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POLYTECHNIC
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LEARNING
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SERVICE
This
book
mjst be
returned on
or
before
the date last
stamped
W-
NOV.
:,.;
C
L
P
P
112494
SISENBERG,
J.M.
and
GKEINER,
W.
Nuclear
Models
Nuclear
Theory
1.
539.74
EIS
i
1
-i
a
n
a
A/C
112494
30107
000
596 541
-*L
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NUCLEAR
THEORY
VOLUME 1
Three
books
by
J. M.
EISENBERG
andW.
GREINER
published
by
North-Holland
Publishing
Company:
VOLUME
1
NUCLEAR
MODELS
VOLUME
2
EXCITATION
MECHANISMS OF
THE NUCLEUS
VOLUME
3
MICROSCOPIC
THEORY OF THE
NUCLEUS
NUCLEAR
MODELS
Collective and
Single-Particle
Phenomena
Judah M.
EISENBERG
Professor
of
Physics
University
of
Virginia,
Charlottesville, Virginia, U.S.A.
Walter
GREINER
Professor of
Theoretical
Physics
Johann
Wolfgang
Goethe
Universitat, Frankfurt am
Mai
Germany
SECOND.
REVISED EDITION
SSc
1975
NORTH-HOLLAND PUBLISHING
COMPANY
AMSTERDAM
•
O
AMERICAN ELSEVIER
PUBLISHING COMPANY,
INC.
NEW
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©
North-
Holland
Publishing
Company,
1975
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 the prior
permission
of
the
copyright owner.
Library
of
Congress
Catalog Card
Number;
72-102719
North-Holland
ISBN:
7204
0326
x
f
American
Elsevier
ISBN: 0444
10790
S
Published
by
NORTH-HOLLAND
PUBLISHING
COMPANY
-
AMSTERDAM
NORTH-HOLLAND
PUBLISHING
COMPANY, LTD.
-OXFORD
Sole
distributors for the
U.S.A. and Canada:
AMERICAN
ELSEVIER PUBLISHING
COMPANY, INC.
52 VANDERBILT
AVENUE
NEW YORK, N.Y.
10017
Mm
1970
td,
revised
edition 1975
Printed in The Netherlands
ACCESSION
NO.
112494
CL,
2
6
APR
1978
N
flic&mftL
Contents
Preface to
the
first
edition
Preface
to
the
second edition
Introduction
1 Varieties
of collective
motion
1.1.
The quadrupole
moments
of
nuclei
1.2.
Electromagnetic
transitions
1.2.1.
Preliminaries
on
radiative transitions.
1
.2.2,
Discussion
of
empirical
data
1,2.2a.
Electric
dipole
transitions
1.2.2b. Electric
quadrupole
transitions
1.2.2c.
Low-energy
spectra (vibrations'and
rotations)
.
References .
2
Collective
coordinates
2.1.
The surface variables
c&n
2.2.
The
collective
variables for density
vibrations
gW
2.3. Canonically conjugate
momenta and phonon
operators for the
lective
variables
References
3
The
structure
of
the
collective
nuclear Hamiltonian
3.1.
The
nuclear
quadrupole
surface
motion
3.2.
Harmonic
vibrations
of the
nuclear surface
3.2.1. Construction of angular
momentum
etgenfunctions ..
3.2.2.
Multipole operators in
collective variables
3.2.3.
Construction of
a
general
angular
momentum
operator in
lective variables
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VI
CONTENTS
3.3.
Simple
applications of
the
harmonic
oscillator model
62
3.3.1. The mean square deformation
62
33.2. The
mean square charge
radius
of excited states
63
3.3.3.
Electromagnetic decay of the quadrupole
vibrational
states .
67
3.3.4.
Quadrupole
moments
of
excited
states
68
References
69
4
The collective
potential
energy
surface
70
4.1.
The
intrinsic
coordinate
system
71
4.2.
The geometric symmetries
in
the
collective
quadrupole
energy
sur-
face
79
4.3.
A schematic model for
the
collective
quadrupole
energy
surface . .
82
4.3.1. The case
Ci>G
82
4.3.2.
The case C
2
<
0.
87
5
Quantum mechanics
of
the rotator
91
5.1. The Euler
angles,
the
space-fixed
coordinate system, the
body-fixed
coordinate system
and
the
system
of
instantaneous rotations
...
92
5.2.
The
components of
the
angular momentum
with
respect
to the
labo-
ratory and to the intrinsic
body-fixed
systems 96
5.3.
The
time
derivatives
of
the
JJ-functions
102
5.4. The collective kinetic energy in terms
of
Euler
angles
and
intrinsic
variables
103
5.5. The
matrix elements
of the components
of the angular momentum
along
the
intrinsic
axes
107
5.5.1.
The eigenvalues of
£
2
112
5.5.2,
The
matrices of
the
angular
momentum
operators
114
5.6.
The
transformation
of the angular momentum eigenvectors
under
finite rotations
115
5.7.
Relation
between
finite and
infinitesimal
rotations
119
5.8.
The
D
l
m
,
m
($,)
as
angular momentum eigenfunctions
122
5.9. The symmetric top
....,.,
124
References
126
6 The
rotation-vibration
model (RVM)
, 127
6.1. The Hamiltonian
of
the
rotation-vibration
model
. 128
6.2. The quantization in
curvilinear
coordinates
136
6.3. Solution of the rotation-vibration
Hamiltonian
143
6.4.
The
symrnetrization of
the
wave
functions
147
6.5.
The effect
of
the
rotation-vibration interaction
152
References
.
.
.'
167
CONTENTS
7 The
asymetric rotator model
(ARM)
7.1.
The
Hamiltonian of
the ARM
7.2.
Electric
quadrupole transition probabilities in the rotational
ban
7.3.
(9-vibrations
of the asymmetric rotator
References
8
Single-particle
models
8.1.
Spherical
single-particle
orbits
8.1.1.
Infinite square
well single-particle Hamiltonian
and
wa
functions
8.1.2.
Harmonic
oscillator single-particte
Hamiltonian
and wa
functions
8.2. The spin-orbit coupling
8.3.
More realistic
single-particle
potentials
8.4.
The
distribution
of
nucleons
8.5.
Pairing and
the
extreme single-particle model
8.5.1,
Predictions
of
quadrupole moments
,
8.5.2.
Predictions
of
magnetic moments
8.6.
Experimental evidence
for
strong
spin-orbit coupling
8,7
Motion of
a spin-J
particle
in
an external
field;
relativistic
contrib
tions
to the
spin-orbit interaction
References
9
The
deformed
shell model
-
the
unified model
9.1.
Weak coupling
9.2.
The
strong-coupling
Hamiltonian
9.3.
Structure
and symmetries of strong-coupling wave
functions
, .
9.4.
The
deformed shell
model
9.5.
Spectra
of odd-^ deformed
nuclei
9.6. Non-adiabatic
effects
9.6.1.
The
Coriolis force
9.6.2.
Departure from
axial
symmetry
9.7. Magnetic
moments of
odd-even
deformed nuclei
9.8. Extensions
of
the
deformed- shell model
9.9.
Odd-odd deformed
nuclei
References
10
Nuclear
hydrodynamics
10.1.
The experimental situation
10.2.
Schematic
survey
of
photonuclear
reactions
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vm
contents
10.3.
The
hydrodynamic
model
283
10.4.
The
classical
dipole
solutions
of
the
hydrodynamic equations
for
a
spherical nucleus
296
10.5.
The
classical
absorption cross
section
301
10.6.
Refinements
of
the
model
304
10.7.
Systematic
classification of
the
eigenmodes
of the
density fluc-
tuations
308
10.7.1. Spherical and
Cartesian
representations
of
tensors
. . .
308
10.7.2.
The
nature ofthe
normal modes
309
10.7.3.
Orthogonality of
the normal
densities
312
10.7.4. Normalization of
the fluctuation densities
313
10.8.
The
energy
of
giant
multipole
resonances
in
terms
of
collective
va-
riables
314
10.9.
Quantum
hydrodynamics; quantization of
giant multipole reso-
nances
317
10.10.
Multipole operators
in
collective
coordinates 318
10.11.
The
y-absorption
cross
section in quantum
hydrodynamics
...
319
10.12.
Microscopic
foundation of
nuclear
hydrodynamics 324
References
330
1 1 The
dynamic
collective model
331
1 1.1.
The form
of
the
interaction
between
surface degrees of
freedom
and density
fluctuations
331
11.2. The
normal
modes
of
density
fluctuationsin
a
deformed
nucleus
-
the
coupling
constants
332
11.3. The
Hamiltonian
for
deformed
even-even
nuclei
342
1
1
.4.
The wave
functions
and
energies
for the giant dipole
resonances
of
even-even
nuclei
345
11.5.
The
wave
functions
and
energies for the
giant
quadrupole
reso-
nances of
even-even
nuclei
350
11.6. The
Hamiltonian
for deformed odd
-A
nuclei
353
11.7. The
wave functions and
energies
of
the giant dipole
resonance
of
deformed
odd-/*
nuclei
354
11.8. The
dipole
and
quadrupole operators in
collective
variables. . .
356
11.9.
Analysis
of
experimental data for
deformed
nuclei ......
358
11.9.1.
Erbium
360
11.9.2. Holmium
.
363
11.10.
The
dynamic
collective
model
for
spherical even-even
nuclei
. .
371
11.10.1. The
structure
of
the
Hamiltonian
372
11.10.2.
The
splitting of
the
giant resonance
in
spherical nuclei
.
374
CONTENTS
11.10.3. The
experimental
absorption
cross sections
11.10.4.
Analysis of
experimental data
for
spherical
nuclei .
References
12 The
application
of
nuclear
models
to
heavy
ion
scattering
12.1.
Deformation
reactions
12.2.
Some useful
preliminaries
12.3. The total
Hamiltonian
function
for
spherical
nuclei
12.4.
The
scattering
of
deformed
nuclei
on
deformed nuclei
12.5.
Quasi-molecular
states; experimental
facts
12.6.
Binding
energies
of nuclei
-
the semi-empirical
mass formula
12.7.
Thomas-Fermi-type
theory
of
finite
nuclei
12.8 Quasi-molecular
nuclear optical
potentials
12.9. The
elastic
^O-
1
*©
scattering
cross
section
12.10. Nuclear
shock
waves in relativistic
heavy
ion collisions
References
APPENDICES
A Energy of
the classical surface
vibrations
of
a
nucleus
in the hydrody
mical approximation
B The
rotation
matrices
C
Diagonalization
of
the
triaxially
deformed shell
model Hamilton
in
cylindrical
coordinates
D
The density distribution in nuclei (derivation of the differentia
equati
(12.35)
to
(12.37)).
.
REFERENCES
INDEX
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Preface
to
the
First
Edition
Nuclear
structure theory has
long
been
one
of
the
most
difficult
a
physics
to learn. This
situation has existed
largely because the tec
which
have been used
in
the
study of
the
nucleus
have
been of
such
d
as
to make it
impossible
to find
extensive
common
themes
among
th
example, nuclear
experiments
have
involved
phenomena
as
diff
(d, n)
reactions
and muon
capture.
Even
those
experimental probe
are
connected
in
their
conceptual
basis, such as
Coulomb
excitation,
nuclear
reactions, and
muonic
atoms
-
all
of
which are basically
magnetic in character
-
have
focused upon
quite different
features
of
dynamics or statics.
As
a consequence,
experimental
information
h
certain degree,
tended
towards
fragmentation
rather
than inte
A
similar
picture emerges in
nuclear
theoretical
work.
Attempts
at
tion by
phenomenological models
have drawn
upon
a
variety of
spe
areas
which
have often
been
quite
far
removed
from
each other:
hydrodynamics,
rotational
molecules,
shell
structure,
supercondu
dispersion
relations, and
so
forth.
In
the face
of this
situation,
most
efforts
to present
nuclear
s
physics
in a
more or
less
global
way
have done so
by
surveying
ke
in a
large
number
of
areas
and
leaving to
the
reader
the hard job'of
out
how the basic
theoretical
concepts
were
welded into
detailed quan
predictions
about
nuclear
properties.
This job
is becoming
incr
difficult
as
the already
vast
and
indigestible
literature of nuclear
is augmented
at a frightening
rate, while some
of
the
very
fundamen
cepts in
the area
continue
to be found
only
in
the
rather
sketchy
accounts of
the research journals.
It appeared
to us
that the t
ripe
to attempt
a
consolidation
of
this
material.
8/18/2019 Eisenberg Grenier Nuclear Theory 1
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XII
PREFACE
Accordingly, we
have
tried
to
set forth
a
self-contained
description
of
nuclear
structure
physics
which
would
seek
out
the
main lines
of
theoretical
advances
and
develop
them in a
closely-reasoned
fashion. It
has
been
our
goal to present
this material
so
that
very
nearly
all
of
the
results which
we
include are
derived here
equation
by
equation.
We
have felt
that a phys-
icist
who
has these main lines of
theoretical thinking
well
under
control
will find
it
relatively
easy to
explore
the
use of
these
essential techniques
in other
nuclear
applications. This
whole
program
is
very
much
aided
by
advances
in
recent years which
have
led to a
greater
understanding
of
the
ways
in
which
apparently
dissimilar accounts
of
nuclear properties
are
in
fact
closely
related
when
one
works
out the
implications
of the
fundamental
concepts
in
a
systematic
fashion.
This
description
of nuclear
structure theory has been
constructed
as
a
series
of three
volumes,
each of
which is intended
to
stand alone, but all
of
which are
mutually
complementary.
The subject
matter
is
organized
by
volume as
follows:
1.
Nuclear models
(collective
and single-particle
phenomena),
dealing
with
the
phenomenological
approaches
to
the depiction
of
collective rotational
and
vibrational
features of
nuclei,
as well as
the
empirically
based
aspects
of
the
spherical and deformed shell
models.
2.
Excitation mechanisms
of
the
nucleus
{electromagnetic and
-weak
in-
teractions), which
treats those
major
techniques used in
studying
the nucleus
which
do not introduce the
complications
of
a strongly
interacting
probe.
3.
Microscopic theory
of
the nucleus, a volume
concerned
with
how one
describes
nuclear
properties
beginning
with
the
fundamental nucleon-
nucleon interaction: it includes
discussions of
two- and three-nucleon
sys-
tems,
nuclear matter,
Hartree-Fock
approaches, the particle-hole
formal-
ism,
pairing, and the
relationship
of these to
the
more phenomenological
approaches.
These
books
have been
written at a level
which
makes them
usable
for
anyone
who
has had
a
conventional
one-year course in
quantum
mechanics
and
who
is
slightly
acquainted
with
nuclear
phenomena. They
may
thus
serve as texts
in an
intermediate
graduate-level
course
in
nuclear
theory,
in
which
case
each
volume would occupy
approximately one semester.
Of
course,
this would require that some
of
the material in each
volume
be omitted
in classroom
discussion.
We
feel
this is as it
should be: In the
past,
in-
structors of nuclear physics
have borne the
burden
of elaborating on the
somewhat skimpy
theoretical
material given
in standard
texts.
To
us
it
makes better sense at this
level
for
the instructor
to focus
on
the
essential
PREFACE
physics
which
is of
interest
for his course,
leaving
to
the
textbook
t
of
filling out
the student's
information
and
preparing
him
for
rea
the
nuclear
research
journals.
We
hope that
these
books
may also
similar
purpose
for research
workers
in
nuclear
physics,
providin
with background
material
in
areas
relating
to their primary
researc
ests.
In
particular, we
would
anticipate that many
nuclear
experimen
will
feel
more
at home
with
a book which puts
in
the
intermediat
than
they
will
with one which
leaves
them out
-
the
details
are
left out
sibly
to
simplify the account,
but
their
omission usually only succ
mystifying.
The present volume
is the first
in this series.
It develops
the
con
collective coordinates
and the
construction
of the
collective
nuclear Ha
nian, and
then discusses the
collective
potential energy surface. The q
mechanics
of
the rotator
is
presented as
preparation
for
the detaile
ment of
the
rotation-vibration
model and the
asymmetric rotator
The
relationship
between
these
models is also
explored
at some
leng
spherical and
deformed
shell models and
the
unified
model
are intr
so
that
the interaction of
single-particle and
collective
degrees
of
f
may
be treated. Then,
returning to
phenomenological
collective
descr
we present
the
subject
of nuclear
hydrodynamics
and
pave
the
wa
discussion
of the
dynamic
collective
model
for
deformed
and spheri
clei.
Lastly, nuclear models are
applied
to
heavy ion
scattering. Ta
gether
these
topics
provide
a
broad
basis for
the
phenomenologic
cription of
nuclear
properties.
They also
set
the stage
for microscop
ries
of the
nucleus
(Volume
3)
in that they
exhibit
the essential
which must
be explained
by
a more
fundamental
theory.
We note that in
the present
volume
those
quantities
which are op
in
a configuration-space
representation
are
denoted
by
a
circumfl
Q,
while
the corresponding
classical
quantity
is
without
special
d
tion
(e.g.
Q).
It is
very
humbling
to
realize
how many
people have
helped us
in
th
ent
enterprise,
and we would like to take
this opportunity to
express
our deep
gratitude. We
are much
indebted
to many
physicists fo
discussions
which
have
led to a
clarification of
the
material in this
b
its companion
volumes. Among them
are
Drs.
H. L. Acker*,
K, Al
M.
Danos,
B. Fink,
B. Fricke,
L.
P.
Fulcher, R. Guy, H.
P. Kelly
Tourneux, H. Marschall, R.
C.
Ritter,
W.
Scheid,
E.
Schopper,
*
Deceased.
8/18/2019 Eisenberg Grenier Nuclear Theory 1
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XIV
PREFACE
Spicer,
H. Stock, H. J. Weber, and W. D.
Whitehead.
One
of
us (W.G.)
would like to
thank
Dr. N.
Cabrera
for
the
hospitality
extended to him at
the
University
of Virginia. We
would
also
like
to express special
thanks
to
Dr.
W. D.
Whitehead,
and
through
him
to the Center
for
Advanced
Studies
of
the
University
of Virginia,
for solving many
difficult
administrative
problems,
thereby making our
collaboration
possible. The Deutsche
Forschungsge-
meinschaft,
the Bundesministerium
fur
wissenschaftliche
Forschung, and
the
Hessisches
Kultusministerium
are also deserving
of thanks in
this
regard.
We would
like to acknowledge the
kind
permission
of the editors
of
The
Physical Review,
Physical
Review
Letters and
Reviews of
Modern
Physics
for permission
to
use
illustrations which
appeared in their journals.
Our task in preparing the manuscript for
press was
substantially lightened
by the
unfailing
cheerfulness
and
patience
with
which
a number
ofpeople
un-
dertook
to
type
this material. Chief
among
these
were Mmes.
E.
L5hlein,
M. Knolle, N.
Leider,
A.
McDaniel,
N.
Miller,
B. Paup,
P.
Walker,
and
S. Wigent. Mrs. B. Utschig helped with
obtaining library materials,
and,
together
with
Mrs. L.
Urbanek drew the
figures. We
express here
our
very
sincere
thanks
for all
of
this assistance.
Charlottesville,
Virginia
Frankfurt am Main
January,
1970
J.
M.
Eisenberg
W.
Greiner
Preface to
the
Second
Edition
The
rapid
pace of developments
in
nuclear physics
makes
it exce
difficult to
keep any
comprehensive
account
of
the
field up
to
da
therefore
particularly
welcomed
the
opportunity to
revise
the
volume
for
a
second edition
since, in
addition
to making a
number
o
corrections,
we were able to
incorporate a new section on
nuclea
waves
in
relativist ic heavy-ion
collisions
-
a
new
and
promising
nuclear
study.
We
wish
once
again
to thank
our
many co-workers, in
particular
Stock,
who have aided
us
over
the years
in
the preparation of
this m
and
wlic have
found errors in
the
first
edition
of
this
work which
w
tried to
correct
here.
Charlottesville,
Virginia
Frankfurt am Main
June,
1975
J. M.
Ei
W.
Grei
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Introduction
In
the last
two
decades our
knowledge concerning
the
structure
of
has
been
tremendously
improved
through
the
development
of
models.
Simple mathematical
assumptions
and
the underlying
physical
pictures have
been
very
successful
in obtaining
a
systemati
pretation of
an
enormous
amount of
experimental
data.
Very
often,
continuing
improvements
of
the
models,
it
was
possible not
only
to
qualitative
description of the
observed
phenomena,
but
also to
acco
them
quantitatively.
The
development of
physical
pictures of
atomic
has
frequently been
accompanied
by a steady
exchange
of ideas wit
fields
in
physics,
in
particular
with
atomic,
molecular,
solid state
a
mentary particle
physics
-
even
hydrodynamics
has
played
an
imp
role
in nuclear
physics.
Typical
examples
of
these similarities
b
various
fields are
the
shell models
of
atoms and
of
nuclei, which
l
deep
understanding
of the
single-particle
structure;
the
rotatio
vibrations
in
nuclei,
which have been
formulated
quite in
analogy w
molecular
rotational
and
vibrational
spectra; the
recently
develope
on nuclear
molecules
and
the occurrence
of quasi-molecular
pot
between
ions,
which
are very
similar
to usual
pictures
of two
inte
atomic
molecules.
The
ultimate
goal
of nuclear
structure
physics is,
of
course,
to a
for
the properties
of
complex
nuclei
in
terms
of
the
interaction
betwe
nucleons.
(Naturally,
if
conclusive
evidence
were found
for
the exist
three-nucleon
forces-
or
muIU-nucleon
forces
-
this
would
constitute
and
important
ingredient
in nuclear
physics which
one would
then
w
pursue
further
in
structure
theory.)
Such
a microscopic
description
1
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INTRODUCTION
nucleus
in
terms
of
the
properties
of
its
constituents
defines
the
direction
in
which
nuclear
structure physics
moves
in
its
quest
for
a
fundamental
theory
of
nuclei.
Current
microscopic
investigations in
nuclear
structure
theory
usually
make
contact
with
the
shell model
of
the nucleus.
Very
early work
on
this
model
(Be-28,
Be-30,
Go-37)
foreshadowed
later
developments.
In
particular
the work of
Schmidt
(Sch-37)
on
the
prediction
of
magnetic
moments
of nuclei
in
their
ground
states
was a milestone
for the shell
model.
A
great
breakthrough came
with
the
work
of
Mayer
who
assembled
experi-
mental
evidence
indicating the
existence of
a shell
structure
(May-48,
Fee-49a,b,
No-49).
It led
directly
to
the
dynamical
model
with
strong
spin-orbit
force
developed
by
Mayer, Jensen,
Haxel,
Suess, Feenberg
and
Nordheim,
which
was
very
successful in correlating
spins and
parities of
ground
states
(Fee-49a,
Ha-49,
May-49,
50,
55)*.
Nowadays
nuclear
shell
structure
is
the
foundation
upon which
most current
theoretical
work
in
nuclear physics
is
based.
Nevertheless it
should be clearly
borne in
mind
that
this
description is
still basically
phenomenological,
and
that deeper
micro-
scopic
justifications
for it are still
being
sought.
In
spite
of
the great
successes
of
the
shell model
in the
prediction
of
ground
state
spins and parities,
magic numbers,
alpha
and beta
decay
systematics
and
so
forth,
this
model
still
has exhibited
many
limitations. For
example,
experimentally
observed nuclear
quadrupole
moments
are in
most
cases
much larger
than the
shell model
predictions,
especially
in
the
regions
between
closed shells
(Gr>49,
To-49);
the
transition
probabilities
of
low-
lying states
exceed the
single-particle
estimates,
very
often
by
as
much
as
two orders
of magnitude;
and typical
rotational
and
vibrational
band
structures
occur which
are
not easily
accounted
for
in
the
shell
model.
Finally,
a
further
impetus
for
the use
of
models
other
than
the
shell
model
in
accounting
for
nuclear
properties
came
with the
discovery
(Ba-47,
48)
in
1947 of
the
photonuclear
giant resonance.
This
appeared as
a
strong
broad
peak
in the photofission cross
sections,
and was
soon
recognized
as
a
feature common
to
all nuclei. It was early
understood
to
embody a
collective
vibration
of protons against
neutrons
in
response to the
oscillatory
electric
field of
the
incident photon, and
so
to
call
for a
collective
description.
These observations led
to
the
development
of
various
collective models,
and in fact two
basically different models
emerged.
The
first
of these
deals
with collective
surface
(deformation)
degrees
of
freedom
and thus
with
low-
*
A
fascinating
report
on
the development
of
the physical
concepts
underlying
nuclear
shell theory has been given
by
Jensen in
his
Nobel
ftize address
(Je-64).
INTRODUCTION
energy
collective
rotations
and
vibrations
of
atomic
nuclei.
The need
came about some
time ago with
the
first
attempts
to
explain
the
pheno
of
nuclear fission.
At
that
time
Feenberg
(Fee-39)
and
Bohr
and
W
(Bo-39)
studied the
shape and
stability
of
a
deformed
and
charged
drop. They studied
the total
energy
as
a
function
of
deformation
fr
spherical equilibrium
shape. Feenberg
was
concerned
with
the
stabi
nuclei
against
spontaneous
fission
while Bohr
and
Wheeler
investig
detail
the
theory
of this
process.
Following
ideas
originally
introdu
Weizsacker (Wei-35)
the
total
nuclear
energy
was
considered
as
the
s
the volume
term,
the
surface
energy
produced
by a
uniform
surface
t
and
the
Coulomb
energy.
During
distortion the
former
increases
whi
latter
decreases.
One
could
conclude
from this
that
the
equilibriu
low-energy
shape
of
even-even
nuclei
should
be
spherical.
A
very
important
step
-now
mostly
forgotten
-in
the
developm
the low-energy
collective
model
was
later
taken
by
Flugge
(Fl-41
discussed
surface
vibrations and
rotations
of
nuclei
for
the
first time.
he
recognized
the
role
of the
surface
tension
in
determining
dynam
well
as
static
properties
of nuclei
through
surface
vibrations.
He
also
r
that
nuclear
rotations
may
produce
rotational
spectra.
However,
he
d
yet
recognize that
surface
vibrations
and
rotations
could
be
related
t
other.
The
large
quadrupole
moments
mentioned
above led
Rainwater
(R
in
1950
to the
assertion
that a
single
odd
particle
could
polarize
the
n
core. He in
fact showed
that the
single
particle
could
have
a lower
if
its potential
well was
deformed.
This
suggestion
thus
indicated
f
first
time that
a
nuclear
system of
a
single
particle
(or
perhaps
a
gr
particles)
coupled
to a
core
may
achieve
a
relative
minimum
energ
figuration
if
the
core
is
deformed
from
the
spherical
shape.*
The essential
contribution
of
Bohr
and
later Bohr
and
Mottelson
(
Bo-53)
was to
recognize
that
the
static
shape
and
orientation
of
a
def
nucleus
and the
collective
deformation
variables
of
a
spherical
nucleu
related
to each
other.
As noticed
by
Flugge,
these
variables
could pl
role
of dynamical
variables.
It was
then
an
important
step
from
the
cl
picture
of
Flugge
to
the
quantized
picture
of Bohr.
The
motion
of
the
dual
particle
is
then
strongly
influenced
by
these
quantized
col
variables.
The
dynamics
of
nuclei
in many
ways
prove
to
be
quite
anal
to
those of
molecules.
The
independent
particles
play
the
role
*
The
notion
of a
deformed core
was
also
suggested
ten
years
earlier
by Schmidt
(S
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4
INTRODUCTION
electrons,
and
the
deformed
shape
and
the
orientation
of
the
nucleus as
a
whole
play
the
role
of
the
molecular
(intrinsic)
frame.
Of course,
unlike
the
molecular
case,
the intrinsic
frame
for
nuclei
is
itself
produced by
the
nucleons.
It
should
be
stressed
that
the
introduction
of
collective degrees
of freedom
is by no
means
to
be
considered as
standing in
opposition
to
the
independent
particle
model (Bo-53).
In fact,
the
very
existence of
the
effective field
for
the
independent
particle
model
is
collective in
nature
(see In-62).
This
is
most
clearly
revealed
in
the works
of
Nilsson
(Ni-55),
Gottfried
(Go-56),
and
Moszkowski
(Mo-55),
who
extended
the
spherical shell model
to
deformed
nuclei.
The
work of
Bohr
and
Mottelson
dealt
with
both
collective
and single-
particle
phenomena, and
established
the
important
conceptual bases for
work in
this
area. The
theoretical
implications
of
their
conceptual
framework
were
worked
out and
considerably
expanded
both by
them and by others.
Davydov
and his
school (Da-58)
started
with Bohr's
collective
Hamiltonian,
and
subsequently
investigated triaxial
statically
deformed
nuclei
with great
success.
Many
E2
transition
probabilities
and
deviations
from
the energy
spectra of
simple
rotators
were explained
in
this
way. Soon
afterwards
systematic
solutions
of
the
dynamical model
with /J- and
y-vibrations were
obtained
for an
axially
symmetric
core (Fae-62, 64, 65;
see also Bi-59).
The
great
importance
of
the
rotation-vibration
interaction
in atomic
nuclei
was
stressed and
the
equivalence
with
the
asymmetric
rotator
model,
especially
for the
ground state
and gamma
bands,
was
established.
The role
of
collective
models
in
understanding
the
photonuclear giant
resonance
began
shortly
after its
discovery, and
involved
a
second
and quite
different
class of
collective
models.
Several
theoretical
approaches
were
suggested by
Goldhaber and
Teller (Go-48)
to
explain
this
phenomenon.
They
involved
collective
dipole effects to
account
for some
of
the
properties
of
the
photonuclear
cross
section
as a
function
of energy.
In one of
these
models
rigid
proton
and
neutron
spheres
were
introduced,
each of
which
executes
harmonic
oscillations
about
its
equilibrium
position. The
resonance
frequencies in this case
proved
to
be
proportional to
R~*
oc
A~*
t
where
R
is
the
nuclear
radius
and A
is the
nucleon
number.
Another
model they
suggested
was
the
two-fluid
model in
which
compressible
proton and
neutron
fluids oscillate against
each other. The
restoring
forces
were assumed
to be
proportional to
the
density
gradients
and
the
resonance
frequencies
turned
out
to
be
proportional
to R'
1
oc A~*.
This model
has
been
extensively
investigated and
given
a complete
mathematical
description by
Steinwedel,
INTRODUCTION
J.
H.D.
Jensen and P. Jensen and by
Danos (St-
50,
Da-51).
They solv
two-fluid problem
for
a
spherical
nucleus and related
the resonance
e
E
m
to
the
phase
velocity of
second-sound
in
nuclear matter,
u, by
E
m
=
2.08
uhR~
l
.
They thus also predicted
that
the giant resonance
energies were
proport
to A~*.
This
model has been extended by
Danos
(Da-58) and Oka
(Ok-58) to statically deformed
axially
symmetric nuclei.
They predic
splitting
of
the giant resonance
proportional
to
the
deformation
o
nucleus, due
to
the
different periodicities
ofthe
sound waves
along the
and minor
nuclear
axes.
This
has been
observed
experimentally
by
and Weiss
(Fu-58)
and by
Spicer,
Thiess,
Baglin and Allum (Sp-59)
The high-energy
collective
models and the
low-energy
collective
m
coexisted
up
to 1964, when
they
were unified for
investigating
the
structure ofgiant
resonances
(Da-64,
Se-65, Li-66.).
In deformedand
sph
nuclei the giant resonance
sound
waves
are
tuned by
the
surface vibra
Thus, a number
of sattelitcs occur
on
top
of
the
giant resonance which
appreciable
dipole strength
so
that
considerable
substructure
(coll
intermediate
structure)
occurs. This substructure
has
subsequently
observed in
many
nuclei. Of some surprise in this connection
wa
splitting
of the
giant
resonances
in
spherical
nuclei (LeT-65,
We-65,
We
The structure
of
the giant
resonances
in light
nuclei
and in
magic
has
been studied extensively
in the
framework
of
an
extended
shell
m
This
work
constitutes an
alternative
microscopic
approach to
the coll
model
of
giant
resonances
outlined
above.
In fact,
the
description
giant resonances
is
particularly
easily accomplished in
both
the
coll
and the microscopic
approaches. This
makes that
phenomenon
espe
suitable
as a
testing
ground
for understanding the
origins
ofcollective
m
from
a
microscopic viewpoint
—
obviously a very
fundamental questi
nuclear
structure
theory. The
various nuclear
models
described so
far
a wide range
of
applicability in
accounting
for features of
low-energy
nu
reactions,
polarization effects
in muonic
atoms, electron scattering,
p
scattering
and Coulomb excitation. An exciting
recent
development
application for these model
approaches is
in the
area of
heavy
ion ph
Heavy
ion
processes allow
for the
possibility
of
creating short-lived
nu
molecules*. In
fact, quasi-molecular nucleus-nucleus
potentials
ca
derived
and understood
on very plausible
grounds, and
seem to
give
•
Such
effects seem to be indicated in
the
experiments
of Bromley and
his
collab
(Br-60,
Si-67).
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INTRODUCTION
to
such
important
quantities
as
the
nuclear
compressibility
(Sch-68).
Even
more
exciting
would
be the discovery
-
which may
lie
ahead
of
us
in
the
future—
of
superheavy
elements
(Z
as 114
or
164).
We
shall
set
forth
in this volume
a
systematic
theory
of
the empirical
nuclear
models. We
shall try to
emphasize
the
physical picture
behind
the
various
developments and
simultaneously to
give
a
complete
mathematical
description
so
that
one can really
carry
out
extensive calculations of
nuclear
properties,
a
requirement
which
any
theory should meet in order to
allow
for
quantitative
checks.
We
hope
ultimately to
demonstrate
that a great
many
experimental
facts
can be correlated
within
these models
so
that a
vivid
picture
of
the
nucleus
emerges
from
them.
CHAPTER 1
Varieties of
Collective
Moti
Many
features
of
nuclei indicate that nuclear motion
does
not
consis
in
simple single-particle
excitations
as might
be
suggested
by
the
shell
m
Instead there
are
several
typical
effects
which
imply a
collective
mo
that is a motion where many nucleons
move
coherently
with well-d
phases.
In order
to
understand
this
kind
of
coherent
motion of nu
let
us start
by
giving
illustrations
of
some typical
examples.
There
ar
important
types of
collective
motion:
I). The
surface
vibrations
of the
nuclear shape are
a
motion of
nu
from
one
region
of
the nuclear
sphere
into
another
one. The
arr
fig. 1(a)
show
this
collective
ordered
motion.
They
are
also
ind
in fig.
1
(b)
where
the nuclear radius periodically
oscillates about
an
librium value
R
Q
. In
this
case
the
density
in
the nuclear
sphere
also inc
and
decreases
periodically
(compression modes).
Fig.
1
(a).
Schematic figure
of
surface vibrations.
The arrows indicate a
possib
(stream lines)
of
nucleons.
The left-hand figure
shows
the spherical
nucleus. The
hand figure shows
the
distortion
of
the
sphere.
7
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10
VARIETIES
OF COLLECTIVE
MOTION
[CH.
1, §
1
pictures
such as those given
above,
this
already
provides
a confirmation
of
the
collective
nature of the
observed
excitations.
However
there
are some
general
effects
typical
of
collective
spectra
which
enable us
to
see
very
quickly the
degree of the collective
nature
of the excitations
in question.
In
fact,
a nucleus
never
has only
collective
levels
(modes) or
only
single-
particle levels
(modes)
but
always both.
We
will see
later
on that
pure
collective
and
pure
single-particle
modes are an idealization.
In
reality both
types
of motion are
mixed
and therefore
it may
sometimes be
very
difficult
to
identify
the true nature
of
a nuclear
state.
The effects
which give us
some
immediate
indication
of
nuclear
collective
motion are:
1).
The large
quadrupole
moments
observed
in nuclei
far
away
from
the
magic
numbers.
2).
Large
transition
probabilities
between
collective
levels.
3).
Certain
distinguishing features
in
low-energy
excitations
(rotational
and vibrational
spectra),
or
in
high-energy
excitations (giant
res-
onances).
We now
discuss
these
various
points in
detail.
1.1.
The quadrupole moments
of nuclei
The
electric
multipole
operators
Qj_
are
defined as
z
Gi»i
—
Zt
r
*
Y
ifl
(Q
k
),
i---i
(1)
where the
sum
goes over
all
protons
in
the
nucleus.
The
electric
quadrupole
moment
Q
of
a
nucleus in the
state
ty
)U
is
then
defined,
up to
a conventional
factor
(V*)*,
as the
expectation
value of
Q
i0
in
the nuclear state
\jM}
=
\jj},
i.e.
Q
=
(W*
=
(¥*)*
JK
Qto
^,dx.
(2)
The
quadrupole
moment
measures
the deviation of
the
nuclear
density from
spherical
symmetry.
More precisely,
if one
expands
^jjifj^r) into spherical
harmonics
JfiM*)
=
Mr)+M
r
)
Y
>
+Mr)
Y20
+
A
3
(r)
Y
3Q
+
• •
(3)
then
the
quadrupole
moment
measures
the fraction
of
the nuclear
density
which
is
proportional
to
Y
zo
.
The dipole and the
octupole moments
measure
the
fraction
of
nuclear
density
proportional
to
Y
lQ
and
Y
30
respectively.
CH.
1
.
§
1
]
QUADRUPOLE
MOMENTS
OF NUCLEI
If
an odd-proton nucleus is
represented by
a
single
proton which
mov
the
nuclear
field
produced
by
all
particles,
the
nuclear wave
functio
the
form
*>
=
n
(0
E
i
J
\M
o
«)
**,
X*
•
Here
w„(r) is the radial wave
function
of the proton
and
(l$j\ticrm
Clebsch-Gordan
coefficient
which
couples
the
angular
momentum
fun
Y
tlt
and
the
spin
function
Xt*
t0
the total
angular
momentum
\jm}.
I
extreme
single-particle model
(see Chapter
8)
it
is
assumed
that pai
nucleons
couple
to angular
momentum zero and
are
incorporated
core of
the
nucleus.
The last odd
(unpaired) particle
alone
deter
the
main features of the
nucleus as
long
as
the
excitation energies
ar
too
high. These points will
be
discussed
in much
more
detail in
Chap
and
9.
With
the
expression
(4)
one obtains for the
quadrupole
mo
(2)
the single-particle
value
a
'--
2
W)/ '
(r,r
'
Wd
-
The
last
factor
is
the
mean
square
radius
of the
nucleon.
Eq.
(4)
has
derived
for
an
odd proton. For
an
odd-neutron
nucleus,
there can
exist
a
quadrupole
moment
due to the
recoil
motion
of
the rest
o
nucleus:
a neutron
located at r„ with respect
to the center
of
mass
to a
charge
Z
provided
by
the
rest of
the nucleus
at a distance
rJ(
from the
center
ofmass (see fig.
5).
The
effective quadrupole
moment
i
Fig.
5.
Odd-neutron
nucleus.
The
positions of
the
odd
neutron, r,,
and of the
rJ(A
—
1),
with
respect to
the
center
of
mass (C.Q.M)
aic indicated.
8/18/2019 Eisenberg Grenier Nuclear Theory 1
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12
VARIETIES OF
COLLECTIVE
MOTION
Q*
a
=
[CH. I,
§
1
(6)
(A-
if *
If
we
consider
a
homogeneous
nuclear
charge
distribution
we
have
=
|J?
Z
where
R
is
the nuclear
radius, R
=
r A*.
Consequently
one
obtains
values
for
gip
(for
odd
proton
nuclei)
which
increase
like
A*
and
are of
the
order 10
-26
cm
2
to
5xl0
25
cm
2
.
For
single-neutron
nuclei
the quantity
given in
eq.
(5)
would
behave
roughly like
A~*
and
would
have values
between
10
28
cm
2
and 1(T
26
cm
2
.
i
pfeffftf
against
Z
ZorN
Fig. 6.
Experimental
values
of
electric
quadiupolc
moments
are
plotted
as
a function
of
the
proton or
neutron
numbei
. Zero
quadrupole
moment
indicates
a
spherical
shape
for
the
corresponding
nucleus.
Large
quadrupole
moments
indicate
nuclear
asymmetry.
The
positions
cf
the
magic
numbers
are
shown.
The
data
are
from
Nu-66.
We
have to
compare
these
results
with
the
experimentally
obtained
quadrupole
moments
shown in fig.
6.
Several
discrepancies
are
imme-
diately apparent.
First,
the
signs
of
the
single-particle
quadrupole
moments
of
eqs.
(5)
and
(6)
are
all
negative
while the
experimental ones
are
predom-
inantly
positive
and
only a
few nuclei just
above
magic
numbers
have
negative
quadrupole
moments.
Second,
the
predicted
values
are
much
too
small.
Third,
when
Q
is fairly
large there
is
no
difference
in
the
order
of
magnitude
of
the
moments
of odd-proton
and
odd-neutron
nuclei.
These
ch.
l,
§
l]
QUADRUPOLE
MOMENTS
OF NUCLEI
discrepancies
indicate that a
large
number
of
protons,
and
not
only
one, may contribute
to the quadrupole
moments.
The odd neutron w
give
only
a
small
correction
to
the
large moment
of
the protons.
Ho
one might raise
the
objection that
a
more sophisticated single-
model
may
perhaps
come closer in
its predictions
to
the
exper
observation.
The
fact
that in the
individual
particle
model
the
momentum of
a
nucleus
is shared
among
all particles
outside
close
at
once
points to an explanation
of the fact
that
odd-neutron nucl
electric
quadrupole
moments.
Let us
therefore
consider
the more
case
of
n
equivalent
protons, each in
the
single-particle state
wit
angular
momentum
j.
For
the calculation of
Q
we
use the
correct
symmetrized
wave
function
corresponding
to
a
total
angular
mome
with
(n
—
1
)
nucleons
coupled to zero angular
momentum (seniority
c
scheme).
In section
8.5
we obtain
the following result
for
Q
«-*('-£?)
with
1^
n
£
2/.
In
particular
for
a
hole (n
=
2/)
we see
that
Q
=
For all
values
of n other
than
1 or
2/,
Q
is
less than
g
sp
. It is negat
a shell
more
than
half filled. This
is only in fair agreement
with
obser
(fig.
6}
since
there is an
apparent
preponderance
of
positive
quad
moments.
Again,
the
order of
magnitude
of
the large
moments
i
can
not be
explained. All
these
effects
show that
the
simple single-
model
is much
too
naive and
that quite
a large
number of
particles
involved
in
cooperative
modes
of
motion of an
aspherical
character.
One
can
estimate
the number of
particles
contributing to
th
quadrupole
moments
by considering
the
model of a uniformly c
spheroid.
Let us
approximate
the
shape of
the
spheroid
by
specif
surface
in
spherical
coordinates
through
R
=
R
a
(t
+
f}
Y
in
),
where
R is the
equivalent
radius
of a sphere with the same vo
lowest order
in
j?
. We
obtain
3Z
**
Go
=
(¥*)*
4itR
3
MJ»
dQ
4jiR
J
w3(5Tt)-*J?
2
Z&
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14
VARIETIES OF
COLLECTIVE
MOTION
[CH.
1,
§
I
From
the
experimental
values of
Q
(fig.
6)
one obtains
values
fi
of
the
order
j8
=
0.2
to
0.3. These values
of
j3
are
quite large,
and
it would
therefore
appear
that
in
many
nuclei
a large number
of
nucleons cooperate
to
produce
spheroidal
density
distributions: The
volume in excess
of
the
inner
sphere
(see
fig.
7)
of
the
quadrupole
ellipsoid is
6V =
(5tc)*
/?
3
#o
and
consequently the number
of nucleons
in this
volume is dA
=
(AfV)dV
=
^(Sfn^A^o
where V
is
the
volume and A
the
number
of
nucleons of the nucleus. We
realize
that
dA/A
as
O
.
Fig.
7.
The
shadowed
portion
characterizes the
excess volume
due
to
collective motion.
Another
striking
feature of
the
graph of
Q
in
fig.
6
is the
occurrence of
zeros
for certain N,
Z values of
the nucleon numbers.
The quantity
Q
faUs
from
high
positive values through zero to
small
negative
values. The latter
are
not inconsistent with the single-particle
moments.
From
the
negative
values,
Q
rises
again
rather abruptly
to
large
positive
values.
The
zeros
occur for
values of
N or Z
=
8,
14, 20, 28, 40, 50, 82,
126. The
nuclei
with
these
values
for Z or
N
are
therefore spherical
and
if one or
two
nucleons
are
added, the spherical
core is
apparently
not
greatly
distorted.
The
predictions
of
the individual
particle
model
in
eq. (7)
seem
to
hold
for
the
few
nuclei
in
the
neighborhood
of
these
magic
nuclei.
For example,
the
opposite signs
ofthe
quadrupole
moments
of
holes
and
particles
are
clearly
reproduced
by
the experiments.
But
as the
number
of
extra
nucleons
is
increased it seems that
the
nucleus
takes
on
a
spheroidal shape.
A
few words
are in
order
concerning
the
various methods
for
determining
nuclear
quadrupole moments
experimentally
(Ko-56,
EI-
59a,
El-61). This
may
be done for
instance by
observing
deviations
from
the
interval
rule in
energy
levels
of the
outer
electrons.
The interval
rule is
based
on
the fact that
the
interaction
energy
between electrons and
nuclei
is
proportional
to
I
-
J
where 1 is the nuclear
spin and
J
is the
electron
spin. This
interaction
is
due
to magnetic coupling. The
total angular
momentum
F of an atom
CH.
I,
S
1J
QUADRUPOLE
MOMENTS
OF
NUCLEI
is therefore
given
by
and the rules
for
quantization
of angular
momenta
apply.
The
possible
of
Fare then
F =
(/+/),
(7+J-l
),-•,(/-/)
if
/£/,
F
=
(I+J),
(I+J-
1).
•.(/-/)
if
/£/.
In
the
first
case
we
have
(2/+1)
values,
in
the
second
case
(2/+1)
v
Because
of
the coupling
of the
nucleus
and
the
electrons
due
to the
mag
dipole-dipoie
interaction the
energies
of the
states
(11)
are
not the
Since
the
nuclear
magnetic
moment
is
so
very much
smaller
tha
electronic
one the
coupling
is
weak,
and
this in
turn
leads
only
to
differences
between
the energy
levels
of
these
states.
The
electron
le
angular
momentum
J splits into
either
(2/+1)
or
(2/+1)
closely
s
levels
depending
on
whether
/
g
/or /
g
J. This
is
known
as
the
hyp
structure
of
the
level. It
is
to be
distinguished
from
the
fine
structur
level,
which
is
due
to
the
coupling
of
the
orbital
and
spin
angular
mom
of
an
electron.
Each
fine-structure
level
can
have
hyperfine
stru
As
mentioned,
the
energy
of
interaction
due
to the magnetic
coupl
proportional
to the
scalar
product
(rotational
invariant)
of the
an
momenta
/
and
J,
#-Ai-l.
The
expectation
value
of this
energy
in an
atomic
state
|
F(/,
/)>
F(/,
J)
indicates
that
/ and
/
are
coupled
to F,
is
m iA
=
HF(F+l)-/(/+l)-/(/+l)).
Substituting
the
various
values
(11)
for
F into
eq.
(13),
we
see
tha
differences
between
successive
states
of
a
multiplet
are
A(l+J),
A(l+J-1),
A(I+J-2),
• •
•,
A{I-J).
These
energy
differences
are
thus
proportional
to
the
values
(1
1)
of
F.
is
known
as
the
interval
rule.
Note
that
if / <
/ it is
immediately
pos
to
determine
the
nuclear
spin
/
by
merely
counting
the levels
into
wh
given
/-level
is
split.
If
/ >
/,
then
/ can
be
found
by
counting
the
hype
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16
VARIETIES
OF
COLLECTIVE
MOTION
[CH.
1, §
1
structure
levels
and
F
can be
found
from
the
relative
separations
of the
levels,
by the
use of
the
interval
rule.
Hence
/
can
again
be
determined. If,
however,
in
addition
to the
magnetic
coupling
there is
also
an
electric
coupling
due
to
the nuclear
quadrupole
moment
(the
quadrupole
interaction
energy
is
proportional to
Qfr
3
)
this
will
lead to
deviations
from
the
interval
rule
from
which
the
quadrupole
moment
can
be
determined. If a
transition
takes
place
between
two
levels each of
which has hyperfine
structure, then
the
corresponding
splitting
of the
spectral lines
will
be
much
more
com-
plicated
than
the splitting of
either
of
the two
levels.
In
this
case
it
is
necessary
to
select
two
electron
levels, one
of
which
has
much more
widely
spaced
hyperfine
structure
than the
other.
In
that
case
the
splitting
of the spectral
line will
be
due
essentially
only
to the
splitting
of
that
level.
A
very accurate
measurement
of
the
quadrupole moment
became
avail-
able
through
the
exploitation of
muonic atoms
(Ack-65a,
b,
Se-67). If a
muon is
captured by
a nucleus into
an
atomic
orbit
through
the
ejection of
an electron, the
muon will
have an
energy spectrum
similar to that
of
a
one-
electron
atom.
Because
of
its
larger mass
(/n„
= 270 m
t
)
its
Bohr
orbits
are
partly inside the
nucleus and
therefore the
muon is a good
probe
for
measuring
nuclear
properties. The muon
is captured in a
high-lying
orbit
from which
it
cascades down to
the
Is
state. The
emitted
X-rays can
be very
precisely
measured.
From the
splitting
of the
2p
i
-*
2p
f
u-meson-level
which is
additional to the
fine
structure, one
can
obtain
even
the
signs
of the
so-called intrinsic
quadrupole moments
of
even
nuclei
*.
A
very
modern
method for the
determination of the transition
quad-
rupole moments
as well as
the
quadrupole moments in
excited
nuclear
states
is
available in
Coulomb
excitation (Br-57a,
deB-64,
65,
Ei-64,
Dou-67,
Gr-68a).
A
low-energy
heavy ion
approaching
a
nucleus
excites
the
nucleus
Fig.
8. Processes occuring
in
Coulomb
excitation
of
the
first
2*
excited
state
of
a
nucleus.
The
direct
tiansition
is indicated by
a
double
arrow.
The
reorientation effect
(Ip*')
is
also
indicated.
*
For
more
details
see Chapter 9 of
Volume
2
in this
series.
ch. 1,§1]
QUADRUPOLE
MOMENTS OF
NUCLEI
due,
for
example,
to
the
quadrupole
part
of
the
Coulomb
interaction.
excitation
in
question
may
take
place, say, from
the
+
ground
sta
the
2
+
first excited
state.
Possible
mechanisms
for this
reaction are
fig-
8):
a),
direct
E2
(electric
quadrupole)
transitions,
b). E2
transitions to
a
higher
state,
then
dcexcitation
to
the
first 2
+
c).
reorientation of
the
nucleus
in
the
2
+
state
(relating to the
quadr
moment
in the excited
slate, see
Br-56,
Ei-64,
MacD-64,
E
The
reorientation of the
nuclear
axis
is
caused by
the
electric field
bombarding
particle
which
has excited
the nucleus.
The change
in nu
spin
direction
affects
the
angular
distribution
of the emitted
X-rays o
scattered
particle.
For
low-energy
bombarding
particles
with small
mass
and
charge
on
direct
transitions are
appreciable.
But
for
heavy
charged
particles
the se
order
effects
begin to
contribute.
Using
second-order
perturbation
theo
transition
probability
has the
form
P
*
|
+ fl|
2
,
where
the
second
term
with
the
factor
a
describes
the
reorientation
in
which
the
nucleus
is
first
excited
into
the
state
|2Af
>
and
then
re
ted
into the
state
|2M>.
If the
transition
operator
is
the
Q
2a
comp
of the quadrupole
operator
in
eq.
(1),
then
this becomes
P x
\aM\Q
2O
\0y\
2
+2ai2M\Q
2o
\0'>
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,6
VARIETIES
OF COLLECTIVE
MOTION
[CH.
1, §
2
CH.
1,
§ 2]
ELECTROMAGNETIC
TRANSITIONS
1.2.
Electromagnetic
transitions
1.2.1.
Preliminaries
on
radiative transitions
The
single
particle
model
fails
to
explain
the magnitude
of
many
nuclear
electromagnetic
transitions
for
reasons similar
to
those which account
for
its
inability to predict
static quadrupole
moments. It
is very
suggestive
and
plausible
that for
any
transition
in which
many
nucleons
contribute
in
a
coherent
way (many
nucleons
jumping
simultaneously)
the
transition
probability
will
be
much
larger
than
for
those transitions
where
only one
nucleon
changes
its
configuration
(single-particle
transitions).
These
ideas
have
to
be
formulated
quantitatively,
and
in particular
the experimental
y-ray
matrix elements
must
be
compared
with
the
simple
single-particle
predictions.
The results of such
a comparison
will
be
surveyed
here.
In this
way it is again
possible
to
exhibit
the collective
properties
of
nuclei.
The mathematical
details
of
the
theory of
radiative
transitions
are de-
scribed
extensively
in
Volume
2
of
this
series.
It
is
therefore
not
necessary
to recapitulate
radiation
theory here.
Instead
we shall
give
directly
the
formulas
for
electromagnetic
transition
probabilities
and
begin
our
discus-
sion
at
this
point. The transition
probability
for
emission of
a photon
of
energy hca,
angular
momentum
A, and
of
electric (a
=
e)
or
magnetic
(o
= m)
character,
with
a
nucleus
going from
an initial
state
|i>
to a final
state
|f
>
is
Here
8
X„
stands for
the
electric
{Q
Xlt
)
or
magnetic
(iv/^)
multipole
operators
which
are
given by
*
4U0
=
&,
=
I
for/
V(Oj)
+
ifcjK»(
8/18/2019 Eisenberg Grenier Nuclear Theory 1
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VARIETOS
OF
COLLECTIVE
MOTION
Ich.
1,
§
2
We
consider here
only
the
significant mullipofes
for
a given A
J and limit
our discussion
to
/
=
AJ
=
A/.
Such
a
transition
is the
only
one possible if
J
t
= as
in the
example
given in fig.
9,
where
a
transition
from
a state
J= JU-1T
E{»
0*
Fig.
9. The decay
of
a
level
withangular
momentum
a
and
paiity
(—
1)*
to
a
+
state
occurs
only through
LU
radiation.
with
J
=
A and
parity
(— l)
A
to a
/
=
+
state is
shown.
In this case
only a
multipole with
X
=
J will
contribute
to the
transition.
Such
situations
occur
frequently
in
nuclear physics especially
for ground
state
transitions
in
even
nuclei,
since
these
nuclei
have spin
zero
in the
ground state. It
is straight-
forward
to
compute
expression
(21)
for
these cases explicitly,
and one
obtains
for the
single-particle
transition
probabilities
where
;.[(2A+1) ]
J
(23)
is the radial
matrix
element,
and
5 is
a combination
of
Clebsch-Gordan
coefficients
which,
in
these
cases,
reduces
to
Kti
'
°
(2J
l
+
l)(2/
>
) A (J
<
-i)
(24)
J
>
and
J
<
are
the larger
and
smaller
values of
J
t
and
J
f
respectively.
It
is
our
aim
to
obtain
order-of-magnitude
expressions
for
the
single-
particle
values
given
in eq.
(22).
We therefore replace the integrals in
eq.
(23)
by
rough
approximations.
Taking
the
radial functions
u(r)
constant
and
normalized
to
unity
over
the nucleus, we
get
*-&)•
(25)
CH. 1,
§ 2]
ELECTROMAGNETIC TRANSITIONS
The
statistical
factor
S
and the
nuclear radius R are now
the only
pa
eq.
(22)
which refer to a specific nucleus.
In
order to define conve
units
for
measuring lifetimes
we
set S
—
I
and
obtain
T
W
(EA)
=
2(A+1) /
3 \
2
e*_
{a>R\
a[(2a+1) ]
2
U
+
3/
hc\ c
)
u
a
sec
The
subscript
W
stands for Weisskopf who
first
introduced these
(Wei-51).
We have
used
for the charge the
unit charge
e.
Naively
one
expect
that this is true for single protons and that single
neutrons
charge
zero.
However, if
one
takes
recoil
effects into
account both par
acquire
an
effective
charge
which
depends
on
both
the
multipolarity
transition and the nature
of
the single-particle
wave function. We
d
distinguish neutron from
proton transitions, since
we
wish
to
use
standards for
comparison
of
all experimental transitions regardless o
model for
the
nuclear
states
involved.
A universal value
is
required
f
nuclear
radius
and it has been taken arbitrarily as R
as
1.2 A*
fm.
It
is
often
useful
to quote
the numerical results in the
form
of
the
wid
y-decay, denned
as
7
=
hT.
If the y-ray
energy is
£
Y
,
then F
yW
(EA)
is defined in analogy to eq.
by
use
of
eq.
(26)
and is proportional to
A
n
E
y
u+l
.
In
order
to h
feeling
for the
order
of
magnitude of f
?w
,
table 1
gives
some
nume
values
for
various
multipole
transitions. If
E
?
is 1
MeV,
the values
fo
in table
1 are electron
volts.
In
comparing
with
experiments
one has
the
partial
y-ray width.
Therefore
corrections have to
be
made for
peting
processes such
as
internal conversion
or,
if the
nuclear
stat
above
the
particle-threshold,
for
particle
emission.
Table I
Weisskopf
estimates
f