Atlas of Neutron Resonances: Resonance Parameters and Thermal Cross Sections

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

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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