MULTISTEP PROCESSES IN NUCLEAR

TRANSFER REACTIONS

CHARLES H. KING, JR.

1973

Multlstep Processes in Nuclear Transfer Reactions

by

Charles H. King, Jr.

B.A., Northwestern University, 1967 M.Phil., Yale University, 1969

A Dissertation Presented to the Faculty of the Graduate School of Yale University in

Candidacy for the Degree of Doctor of Philosophy

1973

To the memory of Mrs. C.T. King and Mrs. T.N. Nickles

and to my family:Mom, Dad, Tom, Betsy, Dave,

David Allen, and Cindy

Abstract

A study was made of multistep nuclear transfer reaction processes involving Inelastic excitations in the target and residual nuclei by investigating the reactions ^®6w ( p ,t)^®4W and W(p,d)*®^w. These reactions were expected to be in­fluenced by such multistep processes, since the tungsten nuclides possess strongly collective excitation modes of a rotational character resulting from their non-spherical equi­librium shapes. Complete angular distributions for these reactions at 18 MeV incident proton energy were measured in the Yale multigap magnetic spectrograph. Data were extracted for (p,t) transitions to states in ^®4W up to 1225 keV exci­tation energy and for (p,d) transitions to states in upto 350 keV excitation energy. In both reactions several examples were noted of transitions involving transfer of the same orbital angular momentum but possessing significantly different angular distribution shapes, and this effect was interpreted as a possible result of the presence of multistep processes.

To confirm this interpretation, coupled-channel Born approximation (CCBA) calculations using the source-term method of Ascuitto and Glendenning were performed for (p,t) transitions to members of the ground and first-e:cited K=0+ bands in atl<j for (p>(i) transitions to members of the1/2“ [510] and 3/2“ f 512] bands in 1®^W. The intrinsic states of the bands in 1 ®^W were determined in the quasiboson ap­proximation using a pairing-plus-quadrupole residual inter­action, and the two bands in were assumed to be singlequasiparticle bands mixed by Coriolis coupling. The CCBA calculations,which include the effects of inelastic excita­tions to all orders but treat the transfer process to first , order, were found to reproduce the experimental differential cross section much better than calculations using the distor­ted-wave Born approximation, which neglects inelastic effects. It is indicated that many of the remaining discrepancies between the CCBA calculations and the experimental data can be attributed to uncertainties in the optical-potential par­ameters and in the parameters used to determine the nuclear structure of the states. It is shown that the inelastic processes affect not only the magnitudes but also the shapes of the angular distributions in such a way that extraction of angular momentum information on the basis of characteris­tic differential cross section shapes is questionable. It is found that for the strongest transitions (peak cross sections £ 250 pb/sr), the multistep effects are often small enough that they can sometimes be simulated in the DWBA by adjusting the optical parameters, but for most other tran­sitions the effects are so strong that the CCBA is needed in order to extract meaningful nuclear structure information from reactions on deformed nuclei.

Acknowledgements

I am pleased to acknowledge the guidance and support

of Dr. Nelson Stein, who served as my research adviser

during the course of the work described herein. His ability

to focus on the essential physics of a problem and his in­

sight into the basic structure of nuclei have provided a

constant example for me. In addition, I am grateful for

the freedom he has allowed me in my work and for the cheerful

manner in which he has borne my idiosyncracies.

I would also like to express my appreciation for the

continuing advice and assistance of Professor Robert J.

Ascuitto, who suggested this problem and without whom any

detailed understanding of the results would have been im­

possible. The energy and enthusiasm he has brought to the

problem, the many long hours he has devoted to making the

calculations work, the free access he has given me to his

computer programs, and his encouragement and friendship I

will never forget. His vast understanding of nuclear

reactions and the honesty and meticulousness with which he

carries out his work have made a lasting impression on me.

I am grateful for the opportunity I have here to thank

Professor D. Allan Bromley, who not only, as Director of

the A.W. Wright Nuclear Structure Laboratory, provided the

facilities and secured the essential financial support for

this work, but also, in the absence of Dr. Stein, graciously

served as my mentor. The excitement he brings to nuclear

physics and his wide-ranging knowledge of physics in general

have long been an inspiration for me, and the time he has

devoted to reading this dissertation and the invaluable

advice he has given me in writing it are sincerely appreciated.

It is also a pleasure to thank Dr. William D. Callender,

who generously contributed his time and skill to the comple­

tion of the experimental aspects of this work, Dr. Terrance

P. Cleary, who was a tremendous help in matters experimental

and a constant friend and sounding board in other matters

as well, Dr. Lance J. McVay, whose help with all aspects of

this work are much appreciated, and Dr. Bent Sorensen, for

his help with calculations.

I would also like to thank my many friends and col­

leagues during my graduate career. The illuminating discus­

sions I have had with Drs. Edward Ayoub, Lowry Chua, George

Holland, Charles Maguire, Paolo Maurenzig, Ravinder Nath,

Daniel Pisano, and Subodh R. Shenoy have been especially

useful in this work. In addition I am grateful to all

those associated with the A.W. Wright Nuclear Structure

Laboratory for the help they have provided over the years,

but especially to Kenzo Sato and Philip Clarkin for their

superior skill in keeping the accelerator working, to Hana

Novak and Paul Prinz for the extra effort they gave in pro­

viding the illustrations for this dissertation, to Bernadette

Kennedy, Muriel Wright, Karen Miller, and Dorothy Purcell

for a superb job of scanning the photographic plates, and

Harriet Comen and Mary Ann Thompson for expert secretarial

assistance. I am also indebted to Cynthia Ellis for the

excellent and intelligent typing she provided for this

dissertation, which the results clearly indicate.

I would like to thank the staff of the New York Univ­

ersity Computing Center for providing the facilities neces­

sary for the calculations and the United States Atomic

Energy Commission for the financial support of this research.

Finally, I would like to express my appreciation to my

parents, whose support both moral and financial at critical

moments was invaluable.

Table of Contents

Abstract page

Acknowledgements

1. Introduction 1

2. Theoretical Foundations: Nuclear Reactions2.1 Introduction 1 12.2 The Nuclear Collision Problem 112.3 The Optical Potential 152.4 The Distorted-Waves Approximation 202.5 Properties of the Distorted-Wave Born

Approximation 222.6 The Coupled-Channel Born Approximation ;34

2.6.1 The CCBA Transition Amplitude 342.6.2 The Source-Term Method 392.6.3 Solutions to the STM Equations 43

2.7 Discussion of the Approximations 46

3. Theoretical Foundations: Nuclear Structure3.1 General Considerations 523.2 Rotational Motion 543.3 The Independent-Partide Hamiltonian 623.4 The Residual Interaction 693.5 Coriolis Coupling 803.6 Applications to Reactions 8 6

3.6.1 Inelastic Matrix Elements 8 63.6.2 The Nuclear Form Factors 89

3A. Coriolis-Coupling Matrix Element 943B. Inelastic Form Factor for Odd-Mass Nuclei 963C. Form Factor for Single-Nucleon Transfer

from an Even-Even Deformed Nucleus 97

4. Experimental Procedure and Presentation of Data4.1 General Description 1004.2 The Beam 1014.3 The Target 1034.4 Particle Detection: The Multigap Magnetic

Spectrograph 1054.5 Determination of Absolute Cross Sections 1094.6 Presentation of Experimental Results 111

4.6.1 186w(p,t)184w 1114.6.2 186w(p,d)185W 1164.6.3 Discussion 120

5. Theoretical Analysis: *-®8 W(p , t) 84W5.1 Introduction 1 2 25.2 Determination of Parameters 1235.3 The Ground-State Rotational Band 1265.4 The B-Vlbrational Band 1315.5 The y-Vibrational Band

Theoretical Analysis: *8 8 W ( p ,d)

133

6 . 1 Introduction 1366 . 2 Determination of Parameters 1386.3 The Form Factors 1436.4 Pure-Band Calculations 1486.5 Inclusion of Coriolis Coupling 1566 . 6 Sensitivity of the Calculations to

Factors and Optical ParametersForm

1596A . The Effect of Coriolis Coupling on

Calculations of Transitions to a State

CCBAPure

167

Summary and Implications 170

References 175

CHAPTER 1

INTRODUCTION

"It must certainly be true for some levels In all nuclei, and all levels in some nuclei, that the usual treat­ment of particle-transfer reactions, which neglects inelastic effects, is invalid."

- R.J. Ascuitto and N.K. Glendenning (As 69)

Perhaps the most significant characteristic of the

atomic nucleus, from the viewpoint of those wishing to study

its properties, is its extremely small size. An object with-13dimensions the order of 1 0 cm must clearly be studied

via indirect means. Following the pioneering work of

Rutherford in the early part of this century (Ru 11, Ru 19),

an important technique used for the study of nuclear proper­

ties has been the observation of its interactions with sub­

atomic projectiles. However, in order to infer information

about the structure of nuclei from such scattering experi­

ments, it is necessary to understand the detailed mechanism

of nuclear reactions.

Originally, most descriptions of nuclear scattering

events were based on one of two extreme models. In one

extreme, known as the compound-nuclear mechanism, the nucleus

is pictured as absorbing the projectile, thereby forming a

"compound nucleus" (Bo 36, Bo 37). In the compound nucleus

the energy of the incident projectile is quickly shared

through collisions by large numbers of nucleons. The assump­

tion is made that so many nuclear interactions occur, and

hence so many degrees of freedom are excited, that the compound

nucleus approaches statistical equilibrium and no correla­

tion with the original entrance-channel motion remains. On

the average, no particle has sufficient energy to escape

from the compound nucleus; but, after a time which is ideally

several orders of magnitude longer than the nuclear transit

time (the time necessary for a medium-energy projectile to

1

'V - 2 2traverse the nucleus = 1 0 sec), the compound state even­

tually decays either by particle or gamma-ray emission.

Experimentally, a reaction through an isolated compound state

is characterized by a cross section with rapid energy dependence

near the resonance energy of that state and outgoing-projectile

angular distributions which are symmetric about 90° in the

center-of-mass reference frame. As the energy of the compound-

nuclear system is increased, the density and width of the

states increase dramatically, particularly for heavy nuclei.

Thus, many descriptions of reactions from the compound-

nucleus viewpoint are concerned with averages over large num­

bers of resonances and involve statistical models for the

average formation and decay probabilities (La 58, Vo 6 8 ).

However, not all nuclear reactions can be well described

in the compound-nucleus limit. Many processes were found to

have cross sections that vary slowly with energy and final-

projectile angular distributions which are forward-peaked

(La 35, Bu 50, Ho 50). It was suggested (Op 35, Bu 51) that

such reactions could be explained in terms of a direct-

reaction mechanism. The term "direct reaction" is used very

loosely in the literature; however, it usually refers to a

transition resulting from a first-order nuclear interaction

and thus exciting a single nuclear degree of freedom. Such

reactions take place in a time on the order of the nuclear

transit time and tend to have a weak energy dependence. Two

examples of direct reactions are the direct transfer and the

direct inelastic. These are illustrated schematically in

2

Figure 1-1 and compared with the compound-nuclear mechanism.

The direct transfer process, ideally, involves the removal

of one or more nucleons from the target nucleus to the pro­

jectile ("pickup") or vice versa ("stripping") without dis­

turbing the motion of any other nucleons in the system. On

the other hand, a direct inelastic process refers to the

inelastic scattering of the projectile, while exciting a

single degree of freedom of the target nucleus. This can

involve either the excitation of a nucleon from one orbit

to another or, as depicted in Figure 1-1, the excitation of

surface oscillations or rotations of the nucleus as a whole.

Such direct reactions result from relatively weak coup­

ling between the nucleus and projectile. Thus, it was sug­

gested by Butler (Bu 51) that these reactions might be des­

cribed using the plane-wave Born approximation (PWBA), and

this treatment yielded the first detailed understanding of

the experimental results. The plane-wave approximation

depends on the assumption that the various reaction processes

are completely independent, and this applies reasonably well

at very high energies. However, within the energy range of

most nuclear experiments, only qualitative agreement with

the data can be obtained.

To achieve a better description of the experimental

data, the effects of the neglected processes must be included,

at least indirectly. The most important of these processes

is elastic scattering. Therefore, it was suggested (Ho 53,

To 55) that a better approximation could be obtained by

3

Figure 1-1. Schematic plctorlallzatlons of nuclear re actions proceeding via the compound-nuclear and direct mechanisms.

Figure 1-1

• »■ 0 ° * o • • ° m O° o ? ° oo • ° 0° O M• 0 V 0V

compoundnucleus

COMPOUND

DIRECT TRANSFER

0°M — *> $ # < *> .

DIRECT INELASTIC

replacing the plane waves with waves distorted by a complex

potential. The real part of this potential describes the

elastic scattering, and the Imaginary part describes the loss

flux into all the other possible reaction channels. This

technique, which is known as the distorted-wave Born approx­

imation (DWBA), provided a considerably improved description

of the experimental direct-reaction data.

The simple division of nuclear processes into compound-

nuclear and direct has been quite useful in the study of

nuclear structure and has led to the determination of a large

number of nuclear properties. This picture has long been

recognized as a caricature of the actual situation, however,

and in retrospect, it seems that its somewhat surprising

success resulted partly from the fact that It was applied to

a limited range of situations. Many early experiments anal­

yzed with the DWBA, for example, considered mainly reactions

on spherical nuclei with relatively weak modes of excitation.

In such situations, the more complex processes generally

make only a small contribution to the transition and can

usually be simulated with sufficient accuracy through the

imaginary part of the distorting potential.

In recent years there has been increasing Interest in

those processes which are neither completely direct nor

completely compound-nuclear. The present dissertation is

concerned with a category of processes which excite

only a limited number of nuclear degrees of freedom, as in

the direct mechanism, but involve Interactions higher than

4

first-order. Since they can be loosely pictured as proceeding

through several direct-reaction steps, they are known as

"higher-order direct" or "multistep" processes. A few exam­

ples of multistep processes are illustrated schematically

in Figure 1-2. Perhaps the simplest is multistep inelastic.

The example shown in Figure 1-2 is one in which the reaction

proceeds by successive excitation of different rotational

degrees of freedom. Closely related to this is the "transfer-

plus-inelastic" mechanism. This process involves direct

transfer accompanied by inelastic excitation of the nucleus

by the incident and/or outgoing projectiles. The addition

of the inelastic scattering allows transfer from excited

states of the target nucleus in addition to the ground state

as in a pure direct process. Other examples of multistep

processes illustrated in Figure 1-2 are the multiple transfer

mechanism, which proceeds via several transfer steps, and

the exchange mechanism. The latter is perhaps not strictly

a higher-order process, since it can be treated in the DWBA

if the scattering amplitude is properly antisymmetrized.

Nevertheless, this antisymmetrization is usually ignored in

the analysis of direct reactions, and the exchange mechanism

is often characterized as a higher-order process.

Inelastic excitation of strongly collective states in

heavy nuclei is generally much stronger than the transfer

mechanism, since collective states involve the coherent motion

of large numbers of nucleons. As a result, multistep pro­

cesses involving collective excitations will often be the

5

Figure 1-2. Schematic plctorlallzatlons of various multi- step nuclear transfer reactions mentioned In th.e text.

Figure 1-2

M U L T IS T E P IN E L A ST IC

IN E L A S T IC + T R A N S F E R

M U L T IP L E T R A N S F E R

EXCHAN GE T R A N S F E R

•o

most important, and it is natural to attempt to understand

these mechanisms first. Multistep inelastic scattering to

collective states has now been extensively studied (Gl 67),

and it has been found that such processes can be treated in

principle to all orders with considerable success by solving

a set of coupled differential equations. In addition, by

using such an analysis, a number of new types of nuclear

information have been extracted from inelastic scattering

data. For example, higher-order moments of the mass distribu­

tion of nuclei which have permanent deformations have been

measured in this manner (He 6 8 , Ap 70).

It was realized aome time ago by Penney and Eatchler

(Pe 64) that the transfer-plus-inelastic processes could be

treated theoretically by combining the coupled equations

used to describe multistep inelastic scattering with the DWBA

to form the "coupled-channel Born approximation" (CCBA).

Unfortunately, the numerical evaluation of the CCBA transition

amplitude in the original formulation proved to be extremely

difficult, and most of the original calculations were performed

by making a number of rather severe approximations, such as

the treatment of the Inelastic interactions only to first

order (la 6 6 ). An alternative model proposed by Kozlowsky

and de-Shalit (Ko 6 6 ) and by Levin (Le 6 6 ) also treated the

inelastic effects to first order only and, in addition, ig­

nored the entrance-channel effects entirely. The preliminary

indications from these approximate calculations and the few

available complete calculations (Du 6 8 , As 70a) were that the

inelastic-plus-transfer processes are either small or result

in angular distributions indistinguishable from those of a

pure direct process (G1 69). Thus, many experimentalists

continued to ignore these effects and attempted to infer

angular momentum and structure details of states in highly

collective nuclei from DWBA analyses of transfer reactions

(El 69, Oo 70).

Meanwhile, a number of experiments with single-nucleon

transfer reactions of relatively small cross section were

revealing anomalies both in the shape and magnitude of the

angular distributions which could not be explained by the

DWBA (Bo 64, Be 65, Co 6 6 , Si 6 6 , De 67, Co 6 8 ). Although

some of these effects were interpreted as resulting from

inelastic processes, none of the calculational methods then

in use could account for them in detail. When it was finally

noted by Ascuitto and Glendenning (As 69) that the CCBA can

be treated via the solution of two sets of coupled equations

without the necessity of evaluating overlap integrals, large-

scale calculations of inelastic processes accompanying trans­

fer reactions became feasible without the necessity of gross

approximations. This "source-term" method has been used for

calculating (p,t) reactions on the collective rare-earth

nuclei, and these calculations reproduced extremely well

(As 71, As 72) certain anomalous angular distributions observed

experimentally (Oo 70, De 72, Oo 73). However, additional

experimental data suitable for use as a test of the CCBA

were largely unavailable! and thus the present investigation was

7

undertaken in an effort to provide such tests.

In the search for adequate test cases for the CCBA it is

important to satisfy certain conditions. First of all, the

case should be one in which the inelastic scattering is stron

enough for the inelastic-plus-transfer to be at least compar­

able to the simple direct transfer process. Next, it is im­

portant that the structure of these states be understood well

theoretically so that structure uncertainties do not mask re­

action effects. Finally, it is advantageous to select a case

in which the transfer part of the reaction mechanism can b e :

well described in terms of a direct process. This will pre­

vent the transfer-plus-inelastic effects from being obscured

by the presence of other strong higher-order effects, such as

multiple transfer, compound-nuclear mechanisms, etc.

These conditions are well satisfied by the ^^^WCpjd)186and W(p,t) reactions, the ones chosen for study in the

present investigation. The tungsten nuclei are heavy nuclei

with collective excitation modes resulting from the presence

of rotational bands. Such rotational excitations are among

the strongest and best understood theoretically of all the

nuclear collective excitations (Na 65). The (p,d) and (p,t)

reactions, which involve the pickup of one and two neutrons

respectively, are among the simplest and most commonly mea­

sured of all reactions used in the study of nuclear structure

They form an instructive contrast, since the single-nucleon

transfer reactions are sensitive only to single components

of the nuclear wavefunction while the two-nucleon transfer

8

reactions can be sensitive to coherent combinations of large

numbers of components (G1 65).

It was decided to perform these reactions at an incident

proton energy of 18 MeV, using the Yale MP tandem accelerator,

which can provide such a beam of protons with high energy

resolution. In addition, the Yale multigap magnetic spectro­

graph provides an excellent detection instrument for the

outgoing deuterons and tritons in these reactions. It can

provide a reliable complete angular distribution, since the

twenty-three gaps in its toroidal magnet allow simultaneous

detection of particles at twenty-three angles from 0° to 172.5°.

Its high-resolution capabilities are extremely important be­

cause of the high density of states in deformed nuclei such

as and . Finally, as an additional bonus, its

broad-range momentum-detection characteristics allow simul­

taneous detection of both deuterons and tritons from these

reactions, so that the two reaction experiments can be performed

at the same time.

The description of these experiments and their analysis

is presented in the succeeding chapters of this dissertation.

In Chapter 2 the theory of transfer reactions is described

in more detail and the advantage of the source-term formula­

tion over the more conventional formulation of the CCBA is

demonstrated. Chapter 3 presents the detailed theoretical

methods required to describe the structure of deformed nuclei

such as the tungsten isotopes and applies them in determining

the quantities necessary for a CCBA calculation. In Chapter

9

4 the experimental methods used In this work are described

and the data are exhibited, and Chapters 5 and 6 contain1 ftfi 1 8 6the results of a CCBA analysis of the W(p,t) and W(p,d)

reactions, respectively. It is shown that the CCBA provides

an excellent description of much of the data and is a consider

able improvement over the more conventional DWBA. In addition

it is demonstrated that the inelastic processes alter not

only the magnitude but also the shape of the angular distri­

bution predicted for a purely direct process and that when

the Inelastic effects are strong, these shapes are not neces­

sarily characteristic of the orbital angular momentum

transferred in the reaction, as is usually the case for a

direct reaction. Finally, in Chapter 7 the principal results

of the analysis are summarized and implications are noted

concerning the use of the CCBA as a useful theoretical tool

for determining nuclear structure from experimental reaction

s t udies.

10

CHAPTER 2

THEORETICAL FOUNDATIONS: NUCLEAR REACTIONS

"Our knowledge being so narrow...It will perhaps give us some light... if discovering how far we have clear and distinct ideas, we confine our thoughts within the contemplation of those things that are within reach of our understanding, and launch not out into that abyss of darkness (where we have not eyes to see, nor faculties to perceive anything) out of a pre­sumption that nothing is beyond our comprehension."

- John Locke, Essay Concerning Human Understanding, Book IV, Chp. Ill, 22.

2.1 Introduction.

There are several theoretical frameworks which have

been devised for the study of nuclear reactions, although

most fall into one of two categories. One type of theory

utilizes the short-range character of nuclear forces by

formulating reaction theory in terms of matching conditions

for the total scattering wavefunction at the boundary of

the interaction region (La 58) . The behavior of _.he sys­

tem in the internal region can then be described in terms

of a discrete set of eigenstates. In the other category

of reaction theories, the dynamical aspect of nuclear

reactions are emphasized by considering explicit solutions

of a Schrtidinger equation simplified by the use of approx­

imate models for the actual system Hamiltonian (Fe 58, Au

70). Although both approaches are general, the former is

most applicable to the description of processes which are

explicitly many-body in character, such as the formation

of compound nuclei, where the discrete spectrum of eigen­

values can be related to the resonance energies. On the

other hand, since the dynamical behavior of direct and

multistep processes is relatively simple, the latter approach

is usually most appropriate in these cases. It is this

approach which is followed in the present chapter.

2.2 The Nuclear Collision Problem.

The basic problem of nuclear reaction theory is to

describe the result of the collision between a projectile a

and a target nucleus A. In this collision numerous possible

11

final states can occur:

a + A -*■ a + A

-+ b + B

-*■ c + C

d + e + D + E

etc .

That is, the final state of the reaction can consist of

several possible partitions of the total system of particles.

These partitions can contain more than two members, but for

present purposes only two-body final states need be consid­

ered .

If the total Hamiltonian of the system is H, the

SchrBdinger equation for the collision process may be written

( H - E ) = 0 (2-1).

To make the partitions of the system meaningful, it must

be assumed that the interaction potential between the vari­

ous members of a partition has a finite range so that at large

relative distances the subgroups are non-interacting. Then,

for each partition a, the Hamiltonian may be separated into

two parts,

H = K + V (2-2) ,a awhere is the Hamiltonian which describes the system out­

side of the range of the interaction potential V . Eachclpartition may be in various states. For example, in the

partition b+B, the projectile b and the target B, besides

being in various states of relative motion, may each possess

various states of internal motion. Each such possible com­

12

13

bination is called a channel. The eigenstates of K forSi

partition a in channel a are defined by

K <P = E <j> (2-3)a aa raa v '

To relate the reaction mechanism to a physically

measureable quantity, it is convenient to define (Go 64)

the transition amplitude between the channels (a,a) and

(b, 3) .

:ac.bB 5 * *bS I Vbl =• (2‘4a)

5 •= " V I v »l K a > <2 - 4 b >-

.<+>where 4laQL is the eigenfunction of H with a plane wave in

channel a of partition a and outgoing spherical waves, and

is the eigenfunction with incoming spherical waves and

a plane wave In channel 6 of partition b. Equation (2-4a)

is known as the "post" form of the transition amplitude,

and Equation (2-4b), the "prior" form. The quantity measured

in a reaction experiment is, of course, the cross section.

It is well known from reaction theory (Go 64, Au 70) that

the transition amplitude and cross section for the process

(a,a) -*■ (b,3 ) are related by

l°acbbg . M PoMB . H , I. (2_5)da "aB (2Trb2) 2 ka aa-bB

Here y and y Q are the reduced masses of channels ct and 3 (OC pm mA +respectively (y = — -- ), and ftk (Ak,) is the relativer ■ ' ' a m +m ' ’ a b 'a A

momentum of projectile a (b) with respect to nucleus A (B).

The quantity is a statistical weighting facto-: resulting

from antisymmetrization.

For the stripping reaction A(a,b)B, that is,

A + (b+x) -*■ (A+x) + b

with b+x = a and A+x = B, it is standard to use the post

form of the transition amplitude, Equation (2-4a). Then

V = V = V + Vb A+ x ,b Ab xb

where V denotes the interaction potential between nucleus

i and nucleus j. Thus,

T stripping = ,v +v | (+)>aa,b8 b3 Ab xb* aa

For the pickup reaction

(B+x) + a -► B + (x+a)

with B+x = A and x+a = b, it is customary to use the prior

form of the transition amplitude so that

tPickup = <1j;(~) I v +v I d) > (2 -6 )aa,b$ b3 ' Ba xa* aa

For example, in the case of the (p,d) reaction on nucleus

A, x represents a neutron, and a, a proton so that

15

2.3. The Optical Potential.

The exact wave function ip describes all the allowed

processes for the system. It contains not only the direct

and multistep mechanisms but also compound-nuclear and other

more complex processes. Obviously, a practical calculation

requires a drastic truncation of 4>. This truncation can be

taken into account, as noted by Feshbach (Fe 58, Fe 62), by

replacing the exact interaction potential with a complex ef­

fective interaction potential.

To achieve the truncation of ip it is convenient to define

the projection operators P and Q such that P projects the full

wavefunction onto the truncated space to be used in the cal­

culations and Q is defined by

Then, it is possible to break up the original equation (2-1)

into a set of coupled equations

(2-7a)

(2-7b)

Here v = QHP

and tp = P'P

with the other quantities defined similarly.

One can derive a formal solution to equation (2-7a) by

using the operator representation of the Green's function:

The term +ie has been added to the denominator of the Green's

function in the usual way to assure outgoing-wave boundary

conditions. No homogeneous term exists in the solution,

since for the channels of the space defined by Q tiere should

be only outgoing waves. An equation for ipp, the truncated

wavefunction, can be obtained by inserting the solution for

into equation (2-7b) , which yields

(E-K-V) # = 0 (2-8)

'V 1where V E V „ + V --------- V (2-9)PP PQ E-HpQ+ie QP v '

Here K and V are defined as in equation (2-2), and Vpp , Vpq>

and V q p are defined in the same way as Hpp, Hpp, and H^p.

Equations (2-8) and (2-9) constitute a description of

the collision problem which is fully as exact as the original

version in Equation (2-1). Of course, this formulation is no

less difficult to solve than the original one, since in order

to determine the effective potential If, it is necessary to

have a description for the system in all channels of the Q-

space. Nevertheless, this new formulation expresses in a

very convenient form the effect of the neglected channels

on those to be considered. These neglected channels enter

through the second term of the quantity If.

Instead of trying to calculate , it is convenient to

substitute a simplified model potential U, sometimes known as

the "optical potential," which is expressed in terms of an

experimentally determined set of parameters. As a result

of the second term in Equation (2-9), this optical potential

should be complex-valued, energy-dependent, and non-local.

In the usual formulations the first two of these requirements

are maintained, but because of the difficulty of working with

non-local potentials, a local version of U is typically sub­

stituted. This has generally been an adequate approximation

(Pe 62), although the non-locality can have significant

effects in some cases (Pe 63a, Au 65). The optical potential

is also commonly chosen to be a one-body operator, and thus

processes which are explicitly many-body in character, such

as compound-nuclear reactions and projectile breakup, must

either be neglected or treated only in an average way. The

adequacy of this approximation is considered in Section 2.7.

The most common application of optical potentials corres­

ponds to the extreme case in which P projects out only the

elastic channel. For this case the optical potential has

been in use since the early 1950’s (Wa 54, Fe 54). The form

of the elastic-scattering potentials is typically based on

the shape of the nuclear surface. This choice is motivated

by the short-range character of the strong interaction. Thus,

the real part of the potential has a Woods-Saxon (Fermi) shape

(Wo 54) :

V [ 1 + exp ((r-R )/a ) ] - 10

Here r represents the radius from the center of the nucleus,

17

and V, , and a are parameters determining the well-depth,

half-radius, and diffuseness, respectively, of the potential.

The imaginary part of U, which represents the removal of

flux out of the elastic channel into the neglected channels,

has been found to be surface-peaked for some projectiles

(Pe 63b, Au 70). Therefore, the imaginary part is often

assumed to have a shape corresponding to the first derivative

of a Woods-Saxon potential or the sum of this with the ordin­

ary Woods-Saxon. A spin-orbit term is frequently added to

this potential for projectiles with spin equal to 1 /2 , al­

though it is usually neglected for simplicity with particles

of higher spin. In this form the elastic-scattering optical

potential has been applied quite successfully to a large body

of scattering data for a wide variety of projectiles and

energies (Au 70). Examples of parametrizations for this po­

tential will be given in Chapters 3, 5, and 6 .

It should be mentioned that the parametrization which

characterizes the optical potential is by no means unambiguous.

Frequently, a large range of parameters will give an accept­

able fit to the elastic-scattering data, even if the same

potential forms are used (Au 70). Some of these ambiguities,

but by no means all, can be eliminated 'by using optical po-r

tentials which describe the elastic scattering throughout

wide ranges of nuclear masses and projectile energies with

smooth variation of the parameters. It has been possible,

particularly for light projectiles on heavy target nuclei to

find smooth parametrizations which satisfy these requirements

18

reasonably well (Pe 63, Pe 63b, Be 69a, FI 69).

A specific parametrization of the elastic-scattering op­

tical potential can be chosen, of course, to describe elastic

scattering to arbitrary accuracy; however, this potential

gains significance only if it can be used as a substitute for

the exact effective interaction ^ in approximate descriptions

of other reaction channels as well. This will clearly not be

a good model for V in general whenever there is strong coup­

ling to important channels in the Q-space. Even parameters

which have been chosen specifically to describe elastic

scattering at the proper energy cannot possibly account for

strong transitions through the Q-space into alternate reaction

channels. A situation in which the elastic-scattering optical

potential will probably be inadequate is the description of

reactions on highly collective nuclei where coupling to in­

elastic channels is very strong. The concept of the optical

potential can be maintained, however, if the space of the

operator P is increased to include all the important inelastic

channels (Bu 63, Ta 65, Gl 67). Such "coupled-channel" (CC)

optical potentials are more complicated than the elastic-

scattering parameters, since they are in reality potential

matrices whose off-diagonal matrix elements can couple various

inelastic channels. It Is evident that the diagonal compon­

ents should have forms similar to the usual one-channel

(elastic-scattering) optical potentials. The off-diagonal

elements must be determined from a specific model for inelas­

tic scattering. In the case of inelastic scattering to rota-

tional and vibrational states, it has been found possible to

parametrize the potential matrix with considerable success

by basing it on the elastic-scattering potential, as is

described in Section 3.6.1.

2.4. The Distorted-Waves Approximation.With the use of optical potentials it is possible to

simplify the reaction problem substantially. Consider the

reaction A(a,b)B with the incoming channel labelled by a

the outgoing by 8 . Suppose that there exists a potential ,

such as the optical potential, which can describe in an average

way the scattering of b from B. The eigenfunctions corres­

ponding to this potential, which are usually known as distor­

ted waves, are defined by

20

< V V E> 5 b(s = 0 (2'10)

Equation (2-1) can now be written in terms of this potential:

(E-K„—0b ) *b|J = (Vb-Ub) (2-11)

Then, solving equation (2-11) formally for and inserting

this into equation (2-4b) for the transition amplitude, one

can show (Au 70) after some manipulation that

W e ' |vb-ubl * £ h <2-12a>

If the potential U , which describes the a+A scattering, isclused, a "prior" form for the transition amplitude can also be

derived (Au 70):

Taa,b6 ‘ lVa-“al <2 ' 1 2 b >

As before, the + (-) superscript on the £'s refers to the solu­

tion of equation (2 - 1 0 ) with outgoing (incoming) scattered waves.

The relations (2-12), which are special cases of a re­

lation originally derived by Gell-Mann and Goldberger (Ge 53),

are exact expressions for the transition amplitude and hold

regardless of the form of the potentials U. However, if U

is chosen judiciously, the form of T can be greatly simplified.

For transfer reactions this can be seen by considering the

example of the pickup reaction (similar results hild for

stripping reactions). The reaction is A(a,b)B, that is,

(B+x) + a ■* B + (x+a)

where A = B+x and b = a+x. The form of the transition amp­

litude was given in equation (2-6). Using equation (2-12b),

and neglecting for the moment questions of symmetrization of

the interaction potential,

Tac,b8 ' ^ b i ' lVBa+V,a-DJ <2-13>

If U is chosen so that it cancels large portions of V„ , a a o x - Bareasonable approximation for the transition amplitude is

T = <)i| ( — |vaa, bg vbg * xa* afi

Finally, if a truncated form of \|> is used as described in

the last section and is approximated by the eigenfunction of

the optical potential for the scattering of b on B, then

21

22

V.bB - <5W > |v*al 5 }> (2-U)

Equation (2-14) is sometimes known as the distorted-

waves approximation for the transition amplitude. It takes

on various forms depending on the choice of the P—space for

the optical potentials. In the description of direct trans­

fer processes, and are chosen to be elastic-scattering

optical potentials. In that case expression (2-14) corres­

ponds to the distorted-wave Born approximation.

2*5. Properties of the Distorted-wave Born Approximation.

A number of excellent reviews (To 61, Au 63, G1 63,

Sa 6 6 , Au 70) of the application of the DWBA to nuclear

transfer reactions have been written. However, it is use­

ful here to summarize briefly the EWBA formalism and to

indicate a few general conclusions about the characteris­

tics of those reactions which proceed via a simple direct-

reaction mechanism. The pickup reaction A(a,b)B is taken

as an example.

Since in the DWBA the distorted waves describe only a single­

channel (elastic scattering), they may be written as simple

products of the target, projectile, and relative motion wave-

functions. In the present discussion the target and projectile

internal wavefunctions are denoted by $ and the relative

motion wavefunctions by X- Thus, using the notation of

Section 2.2

23

H = H a + H + T + V A a a a

= *8 + Hb + Tb + Vwith defined as the internal Hamiltonian of nucleus A

and where T& and are the kinetic energies of relative

motion of the A+a and B+b systems, respectively. Then,

A ^i3! -*■*a <A) - 0

a 6 iSi -<Ha-EB 1 S1) ®a <a> ' °>

and similarly for B and b. The quantities & and & represent

all quantum numbers besides the spins J and s necessary to

define the states, and A and a represent the internal nu­

clear coordinates. Also, the relative motion functions are

defined by

< V V Ei> *ai ' 0

with

< V V Ef> xbf - o

= E® j + Eg s + E*.a f j f e f s f f

In this notation the elastic-scattering distorted waves be­

come

In determining the transition amplitude, these expres­

sions for the distorted waves should be antisymmetrized and

the interaction potential symmetrized. However, the usual

DWBA treatments drop all terms involving exchange of particles

between the projectile and target. (The importance of the

exchange terms is considered in Section 2.7.) Then the

transition amplitude may be written as

Ta??bf - x ‘ ;> * < V * >

a J B S a.J B.Sx <$ f f $ f f |v | $ 1 1 $ 1 i>B b 1 ax A a

x (2- 15)

m r + m r ... 2 - a a x xwith R =- :------ ,m + m a x

mx -*■- --- r ,a m + m. x x A-+ _ -*■

and r = r - r ,ax x a

and where m , m , and m. designate the masses of x, a, and A, X A

25

respectively. (The coordinate system used is displayed in

Figure 2—1.) The matrix element in Equation (2-15) corres-

and a. For simplicity, spin-orbit coupling is ignored in

this discussion, but its inclusion is straightforward (Sa

between the centers of mass of nuclei a and x. When a and

x contain more than one nucleon, the individual interactions

between the nucleons should be considered. In the case of

the (p,t) reaction, Bayman (Ba 70, Ba 71, Ba 73) has recently

performed calculations which treat explicitly the interactions

between the proton and each of the two neutrons. The results

of this calculation were found to be in excellent agreement

with those of one using only an interaction between the

proton and the dineutron center of mass. Thus, for the (p,t)

reaction the above approximation seems very good.

The transition amplitude can be written more explicitly

by defining "parentage expansions" for the nuclei A and b,

relating them to B and a:

ponds to integrations over the internal coordinates 5, x,

64) .

The assumption is usually made that the interaction

potential V depends only on the relative coordinate rSi X dX

aa* v

(2-16a)3 ' B f v

2 E. * XaSS' Xas B ' v

(2-16b)

Figure 2-1. Illustration of the coordinates used in de scriptlons of the pickup reaction A(a,b)B with b = a+x an A=B+x.

Figure 2-1

COORDINATE SYSTEM FOR PICKUP REACTIONS

A (a ,b ) B

26

.voHere, (x) represents a wavefunction for the transferred

group x, which can be in various configurations (va), in

general, within the nucleus A and the projectile b. TheA

symbols J, S , I , A, and O are angular momenta, and r denotes

angular coordinates; that is, r = r/|r|. The brackets

designate angular-momentum coupling; for example,

1 V V E t .V n ? * r ° * -X Qi/) in m . iv YA m^ m^ Z o J A

z o jwhere C is a Clebsch-Gordan coefficient.

Inserting these expressions into Equation (2-15) and

integrating over B, a, and x yields

TaiWbf = S [(2L+1)(2S+1 ) ] 1 / 2 C f 1 C 1 f * LSJ M fMMi m immf

x CL Sd r ” (2-17a)m mM LSJL

»her* rLSJ E H l ° xs] v (2-17b)JtaAv A z Lmi,m A

with W(AaLJ;sA) representing a Racah coefficient (Ro 57) and

m£,mA = . ? ? < - ) * * * ( + ) * ? “ I (: . y mA* :Z Acs Jv " a drx Xbf <kb ’R) Xal a a ' YJI (rx' Y A ( xa}

a a v 8.Bfv _X F£ct J <r*> h U <rax> Vaa <r,*> (2-17c)-

In these expressions, J, L, and S are known as the transferred

total, orbital, and spin angular momenta, respectively. The

transferred orbital angular momentum is related to it, the

orbital angular momentum of x within the nucleus A, and

to its orbital angular momentum within the projectile b,

by the relation u t = L. The nuclear structure information

of the reaction is contained in the coefficients E and F in

the parentage expansions. In this dissertation, the former

is called the "projectile form factor," and the latter, the

"nuclear form factor" or simply the "form factor." This lat­

ter term is also applied in the literature of nuclear reactions

(Au 70) to the entire combination of quantities multiplying

the distorted waves in expression (2-17c).

The cross section can now be calculated using Equation

(2-5). For the usual case of unpolarized targets and beams,

one sums over the final spin projections and averages over

the initial so that

27

da - I |tDWBA 1 2 ( 2 - 1 8 a >dfl = (2 .^+ 1 ) MiM f

ViVf V X 2 -l®b)w i t h K 5 7 2 **1‘‘ T 7 k a

Using Equations (2-17) and the orthogonality property of

Clebsch-Gordan coefficients, one obtains

da 2S +1 M

Thus, in the final cross section the sums over the transferred

angular momenta are incoherent. If spin-orbit coupling is

included, only the sums over J and M will be incoherent (Sa

64) .

For the purposes of nuclear spectroscopy, transfer reac­

tion experiments are basically concerned with measuring the

nuclear form factors. In this application it is obviously

important to use reactions for which the projectile form

factors are well-known. For this reason most nuclear reaction

experiments have involved the measurement of cross sections

for light-projectile reactions; that is, reactions involving

projectiles with four nucleons (alpha-particles) or less.

Not only are the wavefunctions of these projectiles reasonably

well-known, but also they are extremely simple since, to a

good approximation, all the nucleons are in relative S-states

(zero orbital angular momentum) (Au 70). This implies that

A=0 and a=S, thereby greatly simplifying expressions (2-17).

On the other hand, the fact that the transferred particles

in multinucleon transfer reactions are in relative S-states

limits the configurations in the wavefunctions of nucleus A

to which the reactions are sensitive. This is not the case

for reactions using more complex nuclei ("heavy ions") as

projectiles (Br 67).

It is evident that the exact determination of the tran­

sition amplitude requires the evaluation of a complicated

six-dimensional integral (Equation 2-17c), which will not

factor, in general, since the distorted waves, the interaction

28

potential, and the form factors are functions of different com­

binations of the integration variables r arid r . Therefore,a xseveral different approximation schemes have been devised

for the simplification of this expression. The most common

is the "zero-range" (ZR) approximation, which is equivalent

to the assumption that particle b is emitted at the same

point at which particle a is absorbed. This implies r =r ,cL X

so that-*■ m

29

R = r , - ' - A r »a a m + m . ax A

and Equation (2-17c) reduces to a three-dimensional integra­

tion. For reactions in which A=0 only, such as the light-

projectile reactions to a good approximation,

r * ( z r ) = ----------- i r ^ - > DLSJ / 2 S + 1 /2L+1 mx A 0

* a XbS ^kb ’ra' FLSJ *ra* YL ^ra^

X ^ r f a - r S T ?a> <2-19a)x A

“iafwith V (r ) E * (r ) = D <5(r ) (2-19b).ax ax OSS ax' o ax

The zero-range approximation is applicable mainly to

light-projectile reactions, such as the (p,d) and (p,t)

reactions studied in the present investigation, since the

ranges of the wavefunctions and/or internal interactions for

these particles tend to be small (Au 70). Comparisons of

ZR calculations with complete evaluations of the finite

range integral (Equation (2-17c)) indicate that for light-

projectile reactions at medium energies the zero-range

approximation usually reproduces the relative finite-range

cross sections very well (Dr 64, Di 65, Ba 71, Ba 73). Thus,

the ZR approximation has been used in all calculations per­

formed in the present study.

It is now possible to indicate some general predictions

of the DWBA. There are two important quantities which can

be extracted from a differential cross section experiment:

1 ) the strength of the transition and 2 ) the shape of the

experimental angular distribution. As can be seen from

Equations (2-17) and (2-18c) the strength of the transition

for a simple direct process is related to the magnitude of

the nuclear form factor for the transfer process. Thus,

from the measured strengths one can infer information concern­

ing the relationship between the states in nucleus B and the

ground state of nucleus A. For example, in single-nucleon

transfer reactions, such as (p,d), one would measure the

extent to which the states are related by the addition (single­

particle component) or removal (single-hole component) of a

nucleon. Similarly, reactions, such as (p,t), which involve

the transfer of more than one nucleon can be used to measure

the relationship between nuclear states based on configura­

tions of two or more particles. For these reactions, however,

many components of the nuclear wavefunctions can contribute

to the transition, and those transitions between states wi'tlt.

30

coherent multinucleon configurations are especially favored.

For example, in (p,t) the transitions between states pos­

sessing large numbers of coherent two-neutron configurations

have the strongest cross sections, as is shown in Chapter 3 .

The importance of the shapes of angular distributions

results from the fact that in the DWBA these shapes are pre­

dicted to be generally characteristic of the transferred

orbital angular momentum L with only a weak dependence on

the structure of the nuclear states. Hence, the angular dist­

ribution shapes can be used to determine the L-values and

thus information concerning the spins of the nuclear states.

The characteristic dependence of the differential cross section

shapes on L can be understood from semi-classical arguments

(Au 63). Classically, since the nuclear forces are short-

ranged, for any L which is greater than zero the projectile

in a transfer reaction will usually be able to Interact with

the target to produce a transition only for scattering angles

beyond a certain minimum value. This minimum angle is de­

termined to first order only by the size of the nuclear radius,

the magnitude of L, and the Q-value of the reaction. Thus,

for a given nucleus the angular position of the first diffrac­

tion peak is expected to be characteristic of L. Indeed, a

slightly more detailed argument (Au 63) indicates that the

shape of the angular distribution beyond the first maximum also

tends to be characteristic of L. A systematic dependence on

the total transferred angular momentum J for transfer reac­

tions on medium-weight nuclei (mass number 40<A<100) has also

31

been observed experimentally (Le 64, Sh 64, Sc 6 6 ), but this

effect has not yet been fully explained (Au 70).

The characteristic L-dependence would be broken if

there were any strong influence on the shape of the angular

distribution from the details of the nuclear form factor.

This influence is reduced because of the tendency (Au 61,

Ho 6 6 ) for contributions to the DWBA integral to arise

mainly from those partial waves in the decomposition of the

incoming and outgoing distorted waves which correspond to

grazing collisions with the nucleus. Thus, for transitions

with "good angular-momentum matching," that is

I k R-k, R| = L,1 a b 1

with R representing the nuclear surface, contributions from

the nuclear interior tend to be surpressed (Ma 69a) and only

that portion of the nuclear form factor near the nuclear

surface contributes significantly to the final result. In

this region, the radial dependence of the form factors usual­

ly are determined principally by the binding energies of the

nuclear states. This characteristic has led some researchers

(Bo 64, Be 65, Co 6 6 , De 67, Co 6 8 ) to label any experimental

angular distribution which cannot be accounted for simply in

terms of binding-energy or angular-momentum mismatch effects

as "non-stripping" transitions and attribute these to the

presence of higher-order processes. However, it must be

emphasized that cases arise, particularly for weak transitions,

in which the radial dependence of the form factor near the

32

nuclear surface does depend on the nuclear structure, and

this will lead to uncharacteristic angular distributions even

within the DWBA. Examples of this effect will be given in

Chapter 6 . Therefore, the appearance of non-stripping tran­

sitions cannot be taken as proof of the presence of higher-

order piocesses without further analysis.

In summary, the DWBA predicts that for most direct

transfer reactions 1 ) the strengths of the transitions will

be directly dependent upon the relationship between the

structure of the final state reached and that of the target

state, and 2 ) the shape of the angular distribution will

be directly related to the transferred orbital angular momenta

and hence to the spins of the nuclear states. These general208 208 properties are apparent in the Pb(p,t) and Pb(p,d)

reactions, as can be seen in Figures 2-2 and 2-3, which show

angular distributions obtained by Holland (Ho 70) and Whitten

et a l . (Wh 69). Transfer-reaction transitions to low-lying208levels in nuclei near Pb are expected to proceed mainly

via the direct-reaction mechanism, since any strongly col­

lective levels in these nuclei are fairly high ir. excitation

energy. A comparison of angular distributions corresponding

to the same transferred angular momenta Indicates that

the expected similarity in these angular distributions holds

to extremely high accuracy. As predicted, the dependence on

the Q-value, or, equivalently, the binding energy of the final

state is very mild. An interesting example is provided by the

(p,t) transitions to the first two 2+ states in Z^ P b . These

33

Figure 2-2. Angular distributions obtained by Holland (Ho 70) for the reaction ^ ® P b (p, t) ^ P b at an incident proton laboratory energy of 20 MeV. This figure is tak­en from reference Ho 70, and the curves represent the results DWBA calculations discussed in that reference.

Figure 2-3. Angular distributions obtained by Whitten208 207et a l . (Wh 69) for the Pb(p,d) Pb reaction at an

incident laboratory energy of 22 MeV. The curves rep­resent DWBA calculations of Whitten £t _al. with (solid line) and without (dashed line) lower cutoffs in the radial integrals. As discussed in Section 2.7, the im­provement in the fits using the radial cutoff is perhaps an indication of the importance of the breakup of the deuteron into continuum states during the reaction.

(JS

/qr/)

up/-op

F i gu re 2 -2a

Figure 2-2b

do7d

ft(m

h/sr

)

F i gu re 2-3

J I I I I I

Pb208(p,d)Pb207Ep = 22.00 MeV tz » 1/2" GROUND STATE. • 894-keV 3/Z"LEVEL

0.020° 30® 60® 90® !20® 150® 180

®LA8

states are known to have very different structure (St 6 8 b ) ,

the lowest 2+ being much more collective than the second.

It is evident that these structure differences alter the

strengths of the (p,t) transitions but have almost no effect

on the shapes of the angular distributions.

2.6. The Coupled-Channel Born Approximation.208For spherical closed-shell nuclei such as Pb it is

perhaps not surprising that the direct-reaction mechanism

is appropriate for describing transfer reactions to low-

lying states. However, there are large numbers of nuclei,

among them the deformed nuclei, which are dominated by low-

lying collective configurations. For reactions on these nuclei

it is important to consider the possibility of the higher-

order inelastic-plus-transfer mechanism.

2.6.1. The CCBA Transition Amplitude.

A natural method for including explicitly the effect

on transfer reactions of inelastic scattering in the initial

and final nuclei can be derived by returning to the distorted-

waves approximation for the transition amplitude (Equation

(2-14)). When the distorted-waves are chosen to be eigen­

functions of elastic-scattering (one-channel) optical potentials,

one obtains the DWBA. These potentials enter the approximation

in somewhat different ways. In the case of pickup reactions

the optical potential for the entrance-channel scattering,'V/U enters through the assumption U = V , which is used to & s i5d

simplify the exact interaction potential. The exit-channel

optical potential, U , , is chosen as the potential which rep-D

34

resents the truncation of the scattering wavefunction •b pSimilar remarks apply to the stripping transition amplitude.

If Ua and are chosen to be coupled-channel optical

potentials describing not only elastic but also inelastic

scattering, then thn inelastic scattering is included in the

transition amplitude to all orders. In any practical cal­

culation it is impossible to describe Inelastic excitations

to all nuclear states. Thus, the P-space of the potentials

is chosen to include only those states which are coupled

most strongly by inelastic scattering to the states of

interest. It is hoped that the weaker Inelastic couplings

are described with sufficient accuracy through the imaginary

part of the optical potentials. This technique was originally

formulated by Penney and Satchler (Pe 64) and has become

known as the coupled-channel Born approximation (CCBA).

The decomposition of the distorted waves £ into internal

and relative coordinates is more complicated in the CCBA than

in the DWBA, since these coordinates are no longer separable.

Tha distorted wave now Includes all inelastic channels i:

£ (+) - S 4 1 xl( J (2-20).a A a a

The relative wavefunctions are therefore solutions to a set

of coupled equations:

These equations are derived by inserting expression (2-20)

into the distorted-wave equation (2 — 1 0 ) and then by multiplying jllf

on the left by $ $ . Similar equations hold for the dis-A 3,

torted wave of the final partition. The appropriate

boundary conditions used for the solution of Equations (2-21)

are that have an incoming plane wave in the elastic

channel a and outgoing spherical waves in all channels and

that have an outgoing plane wave in the final state

channel B and incoming spherical waves in all channels. The

final CCBA transition amplitude becomes

_ CCBA r .->■ £<">* » f jTa a ,b B = tS£ / d r a ( r i < b | <kb ’ R)

x <4b < i v i (2 - 2 2 ) -

36

The difference between those processes which are includ­

ed in the DWBA and those in the CCBA can be seen from a com­

parison of Equations (2-22) and (2-15). The summation in

Equation (2-22) indicates that in addition to the direct

transfer process from the ground-state of nucleuc A to a state

in nucleus B, as in the DWBA, transfers are allowed in the

CCBA from excited states of A. Also, the transition to a

state in B can arise from transfers to another state in B,

followed by inelastic scattering (the inelastic-plus-transfer

process). These differences between the CCBA and DWBA are

illustrated in Figure 2-4. Although the CCBA is a more

complete model for the transfer process, it is also more dif­

ficult to evaluate. In the DWBA, all that is required is a

Figure 2-4. Comparison of the processes Included In the DWBA, the CCBA, and the core-excltation and perturbation approximations for pickup reactions between nucleus A to nucleus B. The wavy lines denote inelastic scattering, and the solid lines, particle transfer. A single line between two states indicates that the transition is treat ed to first order only, and two lines indicate that the transition is treated to all orders.

A BDirect (DWBA)

A B

Core Excitation

CCBA

Perturbation Method

determination of the distorted waves from two uncoupled

equations and then the execution of a single overlap inte­

gral. In the CCBA, the distorted waves must first be deter­

mined from two sets of coupled equations, and then overlap

Integrals must be performed for each pair of allowed incoming

and outgoing channels.

A number of approximate methods have been introduced to

avoid the exact evaluation of the CCBA transition amplitude.

One approximation, known as the "core-excitation" method,

was suggested by Kozlowsky and de-Shalit (Ko 6 6 ) and inde­

pendently by Levin (Le 6 6 ). In this method, the scattering

wavefunction *s truncated to an elastic-scatteringb peigenfunction 5°^ as in the DWBA. The multistep processes bpresult from the potential

37

V ’ 2 V„ - U Ba Ba a

in Equation (2-13). V ’ is then approximated by a potentialBabV ’ which couples the elastic channel for B+a scattering to d aonly the most important inelastic channels, and the inelastic

scattering is treated as a first-order perturbation. Thus,

the transition amplitude becomes

Core Ex. DWBA <ro(“) I . I fro(+)>act,b3 a a ,b(3 T 1 Ba 1 S a

where is the distorted wave for the B+a elastic scat-aatering. It can be seen that in this approximation the in­

elastic effects are ignored completely in the exit channel

(entrance channel in stripping) and taken only to first order

in the entrance channel. This is illustrated in Figure 2-4.

For most transfer reactions inelastic scattering is expec­

ted to be of similar strength in both the target and residual

nuclei. Thus, the core-excitation approximation is difficult

to Justify. A more symmetrical treatment, often known as

the perturbation method, was introduced by Iano and Austern

(la 6 6 ) and has since been used by a number of different

groups (Ku 69a, Lu 71, Bi 72). This consists in using a

first-order perturbative solution to the coupled equations

( 2- 21 ) :

r( + > 1 rO ( + ) + r l ( + )aa ^aa aa

The zero-order solution, is the usual elastic-scatteringact

distorted wave, and is the first-order perturbation.

Then, the expression for the transition amplitude is

T 1 T DWBA + T 1aa,b@ aa,b@ aot,b@

where T 1 = <£*«"* |v | S°(+)> + l v I S l(+)>*aa,b$ bp 1 xa 1 aa b£ xs ad

As in the core-excitation technique, the perturbation method

treats the inelastic coupling only to first-order, but now

it is treated equally in the entrance and exit channels.

This is shown in Figure 2-4.

Since the inelastic coupling is taken only to first-

order, the perturbation technique is really applicable only

in cases where the inelastic scattering is very weak. For

strongly collective nuclei, such as those with rotational

38

bands, the inelastic scattering is usually much stronger

than the transfer process so that it is hard to justify

treating them to the same order of approximation. A direct

comparison (la 69) of the CCBA and perturbation approximation

indicates that the perturbation approximation yields poor

fits to the CCBA angular distributions for rotational nuclei.186Thus, for the reactions on the rotational nucleus W con­

sidered in the present investigation, both the core-excitation

and perturbation approximations are inadequate for describing

the Inelastic processes.

2.6.2. The Source-Term Method.

In order to include the Inelastic-plus-transfer proces­

ses in the description of reactions on strongly deformed

nuclei, it is evident that an evaluation of the complete CCBA

transition amplitude is necessary. Because of the large

numbers of overlap integrals required, computer calculations

using the CCBA formulation of Equation (2-22) can be long and

costly, although a number of such calculations have recently

been made (Du 6 8 , Sc 69, Du 70, Sc 70, Ta 70, Ud 71, Br 71a, Br

71b, Sc 72, 01 72, Ya 72). There is, however, another approach

to the problem which eliminates the need for overlap integrals

entirely. This technique, introduced by Ascuitto and

Glendenning (As 69), is known as the "source-term method"

(STM) and has been adopted for the calculations performed in

the present investigation.

In the STM, the reaction problem is expressed entirely

In terms of two sets of coupled equations. These equations

39

can be heuristically derived in the following manner for the

example of the pickup reaction A(a,b)B. An operator P is

introduced to project the total wavefunction of the system \p

onto the two partitions A+a and B+ b :

P t (+> = *p (+>

i i i (+) f f f (+)= \ *A *a Xa + 2f *B $b *b <2“23>

This wavefunction satisfies Equation (2-8) of Section 2.3r\j r\j

which involves the effective interaction V. Then if V is

approximated by

40

V = U + U, + V (2-24a)a b x

and if the important orthogonality assumption

<$* S 1 I $ [ > = 0 (2-24b)A a B b

is made, then it can be shown (St 6 6 , Ra 67, Au 70) that the

following coupled equations result:

■I -I i -t j( K - E + <4> $ |u I $ >) X a A a 1 a' A a “a

j j j i j i • i (4")= - s <$. * Iu I > xi?Si» A a 1 a 1 A a a

(+)- 2f <4i i v: J 4 ( 2 - 2 5 a )

41

(Kb - E + <*J |uj *J>) X£

= - 2 <$*• |u | $>*’ > x VB b b B b b

if f- Z <$_ $, |V I $X> X (2-25b)B b 1 x a 1 A a *>

These equations are similar to Equations (2-21), which des­

cribe inelastic scattering. However, in addition to the

inelastic couplings (the first terms on the left-hand sides

of the equations), terms involving the potential V also

appear, representing the particle transfers. Thus, in Equations

(2-25), both the inelastic and the transfer processes are

included to all orders. In the CCBA, the transfer process

is assumed to be weak relative to the strongest inelastic

processes. Thus, if the projection operator P is chosen to

exclude all but the strongest inelastic processes,

. f( + ) < (+),lxb I « lx* I ,

and th®n Equation (2-25) may be approximated as

1 j j (+)(K - E + <$* (&1 I u I $ >) X a A a 1 a 1 A a Aa

1 i i , i ' i »= < * l *a |U J ** > Xa (2-26a)

r t f t(Kb - E + < *b if | u j *B ib>) Xb

- - £ < i i ij |U | t]!' i f ’> x f <+> E o f , B b 1 b 1 B b b Pfi

(2-26b)

Pf l = <4B *b l V* J * 1 * a > Xa<+) ( 2 ' 26c)

Equations (2-26) are the equations of the STM, and

include the inelastic scattering to all orders with the

transfer process taken as a first-order perturbation intro­

duced via the inhomogeneous "source terms" in the final-

partition Equations (2-26b). The initial-partition Equations

(2-26a) are now identical to the inelastic coupled-channel

Equations (2-21). When these equations are solved with

boundary conditions of Incoming waves only in the elastici< + >channel, the solutions X can be used to construct the sourcea

f (+)term for the final-partition equations. The solutions

to this set of equations correspond to outgoing waves only

and have the important and well-known property (Go 64) that

asymptotically their overlap with a free-wave solution is

proportional to the S-matrlx element for the transition,

Saa,b(i' uherc

Eaa,b8 ^aa.bfJ Taa,b8

Thus, in the source-term method the transition amplitude,

and hence the cross section, can be obtained by the solution

of only two sets of coupled equations without the need for

evaluating overlap integrals.

The above derivation is not intended to be rigorous and

indeed serious objections can be raised concerning the as­

sumption (2-24b), since the states | $*> and | a r e

eigenstates of two different Hamiltonians H. + H and H_ + H, ,A d B D

42

43

and thus are not orthogonal In general. Attempts have been

made to determine the validity of this assumption (Ra 67,

Oh 69, Oh 70, Oh 70a, Ha 71). However, for present purposes

the significant point is that the CCBA transition amplitude

(2-22) can be shown (G1 71) to be exactly derivable from

Equations (2-26), and hence these equations form an equiva­

lent and considerably simplified formulation for the CCBA.

Since its introduction the STM has been used in several cases

(As 70a, As 71, As 71a, G1 71, Ma 71, Du 71, As 70) in ad­

dition to the present work. Some of the earlier applications

to (p,t) reactions on deformed nuclei are presented in

Chapter 5.

2.6.3. Solutions to the STM Equations.

Practical solutions to the STM coupled equations require

the reformulation of the problem in terms of a partial-wave

expansion. The partial-wave expansion which is convenient

for the solution of coupled equations is somewhat different

from that ordinarily employed in evaluations, for example, of

the DWBA integral, since the expansion is in the total angular

momentum of the system rather than the orbital angular mo­

mentum of the projectile. The following discussion summarizes

the technique used by Ascultto and Glendenning (As 69).

With I defined as the total angular momentum of the

system, M its projection, and tt the total parity, one can

define the quantities

(r',A,a)aSa (a)] (A)] MI

where & denotes the orbital angular momentum of particle a.

Thu8, the eigenfunction of the total Hamiltonian H may be

written

^ <*.•> (2 -2 8 ) .a i

The coupled equations (2-26a) can then be expressed as a set

of equations for the radial functions w:

(Ti + U i r <ra ’> - Ei> - r 1 (ra'>

= - E uJJ? (r ') w*?1 (r ') (2-29a)i * i

with (ra *) = <*1“ I (ra ',A,a) |Ua | (ra ',A,a)>,

* 2 j 2 A (£ + 1 )and Ti E 2 ^ r '*a a a

In a similar fashion the final-partition equations can be

expressed as

(Tf + ujj1 (R) - e J) w ^ 1 (R)

= - S (R) wjT1 (R)

- E ■ pJJ (R) (2-29b)i

andaTTl * i v

p " - » <* f ; x < i .* ,S ) |V„ ( r ax)i 1,1 r , 3 >

44

(2-29c)

45

In this matrix element, integration is implied over all in­

dependent coordinates except R. The numerical solution of

Equations (2-29) hat. been described elsewhere (Gl 67, As 69).

As mentioned in Section 2.5, the ZR approximation

(Equation (2-19b)) is known to provide a very good descrip­

tion of (p,d) and (p,t) reactions, and thus has been adopted

for the calculations of the present investigation. If this

approximation is used and relative S-states are assumed for

the nucleons in the projectiles, then it has been demonstrated

(As 70, Gl 71) that the source term can be written,

where

(2-30b) .

In this expression J and S are the total angular momentuma aand spin, respectively, of particle a and

j = 2 j + 1

The last three quantities in Equation (2-30b) are respectively

the 3-j, 6 -j, and LS-jj coupling coefficients (Ed 60). In

order to solve these equations, it is first necessary to

evaluate the inelastic matrix elements and and theaiaf 11

form factor • These depend on the specific properties

of the initial and final nuclei, and the methods used to

evaluate them in the present case are discussed in Section

3.6. A computer code LISA has been written by R.J. Ascuitto

(As 70b) to solve Equations (2—29) using the ZR source term

(2-30), and this program has been used in the present work.2.7. Discussion of the Approximations.

The CCBA explicitly includes the multistep inelastic-

plus-transfer processes neglected by the DWBA. There are

other processes, however, which are ignored in both approxi­

mations, and it is useful to estimate their importance in the

reactions chosen for the present investigation. At least

three basic approximations are required for the derivation of

Equation (2-22) :

1. The neglect of the antisymmetrization of the

distorted waves and of the symmetrization of the

interaction potential.

2. The use of one-body optical-potential wavefunctions.

3. The treatment of the transfer interaction to first-

order only.

The proper treatment of the identity of the nucleons in

the total scattering system requires the addition of several

more terms to the transition amplitude. These "exchange terms"

involve the other possible permutations of particles between

the projectiles and target or residual nuclei. Physically,

the exchange terms for the reaction A(a,b)B can be considered

46

to involve the capture of the projectile a by the nucleus

B while b is emitted from nucleus A. They are classified

into two basic categories depending on the part of the sym­

metrized potential involved; "knockout scattering" occurs

through the interaction between the nucleons of the two pro­

jectiles a and b, and "heavy-particle stripping" results from

the interaction between the nucleons of a and all nucleons of

A except those which compose b. Knockout scattering can

sometimes be important for light-projectile reactions, such

as single-nucleon-induced inelastic scattering (Am 67, At 6 8 ,

Ag 6 8 , As 71a) and certain reactions on light nuclei (Ba 67);

however, it is expected to be much weaker than the direct

term for transfer reactions such as (p,d) and (p,t) on heavy

nuclei, because the process requires more complicated nuclear

parentage (Au 70) . Heavy-par t i d e stripping (pickup) can

sometimes be important for reactions in which the projectile

and target have similar mass (Oe 6 8 , Ch 73); but for reactions

on heavy nuclei, because of the approximate orthogonality

between the optical-potential wavefunctions of the projectiles

and those of their bound states in the target, it is expected

to be an unimportant effect (Au 70).

The replacement of the effective interaction by a one-

body optical potential essentially eliminates all processes

which are explicitly many-body in character. One important

many-body process is compound-nucleus formation. This can be

a significant effect for low-energy reactions on light nuclei,

because the density of compound resonances is low in such

47

cases and there are only a small number of open final channels.

Thus, contributions to any given channel from compound-nucleus

decays can be very large. For 18 MeV protons incident on a186nucleus such as W, however, the number of open final chan­

nels is large and the decay contribution to any one will gen­

erally be small. In addition, the energy spacing of the

resonances is much narrower than the energy spread of the beam,

so that the contribution to the cross section from interference

between compound-nuclear and direct or multistep processes

averaged over this energy spread is expected to be quite small

(Au 70). Thus, for such reactions the optical parameters are

generally considered to represent the effective interaction

averaged over the energy spread of the beam. In that case,

the compound-nuclear contributions can usually be treated as

effectively incoherent from the direct and multistep contri­

bution, and since the former contribution is generally small,

it can be ignored.

Exceptions to this occur when there is any strong cor­

relation in the compound-nuclear contributions over an energy

range larger than the beam width, resulting in "intermediate-

structure" resonances such as the well known isobaric-analogue

and giant-dipole resonances (Fe 67). When these intermediate-

structure effects are present, important contributions to

reactions from the compound-nuclear processes are known (St 6 8 )

to occur and can even lead to significant interference effects

(Ay 73). Experiments (Sc 69a) with proton scattering on

tungsten have not revealed any strong intermediate structure

48

resonances near 18 MeV, and thus the neglect of compound-

nuclear effects in the present instance seems justifiable.

There is another important group of many-body processes

of a simpler character than compound-nucleus formation which

are also neglected by using one-body optical-potential wave-

functions for the projectiles. These are processes in which

the projectile Itself is excited during the reaction. Mutual

target-projectile excitation is known to be a significant ef­

fect in some heavy-ion reactions (Br 67). However, most of

the lighter projectiles are reasonably tightly bound and have

no low-lying excited states. Therefore, light-projectile

reactions do not usually involve projectile-excitation proces­

ses. The one important exception is the deuteron, which can

easily break up into continuum states because it is so weakly

bound.

A number of investigations of reactions involving deu-

terons have indicated that contributions from the nuclear

interior must be arbitrarily suppressed by methods such as

cutoffs in the radial integrals (Wh 69, Ha 71a). It is pos­

sible that this phenomenon results from deuteron breakup,

since one would expect the largest effects of many-body pro­

cesses to come from interactions in the nuclear interior.

Thus, the one-body optical potential wavefunctions can be very

poor approximations to the actual wavefunctions in this region

(Ma 69a). A complete analysis of the breakup process is a

complicated three-body problem (Du 6 8 a). Recently, however,

Johnson and Soper (Jo 70) have developed a method in which

49

the breakup effects are approximated by suitable choices of

the deuteron optical-potential parameters.

By using the Johnson-Soper model it has been found that

the experimental angular distributions can be explained with­

out the need of arbitrary nuclear-interior suppression pro­

cedures (Ha 71a, Sa 71, Ma 73). As is indicated in Chapter 6 , 186the W(p,d) transitions have been successfully analyzed in

the present investigation using conventional prescriptions

for optical-potential parameters, and thus no attempt has

been made to use the Johnson-Soper prescription. It is

possible, however, that the particular deuteron parameters18 6which have been chosen for the W(p,d) analysis simulate the

effects of the deuteron breakup in a manner similar to those

of the Johnson-Soper type and that this accounts for the

excellent results obtained. Nevertheless, this is a point

which should be investigated in the future.

The third major approximation which has been used is the

first-order treatment of the transfer interaction V . Ina Xthe last few years, several calculations have been performed

which include processes involving higher-order effects of

V . These correspond to the multiple-transfer processes axmentioned in Chapter 1. Some calculations have been performed

by solving sets of coupled equations similar to Equations (2-

25) (St 6 6 , Ra 67, Oh 70, Oh 70a, As 71a) and some by eval­

uating higher-order terms in the distorted-wave Born series

(To 72, Sc 72a). There are indications from these calcula­

tions that in certain cases large effects from multiple trans-

50

fer processes exist. These are cases either in which a trans­

fer transition is particularly strong so that multiple trans­

fers such as (d,p) + (p,d) become significant (St 66, Ra 67,

Oh 70, Oh 70a, As 71a), or in which the one-step transfer is

weaker than an alternate two-step transfer (To 72, Sc 72a),3 3as, for example, ( He,t) compared to ( He,a) + (a,t). How-

186 186 ever, the W(p,d) and W(p,t) transitions observed in the

present investigation are of moderate strength and there appear

to be no strong alternative two-step transfer processes. There­

fore, it seems that in this case these multiple transfer ef­

fects can be ignored.

In summary, then, it is believed that in the reactions186 186W(p,d) and W(p,t) the higher-order effects other than

the inelastic-plus-transfer are small. This implies that the

use of the CCBA in the present investigation is well-founded.

51

CHAPTER 3

THEORETICAL FOUNDATIONS: NUCLEAR STRUCTURE

"At least I'm sure It may be so In Denmark."

- William Shakespeare, Hamlet Act I, Sc. 5 .

3.1. General Considerations.

The bound-state problem for the deformed nuclei in

the rare-earth region of the periodic table is an inter­

esting one and has been the subject of considerable research

over the last twenty years. The most striking characteris­

tic of the rare-earth nuclei and the feature which makes

them particularly useful in the study of multistep reac­

tion processes involving inelastic scattering is their large

collectivity. That is, the internal motion of these nuc­

lei consists mainly in coherent rotations and vibrations

of the entire system of nucleons. The independent nucleon

motion, which is such an important feature of the nuclei

near closed nucleon shells, tends to be subordinated.

The reason for the break-down of the independent-

particle picture is that the nuclear interactions can be

described only to a first approximation as an average

independent-particle potential. In any shell model of a

many-body system, there remain residual interactions between

the particles. These interactions will tend to have only

a small effect on the lowest energy states in nuclei near

closed shells, because of the large energy gap in the inde­

pendent-particle spectrum. However, as more and more par­

ticles are added outside of the closed shell, the residual

interactions can become extremely important for all states.

It is convenient when discussing the residual interactions

52

to distinguish between their long-range and short-range

components. In a multipole expansion of a two-body force,

that is, (for spin-independent interactions)

V (r ].»r 2) ” 2^ vx (ri*r2) Yl,J(rl) Y*y (r2) *

these correspond respectively to the low-multipole and high-

multipole terms (Mo 60). It is clearly the long-range (low

multipole) component of the force which must be the source

of any collective motion of large numbers of nucleons.

Whenever there are unfilled shells these long-range inter­

actions also lead naturally to deformations of the nucleus.

Each particle outside of a closed shell will tend to distort

the average nuclear potential away from sphericity and, through

the long-range force, will attract the other particles to

follow orbits which lie as close as possible to that of its

own, causing an "aligned coupling scheme" (El 58, Mo 60,

Mo 60a, Na 65) .

Experimentally, however, it is found that large perman­

ent deformations occur only for nuclei with relatively large

numbers of nucleons outside of closed shells (Na 65). Nu­

clei with nearly closed shells tend to be mainly spherical

and any collective motion observed consists principally in

small vibrations about this spherical shape. Therefore,

there must be some restoring force competing with the long-

range interactions and causing a tendency toward sphericity.

The source of this resistance to deformation is the short-

range component of the nuclear residual interactions. Where-

53

as the long-range force leads mainly to forward scattering

between nucleons and thus to a stable deformation of the

nuclear system, the scattering of nucleons resulting from

the short-range force tends to be isotropic and therefore

favors spherical shapes (Mo 60). These short-range forces

also lead to another important effect observed in nuclei,

that is, the tendency of nucleons to form pairs coupled to

zero angular momentum -- the so-called "seniority coupling

scheme" (Ra 43, Mo 60, Na 65). Despite the presence of the

short-range interactions, the long-range forces, because

they produce collective nuclear motion, dominate far away

from closed shells, and thus permanent deformations are pos­

sible. This is precisely the situation for the rare-earth

nucle1.

3.2 Rotational Motion.

Since the rare-earth nuclei have permanent deformations,

they are subject to rotational motion. For such massive

nuclei, this motion will be slow compared to the motion of

the individual nucleons. Thus, the description of the sys­

tem can be simplified by assuming that all nucleon motion

follows adiabatically the rotation of the nuclear surface.

This approximation, originally Introduced by Bohr and Mottelson

(Bo 52, Bo 53), is known as the "strong coupling" approxima­

tion. Using this assumption, the total Hamiltonian of the

system may be written

54

H = H. + H „ int rot

H^nt describes the motion of the individual particles within

the rotating frame of the nucleus, and consists of an aver­

age independent-partide field plus residual interactions.

It is advantageous to allow the field to become deformed to

match the nuclear shape, since this absorbs a major part of

the long-range residual interaction into an individual-particle

Hamiltonian. The quantity H is the kinetic energy of ro-J rot

tation of the nucleus as a whole, and thus,

Hr°t =i = i 2 ^ R±2»

where it is the angular momentum associated with the motion of

the body-fixed axis and the are the moments of inertia of

the nucleus.

A considerable simplification results if the nucleus is

assumed to be axially symmetric, which seems to be an excel­

lent approximation in most cases (Na 65). Then the moment

of inertia about the symmetry axis, ,9g , equal to zero

(since no rotation can be defined quantum mechanically about

such an axis), and the other two moments of inertia are equal

to each other ( = J) , so that

H *- = ^ = 9“"o (R 2+R 2)rot 2*9 2*9 i i

If J represents the total angular momentum of the system,

and 3", the angular momentum associated with the motion of the

particles within the body-fixed system, then R = J-j and

55

56

H = HP + HR + HC (3-la)

where

(3-ld)

(3-lb)

(3-lc)

and J J ± iJ+ 1 2

Use has been made here of the fact that R = 0 and thus3

J 3 = j . Hp describes the motion of the particles within

the body-fixed frame, and H , the rotation of that frame.R

and rotational motions and thus represents the lowest-order

non-adiabatic contribution to the Hamiltonian. This contri­

bution is of a kinematic nature, and because of its similarity

to the classical Coriolis-force potential, this quantity is

known as the "Coriolis-coupling" Hamiltonian.

Ignoring the Coriolis coupling, the total wavefunction

of the system may be written as a product of the intrinsic

and rotational wavefunctions, that is, the eigenfunctions of

H and H . The resulting approximate eigenstate is sometimes * Rknown as the "adiabatic" or "Bohr-Mottelson" wavefunction

(Bo 52, Bo 53). The eigenfunctions of H are the rotation

functions DM ^ and depend on the Euler angles 0, <p, and ip which

define the orientation of the body-fixed axis relative to

the space-fixed axis. The quantum number K is the eigenvalue

The quantity H , results in a coupling between these particle

of the projection of J along the body-fixed symmetry axis.

The intrinsic wavefunctions may be defined by

HP XK “ EK XK <3-2>-

Then, the total wavefunction which is properly axially and

reflection symmetric (Bo 52, Bo 53, Pr 62) is

$JMK = [167T‘ (1+6K0)] ,Z {DMK XK + (-1)J K DM-K R i XK } (3'3)»

where R denotes a rotation of 180° about the 1-axis. Thelenergy eigenvalues of these wavefunctions are easily seen

(Bo 52, Bo 53, Pr 62) to be

2E = 2J - 2k21 + E k (3-4).

Thus, for a given intrinsic state X , there exists a set of

states with J = K, K+ l , K+ 2 , ... forming a rotational band

with energy spacings given by a J(J+1) rule. For the spe­

cial case of K=0, it is clear from the form of the wave­

function in (3-3) that only the even or odd angular momentum

states of the band are allowed, depending on whether the

parity of the intrinsic state is even or odd, respectively.

If the strong-coupling approximation holds, there should

be a rotational band for each mode of excitation of the nuc­

leus within the body-fixed system. Experimentally, the

assignments of observed states to various bands is based on

measured excitation and decay probabilities (Na 65) and

57

comparisons of energy level spacings with those jiven by

Equation (3-4). The energies, spins, and rotational band

assignments for the known states in 184W below 1250 keV

excitation are shown in Figure 3-1. In even-even deformed

nuclei such as ^84W, the lowest intrinsic state has K77 = 0+ ,

just as the ground states of even-even spherical nuclei 7T +always have J = 0 . This results from the short-range

part of the residual interaction, which favors the pairing

of nucleons to zero angular momentum. As with spherical

nuclei, the first excited intrinsic states of even-even

deformed nuclei are usually of a collective character, re­

sulting from the action of the long-range components of the

residual interaction, and can be thought of as vibrations

of the nuclear matter about the basic deformed shape. Thus,

as can be seen in Figure 3-1, above the ground-state rota- 184tional band in W are three vibrational bands. The names

of the B- and y-vibrational bands are vestiges of the time

when the deformed nuclei were approximated as ellipsoids

defined by two parameters, B and y (Pr 62). The B~vibrations

can be thought of as periodic variations in the eccentricity

of the ellipsoid, while the y-vibrations represent vibrations

away from axial symmetry. Since the shapes of the tungsten

nuclei contain not only a quadrupole (approximately ellipsoidal)

but also a hexadecapole moment (see Section 3.3), it is clear

that this physical model is only approximate. The octupole-

vibrational band is, as its name suggests, based on vibrations

of octupole character. The models used for the intrinsic

58

states of the W bands are described in Section 3.4.185In odd-mass deiiormed nuclei such as W, the lowest

Intrinsic states can usually be described as basically a

single particle added to or a single hole in the intrinsic

state of one of the neighboring even-even nuclei. This is

modified somewhat by the short-ranged residual force, as is

described in Section 3.4, so that these states are called

"single-quasiparticle" states. The lowest rotational bands 185in W, which are displayed in Figure 3-2, are single­

quasiparticle bands. These bands are labelled by [N A].

The quantities in the brackets are additional quantum numbers

used to identify the intrinsic state and are called "asymp­

totic" quantum numbers. They are defined in Section 3.3.

An examination of the experimental energies of the

states of the rotational bands shown in Figures 3-1 and 3-2

reveals some significant deviations from the simple energy-

level formula of Equation (3-4). This is particularly true

of the two lowest bands in 1/2 [510] and 3/2 [512],

Part of the reason for these discrepancies is the neglect

of the Coriolis-coupling term H£ in solving for the wave-

functions (3-3). For these wavefunctions, the angular momen­

tum projection K is a good quantum number, but when Corlolis

coupling is included, this symmetry is broken. The Corlolis

term will mix wavefunctions $ TVf„ and $ , where I K—K * I = 1.J M K J M K

Such mixing is generally weak for heavy nuclei unless the

two bands are very close in energy. However, this is precise-185ly the case for the two lowest bands In W, so that It Is

59184

Figure 3-1. The energies and deduced quantum numbers of184the lowest states in W based upon current experimental

information. (See Section 4.6.1.)

Figure 3-2. The energies and deduced quantum numbers of185the lowest states in W based upon current experimental

information. (See Section 4.6.2.)

EX

CIT

ATI

ON

EN

ERG

Y (k

eV)

Figure 3-1

1200

1000

800

600

400

2 0 0

LOW-LYING LEVEL STRUCTURE OF ,84W

1221 3

— 1134 4+ 1121 2 + LL?P— , 2K=2

Octupole_ i 0 0 6 _ 3+ 1002 Q+ Band

K= 09 0 3 2 + £ -B a n dK = 2

/ - B a n d748

3 6 4 ^

I 11

— 2 — 0 +K-0

GroundBand

EXCITATION ENERGY ( keV)

ooT

noOO

0 4oo

5OiSroioT 0 4

o>0>

tnV.noi

*

roi

0 4oro

u>\ro

i

!■»

roi

ro

S? ^o ro

to

0 4Nro

oooo

cn\ro

i

0 40 4

roi

r\>♦

2?.cn.

to-o

ro♦

=1it

roi•7? -'Jcnooj

ro

ro

Figure

necessary to include the complication of the Coriolis mixing.

In fact, the first calculations of Kerman (Ke 56) dealing

with Coriolis coupling were applied to the mixing of these 183two bands in W. Coriolis coupling will be described

more fully in Section 3.5.

Not all deviations from Equation (3-4) are attributable

to the effect of the Coriolis term. Another non-adiabatic

effect is the "rotation-vibration" interaction (Bo 52, Bo

53). This can be thought of classically as the excitation

of vibrations of the nucleus by the centrifugal force, and

results in mixing between the vibrational and non-vibrational

bands. Such Interactions between the rotational and vibra­

tional modes of excitation are evident in the ground and184vibrational bands of W. It is not surprising to find

such effects in the tungsten nuclei since they lie at the

edge of the region in the periodic table with strongly de­

formed nuclei and thus may possess less rigid shapes. A

possible way of treating such effects is to use the phenomeno­

logical collective Hamiltonian of Bohr (Bo 52), determine

approximately the parameters involved from the residual inter­

actions (Ba 68), and then solve the resulting problem exactly

(Ku 67). This procedure has the additional advantage that it

can also account for the effects of non-axially-symmetric

deformations, which also lead to band-mixing effects (Bo 53,

Da 58). However, an application of this technique by Kumar

and Baranger (Ku 68) has not been notably successful for the

tungsten nuclei (Gu 71). Therefore, such effects were

60

Ignored in the present work for simplicity. Another deviation

of the energy levels from the J(J+1) rule, known as "Coriolis

antipairing," results from coupling between the rotational

motion and the short-range residual interactions. This ef­

fect will be explained in Section 3.4.

There is one other difficulty with the adiabatic wave-

functions which should be mentioned. The intrinsic wave-

functions Xv are> *n principle, functions of the coordinatesJlvin the body-fixed system of all A particles in the nucleus,

while the rotational wavefunctions depend on the threeMKEuler angles. Thus, if these two wavefunctions are to be

treated as independent, there are 3A+3 independent coordi-

nates--three too many. Several ways of circumventing this

problem have been proposed (Vi 57), but the most common is

to use the Hill-Wheeler integrals (Hi 53, Pe 57, Br 71).

This method consists of treating the basic single-particle

Hamiltonian as a deformed field which may be determined, for

instance, by Hartree-Fock techniques. Then, since the wave-

functions of such a Hamiltonian are not eigenstates of the

total angular momentum operator, states of good angular

momentum are projected out. To a first approximation these

Hill-Wheeler projected wavefunctions will have the same ma-%

trix elements as the adiabatic wavefunctions of Equation

(3-3). The largest deviations in calculated observables

between these two approaches seem confined mainly to light

nuclei (Ri 68). Therefore, for simplicity the adiabatic

wavefunctions have been utilized in the present work.

61

62

3.3. The Independent-Partlcle Hamiltonian.

The intrinsic Hamiltonian Hp consists of an average

independent-particle part plus residual interactions,

HP ■ H0i + VRes <3-5a>-

where the independent-particle Hamiltonian contains the

kinetic energy plus an average field U:

V - - 2^7 K + (3- 5 b ) -

This Hamiltonian is expressed in terms of the coordinates

of the body-fixed reference frame. If the average field is

chosen to be non-spherical, with the deformations of the po­

tential corresponding in some way to the deformation of the

nuclear surface, then the major affect of the long-range

residual interaction is incorporated into the single-particle

Hamiltonian, so that V_ , the remaining part of the intrin-K6 Ssic Hamiltonian, is small.

The best choice for the average potential would clearly be a self-consistent field constructed from the fundamental two-nucleon interaction using Hartree-Fock techniques. This

is unfortunately impractical for heavy nuclei, and thus it is necessary to use phenomenological forms Instead. The

original calculations with deformed potentials were performed

by Nilsson (Ni 55) using an anisotropic Harmonic oscillator well. Such a well is inadequate, however, for generating wavefunctions to be used in reaction calculations because of its infinite well depth. Instead, most calculations of wavefunctions applied to reactions have used the Woods-

Saxon (Wo 54) form for the average nuclear potential, since

this simulates the physical nuclear matter distribution.

The Woods-Saxon is also the form used for the optical po­

tential, and thus optical-model studies can be used as a

guide in choosing the well parameters. Such an analysis

(Og 71) yields the following form for U, which has been

used in the present study:

D = V I ? ) + V 80 (?,?,?) + i(l-Tt) V coul (3-6.).

The central potential V has the form

V(r) = -[V -T V (N-Z)/4A] w(r) (3-6b)0 3 1

where w is the Woods-Saxon shape,

w(r) = (l + exp[II(r)/a] )-1 (3-6c) .

The spin-orbit potential V gQ is determined from

vs o ( r , M ) = — v s o (-2, " * ) 2 O • t^V(r) x p/ti] (3-6d) ,

which is analogous to the Thomas spin-orbit term used in

atomic physics. In these expressions, a is the Pauli spin

operator, and T , the third component of the corresponding3

isospin operator. (The convention used is that the eigen­

values of T are +1 for neutrons and -1 for protons.) For3

simplicity the Coulomb potential has been chosen in

the present work to correspond to a uniform charge density

63

64

within the deformed nuclear radius.

For spherical nuclei the quantity II(r) in Equation

(3-6c) is

II , (r) = r - R , sph o

where R^ is a constant corresponding to the point at which

the potential drops to half its maximum value. In order to

introduce deformations into this form, it is necessary to

have a description for the nuclear surface. There are numer­

ous ways to accomplish this (Og 71), but a common method is

to expand the surface in spherical harmonics. For a surface

which is axially symmetric about the z-axis and reflection

symmetric about the x-y plane, this expansion may be written:

R(0) = c R [1 + Z° 6. Y. (0)] (3-7),P 1=2 A A0

X even

where c is a constant depending on the 3^ and is chosen so that4 TT 3the volume of the nucleus is R . The earliest work with3 P

deformed nuclei assumed the nuclear surface to be ellipsoidal,

which corresponds approximately to retaining only the 3 (quad-2

rupole) moment in the sum of Equation (3-7). However, studies

with alpha-partide scattering (He 68, Ap 70) indicate that

it is necessary to add the 3 (hexadecapole) moment term in4order to achieve a good approximation to the nuclear surface

of the rare-earth nuclei. The tungsten nuclei in particular

have been shown to possess a large hexadecapole moment (He

71). There is also some evidence for a 3 (tetiahexaconta-6

pole) term in the rare-earth shapes (He 68, Ap 70), but this

term is usually very small and has been ignored in the pre­

sent study.

The proper method of deforming the nuclear potential,

which amounts to the choice of the quantity II(r) in Equation

(3-6c), is not a settled question, and consequently a number

of techniques are found in the literature. One simple

method (Ne 66, Fa 66) is to use the spherical expression

Hgph* replacing the half-radius R by the expression for

the nuclear surface R(0) (Eq. (3-7)) with R = R . AnotherP ocommon method (Fa 66, Ro 67) is to replace r in bY

r/R(8). A disadvantage of both of these methods is that

they lead to a skin thickness which varies along the nuclear

surface, and studies of muonic atoms that contain deformed

nuclei indicate that this thickness should be constant (Ac

65). Thus, an additional method for deforming the nuclear

surface has been proposed (Be 68, Da 69) which assures the

constancy of the skin thickness to first order by demanding

that the normal derivative of the potential be constant

along the surface.

Nevertheless, because of its simplicity the first of

the above methods has been chosen for the calculations

used in the present study (As 72a, So 72), but with a mod­

ification. As is mentioned in Section 3.6.1, the optical

potentials used to calculate the scattering solutions in

the CCBA must also be deformed. Since R is different, ino

65

66

general, for the potentials of each particle and indeed, in

the case of the projectiles, for the real and Imaginary

parts of the same potential, application of this method can

produce several different potential shapes for the same nuc­

leus. Although there is no evidence that for a given nucleus

the potentials of all particles must have the same shape, it

seems more consistent (As 72) to base every potential on the

underlying mass distribution of the nucleus (Eq. (3-7)).

This can be achieved (As 72) by demanding that

n(r ) = r - r k - cR [1 + E Yx (6)] (3-8)A = 2 oA even

for all potentials used in the analysis. The value chosen

for Rp in the present work is the nuclear radius determined

by Myers (My 70) from a Thomas-Fermi treatment of the nucleus

The quantity r^ depends on the specific potential and is

chosen so that r. + R = R , with R the usual optical­ly P o opotential radius. It has the character of an effective

radius for the interaction of the pafticle with the nucleus.

A number of techniques (Og 71) are also frequently used

in solving the single-particle eigenvalue problem:

„ , H j . IT0 VK " e VKlT ^VK

A very convenient method is to expand the wavefunctions IT4> on a basis of harmonic oscillator functions and then to V K

diagonalize the resulting matrices. This can be done on a

67

spherical basis

(3-9a)

or a cylindrical basis

N N A P zI K ^ N N A> (3-9b)p z

In these expressions, A is the eigenvalue of Z , the compon-3

ent of the orbital angular momentum of the particle along

where N is the total number of oscillator quanta. The quan­

tity v represents all the quantum numbers besides K and IT

needed to label the single-particle state. It is necessary

to limit the sums in Equations (3-9) to a few terms in any

practical solution of the diagonalization problem. Thus,

it is important to choose the expansion which gives the bet­

ter convergence. For non-spherical nuclei the cylindrical

basis is better in this respect (Og 71) and has been adopted

in the calculations used for the present analysis (As 72a,

So 72). The resulting single-particle states are generally

designated by their dominant cylindrical component at large

prolate deformations: | K 7*; N z A > . As is noted in Sectio

3.2, the customary labelling scheme is K U [N N A] withz

the symmetry axis, and N , N , and n satisfyP z

N = 2n + £ = N + N P z

It should be mentioned that there exists another pro­

cedure (Ro 67) for generating the single-particle eigenvalues

to be used in reaction calculations. This consists of making

the expansion

68

and then solving for the u^^(r) numerically. (The spin spher­

ical harmonic g j defined in reference Ed 60.) This

results in a set of equations for the u. . Since the ap-* J

proximation of the actual eigenfunctions <{> by a limitedV K

set of oscillator wavefunctions will be least adequate for

large values of r and since this is the region of greatest

importance for reaction calculations, the coupled-equation

method will often be a much better approximation to the

eigenfunctions for such calculations. Nevertheless, the

diagonalization technique was chosen in the present study

(As 72a, So 72) because of its convenience.7TThe single-particle eigenfunctions <J> are also sensi-V N

tive at large r to their binding energy in the potential.

Thus, for reaction calculations the potential depth is often

adjusted so that the single-particle eigenvalue has the

correct binding energy for the state of interest. However,

this is not a very consistent method, particularly for two-

nucleon transfer reactions where many single-particle levels

are involved. Instead, in the present calculations (As 72a,

So 72) the same potential depth was used for determining all

of the single-particle eigenfunctions involved in the reac­

tion. This procedure worked well for the (p,t) calculations,

but in the (p,d) calculations it was necessary in the region

outside of the nucleus to use a spherical Hankel function

in place of the calculated eigenfunctions, as is explained

in Section 6.3.

3.4. The Residual Interactions.

As a final step in calculating the intrinsic structure

of rotational nuclei it is necessary to consider the effect

of the residual interactions on the deformed single-particle

states determined in Section 3.3. The first problem is to

deduce the form of these interactions. Since it is a dif­

ficult problem to construct the residual force directly from

the two-nucleon force, the standard procedure is to construct

models. As is mentioned in Section 3.1, these forces can

be rather arbitrarily divided into long-range and short-range

components. The short-range components are considered first.

An obvious simple model for a short-range force is a

delta-function interaction. This interaction has the feature

that states with the maximum number of particles coupled to

zero angular momentum to have a significantly lower energy

than all others (Mo 60, Na 65). As a result of this char­

acteristic, it was suggested (Bo 58) that an alternative

model for the short-range forces which would be considerably

easier to handle than the delta-function interaction is the

so-called "pairing force." The pairing Hamiltonian can be

written in second-quantized notation as

69

70

HPair = “ G vv' av+ av+ av ' av' (3-10)

Here and ay are the creation and destruction operators,

respectively, for a particle in orbit |v>, and avT and%

are the time-reversed creation and destruction operators.

The phase of ay is chosen such that

^ t xl/2"K t®VK “ (_1) av -K'

Then, Hpa^r has non-zero matrix elements only between states

with pairs of particles coupled to zero angular momentum,

and the effect of the off-diagonal matrix elements is coher­

ent. Hence, the pairing force simulates the major effect of

the delta-function force. The strength of the pairing inter­

action, G, is usually taken to be constant to simplify the

calculations, but the interaction is assumed to be effective

only for particles near the Fermi surface. The value of G

is different for proton-proton and neutron-neutron pairing

(Na 65, Be 69). This was taken into account in the calcula­

tions used for the present analysis (As 72a) but will be

ignored in the following discussion for convenience. Neutron-

proton pairing is generally ignored for heavy nuclei because

of the large neutron excess (Na 65, Be 69).

An exact solution to the pairing problem is easily ob­

tained only in very idealized cases (Mo 60, La 64, Be 69).

A commonly used approximate solution is based on the work of

Bardeen, Cooper, and Schrieffer (BCS) with the superconductiv­

ity problem (Ba 57). They found that an approximate ground-

71

state solution obtainable in a variational approach is

where UV2 + V = 1and n .

The quantity n is the number of particles in the system, and

£2 , is half the degeneracy of level V. For intrinsic states

ground-state solution, Equation (3-10), has the property

that the total number of particles in each of its terms is

not the same. Thus, it involves a fluctuation in the number

of particles about the average value n and is therefore valid

only in the limit of large n. Consequently, this technique

can only be used for reasonably heavy nuclei and describes

only their average properties.

Physically, the BCS ground-state solution describes a

scattering of particles across the Fermi surface with the

occupation probability of any one given level being equal to

2V^2 . Thus, in order to put the pairing force into a form

suitable to apply the BCS approximation, it is useful to

perform a Bogoliubov-Valatin transformation (Bo 58a, Bo 58b,

of deformed nuclei, is always equal to one. The BCS

Va 58):

a t (3-12a)V

(3-12b)

■j* <b *f"The so-called "quasiparticle" operators cx and <x evident­

ly possess both particle and hole character and have as their

vacuum the BCS ground-state, Using these operators,BC Sa Hamiltonian consisting of the single-particle component

plus the pairing interaction can be written (La 64, Be 59)

72

H = U + H + H' (3-13a)qp»!■«* U 2 <*BCS I H l *K S> C3-13b).

involves terms containing two ex's, and H', terms con-qptaining four ex's. It is customary to neglect H' as a first

approximation, but this causes the number operator N not to

commute with the Hamiltonian. Consequently, it is necessary

to add a term XN, where X is a Lagrange multiplier, and

demand that the ground-state expectation value of the number

operator be equal to n. Then, minimizing the ground-state

energy and neglecting self-energy terms, one finds that

|<(>Bcs> corresponds to the ground-state and

H = 2 . E „ (at, a „ + a t ^ ~ ) (3-13c)qp K>0 VK7T v VKTT VKTT VKtt vKlTV7T

Als°- d vkit - + T r * - ) <3-14a>VKTT

e -Xand Vv 4¥ = 1/2 (1 - -X52L-) (3-14b)

with Ev m = /A‘ + <eVKlI-l)‘ (3-15.)

73

and A = G 2 U V (3-15b) .VKTr v Ktt VKtt v j '

The numbers are the energies of the single-particle

levels |vKTT>. It is clear from the above equations and the

definition of V that the Lagrange multiplier A corresponds

to the Fermi energy of the nucleus and the diffuseness of

the.Fermi surface is proportional to the "gap parameter" A.

The terms U+H can be thought of as constituting anqp

"independent-quaslparticle" Hamiltonian. The term H' plus

the long-range part of the residual interactions then cons­

titute a quasiparticle residual interaction. Thus, it is

apparent that the effect of the pairing force is to alter

the vacuum and change the particle excitations into quasi­

particle excitations. Specifically, this means that the

Intrinsic state of the ground band in an even-even deformed184nucleus such as W should be described to a good approxi­

mation by the BCS vacuum state, which varies smoothly from

nucleus to nucleus. Similarly, the intrinsic states of the

lowest bands in odd-mass deformed nuclei correspond mainly185to single-quasiparticle states. All of the bands in W

of interest in this investigation are of such a nature.

In order to determine and V^, the two relevant quan­

tities describing the structure of the BCS and single-quasi-

particle states, it is necessary to fix the gap parameter A.

Since A corresponds to the Fermi energy, it is clear that E

for the ground-state of an odd-mass nucleus is nearly equal

to A and thus, from Equation (3-15a), A should be related to

the odd-even mass difference (Ni 61). However, when a single­

quasiparticle excitation is added to a BCS vacuum, this re­

sults in the breaking of a pair and the occupation of some

orbital by an odd particle. Because of the Pauli principle,

this orbital is no longer available for the pairing correla­

tion, an effect known as "blocking" (Ni 61, Na 65). The

existence of blocking means that the pairing gap A should

actually be readjusted in odd-mass nuclei from the value in

the neighboring even-even nuclei. The use of different

values of A in the target and residual nucleus, however,

complicates the description of single-nucleon transfer

reactions and thus the effect of blocking has been ignored

in the present calculations (So 72). Another complication

which can influence the calculation of A is the "Coriolis

antipairing" (CAP) effect. This results from the application

of pairing to a rotating system. The pairing force relates

two particles in time-reversed orbits; however, the Coriolis

force has an opposite effect on time-reversed particles and

hence will tend to break pairs (Mo 60b). This leads to a

modification of A which depends on the rotational angular

momentum of the state. The rotational energy must be expli­

citly included in the BCS minimization procedure in order to

account for this effect (Ma 65, Ch 66, Ha 69). The CAP ef­

fect influences mainly the higher angular momentum members

of a rotational band and hence should have little effect on

the transitions of interest in the present work. Thus, it

has been ignored.

74

75

For higher-lying bands in deformed nuclei, such as the

ing two or more quasiparticles, but for these higher-energy

states the quasiparticle residual interaction can no longer

be ignored. Some of the long-range residual force has been

used to produce the deformed average field; however, some

remains, and the simplest approximation is to limit this

remaining part to the lowest non-trivial multipole, the

quadrupole (Ki 60, Ki 63). Neglecting exchange terms and

assuming charge independence, this force can be written in

a convenient product form as

The radial dependence of this force, R(r), has been chosen

for the calculations used in the present analysis (As 72a)

to have a surface-peaked form determined through a self-

consistency condition from the single-particle potential

(Ku 70). Performing the Bogoliubov-Valatin Transformation,

one obtains several terms. It is convenient to neglect all

of these terms except

excited bands in 184 W, one must use intrinsic states involv-

(3-16a)

where q = E <v K tt |R(r) Y y (r)| v 'Ktt> a^ MM • 2VV'

KIT

tVKTT aV , KTT (3-16b)

(3-17a)

where

q = £ <VKtt | R(r) Y y (r)| v 'Ktt> (Uy K>0 2

VV ' IT

„ V . „ + U V „ )vKir v 'K tt v Kir VKtt

(3-17b)

76

wlth Bvv'nw = “ vkw Vi e w

Some of the eerms neglected in going from H to H can be2 2

thought of as renormalizing the vacuum energy U and the

single-quasiparticle energies (Ba 60, Be 63), but this

accounts for only a small portion. An alternative view isa.that is an a l hoc quasiparticle residual force. It is

useful to add to H part of the residual pairing force H 1,

namely

H s - ' G vv i^UvKir Bvvktt ~ VVKir Bvvktt^

X (UV'KW - < ’KW B+0 ' V K W > <3-18>

The reason for retaining the particular terms (3-17)

and (3-18) as the quasiparticle residual interaction becomes

clearer if the total remaining Hamiltonian H is written in

another form (Ho 60):

'VH E U + H + H + Hqp s 2

U + KS>0 Evktt Nvktt " 2G{[k >q Pv v 'Ktt 6v v ,] vtt vv'ir

+ 2 ,K>0 vKir v v 1 ir

' X2P l£>i PVV-KW V'KW|Z ( 3 - 1 9 a )

VV ' TT

77

t 'V't ^w VKIr E < W + “vX* V t <3'19b>

P» V B E < V v M + Bv V K 3 > //2 <3-19<:>

QW x i r 5 < V v x u ' Bvv'Xw>/ i / 2 <3‘ 19d>

e L ' i r . 5 <VK,r l R<r > Y M<r >l v'Kir>

X <\xil VV ' KTT + "v'K. + VVKlr> <3-19e)

By making the further approximation that

[B V V 'K TT * BV V 'K TT = V ^ v ' v ' K 7T TT (3-20), 1 1 1 1 2 2 2 2 1 2 1 2 1 2 1 2

it is easy to show that

<QV V 'K IT ’ PV V 'K IT 1 V 1 ^ v ' v K ^ TT TT1 1 1 1 2 2 2 2 1 2 1 2 1 2 1 2

'XjIn addition, the commutators of the Q's with H are equal to

sums over the P's, and vice versa. Thus, the Hamiltonian

(3-19) with the approximation (3-20) is analogous to the

problem of coupled harmonic oscillators with the Q's and

P's acting as "coordinates" and "momenta," respectively.

The approximation involved in Equation (3-20) is known as

the "quasiboson" approximation, and is valid only in the

limit in which the occupancy of the orbitals is much less

than the orbital degeneracy. Since the degeneracy of all

orbitals in deformed nuclei is only two, the applicability

of the quasiboson approximation for these nuclei is ques­

tionable, but this problem is usually ignored.

The solution of the coupled oscillator problem is well-

78

known. A set of coordinates, ©k and n^, is defined which

decouples the equations:

®k “ K>Q ^ k . v v ' K i r Qv v 'K7t

w ’ 7T

^ k K>0 ^ k . V V ’ KTT PVV, KTT

vv ’ ir

such that

' V “ i ■ 1 ck nk

‘V - -1 Dk 0k

j*Then, an oscillator-quantum creation operator 0^ can be

defined in exact analogy to the ordinary harmonic oscillator

problem:

° l = ^ \ nk

The equation of motion for this operator is

[H, 0+] = Wk 0+ (3-21a)

where u>k = /CfcDk (3-21b)

The quantity u>k is the excitation energy of the oscillator

quantum. In terms of the B's, the operator 0 " can be writtenJx

°I - (xvv'k b v v 'm - Yvv'k B» » ' n > (3'22)VV ’ 7T

By using the relation

[°k’ ° k ,] = 6kk'’

one obtains the normalization condition

y (x Ktt* Ktt Ktt* KitK>Q w ' k W * k ~ W 'k YVV'k; 1 *VV 1 TT

KttThus, Xv v ,k is a measure of the probability that state

k is obtained from the ground-state by the creation of aK ttquasiboson, and Y the probability of obtaining k by

K ttdestroying a quasiboson. Since the quantities will

be non-zero in general, it is clear that the ground-state

of the system, which is the vacuum for the operators ol"K.must differ from the BCS vacuum (Equation (3-11)). This is

because the residual interaction which has been used in

this procedure (Equations (3-17) and (3-18)) connects the

BCS vacuum with the two-quasiparticle configurations and

hence mixes such configurations into the ground-state. In

general, however, these "ground-state correlations" are

small and can be ignored in reaction calculations (As 72d).

The approximation defined by Equations (3-21) and

(3-22) is a special case of a general technique (La 64, Ba

60) known as the "random-phase approximation" (RPA). The

RPA has a definite advantage over perturbation techniques

in the specific case at hand. Because some terms of the

quasiparticle residual interaction have been neglected, the

number operator does not commute with the Hamiltonian.

79

This lack of particle-number conservation generates spurious

modes of excitation corresponding to particle-number fluct­

uation, in a manner similar to the spurious center-of-mass

motion for a non-translationally-invariant Hamiltonian. It

can be shown, however, that in the RPA all such modes sep­

arate out as zero-energy states, which do not mix with the

proper modes of excitation of the nucleus (Ba 60, Ho 60,

Je 67). With a perturbation technique, on the other hand,

these spurious states are mixed with the proper states.

The particular form of the RPA outlined above, using

a quadrupole residual interaction, is applicable to 8-

and y-vibrational bands in deformed nuclei. The B-vibration

case uses the y=0 part of the quadrupole force, and the y-

vibration, the y=±2 parts. In the present analysis,

calculations of Ascuitto and Sorensen (As 72a) were used184for the first B~vibrational band in W. No attempt has

184yet been made to calculate the y-vibrational bands of W.3.5. Corlolis Coupling.

In Section 3.4 it is noted that for most odd-mass de­

formed nuclei of the rare-earth region, the intrinsic states

of the lowest rotational bands can be described to a good

approximation as single-quasiparticle states; that is,

l XvKir> = avKir I < BCS> ’

One effect which can cause a deviation from this simple

picture is band mixing by the Coriolis-coupling Hamiltonian

Hc ■ - 25 < V - + J-3 + > ” -ld>-

80

This operator mixes states $ and $ , if Ik -K'I = 1. InJ r l K J M K

addition, has diagonal matrix elements if K = 1/2, which

perturb the eigenvalues of K = 1/2 bands from the J(J+1)

spacing rule.

As was originally noted by Kerman (Ke 56), the Coriolis

mixing is especially strong between the 1/2 [510] and 3/2 [512]

bands in the tungsten isotopes. The evidence for this in 185 W is based on strong interband transitions observed in

decay studies (Da 69a, Ma 69, Ku 69, Gu 70) and the perturbed

energy-level sequence of the bands. Kerman treated the mixing18 3of these two bands in W by considering the extreme model

in which only two bands are included in the space of the

mixing calculation. The problem then reduces to a simple two-

state diagonalization with eigenvalues (Ke 56)

Ej ■ I {e ;,k + i + e ;,k * [(e ;,k + i - e ],k )! + 4a k(3-23a)

where AR = [(J-K)(J+K+l)] <JMK+1 |H | JMK> (3-23b)

and the E° are the unperturbed eigenvalues:

e °t v E T"n [ J ( J + U " 2K 2 + 6- , (-1)J+1/2 (J + l / 2 ) a ] + E? (3-23c)J I K ^ t / & > 1 / 2 v

The term involving the "decoupling parameter" a applies only to

K = 1/2 bands and represents the effect of the diagonal matrix

elements of Hc . The decoupling parameter is defined by

a = (-1)J“ 1/2 (J+l/2)"- <JMK=l/2 |H I JMK=l/2> (3-23d).c

The mixed eigenfunctions are

81

82

$JM " a $JMK + ^ $JMK+1 (3-24a)

and = 3 $JMK " ° *JMK+ 1 (3-24b) ,

where a 2 + e2 = 1 (3-24 c)

A2and a2 = r , „ > •> >>

[(EJ,K“EJ } + A] (3-24d)

with A = <JMK+1 |Hc | JMK> (3-24e).

Here the symbols > and < refer to the higher and lower energy

solutions, respectively.

In the present analysis, the mixing between the 1/2 [510]_ 185and 3/2 [512] bands in W was calculated using this simple

two-band mixing model. The mixing parameter waS <*eter”

mined by assuming single-quasiparticle intrinsic states for

the two bands. The expression for A in this case is derivedIvin Appendix 3A. The single-quasiparticle states were calcu­

lated in the manner described in Sections 3.3 and 3.4 using

the parameters of the single-particle well and pairing inter­

action listed in Table 6-1. The inertial parameter fi2/2j was

initially chosen to be equal to the average of the inertial

parameters of the ground bands in ^E4W and as determined

from the 0+-2+ energy difference, and the only adjustable

parameter was taken to be the energy separation of the two

unperturbed bands (E 3 /2 3/2~El/2 1/2^*The results of this calculation are listed as set 0 in

Table 3-1, and it is evident that the experimental energies

of the levels are not well reproduced. This perhaps indicates

the neglect of Coriolis coupling to higer-lying bands and

possibly higher-order non-adiabatic couplings as well. The

lowest-order effect of such couplings is to change the mo­

ments of inertia of the bands (Ke 56). Thus, a calculation

was performed allowing the moments of inertia of the two bands

to vary in addition to the unperturbed band spacing but keep­

ing the inertial parameter in the mixing matrix element,2”/2*9cor, equal to the initial value. This three-parameter

variation (Set 1) reproduces the experimental level sequence

very well as indicated in Table 3-1. A higher-order effect

of the neglected couplings (Ke 56) is to include a term in­

volving [J(J+1)]2 in the unperturbed energy formula (3-23c),

but this adds two more adjustable parameters to the calcula­

tion and thus was ignored in order to minimize the number

of adjustable parameters.

It has been found in various Coriolis-coup]ing calcula­

tions for deformed nuclei that it is sometimes necessary to

adjust the strength of the Coriolis matrix element in order

to reproduce the observed population of levels in transfer

reactions (Ca 72) and intensities of interband gamma-ray

transitions (Br 65, Zy 66, Ma 67, St 68c, Ha 69a, Hj 70), even

when some of the weaker couplings to distant bands are in­

cluded explicitly. It has generally been observed that this

strength must be reduced. One explanation for this (St 68c)

may be that the pairing factors U and V used in evaluating

A are not properly determined because of such effects as

blocking and Coriolis antipairing; however, this is yet an

83

Table 3-la. Parameters Used for Coriolis-Coupling Calculation.

84

ft2 ft 2 ft2 E° -E° A .M /, 2j,/2 2 \ u

(keV) (keV) (keV) (keV) (keV)

Set 0 19.42 19.42 19.42 5.0 -18.66 0.095

Set 1 17.40 16.80 19.42 4.5 -18.66 0.095

Set 2 18.60 15.70 16.50 0.0 -15.85 0.095

Table 3-lb. Coriolis-Coupling Eigenfunctions.

Set 0 Set 1 Set 2J a 8 a 6 a 6

1 / 2 “ 1.0 0.0 1.0 0.0 1.0 0.0

3/2“ 0.939 -0.344 0.929 -0.371 0.951 -0.309

5/2“ 0.877 -0.481 0.869 -0.495 0.914 -0.405

7/2“ 0.862 -0.508 0.859 -0.511 0.917 -0.400

9/2“ 0.815 -0.580 0.823 -0.569 0.901 -0.433

Table 3-lc. Coriolis —Coupling Eigenvalues in keV.

Level Experimental* Set 0 Set 1 Set 2

1/2“ 23.6 23.6 23.6 23 . 6

3/2“ 0 -0.8 -0.1 0.0

3/2“ 93 .8 99.2 93 .7 93.6

3 /2< 66.1 79.1 66.8 67.6

5/2> 188.2 204 .2 189.5 188.7

1 V

CM 173 .8 201.5 171.4 170.6

1 A

CSI 334 336.7 335 .9 338 .2

vo N> A 1

302 353.8 302 .4 301.3

9/2 “ ^492 547 .3 497 .8 500.3

* Taken from Casten, et a l . (Ca 72).

Table 3-ld. B(E2) Ratios.

Experimental* Set 0 " Set l^ Set 2t

B (E2 : 5/2“ -»■ 3/2“)---------- 9.4 5.88 5.45 8.85B(E2: 5/2< -*■ 1/2“)

B(E2: 7/2“ -»■ 3/2“ )--------------------- 1.6 1.02 1.05 0.89B(E2: 7/2“ - 5/2“)

t Calculated making approximations described in the text.

* Taken from Kuroyanagi, £t d^. (Ku 69).

86

unresolved question. In order to determine if the present

analysis is strongly influenced by reduced coupling strength,

an additional three-parameter fit to the experimental energies

was performed using an arbitrarily reduced value of A ^ 2 *

The results of this calculation are listed as Set 2 in Table

3-1. Table 3-1 also displays some B(E2) ratios for transitions 185in W predicted from the three sets of coupling coefficients

and the experimentally measured values (Ku 69) for comparison.

Following Kerman (Ke 56), these were calculated assuming the

quadrupole moments of the two bands are the same and neglecting

the single-particle transition strength.

3.6. Applications to Reactions.

3.6.1. Inelastic Matrix Elements.

The expressions developed in the preceeding sections of

this chapter for the nuclear bound-state wavefunctions can3 TT I b TT Ibe used to evaluate the matrix elements U... and in theli f f

coupled equations (2-29) of the reaction problem. First,

however, it is necessary to determine the coupled-channel

optical potentials Ua and . Since in the analysis presented

in Chapters 5 and 6 no inelastic scattering has been allowed

between states with different intrinsic structure, the coupled-

channel potentials are meant to correspond to a truncation

of the space to scattering within a single rotational band.

Thus, a particularly simple procedure is to use the same basic

form as in the elastic-scattering optical potential, to express

the potential in the nuclear intrinsic coordinate system, and

then to incorporate the deformation of the surface in some

manner. The angular dependence introduced by the surface

deformation causes this potential to possess not only diagonal

but also off-diagonal matrix elements, and thus it can des­

cribe both elastic and inelastic scattering. The deformed

imaginary well obtained as a by-product of this procedure is,

of course, meant to account for not only the elimination of

the scattering to higher states of the rotational band which

have not been retained in the calculation but also the elim­

ination of all other reaction channels. It is an open ques­

tion whether the complex well obtained in this fashion is

adequate. Nevertheless, the above model has been applied

very successfully to descriptions of inelastic scattering on

deformed nuclei (Ta 65, Gl 67), and this will be taken as suf­

ficient justification for its use here.

The present parametrization of the coupled-channel op­

tical potential is similar to that used for the deformed

bound-state potential well except for the addition of an im­

aginary part :

U = - V w (r,R ,a) - i W w (r,R ,a) o o

+ 4i a 4“ w <r,R ,a) + V + V (3-25a)D dr o so coul

where w is the Woods-Saxon form defined by Equations (3-6c)

and (3-8). In the analysis performed in the present investiga­

tion, VgQ was non-zero only for protons and in this case the

following form was used:

87

where the Woods-Saxon form here corresponds to a spherical

mass distribution for simplicity. As in the bound-state

calculations ^ c o u 2 was chosen to correspond to a uniform

charge distribution of average radius .

The parameters of this coupled-channel optical potential

should ideally be determined by fitting inelastic scattering

on the nuclide of interest with a set of coupled equations

of the form given by Equations (2-21). However, the inelas­

tic-scattering data is not always available. Therefore, since

the inelastic scattering on spherical nuclei should be weak,

it is sometimes possible to obtain an adequate coupled-channel

potential by using average parameters determined from elastic

scattering on spherical nuclei. The choice of parameters

for the present analysis is described in Chapters 5 and 6 .

In order to determine the inelastic matrix elements it

is convenient to expand the potential in multipoles. In the

body-fixed reference frame, this expansion is

V (r-R(0 )) = V (r-R ) + 2 V. (r) Y (0) (3-26)0 LK LK LK

where V(r-R(0)) = U - V so

The coefficients V depend on the deformation parameters 3LK. A

and derivatives of V(r-R(0)). Expressions for these quanti­

ties have been given by Glendenning (G1 67). After transforming

expression (3-26) to the laboratory frame, it can be shown

(G1 67) that the inelastic matrix elements are

89

<<j>i*I (r’A) I V (r-R (0 ) ) | (r , A) >

= Z (-1) 1 1 J 1 1 > <j | | Y (0) | | j ,>L J± , j± , L 1 L 1

* <(V i • |QL* I V Ji'> (3-27),

where |aJ> represents the nuclear bound-state wavefunction,

and j, the angular momentum of the projectile. The reduced

matrix elements are those defined by Racah (See reference

(De 63)). The quantity <a J ||Q || a.,J ,> is known as the1 1 Li 1 1"inelastic form factor" and QT is definedLi

(dJ; + dJ; )Qtm = z V (r) n ■ 6 (3-28)LM K>0 LK (1 + KO

For even-even nuclei the expression for the inelastic form

factor has been given by Glendenning (G1 67). The expression

for this form factor in the case of an odd-mass nucleus with185mixed bands, such as W, is given in Appendix 3B.

3.6.2. The Nuclear Form Factors.

The nuclear form factor for the source term is defined

in Equation (2-16a) . An alternate way of expressing this

form factor is

FLSJ (rx ) = <$otiJ1M 1K 1 ^ ALSJ (rx ) l $af J f K f ^ J1M i (3_29a)

Here A^ is a creation operator for the transferred group x.

The assumption has been made here that x has only one in­

trinsic configuration in the projectile. For (p,d) reac­

tions

t .-bwhere a (r) is the nucleon field operator. For (p,t) reac­

tions,

4 o w ° < rx> E //dtx d d , dd2 dp Y£ o L (*x > V V *x(p)

x a+ (t ) a+ (r ) (3-29c) .1 2

Here r^, r ^ , , and a are the space and spin coordinates of

the two neutrons in x, and p is their relative coordinate:

p = r - r . As is usual (G1 65), the triton wavefunction is l 2

assumed to have a gaussian form with all the nucleons in

relative s-states, and therefore the dineutron and proton

radial parts are separable. Then,

$x (p) « e“Y P .

For deformed nuclei, using the adiabatic wavefunctions+(3-3) and expressing the transfer creation operators A in

the intrinsic frame (primed coordinates),

alsjm (rx } = ^ dmk alsjk <r x '} *

the form factors take on a rather simple form:

oi ( — 1) " " Jf J Jf ai^iaf^fFLSJ (rx ) = [(l+6Kio )(1+«KfO )]l/2 tC-KiK-Kf £lSJK 6 r » K i"Kf

J -K J JJ -OlxK.afK + (-1) C-K±KKf fLSJK 6K, K±+Kf * (3-30a)

The quantities f and f have the character of form factors

between the intrinsic states:

Ot K Ot KfLSJK < V > = <Xa iK ± lALSJK (rx ?) XCtf Kf > (3~30b>

ct K a K7 l s j k f f s <xa tK± K s j k R,l xa fKf> (3'30c>

These two functions can be simply related depending on the

structure of X and X . Thus, the transitions between all K i Kf

members of the initial and final bands depend, aside from

geometrical factors, on a single form factor f. This demon­

strates why deformed nuclei are so useful for reaction studies:

several transitions can be determined by a single nuclear in­

trinsic wavefunction.

Using the spherical harmonic oscillator expansion (Equation

(3-9a)) for the single-particle wavefunctions, the intrinsic

form factors can be written:

a k a K aiK iafK ffLSJK (rx f) " Z GnLSJK u nL ^rx'^ (3“31)n

For (p,d ) reactions, u (r ') represents an oscillator wave-nL xfunction for the transferred nucleon, and it is shown in

Appendix 3C that for transitions between a K=0 initial state

and a K=K^ final state,

K =0 VpK _ J+K.G n L i j Kf <R ’d) ' (-1) 'f C [ V f '.,V f (3-32)-

91

In fact, the entire form factor has a very simple form:

92Ki=0 v K

FT 1 T

x £ Cn

nL JVfRfTTf UnL (rn ’) (3-33).

As is described in Section 3.3, the quantities actually cal­

culated are not the spherical harmonic oscillator coefficients

These two sets of coefficients, however, can be related via

Equation (3-33) exhibits the form generally associated

with single-nucleon transfer reactions on deformed nuclei

(Sa 58, El 69, Au 70). It depends on the degree V ^ to

which the orbital from which the neutron is taken is filled.

In addition, the strength of the transition to any one member

of the band depends on the spherical expansion coefficients

^v Ktt t*ie intrinsic state. In a DWBA analysis for a reac­

tion on an even-even nucleus, J^=0, and thus the transferred

orbital angular momentum J is equal to the final state angu­

lar momentum . In that case, the strength depends on the

spherical expansion coefficient for only a single J = .

In the case of (p,t) reactions, the situation is much

more complicated, since the coordinates r and r^ in the

expression for A^ (Equation (3-29c)) must be transformed to

the relative and center-of-mass coordinates p and rx (Talmi-

Moshinsky transformation). The oscillator expansion is then

made in these coordinates. This procedure has been described

„ , but rather the cylindrical coefficients Wf f f

93

in detail by Ascuitto and Sorensen (As 72a), and they have

shown that

a K a KGnLOLK (P»0 = Z Y V >>a<b K,K±-Kf gLQLK (3-34a) .

Here S^o l k a comP^-^cate8 expression involving Moshinsky

brackets, and the are the "parentage factors":

K f k *t*eab = <Xa.K ^[dN N A dN N A +K,=k I Xct K * (3-34b>-i i p Z a Pt Z, b a b f fa a B b

The operators d^ create the cylindrical oscillator states

| K77; N N A> of Equation (3-9b). Expressions for the parentage P ^factors for transitions between two BCS ground bands and

between a BCS ground band and an excited RPA 0+ band are

given in reference As 72a.

Expression (3-34a) demonstrates the complexity of two-

nucleon as opposed to single-nucleon transfer transitions.

The two-nucleon transfer case depends on a sum over many two-

nucleon configurations, but the single-nucleon transfer

reactions sample only a single configuration. It can be seen,

however, that if the configurations of the intrinsic states

in (p,t) involve a coherent superposition of two-neutron con­

figurations, the two-neutron transfer cross section can

become particularly strong.

AppendIx

3A . Coriolis-Coupling Matrix Element.

The Coriolis-coupling matrix element is <v'JMK' |HC I VJMK>

where Hc is given by Equation (3-ld),

94

H = - - I2 s (J+j- + J-j + } (3-ld),

and |VJMK> by Equation (3-3). In the case of a single-quasi­

particle band, Equation (3-3) can be reexpressed as

|VJMK> , (||±i) (DjjK

+ d 3_k | ^ cs> (3-3').

This expression has made use of the relation (Pr 62)

R, * V M * * * „ - M ( 3 - 3 5 ) -

The components of angular momenta J and j” in the intrinsic

frame have the commutation relations (Da 65):

[J ,J ] = - i h J1 2 3

and [j ,j ] = +iftj .1 2

Therefore,

J± DMK = t(J*K)CKa+l)]+ (3-36.)

and, using the expansion (3-9a),

3± < m - C m I U « ) ( J « + i ) ] ‘ “S f

where aEJ | 0> = | K7r;ni.j>. This implies that

-L *'+K . tavKTT CVKTT [ (J+K ) (j+ K+1 2 anilj (3 -3 6b )

. K±+ _ __ K±l+ „ 'VK±1+where - U ^ a ^ - .

Then, using the anticommutation relations,

/ K ? Kt-l o r r rn ’fc'j' ’ anAj} " n n ' 6ZZ • jj' K K ' ’

one can show that

K , K±t

^ n ' X . ’ j ' ’ an i l j } = V ' K 1 it 1 UVKTT + Vv ' K ' tt' VVKtt)

X 5nn' 5IZ 1 6j j 1 ' ,K±1 (3_37)

Using Equations (3-36), (3-37), and the relation

fdddtpdip Dm ,k , Dmr = 2J+1 <SM M , 6k k * 6 j j »>

it can be shown after some manipulation that

<v 'JMK' |H I v JMK>1 c 1

- _ JL_ y * cnJlj (U U + V V )2 3 g . v'K'rr vKtt v'K'tt vKir v'K'tt vKttnil j

x {/(J-K)(J+K+l)(j-K)(j+K+1) fiK f K+1

+ /(J+K)(J-K+l)(j+K)(j-K+1) 6r , R-1

+ V , K 5K,i- <-1)J4i <J+^> anllj> » - 38>

"here anlj 5 (3+t ) I

Thus, comparing Equations (3-38) and (3-23), it is evident

that

96

a = — -A r * pTiil j /.j »t + v v tK 2J o . v 1K 1 tt VKtt ^ v ' K ' tt v Ktt + v v ’ K , tt vvktt;n% J

X ✓ U - K M i + K + l ) « K , K+1

3 B . Inelastic Form Factor for Odd-Mass Nuclei,

The inelastic form factor is defined as

<a! J1 llojl v Jlt>

If the single-particle transition strength is ignored, then

K=0 in Equation (3-28). Thus,

QLM = VL0 (r) DM 0 ‘

It is assumed that the states are mixed in general, so that

| aJM> = I A | VJMK> (3-39),VK VK

where | VJMK> is given by Equation (3-3').

Then, using the anticommutation relations for the quasi­

particle creation and destruction operators and the relation

(Pr 62)

T 3 1 1 fiTT 2 -I * ^ 1 ^ -I ’ ^ 3 1/d0d<})di|i Dm ^r Dmq Dm ^ IR = (2j i+1) cMi| n M ± CK O K '

it follows that

97

<ai ji M i lQL M l V V *!■>

Z AaJ A aJ rJ ± ' L Ji rJ i ' L Ji V V K V V M i' M M i K O K1 2

2 J . ,+1 ix ( ) 2 y2 J±+l ; L O ’

Therefore, since (De 63)

<ai Ji M i IQLm I a i ’ Ji' M i»>-

( 1)2^ 3 1 b ^ i

‘ /irTTT V « <a i J i llQi-11 a i' J i ’> ’

it follows that

< a i J ± I IQl I I a ± , J ± , >

u E , a “J k a “Jk Ck 1 ’ O K1 VL0 ( 3 - 4 0 )V V K. l 21 2

3 C . Form Factor for Single-Nucleon Transfer from an Even-

even Deformed Nucleus.

For this case, the single-nucleon transfer intrinsic

form factors can be written

f LSJK (rx * <<**BCS l ALSJK ^ x ’ av f Kf TTf '^BCS'*’

and, using Equation (3-35),

_ V K K -j; V - K

fLSJK (rx ,J " (-1) Tf fLSJK *

98

Then, making the expansion

a ! s j k <v > = e “ „ l <rx ’>n

and comparing with Equation (3-31) , it can be seen that

VfK f .

G = <d> |a a |4> > .nLSJK BCS 1 nLJ V fKfir 1 ®GS

Using the relation based on Equation (3-9a) ,

a+ = 2 CnJlj a K+VKtt VKtt n £ j»

and the anticommutation relations for the creation and destruc­

tion operators, it follows that

G = 6 6 V C (-1)nLSJK JJf K,-Kf V fKfTVf Vf-KfTTf '

However, it can be shown (Pr 62) that

c £JV f « - * »

Therefore,

V K- Jf+K<: 11GnLSJfKf = Wf CVfKfTV£ VVfKflVf (3-32)

Applying Equation (3-41) to evaluate f, it follows that the

complete form factor can be written

vfK J - J + K / J J j \FLSJ = /2(2Jf+l) (-1) ^ Q _K^j TTf VVfKf7Tf

nLJfx 1 CM x ■n u „t (3-33).

n Vf V f nL

If the final nucleus bands are mixed so that the states have

the form given in Equation (3-39), then

a. a J

V K.

99

CHAPTER 4

EXPERIMENTAL PROCEDURE AND PRESENTATION OP DATA

"Canst thou send lightnings, that they may go and say unto th.ee, ’Here we are’?"

- Job 38:35 (KJV).

4.1. General Description.

A number of experimental problems must be solved in186 186 order to obtain data for the reactions W(p,t) and W(p,d)

which can be used for the investigation of multistep proces­

ses in transfer reactions. First of all, the residual nuclei

in these reactions have energy-level spacings w .ich are as

small as a few keV even for low-lying states, as is clear

from Figures 3-1 and 3-2. Thus, high experimental energy

resolution is required. Also, the cross sections for some

of the transitions of interest are less than 10 yb/sr. This

necessitates data collection over long periods of time

(£ 24 hrs.) and hence long-term stability of the experimental

equipment. Finally, because the angular distribution shapes

and relative intensities of transitions can be used as a

means of distinguishing between reaction models, as is indi­

cated in Chapter 2, it is important to obtain accurate rela­

tive differential cross sections.

These requirements have been met by using the Wright

Nuclear Structure Laboratory MP tandem Van de Graaff Accel­

erator to provide the 18 MeV proton beams and the Yale multi­

gap magnetic spectrograph to detect the deuterons and tritons

produced in the reactions. The Yale MP tandem Van de Graaff

has demonstrated a capability of producing beams with an

energy spread (M/E) of better than 0.02% (Ov 69, Le 69),

while the multigap spectrograph is capable of particle de­

tection with energy resolution (AE/E) of better than 0.05%

(Ko 70). Also, it has been shown a number of times (Ko 71,

100

Me 72, Cl 72, Cl 73, Ma 73, Me 73) that these two Instruments,

along with the beam transport system connecting them, are able

to maintain long-term stability. Since the multigap spectro­

graph can collect data at twenty-three angles simultaneously,

it is possible to obtain relative normalization of the data

at the various angles with minimum uncertainty. In addition,

the ability of the spectrograph to detect particles with a

broad range of momenta allows data for both the (p,d) and

(p,t) reactions to be obtained in the same exposure.

4.2. The Beam.

The general configuration of the experimental area of

the A.W. Wright Nuclear Structure Laboratory is shown in Figure

4-1. To produce the necessary proton beam, positive hydrogen

ions are extracted from a duoplasmatron source. After being

converted into negative ions (H ) by exchange reactions with

hydrogen gas, they are injected into the Van de Graaff at an

energy of 300 keV and are accelerated to the terminal, which

has been placed at a large positive potential (about 9 MV for

18 MeV protons). When the H ions reach the terminal, the

two electrons are stripped off by collisions with oxygen

molecules. The resulting H+ ions (protons) are then acceler­

ated away from the terminal, thereby using the accelerating

potential twice. The protons are focussed through the anal­

yzing magnet, which deflects the beam 90° in the horizontal

plane with a radius of curvature of 132 cm. The object and

image of this magnet are defined by rectangular slits. The

horizontal dimensions of these apertures were set to 0.75 mm

101

Figure 4-1. Diagram indicating some of the experimental facilities at the A.W. Wright Nuclear Structure Labora­tory, including those used in the present experiment.

PHYSICAL LAYOUT OF THE WRIGHT NUCLEAR STRUCTURE LABORATORY ACCELERATOR FACIL ITY

c u b ic l e s f o rEXPER IM EN TA L

ELECTRONICS

ILM E N ITECONCRETE

ip ftfT ^ ja -G O N IO M E T E R

QUADRUPOLE

_ E 1 ISOURCE

<S o I

SWITCHING MAGNET I

ANALYZINGM AG NET

MP TANDEM ACCELERATOR

(0.03") for this experiment. A feedback mechanism from the

image slit, which controls current drawn off the terminal

through corona points, allows the energy spread of the beam

to be much smaller in general than that defined by the hor­

izontal slits. Other experiments (Ov 69, Le 69) using the

same aperture size as in the present experiment have achieved

energy spreads below 0.02%, approximately five times smaller

than the resolution determined by the slit geometry alone.

The vertical dimensions of the beam are also critical, since

in the multigap spectrograph the particles are momentum

analyzed in the vertical direction. Therefore, the vertical

object and image slits were also set to 0.75 mm.

Beyond the analyzing-magnet image slits, there are no

more beam-defining slits. This is to prevent degradation of

the beam resolution through slit-edge scattering. Beam

transport from the image slits to the spectrograph is pro­

vided by a quadrupole triplet, a switching magnet, two quad-

rupole doublets, and beam steerers. A set of four ellipsoidal

cylinders ("roller" slits) with a vertical aperture of about

1 mm are placed at the entrance to the spectrograph to

intercept any slit-scattered particles. The beam, when fo­

cussed onto a target at the center of the spectrograph, is

required to have a vertical dimension of less than 0.37 mm

(0.015") in order to achieve an energy resolution for the

reaction products of about 0.05%. The horizontal dimension

is required to be approximately 3 mm (0.120") to assure the

same solid angle in all the gaps. After passing through the

target, the protons are collected in a Faraday cup buried in

102

the wall behind the spectrograph. The integrated beam current

for this experiment was measured to be 10,000 yC.

4.3. The Target.

The target used in this experiment was made of WO^ evap-2orated onto a backing of 10 Ug/cm of carbon. The WO^, which

was purchased from Oak Ridge National Laboratory, was iso-186topically enriched to 97% W. This oxide of tungsten was

used because it has a much lower evaporation temperature

than metallic tungsten (about 1350°C as opposed to 2600°C at

the pressure of approximately 10 torr used in this evapor­

ation). The presence of oxygen and carbon in the target does

not obscure appreciably any of the transitions arising from

the tungsten, since the reactions (p,d) and (p,t) on ^8012and C have highly negative Q-values, and other isotopes

exist as very small percentages of natural oxygen and carbon.

Evaporation of the WO^ powder was accomplished by re­

sistance heating in an open, uncollimated tantalum evaporation

boat. To prevent reduction of the oxide to metallic tungsten,

the boat was lined with 0.002" of platinum foil spot-welded

to the tantalum. The material was evaporated onto a glass

slide at a distance of approximately 1.5" from the boat. This2slide, covered with a layer of 10 yg/cm of carbon and a

release agent to allow the foil to be floated off, was pur­

chased from Yissum Research and Development Corporation. The

resulting target was then floated from the glass backing in

water and mounted on an aluminum target frame of 0.5" diameter.

The thickness of the target was determined by measuring210the transmission of 5.30 MeV alpha particles from a Po

103

source. By comparing the range of the alpha particles in

air with their range when the target was placed in front of

the source and using tabulated values for alpha-particle

stopping powers (Ba 65, Wi 66), the thickness of the tungsten2in the target was determined to be 147 pg/cm . Measurements2of target thicknesses of a few hundred pg/cm by this method

have been found to be accurate to about 10% (Ba 65). The

thickness of the target contributed to the energy resolution

of the experiment through straggling and differential energy

losses of the beam and outgoing particles, but mainly the

latter. In the multigap spectrograph the target is placed

at a 45° angle with respect to the beam. Therefore, the

maximum energy spread due to differential energy loss in

the present experiment, based on tabulated ranges of protons,

deuterons, and tritons (Wi 66), is approximately 4 keV at

angles forward of 90° (transmission through the target) and

12 keV at backward angles (reflection from the target) for186 W(p,d), and approximately 7 keV at forward angles and

18615 keV at backward angles for W(p,t). These contributions

to the energy resolution are quite large, particularly for

the (p,t) reaction; however, considering the small size of

many of the cross sections involved, it was decided not to

use a thinner target in order to keep data-collection time

within reasonable bounds.

For transparent thin films such as those of W0^» the

apparent color in white light is determined by interference

of light reflected from the front and back surfaces of the

104

film (Di 63). Since the target used in this experiment had

a uniform green color, this interference phenomenon can be

used to estimate the uniformity of the target thickness.

The wavelength (A) of the constructively interfering compon­

ent of the white light is directly proportional to the tar­

get thickness (t). Therefore, small variations in this

wavelength (dA) are related to small variations in the tar­

get thickness (dt) by

d A d tA = t *

Assuming (Di 63) that the wavelengths of light which appear

green to the eye are within the range 5250±500 A, an appar­

ently uniform green color for the target implies that the

target thickness is uniform to within 10% of its average

value.

4.4. Particle Detection: The Multigap Magnetic Spectrograph.

The general design of the Yale multigap magnetic spect­

rograph is shown if Figure 4-2. This instrument consists of

a toroidal magnet at the center of which the target is placed.

The maximum magnetic field is 15 k G , sufficient for detection

of momenta corresponding to 85 MeV protons. There are twenty-

three gaps in the toroid through which particles emitted from

the target can pass, thus allowing simultaneous detection of

particles at twenty-three angles. Twelve of these gaps are

at 7.5° intervals in the forward quadrant and eleven at the

same interval in the rear quadrant. The entire toroid can

rotate to three different positions so that the gaps may be

105

Figure 4-2. Top and side views of the Yale multigap magnetic spectrograph, showing some of the details men­tioned in the text.

Figure 4-2

C O IL

II G APS (8 7 .5 -- I6 2 5 * ) /

B E A MIN

YALE MULTIGAP SPECTROGRAPH

placed at three separate sets of scattering angles with

respect to the beam. The present experiment was performed

at the position which gives a minimum angle of 5° and a

maximum of 167.5°.

The geometry of the magnetic field in each gap is of

the Browne-Buechner type (Br 56); that is, the field inside

the gap is uniform with a circular boundary whose radius is

equal to the distance from the target to the edge of the

field (in the present case, 31.0"). This geometry provides

one-dimensional first-order focusing along a hyperbolic

focal surface for a broad range of momenta and with a mini­

mum of defocusing in the transverse direction resulting from

fringing fields. Because of fringing fields and aberrations,

the position of the focal surface cannot be precisely cal­

culated. Therefore, the proper focal surface position has

been determined empirically for each gap (Ko 70). The aberra­

tions are reduced by limiting the vertical acceptance angle of

the particles to 5.2° with slits placed at the entrance to

the magnetic field (these are labelled "a-slits" in Figure

4-2). The Browne-Buechner geometry has the additional pro­

perty that for 90° particle deflection, the second-order

aberrations vanish. For most regions of the focal plane,

however, energy resolution is determined by target thickness

effects and the magnification of the vertical size of the

beam spot on the target. With a sufficiently thin target

and careful focusing of the beam, energy resolutions of bet­

ter than 0.05% have been obtained (Ko 71, Me 72, Cl 72,

106

Cl 73, Ma 73, Me 7 3).

The particles are detected at the focal surface with

photographic plates loaded into portable aluminum holders.

A spring mechanism in the holders presses the plates against

guides machined to match the hyperbolic focal surface of

the magnet. In this experiment Ilford plates with 50 micron

thick, K5 sensitivity emulsions were used. A total of four

feet of emulsion corresponding to two 1" x 24" plates can be

exposed at one time, allowing a momentum detection range of

1.6:1. This broad momentum range has been employed in the

present experiment in order to detect both the outgoing

tritons and deuterons simultaneously. For the field of 10.5

kG used in this experiment, the most energetic tritons from 186the W(p,t) reaction appeared at approximately 110 cm from

the bottom of the exposed emulsion surface, where the solid-4angle is about 3.3 x 10 sr and the horizontal acceptance

angle is about 0.20°; and the most energetic deuterons from the 186W(p,d) reaction appeared at approximately 35 cm, where the

-4solid angle is about 4.3 x 10 sr and the horizontal acceptance

angle is about 0.26°. In principle, these two particles can

be distinguished from the different grain size and lengths

of their tracks in the emulsions; however, to avoid ambiguity,

0.015" of acetate foil was placed in front of the lower 45 cm

of the emulsions. This thickness of acetate is sufficient to

absorb the tritons, but the deuterons can pass through and

enter the emulsion. Thus, on the lower portion of the focal186surface, only the deuterons from the reaction with W should

appear on the plates, while for the upper portion only the

tritons should appear.

107

After their exposure, the plates were developed for thirty minutes at approximately 3°C with Kodak D-19 devel­

oper, fixed for 90 minutes at approximately 10°C with

Kodak Rapid Fixer, and then washed. In all steps, nitrogen

gas was bubbled through the solutions to remove oxygen and

agitate the solution. This, in addition to the low temp­

eratures used, helped produce plates with high contrast

between tracks and background. The developed plates were

scanned microscopically across their full exposed width in

steps of 0.5 mm.

The relationship between the radius of deflection of

the particles in the magnetic field (p) and the position of

their tracks along the photographic plates (D) has been

determined from an elastic-scattering experiment (Ko 70,

Ko 71). This p-D calibration has been found to a good

approximation to be independent of magnetic field (Ko 71).

Hence, from a measurement of the magnetic field, one can

use the calibration to determine the momentum and thus the

energy of the particles. The magnetic field is measured

with an NMR probe in a single gap, and the exact fields in

the various gaps can differ slightly. Therefore, in prac­

tice the "effective magnetic field" in each gap is determined

by referencing to peaks in the spectrum corresponding to

particles of known energy.

The solid angles of the gaps were assumed to follow a

geometrical formula calculated by neglecting the effects of

the fringing fields (Ko 71). The absolute value of the solid

angles as determined by this formula are known to be accu-

108

rate to better than 10% (Me 70), and, on the basis of a

number of experiments which have been performed with the

spectrograph, the relative solid angles of the gaps are

known to be determined by this formula to better than 5%

accuracy (Ko 71, Me 72, Cl 73, Ma 73, Me 73).

4.5. Determination of Absolute Cross Sections.

The principal goal of this experiment was to obtain

accurate relative differential cross sections for the18 6various transitions in the W(p,d) and (p,t) reactions.

However, an effort was also made to determine the absolute

cross sections by normalizing the spectrograph data to the

well-known Rutherford-scattering cross section (Ru 11). This

method avoids uncertainties in the measurements of the

absolute integrated beam current and solid angles and of the

target thickness.

The normalization was accomplished by remeasuring in a18630" Ortec scattering chamber the 18 MeV W(p,d) and (p,t)

reactions at a single laboratory angle (65°). Then, with

the same detector geometry, the beam energy was lowered to

6 MeV, and proton elastic scattering was measured at five

angles between 45° and 70° and was found to follow the Ruther-

ford-scattering angular dependence. The same target was used

as in the spectrograph experiment and the outgoing particles

were detected with a surface barrier counter of 1000 micron

sensitive depth and analyzed via pulse-analysis electronics

consisting of a preamplifier, an amplifier, and a multichannel

analyzer. The energy resolution of this system was about 20

109

keV. The protons from the 18 MeV reactions on W were

distinguished from the deuterons and tritons by using a

detector of thickness sufficient for the deuterons and tritons

to lose all of their energy within the sensitive layer but

not the most energetic protons. The highest energy deuteron

and triton groups were, therefore, well separated in the

spectrum from the protons and could be used to normalize

the detector and spectrograph runs.

In this manner, a normalization of the 18 MeV data to a

known cross section (6 MeV Rutherford-scattering) has been

achieved to within yield statistics (^3%), assuming no var­

iation in the instruments or beam geometry between the 6 MeV

and 18 MeV measurements. The principal instrumental uncer­

tainties are the linearity and stability of the beam current

integrator which have been determined to be generally better

than 15% (Me 70). Changes in beam geometry affect the measure­

ment through the target non-uniformity which, as indicated in

Section 4.3, is probably less than 10% on the average. Thus,

it is estimated that this method has determined the absolute

cross section normalization to better than 20%.

A consistency check for this normalization procedure can

be obtained by calculating the target thickness from the

6 MeV proton-elastic-scattering yield and a measurement of2the detector solid angle. This method yields 166±17 yg/cm

where the assigned error indicates estimated uncertainties in

the solid angle and target angle determinations but not in

the absolute integrated beam current. This value is consis-

110

186

Ill

tent with the value of 147±15 pg/cm based on transmission

of low-energy alpha particles (Section 4.3).

4.6. Presentation of Experimental Results.

4.6.1. W ( p ,t) W .

A sample spectrum of triton tracks is shown in Figure

4-3. In order to determine the triton energies, the measured

magnetic field and p-D calibration were used to identify the

group in each gap corresponding to transitions to the ground 184state of W. Then, the effective magnetic field in each

gap was calculated from the exact position of this group,

assuming a ground state Q-value (-4.480 MeV) equal to that

determined from the 1964 Atomic Mass Table (Ma 65a) . The

effective magnetic fields in all gaps differed by less than

0.2% from the measured magnetic field.

In Figure 4-3, the track density is plotted versus tri-186ton energy in the t + W center of mass system. These ener-

18 6gies have been expressed in terms of Q-values for the W

(p,t) reaction. Some tritons, however, come from reactions

on impurities in the target, and these have been identified

from reaction kinematics based on known Q-values (Ma 65a).

Since the Q-values for (p,t) reactions on many common light

nuclides are highly negative, most observed impurity peaks

came from other isotopes of tungsten in the WO^ used for

making the target and small amounts of platinum which entered

the target during the evaporation process. The presence of

one of these impurities obscured at all angles the peak cor­

responding to the transition to the 6+ state of

2

The energies of the tritons can be expressed alternatively184in terms of the excitation energies of the final states in W

The measured energies for these states below 1250 keV are pre­

sented in Table 4-1. Also shown are the energies and deduced

quantum numbers as determined from decay studies (Ha 64, G1

69a, Ag 70, Ku 70a, Ta 71, Kr 73), neutron capture (Fa 68, Ca

73), inelastic deuteron scattering (Gu 71), Coulomb excitation

(St 68a, Mi 71), and previous transfer reactions (Is 72, Ma

72). As is pointed out in Chapter 3, these states can be

grouped into four rotational bands: a ground-state rotational

band, a y-vibrational band, a 0 -vibrational band, and an

octupole vibrational band (see Figure 3-1). The cross sections

for the various transitions were determined by summing the num­

ber of tracks in a peak and subtracting from this the apparent

background of extraneous tracks. For the triton peaks observed

in this experiment, the background was negligible in most cases.

The average full width at half maximum (FWHM) of the tri­

ton peaks was 12-14 keV. As can be seen from Figure 4-3,

there are two groups of tracks which correspond to unresolved

peaks. The group containing triton tracks resulting from

transitions to the 2+^ , 2 , and 4+ . states clearly has a larger

width than the other peaks in the spectrum. This is because

the 2+g and 4+^ states are separated in energy by nearly the

resolution of the experiment. Thus, it was found possible for

forward-angle spectra to separate these two peaks by means of

a fitting program. The transition to the 2 state is expected

to be very weak, since direct (p,t) transitions to unnatural

parity states are forbidden in the zero-range approximation

112

(Gl 65). Even with the inclusion of finite-range and multi-

step effects, the strength of this transition is expected to be

a small fraction of that to the first natural-parity state in

the band, the 3 at 1221 keV (As 72d). Thus, the 2 state was

ignored in the fitting procedure. This fitting was performed

using the computer program SESAME, written by T.P. Cleary

(Cl 72a), which reproduces the spectrum with a series of peaks

of a given reference shape, varying their separation and magni- 2tudes via a X minimization criterion. In this case, since

the relative energies of the states are well-known, the peak

separations were fixed in the fitting procedure. The refer­

ence shape chosen in each spectrum was that of the 2+^ peak.

At backward angles it was not possible to achieve a consistent

separation of the peaks because the resolution was not as good

and the reference peak contained insufficient counts. The

other unresolved group, corresponding to transitions to the

0+g and 3+^ states, could not be separated by this peak-fitting

procedure because of the close separation of these two states.

However, the 3"*" transition is expected to be a small fraction

of the 0+g transition for the same reasons that the 2 tran­

sition is weak, and thus its contribution has been ignored.

The resulting angular distributions of the tritons ex­

tracted from the spectra are presented in Figure 4-4. As men­

tioned in Section 4.4, the angular spread of each data point is

about 0.20°. The vertical error bars reflect uncertainties in

the relative cross sections from yield statistics, background

determination, fitting, and scanning reproducibility. These

are expected to be the major sources of random errors and, since

113

Figure 4-3. Spectrum of triton tracks observed fromscanning the photographic plate exposed in gap 2 (12.5°).The tracks are plotted versus deduced Q-value correspond-

186ing to the W(p,t) reaction. The shaded areas indicate peaks presumed to result from reactions other than18W t ) 18V

Figure 4-4. Plots of the deduced differential cross sections for the reaction ^88W ( p ,t) leading to 0+ and 3 (4-4a), 2+ (4-4b), and 4+ (4-4c) final states.See Table 4-2 for the numerical values of these cross sections. The lines through the points indicate uncer­tainties in the relative cross sections, estimated as described in the text. In (4-4c) the crosses denote angles at which the cross section of the transition tothe 4+ state may be overestimated by about 10% because

^ + of the nearby 2 g state.

TRAC

KS

PER

AO

Figure 4-3

E X C I T A T I O N ENERGY IN ,8 4 W ( M e V )

Q - V A L U E ( M e V )

E X C I T A T I O N E N E R G Y IN 184 W (MeV)0.6 0.5 0.4 0.3 0.2 0.1 0.0

(js/qt/)

Figure 4-4a

1000

100

,86W(p.t) Eps 18.0 MeV

100

10

10

0* (OkeV)

V *•

i t .

T *

0£ (1002 keV)

3" ( 221

H 1

keV)

V t

0° 40° 80° 120° 160° CENTER OF MASS ANGLE

dcr/

dft

Figure 4-4b

100

186

100

w>\_QJ .

E6

10

10

0.1

E E

■ f i

W(p.t) EP = I8 .0 MeV

2g ( I I I keV)

••

2 + ( 9 0 3 keV)

’♦ T♦ ♦

£ = T 5l i t r t

2 l (1121 keV)

40“ 80* 120° 160”CENTER OF MASS ANGLE

do’/d

n c.

m.

(/^

b/sr

)

F i g u r e 4-4c

100

eV

♦ t

1 0 4* (364 keV)

f

10* *

# ±I

n

4y (1134 keV)

x t :

1

0° 40° 80° 120° 160°CENTER OF MASS ANGLE

Table 4-1. 184Low-lying Levels of W.

114

* E(keV)J r K

Previous Work This Experiment

0 + 0 0 0 + 32 + 0 111 110 + 3

4 + 0 364 364 + 36 + 0 748

9032 + 2 902 + 3

0+ 0 1002

3 + 2 1006► 1003 + 3

2 + 0 1121

2 ~ 2 1130 > 1132 + 5

4 + 2 1134 J3“ 2 1221 1221 + 3

Table 4-2. (p , t) Center-o f-Mas s Differential Cross Sections (-.-I,0 ) in ub/sr.— — — d&6-wcenter- + . + _ _+ - ** urn ,

. + . Tof-mass angle ***

0 , 0g(g•s . )

’ g(111 keV)

’ g (364 keV)

. 2y (903 keV)

0 ’ °0 1 * °8 (1002 keV)(1121 keV)

4 » 2y 3 , 2 (1134 keV)(1121 keV)

5.05 136. 5± 5.3 185.916.7 26.812.0 39.012.3 34.0+2.2 6.8 ±1.0 7 .6±1.0 2.0±0.712.64 106.5± 4.1 154.215.7 26 .413.1 29.711.7 23.Ill.5 7.0 ±0.8 9.010.9 2.710.720.22 590.8118.9 109.214.1 23.011.5 16.411.2 37.211.9 3.4 ±0.6 9.210.9 3.910.627.79 856.1143.6 78.113.2 19.011.3 8.810.8 35.411.8 2.2 ±0.5 7.810.8 4.510.635.36 239.51 8.3 57.812.6 14.811.1 7.110.7 8.310.7 2.4 ±0.5 4.610.6 4.410.642.92 78.61 3.2 43.912.1 8.810.8 11.8+0.9 7.310.7 1.3 ±0.4 4.610.6 3.610.550.48 381.3112.6 31.811.7 4.110.6 11.110.9 30.811.6 0.4810.28 5.010.7 2.210.558.03 483.8115.5 37.511.9 3.310.5 9.210.8 33.411.8 0.5410.23 3.810.5 1.910.465.57 255.61 8.7 48.212.2 3.110.5 7.310.7 14.811.1 0.5910.33 ------------------ 3.110.673.10 137.71 5.1 45.412 .1 2.610.5 9.410.8 10.710.9 0.2710.27 2.710.6 2.210.480.62 192.81 6.8 28.911.6 2.510.5 8.210.7 13.511.0 ----------------------- 3.110.5* 1.210.388.13 231.21 7.9 26.511.5 1.910.5 6.210.7 15.711.1 0.6610.33 1.310.4 1.310.493.13 201.31 6.8 26.111.5 2.410.5 7.210.7 12.711.0 0.2210.22 1.910.4 1.410.4

100.62 125.11 4.6 32.011.7 2.010.5 5.510.6 7.210.7 0.2210.22 2.610.5 1.410.4108.10 101.71 3.9 26.511.5 2.810.5 6.210.6 7.9+0.8 0.3310.22 1.710.4 1.810.4115.57 107.31 4.0 21.811.3 2.610.5 5.610.6 9.610.8 ----------------------- 2.510.5* 2.210.4123.03 103.71 4.3 18.711.6 2.610.8 6.510.7 7.910.8 ----------------------- 2.610.7* 2.010.5130.48 84.01 3.3 21.311.3 2.510.6 4.810.6 7.110.7 ---------- 3.610.6* 2.410.5137 .92 64.11 2.7 21.511.3 3.710.5 4.510.6 5.010.6 ---------- 2.210.5* 1.210.4145.36 55.91 2.5 19.011.2 3.110.5 3.710.5 6.010.7 ---------- 2 .110.4* 1.410.4152.79 58.01 2.7 17.911.2 3.510.5 3.910.6 5.610.6 ----------------------- 2.410.6* 1.410.4160.22 61.21 2.7 14.411.0 2.810.5 2.910.5 5.010.6 ---------- 2.310.5* 1.710.4167.64 64.31 2.8 13.811.0 2.510.5 3 **±0.5 5.010.6 2.210.5* 2.710.5

t Probably contains a small amount of 2 , 2 (1130 keV).* Contains also 2+ , 0_ (1121 keV).

8 +** Probably contains a small amount of 3 , 2 (1006 keV).*** The angular region included at each point is about 0.20°. 115

they are assumed to be Independent, have been added In quad­

rature. Uncertainties in the absolute cross sections and in

the relative cross sections from the solid-angle determination

have not been included, but upper limits on these uncertainties

are discussed in Sections 4.4 and 4.5. The scanning errors

were estimated by having various different regions of the

spectra rescanned. The scanning reproducibility was found to

depend not only on the density of tracks but also on the

specific scanner. Both of these effects were taken into account

in the error estimates. For cases in which the density of

tracks was fewer than 1500 and greater than 20 tracks per half

millimeter, the scanning reproducibility was usually better

than 5%. A list of the measured cross sections is given in

Table 4-2.

4.6.2. 186W(p,d)185W .186The W(p,d) cross sections were determined in a similar

manner to that described in the last section. In Figure 4-5

a sample spectrum of deuteron tracks is plotted versus reaction

Q-value. As in the (p,t) case, the Q-values were determined

by calculating the effective magnetic field which yields a

ground-state Q-value equal to that (-4.989 MeV) based on the

1964 Atomic Mass Table (Ma 65a). In this procedure the peak185corresponding to the transition to the 244 keV state in W

was used as reference, since this transition is much stronger

than the ground-state transition and thus the peak position

was better defined. The deviation between the measured and

effective magnetic fields differed by less than 0.1% in all

gaps. The deduced excitation energies for the states observed

116

in W below an excitation energy of 500 keV are presented

in Table 4-3 together with the energies and deduced quantum

numbers based on decay studies (Ku 69, Da 69a, Ma 69, Gu 70)

and previous transfer reactions (Er 65, Ca 72). As is mentioned

in Chapter 3, these states can be grouped into four single-

quasiparticle rotational bands (see Figure 3-2), with the

two lowest bands, 1/2 [510] and 3/2 [512], being highly ad­

mixed .

The extracted angular distributions are shown in Figure

4-6 and a list of the measured cross sections is given in

Table 4-4. The angular spread of each data point is about

0.26°. As with the (p,t) angular distributions, the vertical

error bars indicate uncertainties in the relative cross sections

from yield statistics, fitting, background determination, and

scanning reproducibility. Peaks which overlap slightly were

separately visually, and estimated uncertainties in this pro­

cedure have been included as a "fitting" uncertainty. Since

the background is much higher for the (p,d) spectrum than for

the (p,t), the background uncertainties are a much larger effect.

In addition, for the strong transitions to the 5/2 (66 keV)

and 3/2 (94 keV) states, the scanning errors are important at

a few angles because of the high density of tracks.

The average FWHM resolution of the deuteron peaks was 15-

17 keV. The peaks corresponding to the 7/2 (174 keV), 5/2

(188 keV), and ll/2+ (197 keV) states are unresolved, as is

indicated in Figure 4-5. The transition to the ll/2+ state is

expected to be much weaker than the other two and has never been

117185

Figure 4-5. Spectrum of deuteron tracks observed fromscanning the photographic plate exposed in gap 7 (50.0°).The tracks are plotted versus deduced Q-value correspond-

186ing to the W(p,d) reaction.

Figure 4-6. Plots of the deduced differential cross sec-186 18 5tions for the reaction W(p,d) W leading to final

states corresponding to a transferred orbital angular mo­mentum L=1 (4-6a) and L=3 and L=5 (4-6b). See Table 4-4 for the numerical values of the cross sections. The lines through the points indicate uncertainties in the relative cross sections, estimated as described in the text. In(4-6c) the cross sections for the transitions to the states

185at 174 keV and 188 keV excitation in W are shown at those angles at which they could be separated.

TRAC

KS

PER

AQ

EXC ITAT ION ENERGY IN l85W(MeV)

Q-VALUE (MeV)

(JS/q

r/) ,UJ'°UP/-°P

F i g u r e 4 -6a

10

1 1 1 1---l8 6 W (p ,d ) E P = I8 .0 MeV

1000

100

10

0.1

M : !E:f:

3/2“, 3/2 [512] (0 keV)---

EEEt:

3/2“, 1/2 [510]- (94 keV]

l/2“ ,l/2 [510] "" (24 keV) ~

0C 40° 80° 120° 160°CENTER OF M A S S ANGLE

F i g u r e 4-6b

1000

100

10

l8®W(p,d) Ep = l8.0 MeV

• - • J -

< > 100

100

10

• ♦

5/2",3/2[5l2]' _(66 keV)__

7/2“,3/2 [512] (174 keV)- +5/2",l/2[5IO](l88 keV)

S 3

M i

7/2“, 1/2 [510]; “ (334 keV)”

•4 7/2“ 7/2 [503]© (244 keV)=]

9/2“ ,3/2 [512] (302 keV)=dP *

0° 40° 80° 120° 160°CENTER OF MASS ANGLE

F i g u r e 4 -6c

l ®®W(p,d) En = l8 .0 MeV

■ft

i

7/2" (174 keVr + 5 /2 "(188 keV)

Im

a s t-q

♦ Total 4> 5/2“* 7/2“

40° 80° 120° 160°CENTER OF MASS ANGLE

Table 4-3. 185Low-lying Levels of W.

tt E(keV)J K

Previous Work This Experiment

3/2~ 3/2 0 0 + 3

1/2" 1/2 24 24 + 35/2" 3/2 66 65 + 33/2" 1/2 94 94 + 37/2" 3/2 174 -'I

5/2" 1/2 188 ' 187 + 3

ll/2 + 11/2 197

7/2" 7/2 244 244 + 3

9/2" 3/2 302 301 + 3

7/2" 1/2 334 333 + 3

13/2 + 11/2 384 1

9/2" 7/2 391 fJ

383 + 3

11/2" 3/2 ^478

9/2" 1/2 ^492

Table 4

center-

-4 . W(p

3~ 3

185H'l U ronfor-nf-Maee Differential

3 ~ 1

) in pb/sr

9' 3l" l 5 3. d^CM

5 1 7" 7 1~ 1of-mass 2 ’ 2 2 ’ 2 2 * 2 2 ’ 2 2 » 2 2 * 2 2 ’ 2 2 ’ 2angle * (0 keV) (24 keV) (66 keV) (94 keV) (188 keV) (244 keV) (302 keV) (334 keV)

5.05 3.7 ±2.2 13.5 ±2.8 87.9± 5.3 202.9± 8.2 5.5±3.2 113.715.0 7.612.1 47.013.412.61 6.6 ±1.1 16.7 ±1.4 88.4± 5.7 521.3±26.6 4 . 8±1.4 109.214.1 6.711.1 40 .212.220.18 6.9 ±1.0 8.2 ±1.1 128.3± 8.0 498.3±26 . 0 13.0±1.6 105.714.0 10.111.3 44 .412.427 . 74 8.4 ±1.0 7.6 ±1.0 250.9±11.3 39 2.1±21.1 19.212.0 147.615.2 10.911.1 69.512.935.30 7.1 ±1.1 24.4 ±1.6 324.0±14.3 636.9±33.8 22.812.1 192.416.5 11.311.3 95 .913.742.85 7.2 ±0.9 25.6 ±1.6 316.7±13.9 694.9±36.1 24.912.1 209.017.1 12.111.5 110.814.250.40 7.2 ±0.9 8.3 ±0.9 310.8±13.6 47 0.0±25 . 4 20.111.7 196.516.7 9.511.4 109.714.257.94 6.6 ±1.3 1.2 ±0.6 282.1±12.4 4 0 3 .5±21.8 19 .6±2.0 198 .716.7 11.111.3 105.614.165.47 3.3 ±0.9 7.1 ±1.4 233.7±10.5 388.7±21.0 16.0±2 .1 179.816.3 9.911.4 105 .814.173.00 1.5 ±0.8 9.3 ±1.2 209.0± 9.4 326.9±12.1 --------- 147.115.2 12.111.3 96.213.880.52 2.6 ±0.9 5.4 ±0.9 201.0±12 .1 243.2±11.2 13.2±1. 6 131.414.7 10.711.2 77 .313.288.02 0.99±0.51 2.1 ±0.5 155.4± 7.2 183.8± 7.5 8.911.2 116.314.3 9.711.1 58.812.193.02 1.4 ±0.45 1.4 ±0.5 128.4± 8.0 154.9± 6.8 7.9±1.0 98.113.7 7.110.9 55.612.4

100.52 1.0 ±0.36 2.7 ±0.6 96.0± 6.0 137.2± 6.2 6 . 2±0 . 8 81.313.2 7.311.0 48.612.2108.00 0.60±0.47 2.0 ±0.6 94.5± 5.9 110.8± 5.3 5.5±1.0 ---------- 6.810.9 43.612.1115.47 0 .49±0 .40 1.9 ±0.6 85.2± 4.2 83.5± 3.8 4.9±1.0 59.412.5 7.210.9 35.911.8122.94 0.1610.19 0. 83±0.36 62.6± 5.2 59.8± 5.3 3 . 2±1.0 49.414.2 5.411.1 28.512.7130.40 0.83±0.58 1.01±0.61 52.5± 2.8 55.8± 2.7 3 . 3±0.7 46.112.1 5.810.9 27.511.4137.85 0.61±0.56 0 .83±0.58 43.7± 3.0 46.2± 2.5 4.5±0.9 36.211.8 4.210.8 23.111.4145.30 0 .16±0.35 1.7 ±0.5 41.8± 2.3 38.7± 2.1 3.0±0.7 33.311.7 5.210.7 21.211.3152.74 0.27 ±0.26 0.95±0.45 37.3± 2.1 30.5± 1.7 3.5±0.8 30.611.6 4.510.7 16.611.2160.18 0.16±0.16 0.50±0.30 28.2± 3.0 30.3± 3.0 2.4±0.8 24.512.3 ________167.61 0.22±0.34 0.83±0.51 30.3± 1.8 26.7± 1.6 2.310.8 26.011.5 5.010.8 13 .611.0

7" 3 11+ 11t Contains small amounts of 2 , 2 (174 keV) and 2 , 2 (197 keV). See text.* The angular region included at each point is about 0.26°.

119

observed in a pickup reaction leading to 185W (Ca 72). Thus,

its contribution to the unresolved group has been ignored. Anattempt was made to separate the other two peaks using pro­

gram SESAME (Cl 72a). The relative energies of the peaks

were fixed in this procedure, and the reference peak was

chosen to be that corresponding to the transition to the

7/2 state at 244 keV. However, an acceptable fit was ob­

tained at only a few forward angles. The results of this

fit are shown in Figure 4-6c. It is evident from this

figure that the 5/2 transition is dominant and it seems

reasonable to assume that at least at forward angles, the

major portion (^90%) of the 5/2 + 7/2 group is contributed

by the 5/2 transition.

4.6.3. Discussion.186 186 The angular distributions for the W(p,t) and W

(p,d) reactions have been grouped according to transferred

orbital angular momenta in Figures 4-4 and 4-6. The contrast

between these and the angular distributions for the same208reactions on the nearby spherical nucleus Pb (Figures 2-3

208and 2-4) is striking. For the Pb reactions, in which the

pure direct reaction model is expected to be reasonably valid,

transitions with nearly the same Q-value and corresponding

to the same transferred orbital angular momenta have nearly

the same angular distributions. On the other hand, for the 186 W reactions, there are a number of cases where strong

differences between transitions of the same transferred or­

bital angular momenta occur. Particularly notable are the

120

differences between the L=4 (p,t) angular distributions in

Figure 4-4c and among the L=1 (p,d) angular distributions in

Figure 4-6a. These are possible candidates for transitions

in which inelastic multistep processes are important. Of

course, there may be other reasons for the observed discrep­

ancies, such as differences in the structure of the states

which are large enough to produce radically different form

factors. Therefore, confirmation of the existence of multi-

step processes requires a thorough theoretical analysis.

121

THEORETICAL ANALYSIS : l 8 6 W ( p , t ) l82,W

CHAPTER 5

How can these things be?"

- John 3 :9b (KJV).

5.1. Introduction.

The first reaction to be considered in the theoretical 186analysis is W(p,t). The (p,t) reactions are somewhat

more complex than the (p,d) reactions; however, this com­

plexity can be an advantage. As is indicated in Section 3.6.2,

(p,t) reactions are sensitive to two-neutron correlations in

the overlaps between the initial and final nuclear states.

Thus, transitions between states which comprise a large num­

ber of coherent two-neutron configurations are especially

favored. This is the case for transitions between BCS ground

states and to a lesser extent for transitions to excited

collective two-quasiparticle states. Since the structure of

such states is complex, calculations of transitions between

them tend to be less sensitive to small uncertainties in their

microscopic description, and thus structure effects are more

easily separated from reaction-mechanism effects. These

characteristics also apply to two-nucleon configurations in

spherical nuclei; however, the deformed nuclei have a definite

advantage, since so many transitions are determined by a single

intrinsic form factor. Thus, although CCBA calculations have

been performed for (p,t) reactions on spherical nuclei (As

70a), the reactions on deformed nuclei pose a much more strin­

gent test for the reaction model.

Another feature of (p,t) transitions between coherent

two-nucleon configurations is their systematic behavior.

Such coherent states tend to vary slowly in properties from

nuclide to nuclide. Thus, changes in the properties of

122

(p,t) transitions as a function, for example, of nuclear

deformation can be observed more readily. These features

are noted in the present chapter by comparing the results186obtained from the specific reaction on W with analyses of

(p,t) reactions on other rare-earth nuclides.

Transitions between ground bands possess another impor­

tant feature. Since the intrinsic states of such bands are

BCS states, their configurations must contain maximal pairing

of neutrons to zero angular momentum. Therefore, zero trans­

ferred angular momentum (L=0) will be strongly favored over

all others. This dominance of a single angular momentum

transfer makes the effects of inelastic processes in the

reaction mechanism easier to extract.

The theoretical calculations presented in the present

chapter were performed by R.J. Ascuitto and B. Sorensen and

have already been published (Ki 72, As 72b).

5.2. Determination of Parameters.

As is indicated in Chapter 3, the structure of the nuclear186states involved in calculations of the W(p,t) reaction is

described by means of Bohr-Mottelson adiabatic wavefunctions.

The intrinsic states were calculated in a deformed single­

particle basis using a pairing-plus-quadrupole residual force.

The single-particle potential-well and deformation parameters

are listed in Table 5-1. (They are defined in Section 3.3.)

The single-particle well parametrization is based on that

used in work in the lead region (B1 60), and the deformation

parameters were extrapolated (Ha 67, Mo 70) from those deter-

123

mined from inelastic alpha-particle scattering on W (He

71). Both quadrupole (f^) an<* hexadecapole (3^) deformations

were assumed for the nuclear-mat ter distribution, and the

charge distribution was taken to have a pure quadrupole de-

formation (32 )• The difference in the deformations between184the ground and 3-bands in W is probably small, since the

moments of inertia implied by the level spacings are nearly

equal. Thus, for convenience, the same deformation parameters

were assumed for both bands.

In the pairing calculation the interaction ;was assumed

constant and was allowed to extend over 21 proton and 23

neutron orbitals. This restriction of the interaction to

approximately 20 levels was earlier found to be optimum (As

72) given the limitations of the constant-matrix-element

approximation. The proton (A ) and neutron (A ) pairing gapsP awere based on the odd-even nuclear mass differences according

to the Nilsson-Prior prescription (Ni 61), and were fixed

by choosing the strength of the pairing force to be of the

form

G = G [1-0.75 T ( ^ - )]/A o 3 A

The values used for A , A , and G are listed in Table 5-1.p n oThe quadrupole force strength X* defined in Section

3.4, was assumed to be charge-independent. Its value, given

in Table 5-1, was chosen so that, on the average, the exci­

tation energies of the low-lying 3-vibrational bands in the

tungsten region were reproduced (As 72b). As a check on the

124182

Table 5-la. Single-Particle Well Parameters.*

V V v r r ao l so o c51.0 132.4 32 1.25 1.25 0.67

Table 5-lb. Deformation Parameters.t

e 0 3 c2 > > 2186W 0.2220 -0.0943 0.270

184W 0.2264 -0.0943 0.275

Table 5-lc Residual Interaction Parameters

186 W184 W

A (MeV) P0.60

0.65

A (MeV) n0.65

0.72

G (MeV) o34.5

34.5

X(MeV~1)

0.88X10-3

0.88X10"3

Table 5-ld. Optical Parameters.*

W WT v r a so soD l . J. 4 . ® »

0 0 c

p 55.6 0.0 14.5 1.25 1.25 1.1267 0.72 0.47 6.2 1.01 0

t 168.8 12.6 0.0 1.16 1.498 1.1267 0.752 0.817 0.0 ---- -

1 I 3* r=R/A , where A is the mass number of the nucleus. Well depth (V) in M e V , radius (r) and diffuseness (a) in fm.

125

so.75

t 8 and 8, are defined with respect to the radius R =6.4285 fm 2 ** p8 c is defined with respect to the radius R c=6.4316 fm.

2 p

values of the quadrupole and pairing strengths, the EO and

E2 transition rates between the B-bands and the ground bands

as well as Intraband transition rates were calculated (As

72b), and were found to agree well with experiment (Gu 71).

The values of the optical parameters are listed in

Table 5-1. (They are defined in Section 3.5.1.) The proton

parameters are based on those used in earlier CCBA calcula­

tions (As 72) with the reaction (p,t), and the triton

parameters are based on parameters determined from the

triton scattering work of Flynn e_t al. (FI 69) on spherical

nuclei. Calculations of (p,t) reactions have been found to

be rather insensitive to the triton parameters (As 72d). The

method used for deforming the proton and triton optical po­

tentials is that described in Section 3.5.1, which treats the

deformations of all parts of the potential consistently.

The intrinsic nuclear radius chosen (11^=6.4285 fm) as well1/3as the Coulomb radius (R =r A ) were based on the formulaec c

of Myers (My 70). The deformation parameters used for deter­

mining the inelastic matrix elements are the same as those

used in the bound-state calculation and are given in Table 5-1.

Again, the same deformations were assumed for the ground and

8-bands of

5.3. The Ground-State Rotational Band.184The calculations of the transitions to the W ground-

state rotational band were limited to a space consisting of

the 0+ , 2+ , and 4+ members of the ground bands of both the

target nucleus and the final nucleus ^ ^ W . Within this

126

Figure 5-1. Intrinsic form factors calculated by Ascuitto186and Sorensen (As 72b) for W(p,t) transitions to the

184ground-state rotational band of W.

Figure 5-2. Comparison of CCBA and DWBA calculations of18 6Ascuitto and Sorensen (As 72b) with the data for W(p,t)

184to the ground-state rotational band of W. The relative normalization of all curves has been maintained.

l86W(p,t)GROUND BAND

da

/dft

c.m

. {fi

b/s

r)

Figure 5-2

1000

100 _ i

100

100

0° 40° 80° 120° 160°CENTER OF MASS ANGLE

space all allowed multipoles of Inelastic excitations, ex­

panded to eighth order in the deformation constants, were

included. In addition, as explained in Chapter 2, all allowed

transfer routes between these states were included to first

order by treating them as source terms in the final coupled

inelastic equations.

The form factors used for the source terms fere deter­

mined from the BCS intrinsic state in the manner described

in Section 3.6.2, retaining quanta up to N=14 in the oscillator

expansion (Eq. 3-9b) of the single-particle orbitals and using

the parameters of Table 5-1. These form factors are displayed

in Figure 5-1. The region immediately outside of the nuclear

surface 0 W . 1 3 fm = r A ) is that in which the coherence ofothe wavefunctions will be most apparent, since this is the

region beyond the last node of the component single-particle

wavefunctions. As expected, the L=0 form factor is considerably

larger than the others near the nuclear surface because of the

favored coupling of neutrons to zero angular momentum in the

ground-band intrinsic wavefunctions. Since the surface region

also has the most influence on the transfer strength, this

means that the L=0 transfers will tend to be important in all

transitions.

The results of the coupled-channel calculation are pre­

sented in Figure 5-2. A DWBA calculation using the same set

of optical parameters is also shown. As can be seen, the

CCBA gives a reasonably good fit to all three transitions in

strong contrast to the DWBA. This is especially evident in

the case of the transition to the 4+ state, for which the

127

data differ from the DWBA calculation by as much as an order

of magnitude at some angles. It should be emphasized that

the relative magnitudes of the calculations for the three

transitions have not been adjusted in any way. The only

normalization used was an overall constant multiplying each

CC and DWBA calculation. The value of this constant was

chosen so that the coupled-channel calculation for the ground-

to-ground transition reproduces the experimental strength.

This overall normalization factor is usually left as an

adjustable parameter, since the absolute theoretical normalization

of two-nucleon transfer reactions is still an unresolved ques­

tion (Ba 71, Li 73). Nevertheless, the reproduction of the

proper relative strengths in the CCBA is strong evidence of

the applicability of the descriptions used both for the

reaction mechanism and the intrinsic structure of the bands.

The small remaining discrepancies between the shapes of the

experimental and theoretical angular distributions are

perhaps mainly the result of inadequacies in the optical para-

metrization.186Further insight into the meaning of the W(p,t) results

can be obtained by comparing them with calculations of (p,t)

reactions leading to the ground-state rotational bands of

other rare-earth nuclei. Figure 5-3 shows data (Oo 70, Oo

73) and calculations (As 72) for the ^^Yb(p,t) reaction at

19 MeV proton incident energy, and Figure 5-4, data (De 72)154and calculations (As 72) for the Sm(p,t) reaction at the

same energy. It is evident that the difference between the

DWBA and CCBA shapes for the transition to the 0+ state is

128

slight in all three cases, indicating that for this transi­

tion the importance of the inelastic processes is small.

This results from the dominance of the direct contribution,

since it involves the strong L=0 form factor.

In contrast to the ground state, the 2+ transition in

all three cases shows a marked difference between the CC and

DWBA calculations. Furthermore, the CCBA in these cases

constitutes a rather good fit to the data, thus indicating

the importance of the contributions from routes proceeding

via inelastic excitations ("indirect routes"). An illustra­

tion of the manner in which the final cross section is pro­

duced is given in Figure 5-5, which shows the partial cross

sections contributed by each of the most important routes for

the case of the ^78Yb(p,t) reaction. Because of the dominance

of the L=0 transfer form factor, the most important indirect

routes involve L=0 in the two-nucleon transfer step:

176Yb(0+ ) -*• 174Yb(0+ ) t 174Yb(2+ ) and

176Yb(0+ ) *■ 178Yb(2+ ) •+ 174Yb(2+ ).

The first of these proceeds via L=0 transfer in the initial

step, and the second Includes L=0 transfer (in addition to

L=0 and L=4) in the final step. It is evident from Figure

5-5 that the contributions from both of these routes is of

comparable magnitude to that from the direct route (solid

line), and all are larger than the data, so that the final

cross section results from destructive interference among the

competing routes. It should be noted that what has been

129

called the "direct route" involves no particle transfers

from excited states but does include inelastic excitations

and deexcitations in the initial and final nuclei. Thus,

the direct route is not Identical to a first-order direct-

reaction transition, and hence the contribution from this

route is somewhat different from the DWBA calculation.

For the transitions to the 4+ states the CCBA again

results in an improvement over the DWBA both in magnitude

and shape, although the most dramatic improvement occurs for186 +W(p,t). Indeed, the 4 angular distributions in the three

cases differ among themselves much more than do the 2+ . It

is tempting to attribute this to the fact that the quadrupole

deformations are rather similar in the three cases (8 =0.2902

for 154Sm, 0.295 for 176Yb, and 0.222 for 186W ) , but the dif­

ferences in the hexadecapole deformations are large (3 =0.05811for ^ 4Sm, -0.052 for ^78Yb, and -0.085 for ^88W ) . However,

it must be emphasized that the L = 4 inelastic form factor

depends on both 8 and the square of 8 (see G1 67). Fur-k 2

thermore, the L=4 two-neutron transfer form factor depends

not only on the nuclear deformations but also on the particular

orbitals which carry the L=4 strength, a feature which can

differ strongly, for example, from ^ 4Sm to ^88W. Also impor­

tant is the fact that for the transition to the 4* state, many

other routes will contribute besides the L=4 direct transfer

and those involving L=4 inelastic excitations (for example,4" *4* + “k’ 4"

0^ + 2 2^ + 4^), although the important indirect routes

will be limited to those involving L=0 two-neutron transfer.

130

Figure 5-3. Comparison of CCBA and DWBA calculation ofAscuitto £jt al_. (As 72) with the data of Oothoudt a l .(Oo 70, Oo 73) for X^Yb(p,t) to the ground-state rota-

174tional band of Y b . The relative normalization of all curves has been maintained.

Figure 5-4. Comparison of CCBA and DWBA calculations ofAscuitto e_t al_. (As 72) with the data of Debenham e_t a l .

154(De 72) for Sm(p,t) to the ground-state rotational band 154of Sm. The relative normalization of all curves has

been maintained.

Figure 5-5. Cross sections for ^2^Yb(p,t) corresponding to the direct (solid line) and selected indirect routes for the 2+ transition. From Ascuitto_et a l . (As 72).

Figure 5-6. Cross sections for X^Yb(p,t) correspondingto the direct (solid line) and selected indirect routes

+

d<r/d

X2c

>m (^

tb/s

r)

Figure 5-3

0° 30® 60® 90® 120® 150® 180®C E N T E R OF MASS A N G L E

dc

r/d

ilcm

(fj

Lb/s

r)

F i g u r e 5-4

0® 30® 60® 90® 120® 150® 180®C E N T E R OF MASS A N G L E

dcr/

dXic>

m (/

ib/s

r)

F i gu r e 5-5

0 * 3 0 * 6 0 ° 9 0 * 1 2 0 * 1 5 0 * 180"CENTER OF MASS ANGLE

(js/qr/ )'u,'° u P/-0

p

F i gu re 5-6

— ------- -------

------- ,7 6 Yb(p ,t) E n = 19 MeiV 4 + -------I i l 1

I

1*

3 i ^

---- \ —

Hi . X

\ M*

^ -----------------% ---- — M

w x

4

1—

i5+ J i /

yi

( 5--------- V J

0 ° 3 0 ° 6 0 ° 9 0 ° 1 2 0 ° 1 5 0 ° 1 8 0 °

CENTER OF MASS ANGLE

Examples of contributions to the 4+ transition for 176Yb(p,t)

are shown in Figure 5-6. The large differences observable

among the ytterbium, samarium, and tungsten 4^ angular

distributions probably reflect the strong differences in the

direct L=4 strengths in the three cases, and these, as has

been mentioned, depend as much on the available single-particle

orbitals as on the deformation parameters. The direct-route186 + contribution in the W(p,t) case is so small that the 4

cross section is determined mainly by indirect routes.

5.4. The g-Vibrational Band.

The calculations of the transitions to the XE4W B-vibra-

tional band were limited to a space consisting of the 0 , 2 ,186and 4 members of the W ground-band and of the same mem-

184bers of the W B-band. This neglects interband inelastic

scattering, which is not expected to be an important effect

since the measured Interband scattering is much smaller than

the intraband scattering. For example, for 12 MeV deuteron184inelastic scattering on W, the cross section for the tran­

sition to the 2+ member of the 3~band is about 0.25% of that

to the 2+ member of the ground-band (Gu 71). As with the

ground-band calculation, the inelastic matrix elements were

expanded to eighth order in the deformation constants.

The form factors used in these calculations are exhibi­

ted in Figure 5-7. These were determined from the first-

excited state of an RPA calculation according to the methods

discribed in Section 3-4, using 14 oscillator quanta in the

single-particle orbital expansion and the parameters of Table

131

5-1. A comparison of Figures 5-7 and 5-1 shows that the

form factors for the ground and 3~bands are quite different.

In particular, the L=0 component no longer dominates in the

8-band as it did in the ground-band. One would therefore

expect differences in the calculated cross sections for the

two bands. The CCBA calculations for the g-band are displayed

in Figure 5-8, and indeed a comparison with Figure 5-2 indi­

cates differences in the transitions, especially those to

the 2 and 4 + states. The DWBA calculations shown were

performed using the same optical parameters and form factors

as used in the CCBA calculations. It is evident from Figure

5-8 that the shapes of the angular distributions are much

better reproduced by the CC than by the DWBA. Thus, as with

the ground-band, the transitions to the g-band show notice­

able effects from the inelastic routes. The 0+ transition

in the 8-band is much more affected by the presence of the

inelastic processes than in the ground-band, which probably

reflects the fact that the L=0 two-neutron-transfer form

factor is no longer dominant.

The CCBA curves in Figure 5-8 have a normalization which

differs from that used for the ground-band curves of Figure

5-2 by a factor of 0.725. This value is very sensitive to

the quadrupole-force strength X, and the reduction in X nec­

essary to make the factor equal to that of the ground-band

increases the calculated g-band excitation energy by less than

100 keV (As 72b). With the value of X listed in Table 5-1

and used in the calculations of Figure 5-8, the calculated

132

Figure 5-7. Intrinsic form factors calculated by Ascuitto186and Sorensen (As 72b) for W(p,t) transitions to the 0-

vibrational band of ^®4W.

Figure 5-8. Comparison of CCBA and DWBA calculations of186Ascuitto and Sorensen (As 72b) for W(p,t) to the 0-

184vibrational band of W. The relative normalization of all curves has been maintained. The absolute normaliza­tion of the curves differs from that used for the ground- state band (Fig. 5-2) by a factor of 0.725.

186

F i g u r e 5-7

R(fm

)

d<r/<

mc.m

. (/ib

/sr)

F i g u r e 5-8

100

0.01

4 0 ° 8 0 ° 120° 160*CENTER OF MASS ANGLE

band-head energy is 1.050 MeV compared to the experimental

value of 1.004 MeV. Deviations of the calculated, energies

of the B-vibrations from the experimentally observed energies

of the order of two hundred keV or less are not uncommon given

the limitations of the model (As 72b).

The quadrupole-force parameter X does not affect the

relative strengths for the various band members, and there

remains a discrepancy in the relative strengths of the 2+ and

0 calculations when compared with experiment, although the

shapes are well reproduced. The poorer quality of the $-band

fits is not too surprising, however, since the 3-band intrinsic

structure has much less two-neutron coherence than that of the

ground-band. Therefore, the (p,t) reaction calculation will

be much more sensitive to the quality of the approximations

used in the structure calculations. As a result, this reac­

tion may be useful as a test of structure calculations for

rotational bands which have less coherent two-neutron intrinsic

structure. A survey of several excited K=0 bands in the rare-

earth region using the same structure and reaction models as

in the present example has been performed by Ascuitto and

Sorensen (As 72b). In these other examples, the relative

strengths of the transitions are also more poorly reproduced

in general by the CCBA than those of the corresponding ground-

bands, although the CCBA angular-distribution shapes usually

reproduce the extant experimental data quite well.

5.5. The y-Vibrational Band.186In this chapter CCBA calculations for W(p,t) transitions

133

to the ground and 0-vibrational bands of W have been pre­

sented. No calculations for the transitions to the y-vibra-

tional band have been performed, since the appropriate cal­

culations of the intrinsic band structure have not yet been

made (but are planned for the near future (As 7Ld)). However,

based on the results presented in Sections 5.3 and 5.4, one

would expect that the angular distributions of the y-band

states will look different in general from the corresponding

states in the other two bands. This results not only from

the different intrinsic structure of the y-band but also from

the fact that the states involved (2+ , 3+ , 4+ ,...as opposed■f -fto 0 , 2 , 4 ,...) are different and hence the inelastic

scattering within the y-band is not the same as that within

the other bands. These expected differences are in fact

apparent in the comparison of the experimental angular distri­

butions of the three bands (Fig. 4-4).

Thus, the existence of two-neutron transfer routes from

excited states allowed in the CCBA results in a new feature:

angular distribution shapes which are not necessarily char­

acteristic of the total transferred angular momentum only

but frequently depend on the intrinsic nuclear structure of

the states as well. As has been mentioned, this is not the208case for (p,t) on a spherical nucleus such as Pb (Fig.

2-2), for which the angular distributions of transitions to

states of the same angular momentum are nearly identical even

if the intrinsic structure of the states is very different

(for example, the first and second 2+ states of ZG^Pb). As

134184

a result, spin assignments for states in deformed nuclei

based on two-nucleon transfer angular distributions (De

72, Oo 73) are highly questionable.

CHAPTER 6

THEORETICAL ANALYSIS : l 8 6 W ( p , d ) l 8 5 W

"Auch ich darf mich so glUcklich nennen Zu schaun, was, Wolfram, du geschaut!" (I too may call myself so happy As to have seen what you, Wolfram, have seen! )

- Richard Wagner, TannhHuser, Act II, Sc. 4.

6.1. Introduction.

When analyses of (p,t) reactions such as those presented

in the last chapter revealed strong effects from multistep

processes, it clearly became necessary to reexamine the applica­

bility of DWBA descriptions of single-nucleon transfer reactions

on deformed nuclei. As is pointed out in Section 3.6.2, the

form factor for a given J-transfer in single-nucleon transfer

reactions depends upon amplitudes tbe expansion of the

intrinsic state on a spherical basis. In the DWBA, the strength

of the transition to a band member J depends only on a single

form factor and hence on a single set of spherical expansion

coefficients Many reaction analyses have simplified

this even further by assuming no n-mixing in the expansion.

Since the spherical expansion is characteristic of a given

intrinsic state, an assumption that the reaction is purely

direct implies that the strengths of the transitions to the

various members of a band can be used as a signature in ident­

ifying the intrinsic state of the band.

This supposed signature has been called a "fingerprint

pattern" by Elbek and collaborators, and has been used by them

and others in investigating the structure of rare-earth nuclei

(El 69). The first-order direct-reaction assumption was just­

ified on the basis of a limited number of relatively complete

experimental angular distributions (Ve 63, Ma 64, Si 66, Ja 67,

Ja 69) which did not indicate any large discrepancies from DWBA

predictions. The use of the DWBA became so accepted that for

experiments which were difficult to perform because of high

136

energy resolution required or the small size of the cross

sections, it became standard practice to measure the cross

sections at only two or three angles (Ke 65, Bu 66, Tj 67,

Br 70, El 70, Ca 72). Although numerous examples were

found which could not be identified by any known intrinsic-

state signature, these were usually explained as resulting

from band mixing (Er 65, Ka 69) or other more complicated

effects.186The analysis of the W(p,d) reaction presented in the

remainder of this chapter together with related work (As 72c,17 2Me 73, Me 73a) on the reactions Yb(d,p) decisively indicates

the importance of inelastic processes in single-nucleon trans­

fer reactions. Some CCBA calculations of such reactions have

already been performed; however, these were either concerned

with light nuclei where other non-dlrect effects can be im­

portant (Sc 70, Ma 71, Br 71a), were applied to strong transi­

tions where the inelastic effects will tend to be minor (Gl

71, Sc 72), or were applied to cases for which the experimental

data was either poor or non-existent (Gl 71, Sc 72). The

present work thus constitutes perhaps the first clear demon­

stration of the existence of inelastic effects in single-nucleon

transfer reactions. This demonstration implies the need for

a complete reevaluation of nuclear structure information and

spin assignments which have been based on single-nucleon-

transfer differential cross sections in regions of the periodic

table away from closed single-nucleon shells.

137

6.2. Determination of Parameters.

The calculations presented in this chapter were per­

formed for transitions to the two lowest-lying bands in185 — _W : 1/2 [510] and 3/2 [512]. Both of these bands were

assumed to be single-quasiparticle bands and were described

using Bohr-Mottelson adiabatic wavefunctions. The intrinsic

structure of the bands were determined by using a deformed

single-particle well whose well parameters are given in Table

6-1. These parameters are the same as those used in the 18 6W(p,t) calculation described in Chapter 5, except for a

reduced diffuseness parameter a. This value was found (So

72) to yield a single-particle level sequence in better accord

with the analysis of Ogle al_. (Og 71) . The mass-distribu-

tion deformation parameters are also the same as used in the

(p,t) analysis; the charge-distribution quadrupole deformation

( $ 2 ) was determined from measured B(E2) values for transitions186in the ground-band of W (St 65, St 68a). These parameters

are listed in Table 6-1. The pairing factors U and V for

the single-quasiparticle intrinsic states were determined by

using 25 single-neutron orbitals and a gap parameter A = 0.59.

In the course of the present analysis it was discovered

that the results are rather sensitive to the deuteron optical

parameters (see Section 6.6). It was at first hoped that an

average set determined from deuteron scattering on spherical

nuclei (Pe 63) could be used. These parameters are listed

as D2 in Table 6-2. As was mentioned in Chapter 5, such aver­

age parameters, which should simulate the proper coupled-

channel set since inelastic scattering on spherical nuclei

is usually relatively weak, were successfully used for the

triton parameters in the (p,t) analysis. However, when the

deformations are added to these parameters according to the

prescription of Section 3.6.1 using the deformation para­

meters of Table 6-1, the scattering calculated with a coupled-186channel (CC) code gives a rather poor fit to the W deuteron

elastic scattering data as seen in Figure 6-1. The proper

coupled-channel optical parameters should, of course, fit

not only the elastic scattering but also the inelastic scat­

tering in such a CC calculation.

It was finally decided to use a set of parameters deter­

mined from a specific fit (Ch 69) of the deuteron elastic 186scattering on W at a deuteron incident energy of 12 M e V .

Since these were determined from a normal one-channel optical-

model analysis, the radius and diffuseness of these para­

meters were adjusted slightly in order to obtain a fit to the

elastic scattering with a coupled-channel calculation. These

adjusted parameters are listed as D1 in Table 6-2 and the

fit is given in Figure 6-1. As a test of these parameters

a CC calculation was made of the deuteron elastic and inelastic 182scattering on W where the data for the scattering to the

2+ state is available (Si 66). The D1 parameters were used

without adjustment and,as can be seen in Figure 6-2, the fit

is quite good. A similar situation was found in the analysis172of deuteron scattering (Me 73, As 73) on Yb which is shown

in Figure 6-3. The parameters labelled "Perey parameters"

are taken from reference Pe 63 and are essentially equivalent

139

Table 6-1. Single-Particle Well* and Deformation^ Parameters.

140

V V v r r0 1 SO 0 c

51 132.4 32 1.25 1.25

3 3 3 c2 it 2

0.222 -0.0943 0.236

1 /** r=R/A , where A is the mass number of the nucleus depth (V), radius (r) and diffuseness (a) in fm.

t 3 and 3 are defined with respect to the radius R2 c * c Pfm: 3 is defined with respect to the radius R =6 P P

.60

Well-

6 .4285 4316 fm.

141

Table 6-2. Optical Parameters.*

V W r r r a a VD 0 0 c so

PI 58 .33 1.26 9.75 1.17 1.32 1.1267 0.75 0.653 0.0P2 54.10 0.0 18.0 1.25 1.25 1.1267 0.65 0.47 0.0P 3 56.78 1.26 9.75 1.17 1.338 1.1267 0.741 0.797 0.0

D1 85 .92 0.0 20.96 1.15 1.31 1.1267 0.892 0.725 0.0D2 104 .0 0.0 13 .5 1.15 1.34 1.1267 0.81 0.68 0.0D3 104.0 0.0 13 .5 1.15 1.30 1.1267 0.81 0.81 0.0D4 113 . 7 0.0 22 .6 1.15 1.36 1.1267 0.901 0.709 0.0

1 /* r=R/A 3, where A is the mass number of the nucleus. Well-

depth (V), radius (r) and diffuseness (a) in fm.

142

Figure 6-1. Coupled-channel calculations for elastic and186inelastic scattering of deuterons on W using various

deuteron optical parameters listed in Table 6-2. The e- lastic scattering cross section has been plotted as its value normalized to the Rutherford scattering cross section. The points correspond to the data of Christensen e_t al.(Ch 69).

Figure 6-2. Coupled-channel calculation for elastic (nor­malized to Rutherford scattering) and inelastic scattering

182of deuterons on W using the D1 deuteron optical parame­ters of Table 6-2. The points correspond to the data of Siemssen and Erskine (Si 66).

Figure 6-3. Coupled-channel calculation for elastic (nor­malized to Rutherford scattering) and inelastic scattering

172of deuterons on Y b . The deuteron optical parameters aresimilar to the D1 ("Christensen") and D2 ("Perey") parame­ters listed in Table 6-2. The elastic scattering data is that of Christensen e_t a_l. (Ch 69); the data for scatter­ing to the 2+ state are that of Burke et al . (Bu 67); and the data for scattering to the 4^ state was taken at Yale (Me 73, As 73).

do-/

d&

(mb/

sr)

Ratio

to

Rut

herf

ord

Figure 6-1

CEN TER OF M ASS ANGLE

doVd

ft (m

b/sr

) Ra

tio

to R

uthe

rfor

d

Figure 6-2

CENTER OF M ASS ANGLE

dcr/

dXi

(mb/

sr)

Ratio

to

Rut

herf

ord

Figure 6-3

to the D2 parameters. The "Christensen parameters" were

adjusted from the analysis of reference Ch 69 in a manner

similar to the D1 parameters. Figure 6-3 clearly shows that

the Christensen parameters give a superior fit not only for

the elastic scattering but also for the inelastic scattering

to the 2+ and 4+ states. The principal difference between

the Christensen parameters (Dl) and the Perey parameters (D2),

in both the tungsten and ytterbium cases, is that the former

have a shallower real and a deeper imaginary well depth.

The reactions were found to be rather insensitive to the

proton parameters, and therefore the average set of Becchetti

and Greenlees (Be 69a), determined from scattering on spher­

ical nuclei, was used. These are labelled PI in Table 6-2.

The spin-orbit part of this potential has very little effect

on the reaction calculation and was neglected for simplicity.

6.3. The Form Factors.18 6The form factors for W(p,d) were calculated by B.

Sorensen (So 72) in the manner described in Section 3.6.2,

using the parameters of Table 6-1. The bound-state calcula­

tion was performed with quanta up to N = 15 in the oscillatorvKexpansion. The coefficients G T T in the expression (3-32)n i l J

for the intrinsic form factor are listed in Table 6-3. Note

that the number of these coefficients decrease as the trans­

ferred orbital angular momentum L increases, since N = 2n + L.

Because the energy eigenvalues of the intrinsic states

calculated in this fashion do not correspond exactly to the

experimental values, the form factors outside of the nuclear

surface do not have the proper radial dependence. This causes

143

the calculated (p,d) angular distributions to be sloped

incorrectly. Thus, spherical Hankel functions corresponding

to the correct binding energy were matched to the calculated

eigenfunctions outside the nuclear radius. Since the spher­

ical Hankel functions are the exact asymptotic negative

energy solutions to the radial Schrildinger equation, this

procedure assures the proper behavior of the eigenfunctions

at large r. An example of the Hankel-function matching in

the case of the L = p form factor for transitions toJ 3 / 2

the 1/2 [510] band is shown in Figure 6-4. Corresponding DWBA

reaction calculations are presented in Figure 6-5. Similar re­

sults are found in CCBA calculations. The effect is clearly

noticeable and the use of the Hankel functions gives a much

better reproduction of the experimental data. The principal

intrinsic form factors for transitions to the two bands are

plotted in Figure 6 - 6 with the Hankel function "tails" added.

A few points should be mentioned concerning these form

factors. It will be noticed particularly in the case of L=1

that the radial dependence for form factors corresponding to

the same orbital-angular-momentum transfer need not be exactly

the same. Indeed, the p form factor for the 1/2 [510] band1 / 2

has one fewer node than the other L=1 form factors. The

reaction calculations for transitions of the same transferred

L can therefore be different even in the DWBA. This is a

result which cannot be obtained by using the less sophistica­

ted form factors customarily employed in an analysis of nu­

clear reactions on deformed nuclei. The usual procedure is

144

to determine the radial dependence of the form factor from a

numerical diagonalization of a spherical Woods-Saxon well

and then to fix the strength of the form factors by multi­

plying these spherical eigenvalues by the coefficients VKG determined from a bound-state calculation with fixed n L J --------

N. An example of the radial dependence of such form factors

is shown in Figure 6-7. As is clear, for these form factors,

the character of the intrinsic state determines only the

strength and not the radial dependence. The radial depen­

dence is determined entirely by the L-value, except for a

negligible J-dependence entering through a spin-orbit part

of the Woods-Saxon potential. Therefore, the shapes of

angular distributions calculated in the DWBA using the con­

ventional form factors will be virtually the same for all

transitions of the same transferred orbital angular momenta.

Examples will be shown later in this chapter.

The L=5 form factors illustrated in Figure 6 -6 d indi­

cate an inadequacy in the form factors used in the present

analysis. It is seen that these form factors possess nodes

outside of the nuclear surface. This is clearly unphysical

and results from the fact that for higher L-values the num-VKber of expansion coefficients GnLj become too few. Fortun­

ately, the form factors are extremely weak in the region

where these spurious nodes occur, and thus this effect has

negligible influence on the reaction calculation. This in­

adequacy, however, should be eliminated in future reaction

calculations, either by increasing the maximum value of N in

145

1" VK —Table 6-3a. GnLJ For transitions to 1/2 [510).

146

L, J1 1 , 1 / 2 1, 3/2 3, 5/2 3, 7/2 5, 9/2

0 -.00694295 .05674383 -.01984304 .03714601 -.383836951 .05850685 .09258735 -.75591413 .49688678 -.141316012 -.01195354 .76170082 .00562898 .02635344 -.011445613 .04860488 -.07647046 .00808530 -.00537343 .026394454 -.00782241 -.00531350 .06124433 -.04418080 .019547715 -.00498668 -.07764225 -.00221601 -.00490644 .001297256 -.00979648 .01354932 -.00169415 .00021019 -----------7 .00104637 -.00075700 ----------- ------------------- -------------------

L, JL 5, 11/2 7, 13/2 7, 15/2 9, 17/2 9, 19/2

0 .16073709 -.x2992102 .11729969 -.00039061 .010921331 .09661684 -.02349099 .01598121 .00392831 .000828422 .00373796 .00464849 -.00698367 .00466654 -.003001803 -.01606082 .01490891 -.01219643 -.00079411 .000105974 -.01410101 -.00021487 -.00040515 ------------------- -----------56 7

L, J

.00033014

1 1 , 2 1 / 2 11, 23/2 13, 25/2 13, 27/2

0 .00376904 -.00253843 .00045764 -.000365691 .00260705 -.00195813 -.00089830 -.000270462 .00116426 .00019827 ----------- -----------34

— — — “ — — — — —

56 7

In the text, L=l, L=3, and L=5 are sometimes referred to as P, f, and h, respectively.

147+ V)K —Table 6-3b. G _ _ for Transitions to 3/2 [512]nL J

L, J— f1, 3/2 3, 5/2 3, 7/2 5, 9/2 5, 11/2

0 -.02562315 -.01776186 -.02393318 -.44513267 -.151627191 -.09128293 -.95723954 -.37661577 -.15174160 -.074652452 -.40652773 .01394888 -.01374156 -.01692427 -.006229443 .00337676 .00423121 .00061117 .02716897 .011656324 .00744200 .07688642 .03263942 .02202911 .011682605 .04655175 -.00322460 .00324133 .00048580 -.000946696 .00040969 -.00048490 .00090351 ------------------- -------------------

7 -.00037436 ----------- ----------- ----------- -----------

L, J7, 13/2 7, 15/2 9, 17/2 9, 19/2 1 1 , 2 1 / 2

0 -.137573301 -.028204182 .002651723 .015483054 -.000024455 ------------------------------------------------

6 -----------7 ------------------------------------------------

-.11580135-.01780114.00506928.01165972.00045347

-.00125463.00312621.00441462

-.00052196

-.01182244-.00179827.00255543.00018673

.00304642

.00233982

.00075494

nL, J

11, 23/2 13, 25/2 13, 27/2

0 .00222705 .00024603 .000350401 .00188737 -.00059830 .000493202 -.00054122 ----------- -----------3 ----------- ----------- -----------4 ----------- ----------- -----------5 ----------- ----------- -----------6 -------------------7--- ----------- ----------- -----------

In the text, L=l, L=3, and L=5 are sometimes referred to as p, f, and h, respectively.

Figure 6-4. Intrinsic single-neutron-transfer form fact­or corresponding to L =P, .„ transitions to the 1/2 [510]

185 'rotational band in W. The form factor is plotted with and without spherical Hankel functions matched, as described in the text. The absolute values of the form factors mul­tiplied by the radius are shown, the solid lines indicating when the form factor is positive and the dashed lines when it is negative. The indicated point R=rQA 2 corresponds to the radius at which the single-particle potential well (r =1.25 fm) is half its maximum depth and thus corresponds approximately to the radius of the nucleus.

Figure 6-5. DWBA calculations using the form factor of Figure 6-4 with (solid line) and without (dashed line) the spherical Hankel function matched. Similar results are found in CCBA calculations.

Figure 6 -6 . Intrinsic single-neutron-transfer form fact­ors for L=l, L = 2 , and L=3 transitions to the 1/2 [510] and

— 1853/2 [512] rotational bands in W, assuming that the bandsare unmixed. For the meaning of the symbols, see the cap­tion to Figure 6-4. In 6 -6 b, 6 -6 c, and 6 -6 d the form fact­ors corresponding to the same L-value are compared, indica­ting the differences in the strengths and radial dependence.

VKThe G T T for these form factors are listed in Table 6-3. nL J

Figure 6-7. Radial dependence of form factors determined from eigenfunctions of a spherical Woods-Saxon well (fixed N). These are labelled by NL . The strengths of such formJfactors would be determined by multiplying the values shown

VKin the figures by G .nL J

Figure 6-4

R(fm)

da/d

n c

m§ (

arbi

trar

y un

its)

Figure 6-5

Rf(R

» Rt (

Ri

Figure 6-6a

FORM FACTOR SET #1

1 /2 " [5 1 0 ]

6 8 10 12 R(fm )

Figure 6-6b

FORM FACTOR SET *1

R(fm)

Figure 6-6c

FORM FACTOR SET *1

R(fm)

|Rf(

R)|

Figure 6-6d

FORM FACTOR SET *1

R(fm)

Figure 6-7a

Basic Woods-Saxon Form Factors,N fixed

Figure 6-7b

R(fm)

the oscillator expansion or solving for the eigenvalues of

the deformed well numerically as described in Section 3.3.

A comparison of the intrinsic form factors for the two

different bands shown in Figure 6 - 6 indicates that they can

vary strongly from band to band. The major difference is

in the relative strengths of the various components, but with

the present model there can be some variation in the radial

dependence as well. This form-factor variation constitutes a

major difference between single-nucleon transfer reactions and

two-nucleon transfer between the states of coherent two-nucleon

character examined in Chapter 6 . The (p,t) reactions for tran­

sitions between ground bands, for example, depend much more

on the nuclear deformation than on the particular microscopic

configuration of the intrinsic states. The (p,d) transitions

to low-lying bands, on the other hand, depend strongly on the

specific intrinsic orbital involved. In addition, since a

single deformed orbital is involved in the (p,d) transitions

rather than a large number of coherent ones as was seen in the

(p,t) reaction, the (p,d) reactions will be much more sensitive

to a correct calculation of the single-particle eigenvalues.

Therefore, one cannot expect as excellent a reproduction of the

relative experimental (p,d) transition strengths as was observed

for the (p,t) ground-band transitions in Chapter 6 .

6.4. Pure-Band Calculations.186The initial CCBA calculations of the W(p,d) cross

sections were performed neglecting the Coriolis coupling

between the two bands of the final nucleus. This is an in-

148

adequate approximation for the mixed states; however, it can

be shown (see Appendix 6A) that the (p,d) transition to the

pure 1/2 state at 24 keV excitation is completely unaffected

by the Coriolis coupling despite the fact that this coupling185changes the inelastic scattering in W.

The first calculations to be presented used the optical

parameters PI and DI and a space consisting of the 0+ and 2+186 185states in W and the l/2~, 3/2- , and 5/2_ states in W.

The calculations for the 1/2 [510] band are shorn in Figure

6 - 8 and for the 3/2 [512] band in Figure 6-9. These are

compared with DWBA calculations using the same optical para­

meters. The factors F denote the normalization of the various

curves relative to the calculation for the transition to the

3/2 state at 94 keV. The relative normalizations of the CC

and DWBA have been maintained.

It is evident from these figures that the relative

strengths are poorly reproduced, as expected, because of the

neglect of the band mixing. Nevertheless, the calculations

match the shapes of the angular distributions reasonably well.

The strong oscillations observed experimentally in the tran­

sition to the 1/2 state are obtained in the CCBA in marked

contrast with the DWBA. The effect of the inelastic channels

is particularly large for this weak transition. For the

stronger transitions like those to the 3/2 state at 94 keV

and the 5/2 state at 6 6 k e V , where the direct transfer should

dominate, the deviation between the CC and DWBA predictions

is not very large, although the CCBA does achieve a somewhat

149

150

better reproduction of the experimental data in both cases.

The CC and DWBA also give reasonably similar predictions

for the two weak transitions to the 3/2 ground state and the

5/2 state at 188 keV excitation. However, as the normaliza­

tion factors indicate, these calculations should not be taken

too seriously. The weakness of the transitions results from

destructive interference in the direct routes because of

Coriolis coupling, and thus they cannot be reproduced within

the present approximation. In fact the quality of the fits

to the angular-distribution shapes is largely fortuitous,

particularly for the ground-state transition, as will be

demonstrated in Section 6.5 when the Coriolis mixing is

included explicitly.

Since the 1/2 transition is totally unaffected by the

band mixing and the strong transition to the 3 / 2 state at

94 keV only negligibly, one can use these transitions as test

cases to examine in detail the effects of the inelastic pro­

cesses for an ideal case without the complications of band

mixing. These two transitions form an interesting contrast,

since they both correspond to an L = 1 orbital angular momentum

transfer and yet have very different angular distribution

shapes. The partial contributions to these transitions from

the direct transfer route and from all routes proceeding by

a combination of inelastic scattering and transfer are shown

in Figures 6-10 and 6-11. The term "direct route" means here,

as it did in Chapter 5, the direct transfer renormalized by

excitations from and deexcitations back into the initial and

final states. Thus, the direct route contribution is not

exactly the same as the DWBA. A comparison of Figures 6-10

and 6-11 with Figure 6 - 8 shows that the distinction between

the direct route and the DWBA is negligible in the case of

the 1/2 transition, but for the 3/2 transition the contribu­

tion of the direct route has the same shape as the DWBA but

is slightly reduced in magnitude.

From Figures 6-10 and 6-11 it can be understood why the

3/2 and 1/2 transitions have such different angular distri­

bution shapes. The inelastic contributions in the two cases

are similar both in magnitude and in shape. However, for the

3/2 case this contribution is so much weaker than the direct

that it only serves to modulate the angular distribution.

For the 1/2 transition the inelastic and direct contributions

are of equal magnitude and interfere to produce the final

strongly oscillatory angular distribution. It should be em­

phasized that while the inelastic contribution to the 1 / 2

transition is itself oscillatory, these oscillations are al­

most totally out of phase with the experimentally observed

oscillations. It is the interference between the direct and

inelastic contributions which is necessary to achieve the

correct result.

Further insight into the complexity of the inelastic con­

tributions to the 1 / 2 transition can be achieved by examining

them in more detail. Figure 6-12 displays again the total

calculated 1 / 2 transition and the direct-route contribution

along with three of the most important inelastic routes:

0+ 3/2~ % 1/2", 0+ 5/2" % 1/2", and 0+ + 2+ 1/2". It is

151

interesting to note how different these three contributions

are. The 0+ + 3/2 «- 1/2 route proceeds by P^ / 2 transfer

in the neutron pickup and principally quadrupole (L=2 ) de­

excitation in the final-state inelastic scattering. The

important multipole in the inelastic scattering for the

0 5/2 ■«- 1/2 route is also the quadrupole, but the neutron

transfer involves the f^ / 2 ^orm factor. The resultant con­

tribution from this route is highly oscillatory while that of

the route through the 3/2 state is basically flat. On the

other hand, the contribution of the 0 + ■+ 2+ ■+ l/2 ~ route,

which involves both Pg / 2 an<* ^5 / 2 neutron transfer and prin­

cipally quadrupole scattering, is also oscillatory but the

oscillations are completely out of phase with those of the

contribution from the route through the 5/2 state.

The manner in which the inelastic contributions depend

on the multipolarity of the neutron transfer is indicated in

Figure 6-13. The P ^ / 2 trans^er contribution consists almost

entirely of the direct route and looks very similar. The

P 3 / 2 contribution is the largest in magnitude but shows

virtually no oscillation, while the f^ / 2 contr^butfon

strongly oscillatory. The f anc bg/2 contr^but^ons to tbe

1 / 2 transition are extremely weak and have not been included

in the diagram. It is tempting to suggest that the source of

the oscillatory behavior of the 1 / 2 transition is due to the

presence of the f t r a n s f e r s ; however, the situation is clearly

more complicated. For one thing, the detailed structure of the

^5 / 2 osc^ ^ at^oas slightly different from that of the os-

152

cillations in the total 1/2 cross section. In addition,

it is clear from comparing Figure 6-13 with Figure 6-10

that the interference between the P 3 / 2 and f5 / 2 contributions>

which together comprise virtually the entire set of indirect

processes, must result in oscillations almost totally out of

phase not only with the contribution alone but also

with the final cross section itself. This is borne out by

comparing the three inelastic routes in Figure 6-12 as well.

It is only the combined effect of all the important routes

in the reaction which can produce the final interference pat­

tern.

Since it depends so delicately on the interference of

several competing angular momentum transfers in the pickup

process, one might expect the character of the 1 / 2 transition

to change as the reaction energy is varied and different

transferred angular momenta become favored. In addition, the

relative strengths of the inelastic and transfer processes

can change as well. This is illustrated in Figure 6-14, which186displays the predictions for the W(p,d) reaction to the

1/2 [510] band with a proton incident energy of 40 MeV. The

calculation shown is precisely the same as that for Figure

6 - 8 except for the change in the proton energy and adjustments

in the well depths of the parameters PI and D1 to account for

the energy changes. Whereas the classically favored angular

momentum transfer at 18 MeV is L=l, for the reaction at 40

MeV it is L=3. This explains why the 5/2 transition is much

larger relative to the 3/2 transition at 40 MeV than at 18

153

MeV. The effect on the 1/2 transition is even more dramatic.

At 18 MeV the oscillations are basically in phase with those

of the 3/2 transition although much deeper, but at 40 MeV

these oscillations are strongly reduced in size and in the

angular range from 40° to 100° are more in phase with the 5/2

transition.

Another interesting perspective is provided in Figure

6-15, which shows the separate contributions of routes pro­

ceeding via proton inelastic scattering (dash-dotted lines)

and via deuteron inelastic scattering (dotted lines). It is

interesting to note that these contributions are of approx­

imately equal magnitude, indicating that those models which

neglect one or the other of these contributions (Ko 6 6 , Le

6 6 ) are invalid. On the other hand, the detailed structure

of the two contributions is considerably different. The size

of the oscillations in the contributions which involve proton

Inelastic scattering is much larger than those in the contri­

butions involving deuteron scattering, although the phase of

the oscillations is basically the same in the two cases. This

suggests that single-neutron pickup reactions like (d,t),

(3He,a), or ( ^ O , ^ 0 ) which involve projectiles with dif­

ferent masses and inelastic scattering properties than the

(p,d) reaction, might yield a rather different result for

the 1 / 2 transition.

In order to assess the effect of the truncation of the

calculation space on the results, the transitions to the un­

coupled bands were recalculated, again using the PI and D1

154

optical parameters but with the space of the calculation in-+ 186 —creased to include the 4 state in W and the 7/2 and

_ 1859/2 states in W. This calculation is compared with the

experimental data in Figures 6-16 and 6-17. The effect on

the 1 / 2 transition is qualitatively small, although the

agreement in shape between the data and the theory is actually

slightly improved by the inclusion of the additional routes.

The calculations for the strong 3/2 and 5/2 transitions are

also changed somewhat from the results shown in Figures 6 - 8

and 6-9. This occurs because these higher-spin transitions

are more affected by a severe truncation of the calculation

space.

It is interesting to compare the transitions observed 186in the W(p,d) reaction with similar transitions in the

17 2Yb(p,d) reaction to 1/2 [512] ground band. These are shown

in Figure 6-18 and compared with calculations similar to those186described in this section for the W(p,d) reaction. The

1/2 [521] band is relatively pure in ^ ^ Y b and, as is apparent

from this picture, the reproduction of the relative magnitudes

of the transitions within the band is much better than in the

two tungsten bands seen in Figures 6-16 and 6-17. In addition,

as in the tungsten case, the CCBA reproduces the shapes of

the angular distributions extremely well. A comparison be-- 1 7 2tween the 1/2 [521] transitions in Yb(p,d) with those of

_ 18 6the 1/2 [510] in W(p,d) is particularly revealing. Both

of these bands are K=l/2 bands and the two reactions were

carried out at similar proton bombarding energies. Neverthe-

155

Figure 6 -8 . Calculations of W(p,d) transitions to the— 185unmixed 1/2 [510] band in W at a proton Incident ener­

gy of 18 M e V . The CCBA calculation, which was performedusing a space consisting of the 0 + and 2 + states in ^88W

— — — 185and the 1/2 , 3/2 , and 5/2 states in W, is indicatedby the solid line and the DWBA calculation, by the dashed line. Both calculations used the optical parameters FI and D1 listed in Table 6-2. The factors F indicate the relative normalizations of the calculations for different transitions. The relative normalization of the CCBA and DWBA calculations has been maintained. The points shown represent the data of the present experiment. A small amount (^10%) of the measured cross section labelled 5/2 (188 keV) corresponds to the transition to the 7/2 (174keV) state (See Section 4.6.2.).

186Figure 6-9. Calculations of W(p,d) transitions to the— 185unmixed 3/2 [512] band in W at a proton incident ener­

gy of 18 M e V . The CCBA calculation was performed using aspace consisting of the 0 + and 2 + states in ^88W and the

_ _ 1853/2 and 5/2 states in W. The symbols are explainedin the caption to Figure 6 - 8 . The data shown are that ofthe present experiment.

Figure 6-10. Total CCBA calculation (solid line) for the_ 185transition to the 1/2 (24 keV) state in W, compared

with the direct (dashed line) and indirect (dotted line) contributions. The direct contribution corresponds to the direct transfer together with inelastic excitations and de­excitations in the target and residual nuclei but with no inelastic-plus-transfer allowed. The indirect contribu­tion corresponds to all the inelastic-plus-transfer routes. The sum of the indirect and direct contribution amplitudes yields the amplitude of the total cross section. The curves shown in the figure correspond to the absolute squares of these amplitudes.

186

Figure 6-11. Total CCBA calculation (solid line) for anddirect (dashed line) and indirect (dotted line) contri-

_ 18 5butions to the unmixed 3/2 (94 keV) state in W. See the caption of Figure 6-10 for further explanation.

Figure 6-12. Total CCBA calculation (solid line) for andthe contribution of various routes to the transition to

— 18 5the 1/2 (24 keV) state in W. The notation for thevarious routes is interpreted in the diagram in the upper right-hand corner. This diagram is explained in the cap­tion to Figure 2-4.

Figure 6-13. Total CCBA calculation (solid line) for the_ 185transition to the 1/2 (24 keV) state in W compared

with all the partial contributions involving Pj/2 * ^ 3 /2 *and f^y2 neutron transfers.

Figure 6-14. The results of calculations performed in the same manner as those shown in Figure 6 - 8 but with an incident proton energy of 40 MeV.

Figure 6-15. Total CCBA calculation (solid line) forand various partial contributions to the 1/2 (24 keV)

185state in W. The partial contributions are explained in the diagrams in the upper right-hand corner. The sym­bols in these diagrams have the same meaning as those in Figures 6-12 and 2-4.

Figure 6-16. Calculations of W(p,d) transitions to• 185the unmixed 1/2 [510] band in W at a proton incident

energy of 18 MeV. The CCBA calculation was performedusing a space consisting of the 0+ , 2+ , and 4+ states inODW and the 1/2", 3/2", 5/2", 7/2", and 9/2" states in

185 W. The symbols are explained in the caption to Figure 6 -8 . The points shown represent the data of the present experiment. A small amount (^10%) of the measured cross section labelled 5/2 (188 keV) corresponds to the tran­sition to the 7/2 (174 keV) state. (See Section 4.6.2.)

186Figure 6-17. Calculations of W(p,d) transitions to— 185the unmixed 3/2 [512] band in W at a proton incident

energy of 18 MeV. The CCBA calculation was performedusing a space consisting of the 0+ , 2+ , and 4+ states in186W and the 3/2", 5/2", 7/2", and 9/2" states in 1 8 5 W.The symbols are explained in the caption to Figure 6 - 8 .The data shown are that of the present experiment.

172Figure 6-18. Calculations of Yb(p,d) transitions to the 1/2 [521] band in 3-7 Yb at a proton incident energy of 17 MeV. The CCBA calculation was performed using a space consisting of the 0+ , 2+ , and 4+ states in ^7^Yb and the 1/2", 3/2", 5/2", 7/2", and 9/2" states in 1 7 1 Yb. The symbols are explained in the caption to Figure 6 -8 . The calculations are that of Ascuitto et al^. (As 73) and the data shown are that of McVay (Me 73).

186

d<r/

dXicm

(/

ib/s

r)

Figure 6-8

100

1 0 0 0

100

40° 80° 120° 160°CENTER OF MASS ANGLE

Figure 6-9

l8?W(p.d) Ep=18.0 MeV

0 ° 40 ° 80° 120° 160*

CENTER OF MASS ANGLE

do-/d

il c.

m. (

arbi

trar

y un

its)

Figure 6-10

C E N T E R OF M ASS ANGLE

dc

r/d

il c.

m. (

arbi

trar

y u

nit

s)

Figure 6-11

CENTER OF MASS ANGLE

cWdX

l cm

(a

rbitr

ary

unit

s)

Figure 6-12

C E N T E R OF M A S S A N G L E

d<r/d

U c

m (a

rbitr

ary

unit

s)

Figure 6-13

l86W(p,d) Ep= 18.0 MeV l/2- (24keV)

C E N T E R OF M A S S A N G L E

d<

r/d

&c

m (/

ib/

sr)

Figure 6-14

CENTER OF MASS ANGLE

dcr/d

ft c>m

> (a

rbit

rary

un

its)

Figure 6-15

0 . 0 1

4 0 ° 8 0 ° 120° 160°C E N T E R OF M A S S A N G L E

d<r/d

&c.m

. (/

ib/s

r)

Figure 6-16

,8 6 W (p .d ) E P= I8 .0 MeV1 /2 " [5 1 0 ]

(js/q-r/) u,0u

p/^

p

Figure 6-17

l86W (p .d ) E P» I8 .0 MeV

3 /2 " [5 1 2 ]

dcr

/dX

lcm

(/ib

/sr)

l72Yb(p,d) 1/2“ [521] Ep= 17.0 MeV

CEN TER OF MASS ANGLE

less, the angular-distrlbution patterns of the corresponding

transitions In the two cases are quite dissimilar. Not only

are the relative magnitudes different but also the structure

of the angular distributions. In particular, the 1/2 tran-186sition shows strong oscillations in the W(p,d) case, but

— 17 2it is the 3/2 transition which is oscillatory for Yb(p,d).

This results mainly from differences in the intrinsic structure

of the two bands, and illustrates the lack of systematics

which should be expected for single-nucleon transfer reactions

on deformed nuclei.

The results presented in this section illustrate that

caution must be exercised in the extraction of angular momen­

tum information from the shapes of transfer-reaction angular

distributions on deformed nuclei. In both the ytterbium and

tungsten examples, important differences are observable be­

tween angular distributions resulting from transfer of the

same orbital angular momentum. In addition, a comparison of

the ytterbium and tungsten cases indicates that there is no

systematic dependence on the total transferred angular mo­

mentum J as has been observed for reactions on medium-weight

nuclei (Le 64, Sh 64, Sc 6 6 ). Therefore, J-dependent arguments

seem invalid as well.

6.5. Inclusion of Coriolis Coupling.

The calculations presented in the last section indicate

definite evidence for the important influence of inelasticX 8 6processes in the W(p,d) reaction, at least for the weak

1/2 transition. However, for the transitions to the other

states, which are strongly mixed due to Coriolis coupling,

156

the picture cannot be complete without the inclusion of this

coupling in the reaction calculation.

The calculations discussed in this section utilized the

Coriolis coupling coefficient set 1 of Table 3-1. These were

calculated as described in Section 3.5 using the full unatten­

uated value of the Coriolis force strength with a three para­

meter variation to reproduce the experimental energy eigenvalues.

With this coupling included, the form factors for the various

transfer routes will depend on different combinations of the

intrinsic form factors of the two bands (see Appendix 3C). An

example of the influence of the coupling for the direct route

form factors can be discerned from Figure 6-19. A comparison

of these with Figure 6 - 6 shows how the strengths of the direct

contributions are changed. Of particular interest is the strong

decrease in the direct contributions for the transitions to the

3/2 ground-state and the 5/2 state at 188 k e V . The predic­

tions for these states in the calculations of Section 6.4

were much too large in magnitude.

The CCBA predictions with the Coriolis mixing included,

using PI and D1 parameters, and with the calculation space

limited to the 0 + and 2+ states of the ^88W ground band and— — — 185the 1/2 , 3/2 , and 5/2 states of the mixed bands in W

are displayed in Figure 6-20. The DWBA predictions shown used

the same mixing and optical parameters. Comparing this figure

with Figures 6 - 8 and 6-9 indicates that a tremendous improve­

ment in the relative strengths has been achieved by including

the band mixing. Note also that the 1/2 results are complete-

157

ly unchanged as is necessary according to the results of

Appendix 6A. The shapes of the angular distributions for

the transitions to the 3/2~ state at 94 keV and the 5/2~

state at 6 6 keV are not much affected by addition of Corlolis

coupling, since the direct routes are strong in these two

cases even with the inclusion of mixing and thus these routes

still dominate the transitions. The relative strengths of

these two transitions are noticeably improved, however. Even

the now weak 5/2 (188 keV) angular distribution is not much

changed in shape from the calculation of Figure 6 -8 , and

the effects of the inelastic processes are still minor.

The ground-state 3/2 transition is now noticeably dif­

ferent. The DWBA prediction for this transition is altered

from that in Figure 6-9 as well. This results from the fact

that with the intrinsic wavefunctions used in the present

analysis the P 3 / 2 ^orm factors do not necessarily have pre­

cisely the same shape in two different bands, and are indeed

different for the present example as Figure 6 -6 b shows. Their

combination for the direct route consequently produces a

shape somewhat different from that of the standard P 3 / 2 ^orm

factor. This is shown in Figure 6-19a. A standard form-fac­

tor calculation would not produce this result.186The ground state transition is now the first W(p,d)

transition encountered for which the CCBA prediction is much

larger than the DWBA. Figure 6-21, which indicates the sep­

arate contributions from the direct route and the collected

inelastic routes, shows that indeed this transition is domi -

158

direct transitions to members of the mixed 1/2~[510] and_ 1853/2 [512] bands in W. These were determined from the

intrinsic form factors shown in Figure 6 - 6 by using Equa­tion (3-42) and the Coriolis-coupling coefficients Set 1 (See Table 3-1). The symbols are explained in the captionto Figure 6-4. The single-neutron-transfer form factors

186corresponding to transfers from excited states of W have somewhat different forms from these direct-transfer form factors, as can be seen from Equations (3-33) and (3-42) .

18 6Figure 6-20. Calculations of W(p,d) transitions to members of the 1/2 [510] and 3/2 [512] bands in mixedwith the Coriolis-coupling coefficient Set 1. The incident proton energy is 18 MeV, and the CCBA calculation was per­formed using a space consisting of the 0 + and 2 + states in1 0 / _ i p c

W and the 1/2 , 3 / 2 , and 5/2 states in W. The sym­bols are explained in the caption to Figure 6 -8 . The data shown are that of the present experiment. A small amount (^10%) of measured cross section labelled 5/2 (188 keV)corresponds to the transition to the 7/2 (174 keV) state.(See Section 4.6.2.)

Figure 6-21. Total CCBA calculation (solid line) for the— 185transition to the mixed 3/2 (0 keV) state in W, compared

with the direct (dashed line) and indirect (dotted line)contributions. See the caption of Figure 6-10 for furtherexplanation.

Figure 6-22. The results of calculations performed in thesame manner as those shown in Figure 6-20 but with the cal-

+ 18 6culation space increased to include the 4 state in W— — 18 5and the 5/2 and 7/2 states in W.

Figure 6-19. Single-neutron-transfer form factors for

Figure 6-19a

CORIOLIS COUPLED FORM FACTORS

2 4 6 8 10 12 14 16 18r(fm)

Figure 6-19b

Coriolis Coupled Form Factors for Routes 0+—-J ir

R(fm)

Figure 6-19c

Coriolis Coupled Form Factors for Routes 0 +—

2 4 6 8 10 12 14 16 18R (fm )

doVdD

f m

Ifib

/w)

Figure 6-20

l8 6 W (p ,d ) EP» I8 .0 MeVCoriolis-Coupling Coefficient Set I

d<r/d

& c.

m. (

arbi

trar

y un

its)

Figure 6-21

0 ° 4 0 ° 8 0 ° 120° 1 60°C E N T E R OF M A S S A N G L E

(JS/q

ry) '“^

up

z-o

p

Figure 6-22a

l8 6 W (p .d ) E P» I8 .0 MeV

d«r

/dflc

m. (

/ib/s

r)

Figure 6-22b

l8 6 W (p .d ) E ps l8 .0 MeV

nated by the indirect processes. It is interesting that the

1 / 2 transition, whose magnitude unlike that for the ground

state is virtually unchanged by the inclusion of the inelas­

tic routes, displays much more dramatic inelastic effects

than the 3/2 transition because of the interference between

the two types of contributions.

A calculation with the space increased to include the+ 186 — —4 state of the W ground bands and the 7/2 and 9/2 states

185of the W bands is an extremely large one when the band

mixing is included, but it was performed to insure that no

unexpected effects result from these extra transitions. The

results are displayed in Figure 6-22. The DWBA calculations

in both figures again used the same optical parameters and

form factors. It is evident that the changes in Figure 6-22

from the smaller-space calculations of Figure 6-20 are rela­

tively minor. The comparison between the CCBA and the data

for the new 7/2 and 9/2 transitions is also quite good.

The transition to the 9/2 state at 302 keV is interesting,

since it represents another example like the ground state

transition of a cross section dominated by inelastic routes.

6 .6 . Sensitivity of the Calculations to Form Factors

and Optical Parameters.

There remain some descrepancies between the CCBA and

the data both in the magnitude and, to a lesser extent, the

shape of the angular distributions, even for the larger-

space calculations of Figure 6-22. There are several possible

sources for these discrepancies and a few will be explored in

159

the present section.

One obvious uncertainty in the calculations presented

in the last section concerns the precise value of the Coriolis

mixing coefficients. To assess the importance of this un­

certainty, an additional calculation was performed using the

coefficient set 2 of Table 4-1. These were determined using

a somewhat attenuated Coriolis strength. A CCBA calculation

using the optical parameters PI and D1 and the smaller cal­

culation space consisting of only the 0 + and 2 + states of186 — — — the W ground band and the 1/2 , 3/2 , and 5/2 states

18 5in W is displayed in Figure 6-23. It is evident that the

main effect of using the coefficient set 2 when compared to

the equivalent calculation with coefficient set 1 in Figure

6-20 is to alter the normalizations of the 5/2 (188 keV)

and 3/2 (0 keV) transitions relative to the other three.

The 5/2 normalization, in fact, undergoes a dramatic change.

Thus, it seems that an adjustment of the mixing coefficients

of the two bands can possibly bring the CCBA calculation into

better accord with the experimental data. A more sophisti­

cated Coriolis-coupling calculation might be needed, including

the effect of mixing with higher-lying bands and perhaps the

effect of blocking in the calculation of the pairing factors.

However, the determination of the structure of the bands of 185 W was not the principal purpose of the present investiga­

tion. Therefore, no such calculation was attempted.

The Coriolis-coupling uncertainty is obviously not the

only source of discrepancy. Even for the transitions to the

160

relatively pure ground band of ^ ^ Y b seen in Figure 6-18

the agreement between the CCBA calculations and the data is

not perfect. Although it is possible that some of the

discrepancies in this case are caused by Coriolis mixing,

it is more likely that they result from uncertainties in the

intrinsic band structure. An example of this latter effect

is provided by using a more standard type of form factor in

the calculation. Many DWBA calculations of simple-nucleon

transfer reactions on deformed nuclei use form factors based

on spherical Woods-Saxon eigenvalues, suqh as those depicted

in Figure 6-7, with the relative strengths determined from

the spherical expansion or "Nilsson" coefficients for

the eigenvalues of a deformed harmonic oscillator well.

Figure 6-24 depicts the results of a CCBA calculation 18 6for W(p,d) using such form factors and the same space,

optical parameters, and Coriolis coupling coefficients as

the calculation shown in Figure 6-20. The Nilsson coefficients

were taken from the calculation of Chi (Ch 6 6 a) with a harmonic

oscillator well of deformation 6=0.2. This deformation para­

meter 6 is roughly equivalent to the parameter The use

of these simpler form factors clearly has a dramatic effect

not only on the relative strengths of the transitions but

also the shapes of their angular distribution patterns. A

particularly interesting example is the 1 / 2 transition for

which the direct contribution is now clearly reduced from

that of the original calculation. Because the inelastic-

direct interference is necessary to achieve the proper angu­

161

lar distribution pattern, the decrease in (he direct strength

completely destroys this pattern, as is apparent in Figure

6-24.

Another example of the effect of varying the form

factors is provided by Figure 6-25. This figure displays

three separate CCBA calculations for the transition to the

1/2 state. All three utilize the smaller calculation space

of 0+ and 2+ in 186W and l/2~, 3/2", and 5/2" in 185W and

the optical parameters PI and Dl. The solid line is the

same calculation as shown in Figure 6-20, or equivalently,

Figure 6 -8 . The dotted line results from a calculation

using a set of form factors calculated In the same manner

as the original set, but with the hexadecapole deformation

set equal to zero and the quadrupole deformation adjusted

so that the volume Integral of the quadrupole moment of the

proton potential Is the same as in the original set (set # 1 ) .

The dashed lines indicate a calculation with form factors

based on the calculations of Faessler and Sheline (Fa 6 6 ) ,

which also used a Woods-Saxon well with zero hexadecapole

deformation. This well, however,.was diagonalized in a

spherical Woods-Saxon basis, severely truncated to include

only orbitals within a single shell (fixed N ) . The form

factors for the direct P ^ / 2 route these three cases are

compared in Figure 6-26. The deformations used to describe

the elastic scattering in the three cases were the same as

given in Table 4-1.

It is clear that the two calculations using form factors

162

calculated with zero hexadecapole moment are considerably

different from the original calculation. The oscillations

in the calculation using the severely truncated basis are

completely out of phase with those of the original. This

is probably caused by the fact that the sign of the P ^ / 2

form factor near the surface, as seen in Figure 6-26, is

opposite from the other two. Therefore, the interference

of the direct route with the indirect routes will be complete­

ly different. The results presented here indicate that

for a particularly sensitive case like the 1 / 2 transition,

the interference between the direct and multistep routes

can possibly be used as a test of the intrinsic wavefunctions

of the band.

One other variable which can influence the calculations

is the choice of the optical parameters. In Figure 6-27 the

CCBA calculation of Figure 6-20 is shown again and compared

with two other calculations performed in exactly the same

manner but with altered deuteron optical parameters. All

three sets of parameters are listed in Table 6-2. As was

indicated in Section 6.2, the D2 set was unable to reproduce 186the elastic W deuteron scattering data in a CC calculation.

The set D3 was determined from D2 by keeping the well depths

fixed but adjusting the imaginary well geometry to fit the

elastic scattering data. It is evident from Figure 6-27 that

the use of either the parameters D2 or the parameters D3

in the reaction calculation results in a much worse fit to

the experimental data. Not only are the shapes of the angu-

163

lar distribution patterns often in considerable disagreement

with the data, most notably for the 1 / 2 transition, but

also, in the case of the ground-state transition, the mag­

nitudes as well.

Both from the superior fit achieved in the reaction cal­

culation and also from the fit to the deuteron inelastic 172scattering on Yb, it is clear that the deuteron para­

meters with shallower real and deeper imaginary well depth,

as represented by the D1 set, are more accurate. Neverthe­

less, the extreme sensitivity of the CCBA calculation to the

deuteron parameters shows that some of the remaining discrep­

ancies, particularly in angular distribution shapes, can

possibly be eliminated by small adjustments in these parame­

ters. Inclusion of deuteron breakup might lead to improve­

ments as well. As for the proton parameters, a comparison

of the CCBA calculation of Figure 6-20 with a similar calcu­

lation shown in Figure 6-28 using another average proton

parameter set (Pe 63b) listed in Table 6-2 indicates that the

calculations are considerably less sensitive to these para­

meters.

It is interesting at this point to compare the CCBA

calculations of this chapter to what might be called a "stan­

dard" DWBA analysis. Such an analysis would use the spheri­

cal Woods-Saxon form factors of Figure 6-7. The ratio between

calculation and the data, known as the "spectroscopic factor,"

is then interpreted as being related to the Nilsson coefficients

and, in a mixed-band case, the band-mixing coefficients. In

164

addition, the optical parameters chosen are generally those

which fit elastic scattering data on the target nucleus in a

one-channel calculation. A comparison with the data of

the angular distribution shapes resulting from such an an­

alysis for the three L=1 transitions is shown in Figure 6-29.

The proton parameters were adjusted from the PI set to fit

the CC elastic scattering corresponding to the PI parameters

in a one-channel calculation. These have been labelled P3

in Table 6-2. The deuteron parameters were taken from the

one-channel fit to the 12 MeV deuteron elastic scattering of

Siemssen and Erskine (Si 6 6 ) and are listed as D4 in Table

6-2. The prediction for the strong 3/2 transition to the

94 keV state is evidently very good. This results from the

elastic deuteron parameters which simulate the effects of

the neglected channels on the shape of the angular distribu­

tion. (The calculation was found to be reasonably insensi­

tive to the proton parameters.) The angular distribution

shape is also not bad even for the weak ground state tran­

sition. However, the new deuteron optical parameters have

a negligible effect on the relative magnitudes of the cross

sections, and thus the standard DWBA analysis will yield

an incorrect magnitude for this state, as the results of

Section 6.4 clearly indicate. For the weak 1/2 transition,

where the interference between direct and inelastic routes

are important, the inadequacy of a standard DWBA analysis is

clearly visible even in the angular distribution shapes.

Thus, this standard calculation suggests part of the

reason why inelastic effects in single-nucleon transfer

165

Figure 6-23. The results of calculations performed in the same manner as those shown in Figure 6-20 but using Coriolis-coupling coefficient Set 2 (See Table 3-1).

Figure 6-24. The results of calculations performed in the same manner as those shown in Figure 6-20 but using the form factors of Figure 6-7. These form factors were multiplied by the Nilsson coefficients of Chi (Chi 6 6 a) calculated with the parameters 6=0.2, k=0.05, and y=0.45. (See Ch 6 6 a)

Figure 6-25. CCBA calculation for the transition to the 1/2 (24 keV) state as shown in Figures 6 - 8 and 6-20(solid line) compared to CCBA calculations using differ­ent form factors explained in the text.

Figure 6-26. Comparison of p i n t r i n s i c single-neutron-transfer form factors for transitions to the 1/2 [510]

185band in W. These form factors were used in the calcu­lations shown in Figure 6-25. The symbols are explained in the caption to Figure 6-4.

Figure 6-27. The CCBA calculation of Figure 6-20 (solidline) compared with two CCBA calculations using different deuteron optical-potential parameters listed in Table 6-2.

Figure 6-28. The CCBA calculation of Figure 6-20 (solidline) compared with a CCBA calculation using a differentset of proton optical-potential parameters listed in Table 6-2 .

Figure 6-29. DWBA calculation using the form factors o Figure 6-7 and the optical-potential parameters P3 and D 4 . The parameters reproduce the elastic scattering in a one-channel calculation. The curves have been normal ized separately to reproduce the strengths of the exper imental differential cross sections, which are those of the present experiment.

dcr/dQ

c m. (y.b/sr)

Figure 6-23

l8 6 W (p .d ) E P= I8 .0 MeVC orio lis -C o u p lin g C o effic ien t Set 2

(JS/q

W)

'“■I'

yp

/^p

Figure 6-24

,8 6 W (p .d ) Ep= 18.0 MeV H A R M O N I C O S C I L L A T O R N I L S S O N C O E F F I C I E N T S ( S = 0 . 2 )

do-/

d&cm

> (a

rbit

rary

un

its)

Figure 6-25

^®W(p,d) 1/2" (24 keV)

Form Factor Set # I (/32= 0 . 2 2 2 , £ 4 = -0 .0 9 4 3 )

Form Factor Set # 2 ( £ 2 =0 ‘ 2 I 4 , £ 4 = 0 .0 )1 0 Faessler-Sheline Form Factors

(i02=O.2, )S4=0.0, N f ix e d )

0 . 1

0 . 0 1

4 0 * 8 0 °9

120* 160*

c* fn •

Figure 6-26

p 1 / 2 Form Factor for 1/2 [ 5 1 0 ]

8 10 12 14 16 18R( fm)

da/dft

cm (/ib/sr)

Figure 6-27

,86W(p.d) Ep-18.0 MeVCC CALCULATIONS

do-/<&

CJn (fib/tr)

Figure 6^28

,8 6 W (p .d ) E p- 18.0 MeV CC CALCULATIONS

CENTER OF MASS ANGLE

d<

r/d

il

Figure 6-29

reactions on deformed nuclei have not been more apparent

experimentally. For very strong transitions such as that

to the 3/2 (94 keV) state, where the inelastic effects result

mainly in small changes in the angular distribution shapes

with very little influence on the magnitudes, a judicious

choice of optical parameters can often reproduce these ef­

fects reasonably well. Nevertheless, for weaker transitions

such as those to the ground and 1 /2 - states, where the shapes

and/or strengths of the angular distributions are strongly

affected by the inelastic processes, it is clear that these

processes must be explicitly included in any serious attempt

to understand the experimental reaction data.

166

167

Appendix

6A . The Effect of Coriolis Coupling on CCBA Calculations

of Transitions to a Pure State.

The Coriolis coupling of the final states enters a

CCBA calculation through the inelastic and transfer form

factors. Expressions for these form factors with final-

state mixing are given in Equations (3-40) and (3-42). In

this appendix it is demonstrated that the mixing of other

states in the calculation space has no effect on the tran­

sition to a pure state, such as the 1/2 (24 keV) state. 185..in W .

Let f designate a final channel in which the nucleus ois in a pure state. It is evident that the transfer form

factor and hence the source terms for this channel are the

same as in a calculation for which all the final states are

unmixed. However, the inelastic form factors will be dif­

ferent. For simplicity, only two bands are assumed to be

mixed, as in the calculations performed in the present in­

vestigation. Let (vK) label the band in which the pure

state lies, and (vK) , the other band. Then for every mixed

state la J> there exists a conjugate orthogonal state oflthe same spin |a J> such that, with the proper choice of

2

phases

(6 -la)

and A 2 = -A— 1 —V K v K (6-lb)

Here the represent the Coriolis-coupling coefficients

as defined in Equation (3-39).

Let f^ designate a final channel for which the nucleus

is in the state |a J>, and f , a final channel for which1 2

the nucleus is in the state la J>. The inelastic form2

factors coupling f with channels f and f areo 1 2

a, J

0.3

<a J | I Qt | | a J> = B A ‘ o o 1 L 1 1 i oi v K

a 2 J<a J I I Q, I I a J> = B A , _, o o L 1 2 02 v K ’

with B representing a form factor for unmixed bands:

BiJ 5 / n jT r 4 0 K 1 \ 0 (6‘2 ) -

It is clear that B =B . Therefore, the coupled equations0 1 0 2

for channel f can be written o

/ T . TT TI I _ . Hi n « T I 711 P Af + U f f - Ef > w f - - 2 f f w f - 2 P£ io o o o o f y f o i o

0

where

1.1 _ “ .J ' 111 4“2J ’ 1TI,w f , - (Av k w f + Av k » f )1 2

7T Iand U. involves the unmixed inelastic form factor Bf f 0 10 TT IThus, if the new effective channel functions w ^ , satisfy

the coupled equations for a pure band, then the coupled

equations for channel f will be effectively those of a pureoband .

It is therefore necessary to examine the equations TT Xsatisfied by w ^ , . These equations are obtained if one

a x jmultiplies by A the equations satisfied by f and bya,J 12Av R the equations satisfied by f and then adds the two sets

of equations. Using the conditions (6-1) and the fact that

(T^ - Ej ) = (T^ - Ej ), it can be seen that 1 1 2 2

r rp I f t ft 1 TTI rf TI IT I 111 ff n / £ q )f 1 u f ,f* " Ef ,] w f' = f ' f " W f " " \ f ’ i }

where f 1' are defined as effective channels which are suchTTIthat the involve the pure-band form factors B of

Equation 6-2.

Since, the Equations (6-3) are effectively equations for

transitions to a pure band, this demonstrates that the pure-

state channels f are unaffected by the mixing of the otherostates.

CHAPTER 7

SUMMARY AND IM PLICATIO NS

"Nature loves to hide."

- Heraclitus of Ephesus (5th Century B.C.)

The results presented in this dissertation indicate1 Q£ 1 ft 6

that for the reactions W(p,t) and W(p,d) the ex­

perimental differential cross sections can be reproduced

much better by the CCBA, which explicitly includes the

effect of inelastic excitations in the target and residual

nuclei, than by the DWBA, which does not. There remain

some discrepancies between the CCBA calculations and the

experimental data; however, as has been indicated in Chapter

5 and 6 , it is possible that changes in parameters used to

determine the nuclear structure of the states (band mixing

coefficients, for example) and perhaps changes in the

optical parameters as well can eliminate many of these

discrepancies. In addition, other higher-order processes,

such as multiple transfer, may possibly be significant for

the weakest transitions. Nevertheless, the success of

the CCBA calculations in reproducing the principal features

of the experimental angular distributions provides con­

vincing evidence of the importance of inelastic processes

accompanying transfer reactions on deformed nuclei. It

has been shown that these processes affect both the magni­

tudes and the shapes of the angular distributions. The

effects on the shape can vary strongly not only from nuclide

to nuclide but also from state to state within the same

nucleus. As a result, caution must be exercised in attemp­

ting to extract angular momentum information directly from

the shapes of the angular distributions.

The inelastic processes were shown to have some effect

for all transitions. For the strongest transitions (that

170

is, those with peak cross sections larger than about 250

yb/sr), the inelastic effects lead mainly to modulations

in the angular distributions which can sometimes, but

not always, be simulated in the DWBA by using appropriate

elastic scattering optical potentials. For weaker tran­

sitions, the effects of the inelastic processes are so

strong that they must be included explicitly in the reaction

calculations. Even for those transitions in which the

multistep processes dominate in determining the strength

of the cross section, their effect on the shapes of the

angular distributions can often be rather undramatic. The

most interesting and significant inelastic effects fre­

quently occur for those transitions in which the direct

and multistep contributions are of equal magnitude and

interfere strongly.

Although the results presented here indicate the in­

adequacy of the DWBA for describing transfer reactions on

deformed nuclei and the importance of reevaluating infor­

mation which has been extracted from experimental data in

this manner, the use of such data for the determination of

nuclear structure is by no means excluded. The CCBA, when

care is taken to avoid unwarranted approximations in the

analysis, is evidently a very satisfactory model for des­

cribing reactions on deformed nuclei, as the calculations

of Chapters 5 and 6 demonstrate. The information to be

extracted must be of a more indirect character, however.

The determination of quantities totally independent of

models for the nuclear structure seems impossible. Never­

171

theless, a CCBA analysis of data can often be used as

a very satisfactory test of the consequences of such

nuclear models.

In favorable cases, the presence of multistep inelastic-

plus-transfer processes can provide information which is

completely unavailable in a simple direct-reaction mechan­

ism. This occurs especially for those transitions in which

the direct and indirect contributions are of comparable

magnitude. The transition to the 1/2” state at 24 keV186excitation in the W(p,d) reaction, for example, because

of the interference of these contributions, was shown to

be sensitive not only to the relative signs of the spherical

expansion coefficients of the intrinsic state but even to

the deformations of the nucleus. Such extremely sensitive

single transitions will no doubt be rare; however, an

exact reproduction of not only the relative strengths but

also the angular distribution shapes for all transitions

to a band can often be strongly dependent on the model for

the intrinsic state. This occurs both for single-nucleon

transfer reactions to odd-nucleus bands and for two-nucleon4

transitions to the less coherent bands In even-even nuclei.

It is important now to apply these techniques to reac­

tions on other deformed nuclei, at other energies, and

involving different projectiles, in order to gain more in­

sight into the mechanism of these reactions. As part of

such a program, an effort should be made to obtain consis­

tent average sets of coupled-channel optical parameters,

which can be used over a large range of nuclides in order

172

to lessen the effect of optical parameter uncertainty in

the analysis. The lack of such a set is especially appar­

ent in the case of deuterons. For this particular case

it may eventually become necessary to include in some way

the effect of projectile breakup in the analysis. Once

all of these problems have been solved, however, it is

very likely that important new information about the struc­

ture of deformed nuclei can be obtained from CCBA analyses,

although it should be remembered that such calculations

are necessarily much more time-consuming and costly than

those using the DWBA.

With the importance of inelastic processes so evident

for reactions in the deformed region and the applicability

of the CCBA demonstrated, such an analysis should be applied

to reactions on other nuclei with strong collective exci­

tations as well. For these nuclei the analysis will some­

times be much more difficult because of structure uncer­

tainties. Nevertheless, important, though indirect,

information on excited-state parentage is surely obtainable

in this manner.

Finally, the investigation of the importance of other

forms of higher-order processes in transfer reactions such

as multiple transfer, projectile excitation, and exchange

processes should be pursued. Some studies of these mech-

ansims have already begun, as indicated in Chapter 2. These

more exotic processes may become increasingly important to

understand, particularly in view of the recent interest in

reactions involving heavy-ion projectiles. Certainly, the

173

study of nuclear reactions is a fascinating one, which

still has many surprises ahead.

175

REFERENCES

Ac 65 H.L. Acker and H. Marshall, Phys. Lett. 3J), 127(1965) .

Ag 6 8 D. Agassi and R. Schaeffer, Phys. Lett. 26B, 703(1968) .

Ag 70 V.A. Ageev, V.I. Gavrilyuk, V.I. Kuprayshkln,G.D. Latyshev, I.N. Lyutyi, V.K. Maldanyuk, Yu.A. Kakovetskii, and A.I. Feoktistov, Bull. Acad. Sci.USSR, Phys. Ser. 34 , 1892 (1970).

Am 67 K .A . Amos, V.A. Madsen, and I.E. McCarthy, Nucl.Phys. A94 , 103 (1967) .

Ap 70 A.A. Aponlck, C.M. Chesterfield, D.A. Bromley, andN.K. Glendenning, Nucl. Phys. A159. 367 (1970); andA.A. Aponlck, Doctoral Dissertation, Yale Univer­sity (1970), unpublished.

As 69 R.J. Ascuitto and N.K. Glendenning, Phys. Rev. 181,1396 (1969).

As 70 R.J. Ascuitto and N.K. Glendenning, Phys. Rev. C 2 ,415 (1970).

As 70a R.J. Ascuitto and N.K. Glendenning, Phys. Rev. C 2 , 1260 (1970).

As 70b R.J. Ascuitto, LISA, Yale University (1970), un­published .

As 71 R.J. Ascuitto, N.K. Glendenning, and B. Sorensen,Phys. Lett. 34B, 17 (1971).

As 71a R.J. Ascuitto, N.K. Glendenning, and B. Sorensen, Nucl. Phys, A170 , 65 (1971).

As 72 R.J. Ascuitto, N.K. Glendenning, and B. Sorensen,Nucl. Phys. A183, 60 (1972).

As 72a R.J. Ascuitto and B. Sorensen, Nucl. Phys. A190,297 (1972).

As 72b R.J. Ascuitto and B. Sorensen, Nucl. Phys. A190,309 (1972).

As 72c R.J. Ascuitto, C.H. King, and L.J. McVay, Phys. Rev, Lett. 29, 1106 (1972).

As 72d R.J. Ascuitto, private communication (1972).

73

68

61

63

65

70

73

57

60

65

67

68

70

71

73

59

63

65

6869

176

R.J. Ascuitto, C.H. King, L.J. McVay, and B. Sorensen, to be published (1973).

J. Atkinson and V.A. Madsen, Phys. Rev. Lett. 2_1, 295(1968) .

N. Austern, Ann. Phys. ljj, 299 (1961).

N. Austern, Fast Neutron Physics, e d . J.B. Marionand J.L. Fowler (Interscience, New York, 1963), Vol. II, p. 1113.

N. Austern, Phys. Rev. 137, B752 (1965).

N. Austern, Direct Nuclear Reaction Theories (Wiley, New York, 1970).

E.E. Ayoub, Doctoral Dissertation, Yale University (1973), unpublished.

J. Bardeen, L. Cooper, and R. Schrieffer, Phys. Rev. 108, 1175 (1957).

M. Baranger, Phys. Rev. 120, 192 (1960).

P.D. Barnes, D. Biegelson, J.R. Comfort, and R.O, Stephen, Wright Nuclear Structure Laboratory Internal Report //2 5 (1965).

H.W. Barz, Nucl. Phys. A 9 1 , 262 (1967).

M. Baranger and K. Kumar, Nucl. Phys. A122, 241(1968).

B.F. Bayman, Phys. Rev. Lett. 2_5, 1768 (1970).

B.F. Bayman, Nucl. Phys. A168, 1 (1971).

B.F. Bayman and D.H. Feng, Nucl. Phys. A205 . 513 (1973) .

S.T. Belyaev, Dan. Vid. Selsk. Mat. F y s . Medd. 31,No. 11 (1959).

D.R. Bes, Nucl. Phys. 4j , 544 (1963).

T.A. Belote, W.E. Dorenbusch, 0. Hansen, and J. Rapa- port, Nucl. Phys. 7_3 , 321 (1965).

J. Bennewitz and P.K. Haug , Z. Physik 212 , 295 (1968).

D.R. Bes and R.A. Sorensen, Advances in Nuclear Phys­ics, Vol. 2, ed. M. Baranger and E. Vogt (Plenum,New York, 1969) p. 129.

177

Be 69a F.D. Becchetti, Jr. and G.W. Greenlees, Phys. Rev.182, 1190 (1969).

Bi 72 P.K. Bindal and R.D. Koshel, Phys. Rev. C 6 , 2281(1972) .

Bl 60 J. Blomquist and S. Wahlborn, Ark. Fys. ]J , 545(1960) .

Bo 36 N. Bohr, Nature 137 , 344 (1936).

Bo 37 N. Bohr and E. Kalckar, Dan. Vid. Selsk. Mat. Fys.Medd. 14_, No. 10 (1937) .

Bo 52 A. Bohr, Dan. Vid. Selsk. Mat. Fys. Medd. 2^, No.14 (1952).

Bo 53 A. Bohr and B. Mottelson, Dan. Vid. Selsk. Mat.Fys. Medd. 2_7, No. 16 (1953).

Bo 58 A. Bohr, B.R. Mottelson, and D. Pines, Phys. Rev.110, 936 (1958).

Bo 58a N.N. Bogoliubov, Sov. Phys. JETP T_> a n 8 51 (1958).

Bo 58b N.N. Bogoliubov, Nuovo Cimento 7_ (Ser. 10), 794(1958) .

Bo 64 R. Bock, H.H. Duhm, R. Rtidel, and R. Stock, Phys.Lett. 13_, 151 (1964) .

Br 56 C.P. Browne and W.W. Buechner, Rev. Sci. Inst. 2 7 ,899 (1956).

Br 65 R.J. Brockmeier, S. Wahlborn, E.J. Seppi, and F.Boehm, Nucl. Phys. 6 3 , 102 (1965).

Br 67 D.A. Bromley, Proc. Int. School of Physics "EnricoFermi," Course XL (1967) e d . M. Jean (Academic Press, New York, 1969) p. 243.

Br 70 T.H. Braid, R.R. Chasman, J.R. Erskine, and A.M.Friedman, Phys. Rev. C!l_, 275 (1970).

Br 71 G.E. Brown, Unified Theory of Nuclear Models andForces (North-Holland, Amsterdam, 3rd ed. 1971).

Br 71a D. Braunschweig, T. Tamura, and T. Udagawa, Phys. Lett. 35B, 273 (1971).

Br 71b R.A. Broglia, S. Landowne, V. Paar, B. Nilsson, D.R.Bes , and E.E. Flynn, Phys. Lett. 36B, 541 (1971).

178

Bu 50 H.B. Burrows, W.M. Gibson, and J. Rotblat, Phys.Rev. .80, 1095 (1950) .

Bu 51 S.T. Butler, Proc. Roy. Soc. (London) A208, 559(1951).

Bu 63 R. Buck, Phys. Rev. 130, 712 (1963).

Bu 6 6 D.G. Burke, B. Zeldman, B. Elbek, B. Herskind, andM. Olesen, Dan. Vld. Selsk. Mat. Fys. Medd. 35,No. 2 (1966).

Bu 67 D.G. Burke and B. Elbek, Dan. Vid. Selsk. Mat. Fys.Medd. No. 6 (1967).

Ca 72 R.F. Casten, P. Kleinheinz, P.J. Daly, and B. Elbek,Dan. Vid. Selsk. Mat. Fys. Medd. N o - 1 3 (1972).

Ca 73 R.F. Casten and W.R. Kane, Phys. Rev. C7 , 419 (1973).

Ch 6 6 K.Y. Chan and J.G. Valatin, Nucl. Phys. <82 , 222(1966).

Ch 6 6 a B. Chi, Nucl. Phys. R3, 97 (1966).

Ch 69 P.R. Christensen, A. Berinde, I. Neamu, and N. Scintei,Nucl. Phys. A129, 337 (1969).

Ch 73 L. Chua, Doctoral Dissertation, Yale University (1973),unpublished.

Cl 72 T.P. Cleary, W.D. Callender, N. Stein, C.H. King,D.A. Bromley, J.P. Coffin, and A. Gallmann, Phys. Rev. Lett. 2£, 699 (1972).

Cl 72a T.P. Cleary, SESAME, Yale University (1972), unpub­lished .

Cl 73 T.P. Cleary, Doctoral Dissertation, Yale University(1973), unpublished.

Co 6 6 E.R. Cosman, C.H. Paris, A. Sperduto, and H .A . Enge,Phys. Rev. 142, 673 (1966).

Co 6 8 E.R. Cosman and D.C. Slater, Phys. Rev. 172. 1126(1968) .

Da 58 A.S. Davydov and G.F. Filippov, Nucl. Phys. j), 237(1958) .

Da 65 J.P. Davidson, Rev. Mod. Phys. 3J_t 105 (1965).

Da 69 J. Damgaard, H.C. Pauli, V.V. Pashkevich, and V.M.Strutinsky, Nucl. Phys. A135, 432 (1969).

179

Da 69a P.J. Daly, P. Kleinheinz, and R.F. Casten, Nucl.Phys. A123, 186 (1969).

De 63 A. de-Shalit and I. Talmi, Nuclear Shell Theory(Academic Press, New York, 1963).

De 67 D. Dehnhard and J.L. Yntema, Phys. Rev. 155, 1261(1967).

De 72 P. Debenham and N.M. Hlntz, Nucl. Phys. A195, 385(1972) .

Di 63 R.W. Ditchburn, Light (Blackle and Son, London, 1963).

Di 65 J.K. Dickens, R.M. Drisko, F.G. Perey, and G.R.Satchler, Phys. Lett. JJ[, 337 (1965).

Dr 64 R.M. Drlskb and G.R. Satchler, Phys. Lett, j), 342(1964) .

Du 6 8 Y. Dupont and M. Chabre, Phys. Lett. 26B, 362 (1968).

Du 6 8 a I. Duck, Advances In Nuclear Physics, Vol. 1, e d .M. Baranger and E. Vogt (Plenum, New York, 1968) p.343 .

Du 70 0. Dumitrescu, V.K. Lukyanov, I.Z. Petkov, H. Schulz,and H.J. Wieblcke, Nucl. Phys. A149, 253 (1970).

Du 71 A. Dudek and D.J. Edens, Phys. Lett. 36F, 309 (1971).

Ed 60 A.R. Edmonds, Angular Momentum In Quantum Mechanics(Princeton University Press, Princeton, 1960).

El 58 J.P. Elliott, Proc. Roy. Soc. (London) A245, 128 and562 (1958).

El 69 B. Elbek and P.O. Tj^m, Advances in Nuclear Physics,Vol. 3, e d . M. Baranger and E. Vogt (Plenum, New York, 1969) p. 259.

El 70 T.W. Elze and J.R. Huizenga, Phys. Rev. Cl, 328 (1970)

Er 65 J.R. Erskine, Phys. Rev. 138, B 6 6 (1965).

Fa 6 6 A. Faessler and R.K. Sheline, Phys. Rev. 148, 1003(1966).

Fa 6 8 K.T. Faler, R.R. Spencer, and R.A. Harlan, Phys. Rev.175, 1495 (1968).

Fe 54 H. Feshbach, C. Porter, and V.F. Weisskopf, Phys.Rev. 96, 448 (1954).

58

62

67

69

53

63

65

67

69

69a

71

64

70

71

64

67

69

f> 9 a

/ I

180

H. Feshbach, Ann. Phys. (N.Y.) 5, 357 (1958).

H. Feshbach, Ann. Phys. 1J), 287 (1962).

H. Feshbach, A.K. Kerman, and R.H. Lemmer, Ann. Phys. (N.Y.) 41, 230 (1967).

E.R. Flynn, D.D. Armstrong, J.G. Beery, and A.G.Blair, Phys. Rev. 3,82 , 1113 (1969).

M. Gell-Mann and M.L. Goldberger, Phys. Rev. 91,398 (1953).

N.K. Glendenning, Ann. Rev. Nucl. Sci. 1 3 , 191 (1963).

N.K. Glendenning, Phys. Rev. 137, B102 (1965).

N.K. Glendenning, Proc. Int. School of Physics "Enrico Fermi," Course XL (1967), ed. M. Jean (Academic Press, New York, 1969) p. 332.

N.K. Glendenning, Proc. Int. Conf. on Properties of Nuclear States, Montreal (1969), ed. M. Harvey, R.Y. Cusson, J.S. Geiger, and J.M. Pearson (University of Montreal Press, Montreal, 1969) p. 245.

J. Glatz, K.E.G. LBbner, and F. Oppermatn, Z. Physik 227 , 83 (1969).

N.K. Glendenning and R.S. Mackintosh, Nucl. Phys.A168, 575 (1971).

M.L. Goldberger and K.M. Watson, Collision Theory (Wiley, New York, 1964).

S.C. Gujrathi and J.M. D'Auria, Can. J. Phys. 4 8 ,502 (1970).

C. Gdnther, P. Kleinheinz, R.F. Casten, and B. Elbek, Nucl. Phys. A172. 273 (1971).

B. Harmatz and T.H. Handley, Nucl. Phys. 5j>, 1 (1964).

K. Harada, Phys. Lett. 1_0, 80 (1967).

I. Hamamoto and T. Udagawa, Nucl. Phys. A126 , 241(1969) .

P.C. Hansen, P. Hornsk(<j, and K.H. Johansen, Nucl.I’llya . A I 26, 4 6 4 (19 6 '/).

Y. Ilrtlin, IMiys. Rev. C^, 1432 (1971).

181

Ha 71a J.D. Harvey and R.C. Johnson, Phys. Rev. (13, 636 (1971).

He 6 8 D.L. Hendrie, N.K. Glendennlng, B.G. Harvey, O.N.Jarvis, H.H. Duhm, J. Saudinos, and J. Mahoney,Phys. Lett. 26B, 127 (1968).

He 71 D.L. Hendrie e_t a_l. , private communication (1971).

Hi 53 D.L. Hill and J.A. Wheeler, Phys. Rev. 89, 1102(1953) .

Hj 70 S.A. Hjorth, H. Ryde, K.A. Hagemann, G. L^vhfJiden,and J.G. Waddington, Nucl. Phys. A144, 513 (1970).

Ho 50 J.R. Holt and C.T. Young, Proc. Phys. Soc. (London)A63 . 833 (1950).

Ho 53 J. Horowitz and A.M.L. Messiah, Journ. Phys. Rad.14., 695 (1953).

Ho 60 J. HHgaasen-Feldman , Nucl. Phys. 2_8, 258 (1960).

Ho 6 6 M.B. Hooper, Nucl Phys. 7_6, 449 (1966).

Ho 70 G. Holland, Doctoral Dissertation, Yale University(1970), unpublished; and G. Holland and D.A. Bromley, to be published.

Ia 69 P.J. Iano, S.K. Penney, and R.M. Drisko, Nucl. Phys.A127 , 47 (1969).

Ia 6 6 P.J. Iano and N. Austern, Phys. Rev. 151, 853 (1966).

Is 72 A. Isoya, T. Maki, T. Nakashima, N. Kato, Y. Kumamoto,T. Sugimitsu, and K. Kimura, J. Phys. Soc. Japan 3 2 , 8 8 6 (1972).

Ja 67 M. JaskoZa, K. Nyb^ , P.O. Tj^m, and B. Elbek, Nucl.Phys. A96, 52 (1967).

Ja 69 M. JaskoZa, P.O. Tj^m, and B. Elbek, Nucl. Phys.A133 , 65 (1969).

Je 67 M. Jean, Proc. Int. School of Physics "Enrico Fermi,"Course XL (1967), e d . M. Jean (Academic Press, New York, 1969) p. 171.

Jo 70 R.C. Johnson and P.J.R. Soper, Phys. Rev. £1, 976(1970).

Ka 69 I. Kanestr^m and P.O. Tj^m, Nucl. Phys. A138, 177(1969).

56

65

60

63

72

66

70

71

73

67

68

69

69a

70

70a

35

58

64

182

A.K. Kerman, Dan. Vid. Selsk. Mat. Fys. Medd. 3 0 , No. 15 (1956).

R.A. Keneflck and R.A. Sheline, Phys. Rev. 139,B1479 (1965).

L .S . Kisslinger M a t . Fys. Medd.

L.S. Kisslinger 3_5 , 853 (1963).

C .H . King, R .J . Phys. Rev. Lett

B . Kozlowsky an(1966) .

and R.A. Sorensen, 32, No. 9 (1960).

and R.A. Sorensen,

Ascuitto, N. Stein 29, 71 (1972).

A. de-Shalit, Nuc

D a n . Vid. Selsk.

Rev. Mod. Phys.

and B. Sorensen,

. Phys . 22.» 2

D.G. Kovar, C.K. Bockelman, W.D. Callender, L.J. McVay, C.F. Maguire, and W.D. Metz, Wright Nuclear Structure Laboratory Internal Report #49 (1970).

D.G. Kovar, Doctoral Dissertation, Yale University(1971), unpublished.

K.S. Krane, C.E. Olsen, and W.A. Steyert, Phys. Rev. C2, 263 (1973).

K. Kumar and M. Baranger, Nucl. Phys. A 9 2 , 608(1967) .

K. Kumar and M. Baranger, Nucl. Phys. A122, 273(1968) .

T. Kuroyanagi and T. Tamura, Nucl. Phys. A133, 554(1969) .

P.D. Kunz, E. Rost, and R.R. Johnson, Phys. Rev.177, 1737 (1969).

K. Kumar and B. Sorensen, Nucl. Phys. A146, 1 (1970).

A.H. Kukoc, B. Singh, J.D. King, and H.W. Taylor, Nucl. Phys. A143, 545 (1970).

E.O. Lawrence, E. McMillan, and R.L. Thornton, Phys. Rev. 48, 493 (1935).

A.M. Lane and R.G. Thomas, Rev. Mod. Phys. 3^, 257 (1958).

A.M. Lane, Nuclear Theory (Benjamin, New York, 1964).

183

Le 64 L.L. Lee and J.P. Schiffer, Phys. Rev. 136 , B405(1964) .

Le 6 6 F.S. Levin, Phys. Rev. 147, 715 (1966).

Le 69 M. LeVine and P. Parker, Phys. Rev. 186, 1021 (1969).

Li 73 T.K. Lim, Phys. Rev. C7 , 1288 (1973).

Lu 71 V.K. Lukyanov, Phys. Lett. 34B, 354 (1971).

Me 70 L.J. McVay, private communication (1970).

Me 73 L.J. McVay, Doctoral Dissertation, Yale University(1973), unpublished; and L.J. McVay £t cQ. to be published.

Me 73a L.J. McVay, R.J. Ascuitto, and C.H. King, Phys. Lett. 43B. 119 (1973).

Ma 64 B.E.F. Macefield and R. Middleton, Nucl. Phys. 59,561 (1964).

Ma 65 E.U. Marshalek, Phys. Rev. 139, B770 (1965).

Ma 65a J.H.E. Mattauch, W. Thiele, and A.H. Wapstra, Nucl.Phys. 67_, 1 (1965).

Ma 67 S.G. Malmskog and S. Wahlborn, Nucl. Phys. A102,273 (1967).

Ma 69 S.G. Malmskog, M. HBjeberg, and V. Berg, Ark, Fys.4J), 247 (1969).

Ma 69a M.H. Macfarlane, Proc. Int. Conf. on Properties ofNuclear States, Montreal (1969), e d . M. Harvey, R.Y. Cusson, J.S. Geiger, and J.M. Pearson (University of Montreal Press, Montreal, 1969) p. 385.

Ma 71 R.S. Mackintosh, Nucl. Phys. A170, 353 (1971).

Ma 72 J.V. Maher, J.R. Erskine, A.M. Friedman, R.H. Siemssen,and J.P. Schiffer, Phys. Rev. C_5 , 1380 (1972).

Ma 73 C.F. Maguire, Doctoral Dissertation, Yale University(1973), unpublished; and C.F. Maguire £t a .. to be published.

Me 70 W.D. Metz, private communication (1970).

Me 72 W.D. Metz, Doctoral Dissertation, Yale University(1972), unpublished.

184

Ml 71 W.T. Milner, F.K. McGowan, R.L. Robinson, P.H. Stelson,and R.O. Sayer, Nucl. Phys. A177 , 1 (1971).

Mo 60 B.R. Mottelson, Proc. Int. School of Physics "EnricoFermi," Course XV (1960), e d . G. Racah (Zanichelli, Bologna, 1962) p. 44.

Mo 60a B.R. Mottelson, Proc. Int. Conf . on Nuclear Structure, Kingston (I960), ed. D.A. Bromley and E.W. Vogt (U. of Toronto Press, Toronto, 1960) p. 525.

Mo 60b B.R. Mottelson and J.G. Valatin, Phys. Rev. Lett. J>,511 (1960).

Mo 70 P. MHller, Nucl Phys. A142. 1 (1970).

Mt 70 W.D. Myers, Nucl. Phys. A145, 387 (1970).

Na 65 0. Nathan and S.G. Nilsson, Alpha-,, Beta-, and Gamma-Ray Spectroscopy, e d . K. Siegbahn (North Holland, Amsterdam, 1965) p. 601.

Ne 6 6 P.E. Nemirovskii and V.A. Chepurnov, Sov. J. Nucl.Phys. 3 , 730 (1966).

Ni 55 S.G. Nilsson, Dan. Vid. Selsk. Mat. Fys. Medd. 2 9 ,No. 16 (1955).

Ni 61 S.G. Nilsson and 0. Prior, Dan. Vid. Selsk. Mat.Fys. Medd. 3^, No. 16 (1961).

Oe 6 8 W. von Oertzen, H.H. Gutbrod, M. Mdller, U. Voos,and R. Bock, Phys. Lett. 2 6 B , 291 (1968).

Og 71 W. Ogle, S. Wahlborn, P. Piepenbring, and S. Fredriks-son, Rev. Mod. Phys. 4J3 , 424 (1971).

Oh 69 T. Ohmura, B. Imanishi, M. Ichimura, and M. Kawai,Prog. Theor. Phys. 41^, 391 (1969).

Oh 70 T. Ohmura, B. Imanishi, M. Ichimura, and M. Kawai,Prog. Theor. Phys. 4_3 , 347 (1970).

Oh 70a T. Ohmura, B. Imanishi, M. Ichimura, and M. Kawai,Prog. Theor. Phys. 44_, 1242 (1970).

01 72 D.K. Olsen, T. Udagawa, T. Tamura, and R.E. Brown,Phys. Rev. Lett. 2j), 1178 (1972).

Oo 70 M. Oothoudt, N.M. Hintz, and P. Vedelsby, Phys.Lett. 32ji, 270 (1970).

Oo 73 M. Oothoudt and N.M. Hintz, to be published (1973).

35

69

57

62

63

63a

63b

64

62

43

67

68

57

67

1119

58

64

66

71

66

185

J.R. Oppenheimer and M. Phillips, Phys. Rev. 4 8 ,500 (1935).

J. Overley, P. Parker, and D.A. Bromley, Nucl. Inst. Meths . 6J5, 61 (1969) .

R.E. Peierls and J. Yoccoz, Proc. Phys. Soc. (London) A 7 0 . 381 (1957).

F.G. Perey and B. Buck, Nucl. Phys. 32^, 353 (1962).

C.M. Perey and F.G. Perey, Phys. Rev. 132, 755 (1963).

F.G. Perey, Direct Interactions and Nuclear Mechan­isms , e d . E. Clementel and C. Villi (Gordon and Breach, New York, 1963) p. 125.

F.G. Perey, Phys. Rev. 131, 745 (1963).

S.K. Penney and G.R. Satchler, Nucl. Phys. 53,145 (1964).

M.A. Preston, Physics of the Nucleus (Addison-Wesley, Reading, 1962).

G. Racah, Phys. Rev. 6_3 , 367 (1943).

G. Rawitscher, Phys. Rev. 163, 1223 (1967).

G. Ripka, Advances in Nuclear Physics, Vol. 1, e d .M. Baranger and E. Vogt (Plenum, New York, 1968) p. 183.

M.E. Rose, Elementary Theory of Angular Momentum (Wiley, New York, 1957).

E. Rost, Phys. Rev. 154, 994 (1967).

E. Rutherford, Phil. Mag. (6 th Ser.) 2_1 , 669 (1911).

E. Rutherford, Phil. Mag. (6 th Ser.) 3_7 , 537 (1919).

G.R. Satchler, Ann. Phys. 3 , 275 (1958).

G.R. Satchler, Nucl. Phys. 5_5 , 1 (1964).

G.R. Satchler, Lectures in Theoretical Physics, Vol. VIIIC, ed. P.D. Kunz, D.A. Lind, W.E. Brittin (Univ­ersity of Colorado Press, Boulder, 1966) p. 73.

G.R. Satchler, Phys. Rev. C4 , 1485 (1971).

J.P. Schiffer, L.L. Lee, A. Marinov, and C. Mayer- BOricke, Phys. Rev. 147, 829 (1966).

186

Sc 69 H. Schulz and H.J. Wiebicke, Phys. Lett. 29B, 18(1969) .

Sc 69a J.P. Schiffer, P. Kienle, and G.C. Morrison, Nuclear Isospin, ed. J.D. Anderson, S.D. Bloom, J. Cerny, and W.W. True (Academic Press, New York, 1969) p. 679

Sc 70 H. Schulz, H.J. Wiebicke, R. Fttlle, D. Netzband, andK. Schlott, Nucl. Phys. A159 , 324 (1970).

Sc 72 H. Schulz, H.J. Wiebicke, and F.A. Gareev, Nucl.Phys. A180 , 625 (1972).

Sc 72a R. Schaeffer and G.F. Bertsch, Phys. Lett. 38B,159 (1972).

Sh 64 R. Sheer, E. Rost, and M.E. Rickey, Phys. Rev. Lett.T2, 420 (1964).

Si 6 6 R.H. Siemssen and J.R. Erskine, Phys. Rev. 146, 911(1966) .

So 72 B. Sorensen, private communication (1972).

St 65 P.H. Stelson and L. Grodzins, Nuclear Data A L, 21(1965).

St 6 6 A.P. Stamp, Nucl. Phys. 8J3, 232 (1966).

St 6 8 N. Stein, J.P. Coffin, C.A. Whitten, Jr., and D.A.Bromley, Phys. Rev. Lett. 21, 1456 (1968).

St 6 8 a R.G. Stokstad and B. Persson, Phys. Rev. 170, 1072(1968) .

St 6 8 b N. Stein, C.A. Whitten, Jr., and D.A. Bromley, Phys.Rev. Lett. 2j0, 113 (1968).

St 6 8 c F.S. Stevens, M.D. Holtz, R.M. Diamond, and J.O.Newton, Nucl. Phys. A115, 129 (1968).

Ta 65 T. Tamura, Rev. Mod. Phys. 37_, 679 (1965).

Ta 70 T. Tamura, D.R. Bes, R.A. Beoglia, and S. Landowne,Phys. Rev. Lett. J2J5 , 1507 (1970) and erratum Phys. Rev. Lett. 2_6, 156 (1971).

Ta 71 H.W. Taylor, J.D. King, and B. Singh, Can. J. Phys.4.9, 2614 (1971).

Tj 67 P.O. Tjtfm and B. Elbek, Dan. Vid. Selsk. Mat. Fys.M e d d . 36, No. 8 (1967).

To 55 W. Tobocman and M.H. Kalos, Phys. Rev. 97_, 132 (1955)

W. Tobocman, Theory of Direct Nuclear Reactions (Oxford University Press, London, 1961).

M. Toyama, Phys. Lett. 38B, 147 (1972).

T. Udagawa, T. Tamura, and T. Izumoto, ?hys. Lett; 35B, 129 (1971).

J.G. Valatin, Nuovo Cimento 843 (1958).

M.N. Vergnes and R.K. Sheline, Phys. Rev. 132 ,1736 (1963).

F. Villars, Ann. Rev. Nucl. Sci. _7> (1957).

E. Vogt, Advances in Nuclear Physics, Vol. 1, ed.M. Baranger and E. Vogt (Plenum, New York, 1968)p. 261.

M. Walt and H.H. Barschall, Phys. Rev. j 3, 1062(1954) .

C.A. Whitten, Jr., N. Stein, G.E. Holland, andD.A. Bromley, Phys. Rev. 188, 1941 (1969).

C.F. Williamson, J.P. Boujot, and J. Picard, Rapport CEA-R3042, Centre d'Etude Nucleaires de Saclay (1966).

R.D. Woods and D.S. Saxon, Phys. Rev. 95_, 577 (1954).

K. Yagi, K. Sato, Y. Aokl, T. Udagawa, and T. Tamura, Phys. Rev. Lett. 29 , 1334 (1972).

•

J. Zylicz, P.G. Hansen, H.L. Nielsen, and K. Wilsky, Nucl. Phys. ( 4 , 13 (1966).