Direct Nuclear Reactions

387 Pages·1983·7.009 MB·English

by  Norman Glendenning (Auth.)

Direct Nuclear Reactions
NORMAN Κ. GLENDENNING
Nuclear Science Division
Lawrence Berkeley Laboratory
University of California
Berkeley, California
1983
ACADEMIC PRESS
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Library of Congress Cataloging in Publication Data
Glendenning, Norman K.
Direct nuclear reactions.
Includes index.
1. Direct reactions (Nuclear physics) I. Title.
QC79^.8.D57G578 1983 539.7'5^ 82-2^365
ISBN 0-12-286320-8
PRINTED IN THE UNITED STATES OF AMERICA
83 84 85 86 9 8 7 6 5 4 3 2 1
In Memory of Florence and Norman,
Mother and Father
Preface
The spontaneous disintegration of long-lived, naturally occurring isotopes
provides one source of information on nuclei. However, only a limited number of
nuclei are accessible for study by this natural process and then only under a
narrow range of circumstances. On the other hand, nuclear reactipns can be
induced in the myriad of pairwise combinations provided by stable or long-lived
nuclei and over the wide range of energies provided by the accelerators in the
physics laboratories of the world. Reactions therefore provide the greatest
volume and widest range of nuclear data. The energy loss of a beam particle can
be directly interpreted as an excitation energy in the target nucleus. Usually,
however, the data acquire meaning for the structure of nuclei only after they have
been interpreted through a reaction theory. The synthesis of such accumulated
information into a coherent theory of the nucleus is the main subject of nuclear
physics. There are a number of volumes on nuclear structure that review this
ongoing process.
This book is about the theory of direct nuclear reactions. It emphasizes the
microscopic aspects of the reactions and their description in terms of the changes
induced in the motions of the individual nucléons by the reaction. However,
because collective motion can be accurately described by a few collective pa-
rameters, we sometimes depart from a strictly microscopic description; any
account of direct reactions would otherwise be incomplete.
The book begins essentially at the beginning, assuming only a modest knowl-
edge of quantum mechanics and some acquaintance with angular momentum
algebra, and ends by describing some of the most recent topics. The principal
results of the theory are described with emphasis on the approximations involved
to provide a guide to how well the theory can be expected to hold under specific
xiii
xiv Preface
experimental conditions and also to suggest areas in which improvements can be
made. Applications to the analysis of experiments are also emphasized, not only
because reactions are interesting in themselves, but because they can be used to
measure nuclear properties. Indeed, most of our detailed knowledge of nuclear
properties has been discovered by means of reactions.
The goal of the book is thus to provide the novice with the means to become
competent to do research in direct nuclear reactions and to provide the experi-
enced researcher with a detailed discussion of the advanced topics. Topics cov-
ered include coupled equations and the distorted-wave Born approximation, form
factors and their nuclear structure content, the basis of the optical potential as an
effective interaction, reactions such as inelastic single- and two-nucleon transfer
reactions, the effect of nuclear correlations, the role of multiple-step reactions,
the theory of inelastic scattering and the relationship of the effective interaction
to the free one, reactions between heavy ions, the polarizability of nuclear wave
functions during a reaction, and the calculation of components of the optical
potential arising from specific collective or transfer reactions.
Acknowledgments
I began thinking of this project in 1969, at the suggestion of the editors at
Academic Press. However, it was not until early 1978 that I actually began
writing, in response to an invitation from S. K. Bhattacherjee and B. Banerjee of
the Tata Institute of Fundamental Research, Bombay, to give a course at a
summer school in Bangalore, India. I very much enjoyed the summer school and
the hospitality of the organizers and participants. When I had completed the
lecture notes I realized that I had in hand a part of the project that had been
suggested almost 10 years earlier. More years have elapsed, and it is now much
later than my too optimistic estimate of when I would complete the manuscript.
The delay did, however, provide the opportunity of including some very recent
and quite important developments.
The field of direct nuclear reactions is now more than 30 years old. Many
researchers throughout the world have contributed to the development of the
theory and its applications to the interpretation of experimental data, the predic-
tion of new phenomena, and of course the design of accelerators, detection
equipment, and the experiments that actually obtained the data. Of the volumi-
nous scientific literature that documents the achievements in this field, reference
here is directed mainly to the theory of reactions and accordingly does not do
justice to all of those whose research inspired the development of the theory and
made it a fruitful endeavor.
I especially thank B. G. Harvey of the Lawrence Berkeley Laboratory for
nurturing my early interest in this field through his own keen interest and pro-
vocative experiments. I am also grateful to M. A. Nagarajan of the Daresbury
Laboratory for reading parts of the manuscript and for offering many useful
suggestions from his wide knowledge.
xv
xvi Acknowledgments
Two of the chapters, those having to do with the recent advances in relating
the free interaction and the effective interaction in the theory of inelastic scatter-
ing, could not have been written in time for this publication were it not for
extensive help from F. Petrovich of Florida State University. He contributed
theoretical developments before their publication in the journals. I am very
grateful to him for his generosity.
Notational Conventions
sé antisymmetrization operator
A, a intrinsic coordinates of nucleus A and a, respectively, of which there are 3(A — 1)
and 3(a - 1)
a, a', a" different channels in the partition a, with a denoting either a typical channel or
the ground state channel and α', α" channels in which one, the other, or both of
nuclei a or A is excited
a, /?,... frequently used to denote different two-body partitions a + A, b + B,..., of a
many-nucleon system
B, b similar to A, a, but for nuclei Β and b
(±]
χ optical model wave function in channel α in which the relative motion is distorted
α
by an optical potential (the asymptotic condition of outgoing (+) or incoming
( —) waves is indicated)
Ε eigenvalue of H, eigenfunction Ψ, or ψ£±}
E kinetic energy in channel a, eigenvalue of T eigenfunction of exp(ik · rj (E =
a ai a a
ε eigenvalue of H, eigenfunction Φ, and where ε and ε are eigenvalues of H
α a α Α α A
e
and H with eigenfunction Φ and Φ (ε = ε + a)
a Α Λ α Α
Η total Hamiltonian of the A -I- a = Β + b ... many-nucleon system, ordered in
terms of various partitions (H = H+ T+ V = H + T + V = - · ·)
a a a ß ß ß
H , H nuclear Hamiltonians for nuclei A and a
A a
xvii
Notational Conventions
H + H
A a
wave number of relative motion in channel a, (k\ = 2mEJh)
a
2/+ 1
reduced mass in channel α
combinatorial symbol is A/!/[(A7 — n)\n\~\
product (ΦΦ) °f nuclear wave functions in channel α and therefore a function
Α 8
of A and a intrinsic coordinates and of spin and isospin
the many-nucleon wave function in channel α when the relative motion is a plane
(or Coulomb) wave; i.e., Φ exp(ik · r)
α a a
nuclear state vectors or wave functions (if coordinate space representation is im-
plied or specified) for nucleus A and a, respectively they are functions of A and a,
respectively, and of spin and isospin coordinates in general (a sometimes denotes
a light nucleus like 3He)
single-particle spin-orbit wave function
state vector or wave function of the A + a many-nucleon system.
state vector or wave function of many-nucleon system as above, for which the
boundary conditions are specified as follows: there is a plane (or Coulomb) wave
in partition α representing the incident beam in a collision, and asymptotically at
infinity there are outgoing waves in all open channels of all partitions; the wave
function is a function of the coordinates of A + a nucléons, which in the center
of mass frame are taken as A, a, r
a
the integration (|ΨΪΨΰ carried out over all position coordinates and the sum-
mation carried out over all spin and isospin coordinates
as <Ψ|Ψ«> but the relative coordinate is not integrated (unless otherwise defined);
α
the bracket, therefore, is a function of r
e
a spin and isospin summation (ΧΨ?Ψ^) unless otherwise defined
0, φ polar angles of r
relative coordinate between nuclei A and a
kinetic energy (-{h2/2m)V2) of relative motion in channel α
a
optical potential in channel α
shell-model potential acting on nucléon labeled i
radial part of above wave function φ (this can also depend on j if the shell-model
η(ί
potential depends on spin)
interaction other than optical potential in channel a, often a sum of two-body
effective interactions
statistical weight of direct integrals contributing to the amplitude of a
nucléon transfer reaction see (5.32)
nucléon spin eigenfunction (or isospin if τ replaces σ)
two-particle spin eigenfunction with eigenvalues S2 = S(S + 1), S = 0 or 1 and
S = M
2
Chapter 1
Introduction :
Direct and Compound
Nuclear Reactions
A. THE OBSERVABLES
The nature of the experimental observation is the following: A beam
of particles, such as protons, deuterons, alpha particles, or heavier nuclei
(referred to as heavy ions), is accelerated to a desired energy and then
deflected so as to strike a target of known isotopic composition. The energy
and scattering angle of some of the products of the collision are measured.
For example, in a reaction initiated by protons, the proton itself will some-
times emerge, deflected in angle but having the same energy in the center-
of-mass system. This is elastic scattering, and (as we shall discuss) the
measurement of this cross section is important because its analysis yields
the parameters of the optical potential
Sometimes the proton will excite the target nucleus from its ground to
some higher-energy state, thus losing some energy and at the same time
being deflected in angle. This is an inelastic event. The cross section for
such an event yields information on the spin and parity of the nuclear
transition and, in a way that will be described more fully, is sensitive to
the wave functions of the nucléons that are excited. When the detector angle
ι