Theory and Applications of Moment Methods in Many-Fermion Systems

509 Pages·1980·28.247 MB·English

by  J. B. French (auth.), B. J. Dalton, S. M. Grimes, J. P. Vary, S. A. Williams (eds.)

Theory and Applications of
Moment Methods in
Many-Fermion Systems
Theory and Applications of
Moment Methods in
Many-Fennion Systems
Edited by
B.J. Dalton
Ames Laboratory
Iowa State University
Ames, Iowa
S.M. Grimes
Lawrence Livermore Laboratory
Livermore, California
J. P. Vary
AND
S. A. Williams
Ames Laboratory and Department of Physics
Iowa State University
Ames, Iowa
PLENUM PRESS · NEW YORK AND LONDON
Library of Congress Cataloging in Publication Data
International Conference on Theory and Applications of Moment Methods in Many
Fermion Systems, Spectral Distribution Methods, Iowa State University, 1979.
Theory and applications of moment methods in many fermion systems.
Sponsored by the Physics Department, Iowa State University, and others.
Includes index.
1. Fermions-Spectra-Congresses. 2. Spectral energy distribution-Congresses. 3.
Many-body problem-Congresses. 4. Moments method (Statistics)-Congresses
I. Dalton, W. Bill J., 1940- II. Iowa State University of Science and Tech
nology, Ames. Physics Dept. III. Title.
QC793.5.F427I57 1979 539.7'21 80-21054
ISBN-13: 978-1-4613-3122-3 e-ISBN-13: 978-1-4613-3120-9
DOl: 10.1007/978-1-4613-3120-9
Proceedings of the International Conference on
Theory and Applications of Moment Methods in Many-Fermion
Systems - Spectral Distribution Methods,
held at Iowa State University, Ames, Iowa,
September 10-14, 1979.
©1980 Plenum Press, New York
A Division of Plenum Publishing Corporation
227 West 17th Street, New York, N.Y. 10011
Softcover reprint of the hardcover 1st edition 1980
All righ ts reserved
No part of this book may be reproduced, stored in a retrieval system, or transmitted,
in any form or by any means, electronic, mechanical, photocopying, microftlming,
recording, or otherwise, without written permission from the Publisher
PREFACE
The first international conference on "Theory and Applications
of Moment Methods in Many-Fermion Systems" was held September 10 -
13, 1979 at Iowa State University. Manuscripts of the invited
talks presented at this conference are the contents of this volume.
These manuscripts were prepared and delivered to the editors by
the authors; the responsibility for any errors in scientific con
tent is theirs. While we, the editors, have made every effort to
keep this volume as free from typographical errors as possible, we
accept responsibility for such errors as do occur, even those which
may be mistaken for scientific error. All but one of the invited
talks given at the conference are reported here; those authors
apparently felt unable to provide the editors with manuscripts.
The editors. who also served as the organizing committee,
would like to express appreciation to the sponsors of this con
ference: Physics Department, Ames Laboratory, Energy and Mineral
Resources Research Institute, and the Graduate College, all of
Iowa State University, the National Science Foundation, and the
U. S. Department of Energy. Their generosity both in terms of
funding and support made the conference possible.
We should also like to express our gratitude to the Interna
tional Advisory Committee whose prestige lent support to the con
ference and whose advice in topic selection was invaluable. Those
members were:
R. Arvieu Grenoble, France
C. Bender Lawrence Livermore Laboratory
J. B. French University of Rochester
W. Fowler California Institute of Technology
R. Gordon Harvard University
J. N. Ginocchio Los Alamos Scientific Laboratory
J. Huizenga University of Rochester
A. Merts Los Alamos Scientific Laboratory
J. C. Parikh Ahmedabad, India
C. Quesne University Libre de Bruxelles, Belgium
v
vi PREFACE
S. Raman Oak Ridge National Laboratory
R. R. Whitehead University of Glasgow, Scotland
S. S. H. Wong University of Toronto, Canada
lye are also grateful to all the participants in this conference;
credit for its success is theirs.
Finally, we express profound gratitude to our secretary,
Mrs. Pat Thiede, who took care of all the conference logistics,
smoothly handled all the problems of the conference, and typed
all of the manuscripts. Throughout, she had to deal with the
editors which required the patience of a saint.
B.J. Dalton
S.M. Grimes
J.P. Vary
S.A. Williams
CONTENTS
Elementary Principles of Spectral
Distribution ..... 1
J.B. French
Limits of Thermodynamic Models for Nuclear
Level Densities . . .. •••• 17
S.M. Grimes
Statistical Properties from High Resolution
Proton Resonance Reactions • . . • . . 33
G. E. Mitchell
Nuclear Level Densities in Astrophysics 61
S.E. Woosley
Astrophysical Opacities and Moment Methods in
the Interpretation of Spectral
Observation in Atomic Spectra 81
A.L. Merts
Special Topics in Spectral Distributions . . . . . . . . . 91
J.B. French
On the Averages of Operators in Finite
Fermion Systems . • . • . 109
J.N. Ginocchio
Moment Techniques in Atomic and Molecular
Scattering Theory • • . . . . • • . 129
William P. Reinhardt
The Representative-Vector Method for Calculating
Operator Moments . . . • . . . • . . . . . 151
S.D. Bloom and R.F. Hausman, Jr.
vii
viii CONTENTS
Phenomenological Approach to Nuclear Level
Densities ••.••.•••• 167
G. Reffo
Stieltjes-Tchebycheff Moment-Theory Approach
to Photoeffect Studies in Hilbert Space 191
P.W. Langhoff
Polynomial Expansions and Transition
Strengths •••••••••• 213
J.P. Draayer
Moment Methods and Lanczos Methods • . • • • • • • • • •• 235
R.R. Whitehead
Spectral Distributions and Symmetries 257 .
C. Quesne
Calculation of Spin Cutoff Parameters using
Moment Techniques •••.••••• 273
S.M. Grimes
Group Symmetries and Information Propagation • • • • • •• 287
J.P. Draayer
Level Densities in Nuclear Physics • • . . • • • • . . •• 307
M. Beckerman
Spectral Methods Applied to Ising Models . • • • • • • •• 327
B. DeFacio, C.L. Hammer, and
J. Ely Schrauner
The Partition Function as a Laplace
Transform of a Positive Measure in the
Strength Parameter • . . . . • • • 363
D. Bessis
Application of Spectral Distributions in
Effective Interaction Theory • • • 371
B.D. Chang
Beta Decay and Strength Distributions 389
J.C. Hardy
Study of Effective Interactions and Models in
Nuclei Using the Moment Method • • • 405
Jitendra C. Parikh
CONTENTS ix
Realistic Hamiltonians for No-Core Moment
Methods Studies • . • . . . . . 423
J.P. Vary
Fixed-J Moments: Exact Calculations 437
C. Jacquemin
Radial Densities of Nuclear Matter and Charge
Via Moment Methods . . • •• .• . • 451
B.J. Dalton
Sum Rules, Strength Distributions and Giant
Resonances ...•••••.•.•. 463
O. Bohigas
Part ic ipant s 499
Index 505
ELEMENTARY PRINCIPLES OF SPECTRAL DISTRIBUTIONS
J. B. French
Department of Physics and Astronomy
University of Rochester
Rochester, N.Y. 14627
I. INTRODUCTION
It is a common observation that as we add particles, one by
one, to a "simple" system, things get steadily more and more com
plicated. For example if the system is describable in shell-model
terms, i.e., with a model space in which m particles are distributed
over N sinrle-particle states, then, as long as m« N, the dimen-
l:)
sionality increases rapidly with particle number. On the other
hand, for the usual (l+2)-body Hamiltonian, the (m~2)-particle
spectrum and wave functions are determined by operators defined in
the one-particle space (for the single-particle energies) and the
two-particle space (for the interactions). We may say then that
the "input" information becomes more and more fragmented as the
particle number increases, the fixed "amount" of information being
distributed over more and more matrix elements.
On the other hand there arise also new simplicities (Fig. 1)
whose origin is connected with the operation of statistical laws.
There is a "macroscopic" simplicity corresponding to ¢e fact that
the smoothed spectrum takes on a characteristic shape1 defined by
a few parameters (low-order moments) of the spectrum. There is a
"microscopic" simplicity corresponding to a remarkable spectral
rigidity2 which extends over the entire spectrum and guarantees us
that the f1ucutuations from uniformity in the spectrum are small
and in many cases carry little information.
The purpose of spectral-distribution theory, as applied to
these problems, is to deal with the complexities by taking advantage
of the simplicities.
2 J. B. FRENCH
To make clear what we mean by a "spectral distribution" we
consider a few examples:
1) H-Eigenvalue Distribution: PE(x) defines the density of eigen
values of H acting in the model space; for finite spaces we take
JPE(x)dx = 1 so that a dimensionality factor d(m) is needed for the
true state density I(x). Then the moments of PE are ~ = JPE(x)xPdx=
<HP>m where, for any operator G, d(m)x<G>m:: «G»m is the trace
of G over the model space (or over the m-particle space as the case
may be). From the standpoint of statistical mechanics PE(x) and the
partition function Z(8) are related by Laplace transformation, and
therefore carry the same information; PE(x), which is to s~me extent
measurable, is more convenient for us. From p, aA a 6un~on 06 the
p~met~ 06 the ~y~tem, we should be able to determine most
things of interest about the system, just as one can from the par
tition function. The complete evaluation is in general out of the
question but we shall be able to calculate a smoothed version of p,
and deal with the deviations by introducing an appropriate ensemble.
Thus
P = Psmoothed + Pfluctuation (1)
in which our main interest will be in the first term, for which we
shall drop the subscript label. From the smoothed density we can,
as first done by Ratchiffl , recover a (smoothed) spectrum by re
placing the smoothed distribution function F(x) = JXp(z)dz by a
staircase function with jumps of magnitude d-l at x values for which
F is half-integral; an example is given in Fig. 1. An obvious
extension is called for if the spectrum has known degeneracies.
2) Expectation-value Distributions: We might be interested in the
expectation value of an operator K in the ~amiltonian eigenstates,
i.e. in <~E K ~E> = K(E) say, or of H in J eigenstates, etc. The
second example would come up in dealing with the J-decomposition of
the level density.
Figure 1. The upper fieure shows a computer histogram of a (ds)12,
J=T=O spectrum for a "realistic" Hamiltonian. The di
mensionality is 839. The bins are 1.7 MeV wide, and
the Gaussian values are indicated by oval points.
The lower figure shows a comparison between the exact
shell-model spectrum and the smoothed spectrum generated
by using a four-moment distribution. Two segments of the
spectrum are shown, for levels 1-23 and 503-524. In each
case the exact spectrum is to the left.