Eisenberg Grenier Nuclear Theory 1

8/18/2019 Eisenberg Grenier Nuclear Theory 1

1/257

nuclear

theory

EIS


2/257

MCMXCVII


3/257

PRESTON

POLYTECHNIC

LIBRARY &

LEARNING

RESOURCES

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


4/257

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


5/257

©

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


6/257

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


7/257

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


8/257

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.


9/257

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.


10/257

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


11/257

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


12/257

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


13/257

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).


14/257

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


15/257


16/257

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.


17/257

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&


18/257

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


19/257

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'>


20/257

,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»(


21/257

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

Eisenberg Grenier Nuclear Theory 1

Documents

Transcript of Eisenberg Grenier Nuclear Theory 1