Dynamical Collision Theory and its Applications

495 Pages·1991·28.407 MB·English

by  Sadhan Adhikari (Auth.)

Dynamical
Collision Theory
and
Its Applications
Sadhan K. Adhikari
Departamento de Fisica
Universidade Federal de Pernamhuco
Recife, Pernamhuco, Brazil
Kenneth L. Kowalski
Department of Physics
Case Western Reserve University
Cleveland, Ohio
ACADEMIC PRESS, INC.
Harcourt Brace Jovanovich, Publishers
Boston San Diego New York
London Sydney Tokyo Toronto
This book is printed on acid-free paper. @
COPYRIGHT© 1991, BY ACADEMIC PRESS, INC.
ALL RIGHTS RESERVED.
NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR
TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC
OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY
INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT
PERMISSION IN WRITING FROM THE PUBLISHER.
ACADEMIC PRESS, INC.
1250 Sixth Avenue, San Diego, CA 92101
United Kingdom Edition published by
ACADEMIC PRESS LIMITED
24-28 Oval Road, London NW1 7DX
Library of Congress Cataloging-in-Publication Data
Adhikari, Sadhan K., date.
Dynamical collision theory and its applications/Sadhan K.
Adhikari, Kenneth L. Kowalski.
p. cm.
Includes bibliographical references and index.
ISBN 0-12-044273-6 (acid-free paper)
1. Collisions (Nuclear physics)—Mathematical models.
2. Scattering (Physics)—Mathematical models. I. Kowalski, Kenneth
L. II. Title.
QC794.6.C6A24 1991
539.7'5—dc20 90-47161
CIP
Printed in the United States of America
91 92 93 94 9 8 7 6 5 4 3 2 1
To our families,
Ratnabali and Avishek,
and
Audrey, Claudia, and Eric,
and to our parents.
Preface
Collisions among atomic and subatomic particles that can be described in
terms of nonrelativistic quantum mechanics occur throughout nature.
Potentials are used in the Schrodinger equation as phenomenological
representations of the microscopic electromagnetic or hadronic interactions
that mediate the collisions. This model, along with its relativistic general-
izations, has been extraordinarily successful in organizing and explaining a
host of chemical, atomic, nuclear, and hadronic phenomena, and is likely to
remain as useful in the future. Thus, an understanding of the techniques for
calculating the predictions of this model for collision processes is relevant to
much of the physical sciences. The major objective of this book is to provide
an introduction to some of the powerful methods that have evolved in the last
two decades for carrying out these calculations to an accuracy sufficient to test
the model.
The real problem in all of this, of course, is to deal with collisions that
involve bound composites of constituent particles. There are widely differing
philosophies, that we call "dynamical strategies," as to how one should go
about doing this in a way that is both practical yet comprehensive enough to
be scientifically credible. The range of currently popular dynamical strategies
is fairly accurately represented by the extant monographs on collision theory.
If we exclude the comprehensive works, e.g., Goldberger (1964), Joachain
(1975), and Newton (1982) (We use the last name of the first'author and the
xi
xii Preface
year of publication as a labeling for our citations throughout the book.), which
are concerned mainly with the development of the foundations of the subject
rather than its applications, we are left with monographs that represent two
distinct dynamical strategies. The more traditional of these, e.g., Tobocman
(1961), Mahaux (1969), Austern (1970), Jackson (1970), Hodgson (1971),
Wildermuth (1977), Satchler (1980, 1983), Glendenning (1983), and Fano
(1986), adapt strategies that are designed to yield dynamical equations that
are two-body in nature, and therefore routinely calculable. In many cases,
modifications of the basic approximations are introduced in the form of
phenomenological effective interactions. This represents the adaptation of still
another modeling of the collision process rather than a controlled correction
to the original approximation. On the other hand, those books that follow a
few-body point of view, e.g., Watson (1967), Schmid (1974), Thomas (1977),
and Glockle (1983a), basically take all the active degrees of freedom into
account and aim for an exact solution. The latter approach, of course, rapidly
loses its appeal because of sheer complexity as the number of particles goes
beyond three. Both approaches have been very fruitful in their special realms.
In our opinion, an exciting development in the past two decades concerns
the body of work that has attempted to bridge the calculational and intellec-
tual differences between the "conventional" and the "few-body" strategies.
Surely it must be possible to construct controlled corrections to conventional
approximations, such as the coupled-reaction channel and resonating-group
methods, as well as to forge complete sets of collision integral equations
such as those due to Faddeev and Yakubovskii into practical calculational
schemes by a well-articulated series of approximations. Although these hopes
are not yet fully realized, we have attempted to present a point of view that
will provide an introduction to that research by means of a solid grounding
in the realizable means of formulating and solving collision integral equations
from N = 2 on up. It is this point of view that we call "dynamical collision
theory" and which is elaborated upon in further detail in Ch. I.
Two-particle potential scattering is reviewed in Ch. II. Then, in Ch. Ill, we
have an extensive review of a variety of methods for calculating off-shell two-
body amplitudes as well as approximating them by finite-rank forms. In
Ch. IV, the interpretation and applicability of the multichannel, multiparticle
Lippmann-Schwinger equations, which form the basis for the conventional
approaches, are discussed in considerable detail. A self-contained discussion is
given in Ch. V of most of the known ΛΓ-particle connected-kernel integral
equations; their physical predictions are reviewed in Chs. VII and VIII for
N = 3 and N = 4, respectively. We have included descriptions of contempo-
rary field-theoretical and relativistic approaches, such as the Dirac phenom-
Preface xiii
enology for intermediate energy nucleon-nucleus scattering, to the extent
that they fit into our general format. The review of the singularity structure
of multiparticle amplitudes and the associated dispersion-relation techniques
given in Ch. VI is, in part, complementary to that format; on the other hand,
the singularity structure associated with the Efimov effect for N = 3 plays a
crucial role in interpreting the three-nucleon calculational results reported
in Ch. VII. Finally, in Ch. IX, we have described what is known at present
about the relationship between the conventional (optical potentials, multiple-
scattering theories, and the coupled-reaction channel and resonating-group
methods) and the few-body approaches.
A number of important topics have been ignored, such as the Coulomb
problem and configuration-space methods, that lend themselves to the use of
differential equations. At present there is no entirely satisfactory way to in-
corporate unscreened Coulomb potentials into multiparticle collision theory.
We almost exclusively work with collision integral equations, and these
are most efficiently dealt with in momentum space. Undoubtedly, we have
also inadvertently ignored, or worse, not adequately represented, work that
is pertinent to what we do. We sincerely apologize to those authors who
have been dealt with in either fashion.
The book is primarily intended for chemists, physicists, and graduate
students interested in general scattering theory, intermediate and low-energy
hadron and nuclear physics, atomic and molecular physics, statistical
mechanics, and physical and quantum chemistry. Moreover, there are a
number of topics that may prove interesting to mathematicians as well as
particle physicists. Parts, or all, of the book are suitable for use as a primary or
secondary text in advanced graduate courses that involve collision theory.
Within the range of subjects that we discuss, there is considerable variation
in the level of the presentation. There are a number of reasons why this has
occurred that are related to the subject itself, to frequency of use in numerical
calculations, and to expectations about reader interest. Whenever possible, we
have pointed out what appear to be open questions for further research both in
the execution and organization of calculations as well as formal problems. In
some chapters, the gap between what is known and what one would like to
know is all too obvious.
A number of figures in this book are reproduced from other works. We
thank the authors and the American Physical Society, the North-Holland
Publishing Company, and Springer-Verlag for granting us permission to use
copyrighted material from their publications.
It gives us great pleasure to express our gratitude to our friends and
collaborators R. D. Amado, D. Bolle, A. Delfino, L. L. Foldy, A. C. Fonseca,
xiv Preface
T. Frederico, E. Gerjuoy, W. Glockle, R. Goldflam, M. S. Hussein, F. S. Levin,
J. A. Lock, H. P. Noyes, T. A. Osborn, A. Picklesimer, S. C. Pieper, W. N.
Polyzou, E. F. Redish, D. K. Sharma, E. R. Siciliano, P. C. Tandy, R. M.
Thaler, W. Tobocman, and L. Tomio, who have contributed so much in
enhancing our knowledge about the topics covered in this book.
We are grateful for the support given throughout the duration of this
project by W. Gordon, Chairman of the Case Western Reserve University
Physics Department. S. K. A. acknowledges financial support from Conselho
Nacional de Desenvolvimento—Cientifico e Tecnologico of Brazil, which
made the collaboration so pleasant. All of the research reported in the book
involving K. L. K. was funded, in part, by a series of grants from the U.S.
National Science Foundation. Part of the manuscript was written during the
sabbatical leave of K. L. K. at the Argonne National Laboratory; the warm
hospitality of the High-Energy Division is gratefully acknowledged.
We deeply appreciate the encouragement we received from S. R. A. Canuto,
M. A. F. Gomes, and D. L. Karatas. Finally, we thank Mrs. Dorothy M.
Straughter for her patience, enthusiasm, and superb professional effort in
word processing the manuscript.
One of us (S. K. A.) would like to note his new address and affiliation:
Instituto de Fisica Teorica, Universidade Estadual Paulista, Rua Pamplona
145, 01405 Sao Paulo, Sao Paulo, Brazil.
Sadhan K. Adhikari and Kenneth L. Kowalski
Chapter 1
Scattering Theory
1.1. Introduction
Scattering processes occur at every level of natural phenomena. They often
can be described qualitatively in simple, universal terms. It is usually a lot
more difficult to characterize them quantitatively, which is the objective of
scattering or, as it is sometimes called, collision theory. We are concerned with
the development and, especially, the application of that part of this theory
that has nonrelativistic quantum mechanics for a fixed number of constituent
particles as its dynamical basis. This encompasses those models of atomic and
subatomic processes for which the interactions between particles are repre-
sented by potentials. We also consider the relativistic generalizations that
are possible so long as the explicit effects of particle creation and annihilation
are negligible.
The quantum theory of scattering was developed to explain experimental
results in atomic, nuclear, and particle physics. As the styles and agendas of
these fields diverged, so did their terminologies. In what follows, our termi-
nology has been chosen to cover the broadest range of circumstances with
precision.
The mathematical formulation and substantiation of scattering theory
has produced breakthroughs for doing practical calculations such as the
1
2 1 Scattering Theory
Lippmann-Schwinger and Faddeev equations, as well as rigorous work such
as proofs of asymptotic completeness. This effort has been accompanied by an
unfamiliar terminology and a literature guided very often by its own agenda.
Our goal is to render this approach and its results more understandable to
other practitioners of scattering theory. Our objectives in this regard are:
(i) To establish that multiparticle scattering theory is soundly based and
that definitive solution algorithms exist, in the form of linear integral
equations, for calculating any scattering process with a dynamics
defined by the Schrodinger equation;
(ii) To make use of an efficient formal machinery for dealing with multi-
particle continuum configurations; and
(iii) To indicate how conventional low-order approximations can be placed
in a broader framework.
There are a number of interesting and successful approaches in collision
theory that at the present time are better described in their own right rather
than as special cases of a more comprehensive theory. For example, much of
the work on relativistic few-body collisions, Dirac methods in nuclear scat-
tering, and mean-field techniques for heavy-ion collisions is of this kind.
Our remaining objectives involve the description of contemporary work on
some of these and similar topics.
Scattering refers to the evolution, by means of an interaction of finite
duration, of a system from a collection of essentially noninteracting compo-
nents to another collection of this kind. In the absence of external influences,
each of these components is supposed to persist in its free motion for an
interval that is long with respect to the interaction time, and is regarded as a
particle with an internal structure that is of secondary interest asymptotically.
The complete interaction among the free particles is called a collision.1 In
many simple situations, such as two-particle scattering by a short-range force,
the complete interaction characterizing the collision is localized in space and
time. When more than two constituents are involved, this is not true even if all
of the interconstituent forces are of short range.
A scattering, therefore, involves a redistribution, in some manner, of a group
of entities that can be identified with the fundamental constituents of the
model of interest, say electrons and nuclei in atomic physics. The freely
moving asymptotic components may either be these fundamental constituents
or stable composites of them, e.g., atoms or ions in the atomic case. In theories
where there is confinement, such as quantum chromodynamics (QCD), only
composites appear in the asymptotic configurations.
Evidently, scattering encompasses the entire history of a process from the
infinite past to the infinite future. So one can picture the process in a time-
1.1. Introduction 3
independent fashion, if semi-infinite time spans are associated with infinite
reaches of space measured from an interaction region. Most of the scattering
process, whether gauged by duration or distance, is dynamically trivial. Only
collisions engage the dynamics in an interesting way.
We are mainly interested in the means of calculating collision processes by
solving the integral equations for the transition probability amplitudes. These
equations incorporate the assumed interparticle interactions and also the
asympotic boundary conditions. It is this approach to the subject that we call
dynamical collision theory.
Nonrelativistic quantum mechanics has been used extensively and with
considerable phenomenological success for the description of atomic and
nuclear collisions. Within this framework, one can formulate well-defined
dynamical equations with solutions whose general features are understood.
One can also be very specific about how one can go about obtaining these
solutions. This is not the case for the relativistic field theories—quantum
electrodynamics (QED) and QCD—that provide a microscopic basis for
atomic and nuclear collision processes, respectively.
The passage from QED to the description of atomic and molecular
interactions in terms of Coulomb and shorter-ranged potentials is reasonably
clear in principle, yet quite complicated in practice. None except the simplest
features of interatomic forces is presently calculable ab initio. So the modeling
of such interactions by semi-phenomenological potentials, which are then
employed in a Schrodinger or a Dirac equation, represents the dominant style
of atomic scattering theory, and is adhered to in this book.
By way of contrast, it is not yet known how to describe hadronic processes
starting from QCD. Those models of nuclear processes that have required
some statement about the nucleon-nucleon (NN) interaction have usually
employed the device of a potential. This has been very successful in codifying
the general features of the constituent NN collisions. An important advantage
of the NN potential as a repository of the complex interactions among the
constituents of the composite nucleons is the simplicity with which one can
exploit this information: The potential is just inserted into the Schrodinger
equation. We accept this as a means to the dynamical ends pursued in this
book.
Relativistic effects and some features of particle annihilation/creation have
been successfully modeled in ways that lead to dynamical schemes not too
different from what one obtains in nonrelativistic scattering theory. Models for
low-energy to medium-energy pion-deuteron scattering are exemplars of
these approaches. Generally, pions can be absorbed in nuclei so that one is
confronted with manifest particle annihilation. Of course, it is not yet possi-
ble to deal with these processes in a fundamental way. Nevertheless, these