paris) Letters JRC (J.of Radioanal.Chem.) Journal of Radioanalytical Chemistry JRN J.of Radioanalytical and Nuclear Chemistry, Hungary JUELKernfbrschungsanlage, Juelich, reports KAPLKnolls Atomic Power Laboratory reports KE (Kernenergie) KFI KFKI(Kozponti Fizikai Kutato Intezet) Kozlemenyek KFKKernforschungszentrum Karlsruhe reports KKH Kakuriken Kenkyu Hokoko (Japan) KRI (Kristallografiya) Kristallografiya KURRI-TRKyoto Univ.,Reactor Research Institute reports LA-URLos Alamos Scientific Laboratory report series LAMSLos Alamos Scientific Laboratory report series MCMontreal Laboratory, National Research Council of Canada MET (Metrologia) Metrologia MLMMound Laboratory, Miarnisburg, reports MONMonsanto Laboratory reports NAA-SRNorth American Aviation report series NAP (NucLAppl.) Nuclear Applications NASA-TN-D- NASA Technical Note NAT (Nature(London)) NB.GS.COMP. Noble Gas Compounds, Chicago Press (1963) NC (Nuovo Cimento) Nuovo Cimento NCL (Lett.Nuovo Cimento) Lettere al Nuovo Cimento NCS (Nuovo Cimento Supl.) Nuovo Cimento, Supplemento NCSACUSAEC Nuclear Cross Section Advisory Committee reports NEANDC(E)- Reports from EURATOM and member countries to the Nuclear Energy Agency Nuclear Data Committee NEANDC(US)- Reports from the UNited States to the Nuclear Energy Agency Nuclear Data Committee NIM (Nucl.lnstrum.Methods) Nuclear Instruments and Methods, formerly Nuclear Instruments NKA (Nukleonika) Nukleonika NP (Nucl.Phys.) Nuclear Physics NP/A (Nucl.Phys.A) Nuclear Physics, Section A NRDCAERE-Harwell report series NSA (Nucl.Sci.Abstr.) Nuclear Science Abstracts NSE (Nucl.Sci.Eng.) Nuclear Science and Engineering NST (J.Nucl.Sci.and Technol.) Journal of Nuclear Science and Technology, Tokyo NUK (Nukleonik) Nukleonik NWS (Naturwissenschaften) Naturwissenschaften NYOUSAEC New York Operations Office contract reports
116
ORNLPCS PDTUM-FRM PHY PL PL/A PL/B PM PNE PPS PS/A PR PR-CMPR-P= PR/B PR/C PRL PRN PRS PRS/A PS PSS PTE RCA RCNRMP RPIRRL RRP RSI SCI SCP SJA SNP SPD SSC THAI-AEC-
4- Notation and Nomenclature
Oak Ridge National Laboratory reports (J.Phys.Chem.Solids) Journal of Physics and Chemistry of Solids Univ. Munich Physics Department, technical reports (Physica(Utrecht)) Physica (Phys.Lett.) Physics Letters (Phys.Lett.A) Physics Letters, Section A (Phys.Lett.B) Physics Letters, Section B (Phil.Mag.) Philosophical Magazine (Progress in Nuclear Energy) (Proc.Phys.Soc.(London)) Proceedings of the Physical Society, London (Proc.Phys.Soc. (London) Sect.A) Proceedings of the Physical Society, London, Section A (Phys.Rev.) Physical Review Atomic Energy of Canada Ltd. report series Atomic Energy of Canada Ltd. report series (Phys.Rev.,Part B) Physical Review, Part B (Phys.Rev.,Part C) Physical Review, Part C, Nuclear Physics (Phys.Rev.Lett.) Physical Review Letters Physics Reports (Proc.Roy.Soc,London) Proceedings of the Royal Society, London (Proc.Roy.Soc,London,Ser.A) Proceedings of the Royal Society, London, Series A, Mathematical and Physical Sciences (physica Scripta) (Phys.Status Solidi) Physica Status Solidi (Prib.Tekh.Eksp.) Pribory I Tekhnika Eksperimenta (Radiochim.Acta) Radiochimica Acta REactor Centrum Nederland, Petten, reports (Rev.Mod.Phys.) Review of Moder Physics Rensselaer Polytechnic Institute reports (Radiochem.and Radioanal.Letters) (Rev.Roum.Phys.) Revue Roumaine de Physique, formerly Revue de Physique (Rev.Sci.Instrum.) Review of Scientific Instruments (Science) Science (Sci.Pap.Inst.Phys.Chem.Res., Tokyo) Scientific Papers of the Institute of Physical and Chemical Research, Tokyo (Sov.At.Energy) Soviet Atomic Energy, English translation of AE (Sov.J.Nucl.Phys.) Soviet Journal of Nuclear Physics, English translation of YF (Sov.Phys.-Dokl.) Soviet Physics-Doklady, English translation of DOK (Sol.State Comm.) Solid State Communications Thailand Atomic Energy Commission reports
5. Reference Code Mnemonics TRANSU.EL. TIDUCRLUFZ UJVUSNDCWAPD-TMWASHYF YKZAP ZET ZFKZK ZMP ZN ZN/A ZP ZP/A 55GENEVA 57GOLUMBIA 58GENEVA 60KINGSTON
60VIENNA
61BUCHAR
61DUBNA 61SACLAY 62MADRAS
117
The Transuranium Elements, National Nuclear Energy Series, Div. IV, Vol.148, McGraw- Hill (1949) USAEC Div. of Technical Information reports Lawrence Radiation Laboratory reports (both Berkeley and Livermore) (Ukr.Fiz.Zh.) Ukrains'kii Fizichnii Zhurnal, see UPJ for English translation Ustav Jad. Vyzkumu (Czech.Inst.Nucl.Res.) reports Reports to the US Nuclear Data Committee Westinghouse Atomic Power Div.(Bettis) reports USAEC Washington Office reports (Yad.Fiz.) Yadernaya Fizika, see SNP for English translation Jadernye Konstanty (Z.Angew.Phys.) Zeitschrift fuer Angewandte Physik (Zh.Eksp.Teor.Fiz.) Zhurnal Eksperiniental'noi I Teoreticheskoi Fiziki, see JET for English translation Zentralinstitut fuer Kernforschung, Rossendorf, reports (Z.Kristallogr.) Zeitschrift fuer Kristallographie, Kristallgeometrie, Kristallphysik, Kristallchemie (Z.Agnew.Math.Phys.) Zeitschrift fuer Angewandte Mathematick und Physik (Z.Naturforsch.) Zeitschrift fuer Naturforschung (Z.Naturforsch.A) Zeitschrift fuer Naturforschung, Section A, Atoms and Nuclei (Z,Phys,) Zeitschrift fuer Physik (Z,Phys,A) Zeitschrift fuer Physik, Section A, A toms and Nuclei First UN Conference on the Peaceful Uses of Atomic Energy, Geneva, 8-20 Aug 1955 (Columbia Conference, New York 1957) International Conference on Neutron Interactions with Nuclei, New York, Sep 1957 Second UN Conference on the Peaceful Uses of Atomic Energy, Geneva, 1-13 Sep 1958 (Nuclear Structure Conference, Kingston 1960) United Nations Conference on Nuclear Structure. Kingston, Canada, 29 Aug - 3 Sep 1960 (Pile Neutron Research Symp., Vienna 1960) IAEA Symposium on Pile Neutron Research in Physics, IAEA, Vienna, 17-21 Oct 1960 (Research Reactors Conf., Bucharest 1961) International Conference on Physics and Technology of Research Reactors, Bucharest, Rumania, 10-17 Nov 1961 (Slow Neutron Physics Conf., Dubna 1961) Conference on Slow Neutron Physics, Dubna, USSR, 7-12 Dec 1961 (Time of Flight Methods Conf., Saclay 1961) (Nuclear Physics Symp. Madras 1962) Nuclear Physics Symposium, Madras, India, 28 Feb - 2 Mar 1962
118
63ANL
63MANCHST 64PARIS 65ANTWERP
65KRLSRH
66BERKELEY
66PARIS
66SDIEGO
66WASH
67JUELICH
68RIGA
68WASH
69MONTRL 69STUDSVIK
70ANL
70HELSINKI
4- Notation and Nomenclature (Nuclear Physics with Reactor Neutrons Conf., Argonne 1963) International Conference on Nuclear Physics with Reactor Neutrons, Argonne, 15-17 Oct 1963, proceedings published as ANL-6797 (Nuclear Physics Conference, Manchester 1963) Conference on Low and Medium Energy Nuclear Physics, Manchester (Nuclear Physics Congress, Paris 1964) Congres International de Physique Nucleaire, Paris, France, 2-8 Jul 1964 (Nuclear Structure Conf., Antwerp 1965) International Conference on the Study of Nuclear Structure, Antwerp, Belgium, 19-23 Jul 1965 (Pulsed Neutron Symp., Karlsruhe 1965) IAEA Symposium on Pulsed Neutron Research, Karlsruhe, 10-14 May 1965, proceedings published by IAEA as STI/PUB/104 (Radiation Measurements Conf., Berkeley 1966) CEGB Conference on Radiation Measurements in Nuclear Power, Berkeley, UK, 12-16 Sept 1966 (Nuclear Data for Reactors Conf., Paris 1966) IAEA Conference on Nuclear Data for Reactors, Paris, France, 17-21 Oct 1966, proceedings published by IAEA as STI/PUB/140 (Reactor Physics Conf., San Diego 1966) ANS Conference on Reactor Physics in the Resonance and Thermal Region, San Diego, 7-9 Feb 1966 (Neutron Cross Section Tech. Conf., Washington 1966)Conference on Neutron Cross Section Technology, Washington D.C., 2224 March 1966, proceedings published as USAEC report CONF660303 (Neutron Physics At Reactors Conf., Juelich 1967) Neutron Physics At Research Reactors, Juelich, Fed.Rep.Germany, 25-28 April 1967 (Nucl. Spectroscopy and Structure Conf., Riga 1968) 18th Annual Conference on Nuclear Spectroscopy and Nuclear Structure, Riga, USSR, Jan-Feb 1968 (Nuclear Cross Section Tech. Conf., Washington 1968)2nd Conference on Nuclear Cross Sections and Technology, Washington D.C., 4-7 March 1968, published as NBS Special Publication 299 (Intl.Conf. on Properties of Nuclear States, Montreal, Canada, 25-30 Aug 1969 ) (Neutron Capture Gamma-Ray Spectroscopy, Studsvik,1969)International Symposium on Neutron Capture Gamma-Ray Spectroscopy, Studsvik, Sweden, 11-15 Aug 1969, proceedings published by IAEA as STI/PUB/235. (Neutron Standards Symp., Argonne 1970) EANDC Symposium on Neutron Standards and Flux Normalization, Argonne, 21-23 Oct 1970. Published as USAEC report CONF-701002 (Nuclear Data for Reactors Conf., Helsinki 1970) Second IAEA Conference on Nuclear Data for Reactors, Helsinki, Finland, 1519 June 1970, proceedings published by IAEA as STI/PUB/259
4-5. Reference Code Mnemonics 71ALBANY
71KIEV
71KN0X
72BUD
72DENVER
73ASILOMAR
73KIEV
73PARIS
73ROCH
74PETTEN 75HARWELL 75KIEV 75WASH 76LOWELL 77GEEL 77KIEV 77PARIS
119
(Statistical Properties of Nuclei, Albany 1971) Int.Conf. on Star tistica! Properties of Nuclei, Albany, New York, 23-27 Aug 1971, proceedings published by Plenum Press, New York (1972) (Neutron Physics Conf., Kiev 1971)Conference on NeutronPhysics, Kiev, USSR, 24-28 May 1971, proceedings published as 'Nejtronnaja Fizika' in Two Volumes, Kiev 1972. (Neutron Cross Section Tech. Conf., Knoxville 1971)3rd Conference on Neutron Cross Sections and Technology, University of Tennessee, Knoxville, 15-17 March 1971, proceedings published as USAEC report CONF-710301 in 2 Volumes (Nuclear Structure Conf., Budapest 1972) Conference on Nuclear Structure and Study with Neutrons, Budapest, Hungary, 31 July 5 Aug 1972, proceedings published by Plenum Press (1974) (Magnetism and Magnetic Materials, Denver 1972) 18th Annual Conf. on Magnetism and Magnetic Materials, Denver, Colorado, 28 Nov-1 Dec 1972 American Inst.Phys.Conf.Proc. 10 (1973) (Int. Conf. on Photonuclear Reactions and Applications, Asilomar, 1973) International Conference on Photonuclear Reactions and Applications, Pacific Grove, California, 26-30 Mar 1973, proceedings published as USAEC report CONF-730301 in 2 volumes (Neytronnaya Fizika Conf., Kiev, 1973)Second National Soviet Conference on Neutron Physics, Kiev, 28 May - 1 June 1973, proceedings published by F.E.I. Obninsk in 4 Volumes (Applications of Nuclear Data Symp., Paris 1973)Symposium on Applications of Nuclear Data in Science and Technology, Paris, France, 13-16 Mar 1973, proceedings published by IAEA as STI/PUB/343 in 2 Volumes (Physics and Chemistry of Fission, Rochester 1973)3rd IAEA Symposium on the Physics and Chemistry of Fission, Rochester, NY, 13-17 Aug 1973, proceedings published by IAEA as STI/PUB/347 in 2 Vols,, Vienna 1974 2nd Int. Symp. on Neutron Capture Gamma Ray Spectroscopy and Related Topics, Petten, 2-6 Sep 1974 (europhysics Conf. on Nucl. Interactions at Medium and Low Energies, Harwell, 24-26 March 1975) 3rd All Union Conf. on Neutron Physics, Kiev, 9-13 June 1975 4th Conf. on Nuclear Cross Sections and Technology, Washington, D.C., 3-7 Mar 1975 Int. Conf. on Interactions of Neutrons with Nuclei, Lowell, Massachusetts, 6-9 July 1976 (Specialist's Meeting on Neutron Data of Structural Materials for Fast Reactors, CBNM, Geel, 5-8 Dec 1977 ) 4th All Union Conf. on Neutron Physics, Kiev, USSR, 18-22 Apr 1977 Meeting of the Technical Committee on Natural Fission Reactors, Paris, 19-21 Dec 1977, proceedings published by IAEA as STI/PUB/475
120
4- Notation and Nomenclature
78ALUSHTA 3rd Int. School on Neutron Physics, Alushta, USSR, 19-30 April 1978 78BNL 3rd Int. Symp. on Neutron Capture Gamma Ray Spectroscopy and RelatedTopics, Brookhaven National Laboratory, New York, 18-22 Sep 1978 78HARWELL Int. Conf. on Neutron Physics and Nuclear Data for Reactors and Other Applied Purposes, AERE Harwell, UK, 25-29 Sept 1978 78MAYAG Computers in Activation Analysis and Gamma-Ray Spectroscopy, Mayagues, Puerto Rico, 30 Apr - 4 May, 1978 79BOLOGN (Specialist's Meeting on Neutron Cross Sections of Fission Product Nuclei, Bologna, 12-14 Dec 1979) published as NEANDC(E)209 5th Nuclear Cross Section Technology Conf., Knoxville, Ten79KNOX nessee, 22-26 Oct 1979, proceedings published as NBS Special Publication 594 80KIEV 5th All Union Conf. on Neutron Physics, Kiev, 15-19 Sep 1980 81GRENOB 4th Int. Symp. on Neutron-Capture Gamma-Ray Spectroscopy and Related Topics, Grenoble, France, 7-11 Sep 1981 82ANTWER Int. Conf. on Nuclear Data for Science and Technology, Antwerp, Belgium, 6-10 Sep 1982 82KIAMES Topical Meeting on Advances in Reactor Physics and Core Thermal Hydraulics, Kiamesha Lake, New York, 22-24 Sep 1982, proceedings published as NUREG/CP-0034 in 2 Volumes All Union Conf. On Neutron Phys., Kiev, CCP, OCTOBER 2-6 83KIEV 1983 85SANTA Conf. on Nuclear Data for Basic and Applied Science, Santa Fe, USA May 13-17,1985 Int. Conf. on Neutron Physics, Kiev,CCP, Sept 14-18 1987 87KIEV 88MITO Conf. on Nucl. Data for Sci. and Technology, Mito, Japan, 1988 88SMOLEN Symp. on Nucleon Induced Reactions, Smolenice, 1988 90MARSEI Int. Conf. on the Physics of Reactors, Marseille, 1990 91JUELIC Conf. on Nucl. Data for Sci. and Technol., Juelich, Germany,1991 8th Int. Conf. Capture Gamma-Ray Spectroscopy, Fribourg, 1993 93FRIBOU 94GATLIN Conf. on Nucl. Data for Sci. and Technol., Gatlinburg, USA, 9-13 May,1994. Int. Conf. on Nuclei in the Cosmos, Aqui, ITALY,1994 (American 94AQUI Institute in Physics, New York ,1995) Conf. on Nucl. Data for Sci. and Technol.,Trieste, Italy, 19-24 97TRIEST May,1997 10th Int.Symp. on Capt. Gamma-Ray Spectroscopy, Santa Fe, 99SANTA NM, USA, 30 August - 3 September, 1999 02PRUHON 11th Int. Symp. on Capture Gamma-Ray Spectroscopy, Pruhonice, Czech Republic, 2-6 September, 2002. Int. Conf. on Nuclear Data for Sci and Tech., Santa Fe, USA, 26 05SANTA September -1 October, 2005
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