Table Of ContentNUCLEAR STRUCTURE
WILLIAM F. HORNYAK
Department of Physics
University of Maryland
College Park, Maryland
ACADEMIC PRESS New York San Francisco London 1975
A Subsidiary of Harcourt Brace Jovanovich, Publishers
COPYRIGHT © 1975, BY ACADEMIC PRESS, INC.
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Library of Congress Cataloging in Publication Data
Hornyak, W F (date)
Nuclear structure.
Includes bibliographies and index.
1. Nuclear physics. I. Title.
QC776.H6 539.7'4 74-10201
ISBN 0-12-356050-0
PRINTED IN THE UNITED STATES OF AMERICA
PREFACE
This text is an outgrowth of a course conducted by the author for several
years at the University of Maryland. It covers material normally discussed
in courses relating to nuclear structure. The text is basically designed for a
second-year graduate student, preferably but not necessarily having had
some introductory nuclear physics at an elementary or undergraduate level.
The presentation while relying only slightly on such a background does,
however, require a good knowledge of the elements of quantum mechanics
including an introduction to Dirac theory, knowledge commonly gained
from the usual one-year graduate level course in the subject.
One motivation for writing this text has been to present the subject in a
manner offering the realistic possibility that an average student with proper
preparation could, in fact, absorb this material. Perhaps the author should
immediately interject the comment that he is aware of the limited technical
facility with quantum mechanics of the average student. This, of course, is
due to lack of time for the student to absorb the full implications of the theory,
which is unavoidably presented to him in rather concentrated doses. This
"working text" allows for this limitation and is intended for the student.
However, the research scientist will also benefit from numerous current
reviews and up-to-date references.
IX
X PREFACE
A limited number of selected topics are treated with some completeness
using techniques that are only simple extensions of those reasonably expected
to have been introduced in the prerequisite quantum mechanics course.
Perforce this implies the use of somewhat "old fashioned" methods resulting
in some loss of the elegance inherent in current treatments but which have
the advantage of throwing some light on the historical development of the
subject. The development of the newer techniques and their application to
nuclear theory is left to more specialized works. Emphasis is placed on
establishing the basic concepts with specialized and detailed applications
only sparingly considered in contradistinction to the more usual treatment.
Whenever possible, such concepts are reinforced by examples employing
direct calculations with simple model wave functions, operators, etc., partly
to convert in the student's mind his knowledge of elementary quantum
mechanics into a workaday tool. In large part, this is made possible by the
adopted philosophy of relying on (and using) only elementary theoretical
methods in the text.
While the above philosophy may indeed reduce the reflection coefficient
at the student interface, the author is well aware that the price paid may be
to blur the available accuracy of the description of nature by denying the
advantage that the precision of modern techniques is, in fact, designed to offer.
Although a strong effort has been made to minimize this problem, some
ambiguities undoubtedly remain.
The first chapter, Nucleon-Nucleon Forces, is in some ways different in
character from the remaining chapters. In view of the recent encouraging
progress made in relating both the free nucleon-nucleon interaction and
the complex nuclear many-body problems to meson theory, this chapter
attempts to develop a background in the relevant portions of elementary
particle physics adequate to provide the student with a basis for under
standing these relationships. Indeed, it may be argued that nuclear physics
has arrived at the point where the inclusion of elementary particle inter
actions is in fact essential. This chapter inevitably tends to be in the nature
of a survey with a rather large number of reference citations. Introducing
the student to periodical literature may also be a service in itself. A subsidiary
consideration motivating the breadth of this chapter is the recognition that
classes devoted to the topics of this text are also attended by students of
elementary particle physics as well as of nuclear physics. The relevant
connections between these basic fields are generally appreciated by all the
students.
The remaining chapters are more self-contained and generally give a
step-by-step derivation of the important equations (sometimes provided in
an Exercise). The "it can be shown" presentation is kept to a minimum and
over 125 exercises are provided along with references to additional profitable
PREFACE xi
reading. A special effort has also been made to provide numerous reference
citations in those areas in which there is current active research. To provide
coherence and continuity a number of topics are illustrated by selecting
examples from the IP-shell nuclei. The treatment of these somewhat simpler
cases permits a sharper focus on the theoretical point in question without
any particular loss in essential generality.
Somewhat more material is presented than can be thoroughly covered in a
typical semester to allow the instructor the freedom to select his own desired
emphasis. The topics covered—nucleon-nucleon forces, nuclear shape and
nuclear moments, nuclear matter characteristics, single-particle shell model,
individual-particle model, collective nuclear effects, electromagnetic inter
actions with nuclei, and beta-decay—emphasize only topics directly related
to the properties of the nucleus as such. Topics such as ionization, stopping
of charged particles, and molecular beam techniques are not treated.
Virtually no description is given of the experiments or the apparatus used
to obtain the empirical evidence cited, in the belief that devoting valuable
space to such an exposition tends only to emphasize the historical fact that
the development of nuclear physics consists of a collection of topics rather
than a unified discipline leaning on an all-embracing theoretical foundation.
It should also be noted that even a cursory presentation of experimental
techniques would add considerably to the length of the text and would be
perhaps of only limited value since these techniques become rapidly dated.
It is hoped that a prior course at an elementary level would have given the
student some appreciation for the many interesting experimental techniques
that have been employed, even if only in a historical context. Instead, an
effort is made to connect the analyzed empirical data with the relevant theory.
ACKNOWLEDGMENTS
The author is particularly indebted to Professor Edward F. Redish for
his many useful and critical comments. Professors Manoj K. Banerjee and
Gerard Stephenson, as well as other members of the University of Maryland
nuclear physics group, have been very helpful in numerous discussions.
Secretaries Marie Daston and Mary Ann DeMent have tirelessly contributed
their labors. Finally, I am most grateful for the long hours my wife Eva
devoted to proofreading and translating my efforts from "archaic Hungarian"
to English.
xni
Chapter I
MCLEON-MICLEON FORCES
A. INTRODUCTION
A central problem of nuclear physics is to understand the nature of the
isolated nucleon-nucleon interaction and to explain the properties of complex
nuclei in terms of these nuclear forces. The description of nuclear systems can
be attempted in a fundamental or microscopic sense by explicitly accounting
for the motion of each nucleon. This approach is generally quite complex,
occasionally even to the point of obscuring the "physics" of the problem by
the presence of mathematical or calculational difficulties. The advent of fast
computer technology has in many instances materially aided in carrying out
this program. However, the penetrating insight generally offered by closed-
form analytic expressions is seldom available. This situation is hardly sur
prising when the large number of degrees of freedom involved is considered.
Alternately, one might develop relevant macroscopic or many-body con
cepts, models, and parameters in terms of which a satisfactory treatment of
complex nuclei could be sought. The reduced number of collective variables,
the principal economy of this approach, usually results in a more manageable,
if not always transparent, grasp of the problem. When desirable, such model
calculations can be augmented by specific corrections for aspects of the
1
2 / NUCLEON-NUCLEON FORCES
fundamental nucleon interactions that have been either omitted or in
adequately included in the model. These corrections, if small enough, can be
successfully considered in the framework of perturbation theory.
It not infrequently happens that the added understanding offered by a
macroscopic view merits considering the development of suitable models as
an additional valid goal in itself. In a number of the following chapters, our
main purpose will be to dwell on the progress that has been made in this
direction. It should also be pointed out from the outset that, while many
remarkably accurate models for the behavior of complex nuclei exist, the
derivative connection to the nucleon-nucleon interaction in a fundamental
sense is only in its infancy in many cases.
The objective of this chapter is to discuss the basic nature of the nucleon-
nucleon force from an elementary particle point of view. This task is materially
aided by describing the nature of the nucleons in terms of any possible sub
structure they may possess and by discussing the relationship of the nucleons
to other closely associated elementary particles. These topics are discussed
in this chapter largely in a survey format and only to the extent necessary to
provide useful nuclear physics background material. Indeed, a detailed
treatment would require the full theoretical apparatus of elementary particle
physics, which would take us far beyond the intended scope of this text. Such
treatment is left to the numerous excellent texts on the subject and the
conference reports in this rapidly developing field.
Notwithstanding the now apparent composite structure of both the free
neutron and free proton as they are encountered individually in the laboratory,
complex nuclei can be conveniently considered to consist of Z protons strongly
interacting with N— A—Z neutrons. For example, the isotope of beryllium
10Be with atomic number Z = 4 and mass number A = 10 is considered to
consist of 4 protons strongly interacting with 6 neutrons. When 10Be under
goes ß~ radioactive decay to its neighboring isobar of boron 10B (with 5
protons and 5 neutrons)1 by the nuclear emission of an electron (and anti-
neutrino), one of the 6 neutrons is imagined to decay into a proton (viz.,
n->p + e~ + v). Thus the observation of electrons being emitted from the
nuclear interior does not require the prior or steady-state existence of electrons
within the nucleus.
The basic properties of the free neutron and proton are (Particle Data
Group, 1973)
mass:2 m c2 = (938.2592±0.0052) MeV
p
m c2 = (939.5527 + 0.0052) MeV
n
mn — mp = 2.531w ;
e
1 Elements with the same Z but different A are called isotopes. Elements with the same A
but different Z are called isobars.
2 The electron mass is wc2 = (0.5110041 ±0.0000016) MeV.
e
B. FUNDAMENTALS 3
magnetic moment: μ = (2.792782±0.000017)μ
ρ 0
μ = -(1.913148±0.000066)^o
η
where μ = eh/2m c is the nuclear magneton. In addition, both are fermions,
0 p
i.e., have intrinsic spin angular momentum \h and obey Fermi-Dirac
statistics. By convention, both are taken to have the same intrinsic parity
(defined as even). The free neutron is unstable and j8-decays to the proton
with a mean life of (15.6 + 0.2) min providing (2.531 -1)m c2 = 0.782 MeV
e
of kinetic energy for the decay products.
B. FUNDAMENTALS
In this section we introduce a number of concepts that we shall find useful
later. The first of these, the isospin variable, allows for a more compact
formulation of nucleon wave function symmetries and leads to a generalized
Pauli principle in the extended nucleon space-spin-isospin degrees of freedom.
In addition, the fact that isospin in many instances is almost a good quantum
number allows a useful first-order classification of nuclear system character
istics. The symmetry property of nuclear wave functions associated with
parity also plays an important role, since strong interactions between elementary
particles, such as in the dominant nucleon-nucleon interaction, conserve
parity. On the other hand, the fact that weak interactions (responsible, for
example, for nuclear /?-decay) violate parity conservation has far-reaching
consequences.
A brief description of the systematics of the elementary particles closely
associated with the nucleons (such as the π-mesons or pions) is given in the
following subsections. Since the present emphasis is on nuclear physics, a
large amount of important and interesting material relating to elementary
particle physics per se is omitted. Thus many of the fundamental experiments
involving elementary particles, such as the "missing mass" experiments, are
eschewed in favor of those more closely associated with the nucleons them
selves. Because high-energy electron scattering offers particularly striking
evidence for the existence of nucleon substructure, it is discussed at length.
In the next chapter we shall find that such experiments also relate importantly
to our knowledge of the charge density distribution in complex nuclei.
The stage is set for the presentation of modern meson-exchange models of
the nucleon-nucleon force in the final section of this chapter by first discussing
in detail a simplified one-boson exchange model (OBEM) involving only
single pions (ignoring such vital but unfortunately complicated factors as
pion-pion interactions and resonances, etc.). A brief discussion of some of
the more complicated two-pion exchange effects follows.
Finally, the quark model of elementary particles is introduced, not only to
give a coherent and comprehensive view of the possible nature of a more
4 / NUCLEON-NUCLEON FORCES
fundamental underlying substructure of elementary particles, but also to
shed light on the nature of the strong interactions.
Some aspects of the preceding topics, when amenable to simple theoretical
analysis, are explicitly treated; however, much of the presentation is relegated
perforce to a descriptive survey.
1. Isospin
Soon after the discovery of the neutron, Heisenberg (1932) speculated, on
the basis of the similar properties of the proton and the neutron, that they
represent two different charge states of the same particle, referred to generically
as the nucleon. The new internal variable distinguishing this nearly degenerate
mass doublet, called isospin (earlier designations were isotopic spin, isobaric
spin, and i-spin) is associated with a vector operator i.1 We wish to describe
the two-nucleon charge states as discrete projection states of the eigenvectors
of this operator. Thus a formal analogy with the two discrete m-states or
space projection states of the (mechanical) spin-£ operator suggests itself. In
analogy to the Pauli matrices for σ and its relationship to the spin operators
s = ^ήσ, we introduce ϊ = \τ. In terms of Cartesian coordinates in isospin
space (unit vectors ί, % and 3) we write2
τ = τ 1 4- τ 2 + τ 3
γ 2 3
<-(?;> - ( ? - ;> *-(ί-?)· <-■>
It then follows, in analogy to s2 = §·§ = Jft2/, that
where / is the identity matrix (J ?). This can be used to define the isospin
quantum number t, viz., t(t+l) = i, giving t = %.3 The two-component
isospinors π = (£) and v = (?) are immediately seen to be eigenspinors of the
diagonal matrix τ with eigenvalues +1 and — 1, respectively, corresponding
3
to ±i for the eigenvalues of ?. The identification can then be made that
3
nucleons are particles with isospin ·£, the isospin polarization "up" or π-state
representing the proton and the polarization "down" or v-state representing
the neutron.4 One can then write proton and neutron wave functions in terms
1 We shall employ a tilde over a symbol to call attention explicitly to an operator quantity
when required for clarity.
2 Following general custom, a caret over a symbol will be used to designate an appropriate
unit vector.
3 The solution / = — f is unacceptable, since t ^ 0 is required by convention.
4 This convention conforms to modern usage, while earlier literature in nuclear physics
generally used reversed designations.
